Blejske delavnice iz fizike Bled Workshops in Physics ISSN 1580-4992 Letnik 20, St. 2 Vol. 20, No. 2 Proceedings to the 22nd Workshop What Comes Beyond the Standard Models Bled, July 6-14, 2019 Edited by Norma Susana Mankoc Borstnik Holger Bech Nielsen Dragan Lukman DMFA - zaloZniStvo Ljubljana, december 2019 The 22nd Workshop What Comes Beyond the Standard Models, 6.- 14. July 2019, Bled was organized by Society of Mathematicians, Physicists and Astronomers of Slovenia and sponsored by Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Beyond Semiconductor (Matjaži Breskvar) Scientific Committee John Ellis, King's College London / CERN Roman Jackiw, MIT Masao Ninomiya, Yukawa Institute for Theoretical Physics, Kyoto University Organizing Committee Norma Susana Mankoč Borštnik Holger Bech Nielsen Maxim Yu. Khlopov The Members of the Organizing Committee of the International Workshop "What Comes Beyond the Standard Models", Bled, Slovenia, state that the articles published in the Proceedings to the 22nd Workshop "What Comes Beyond the Standard Models", Bled, Slovenia are refereed at the Workshop in intense in-depth discussions. Workshops organized at Bled > What Comes Beyond the Standard Models (June 29-July 9,1998), Vol. 0 (1999) No. 1 (July 22-31,1999) (july 17-31, 2000) (july 16-28, 2001), Vol. 2 (2001) No. 2 (july 14-25, 2002), Vol. 3 (2002) No. 4 (july 18-28, 2003) Vol. 4 (2003) Nos. 2-3 (july 19-31, 2004), Vol. 5 (2004) No. 2 (july 19-29, 2005), Vol. 6 (2005) No. 2 (September 16-26, 2006), Vol. 7 (2006) No. 2 (july 17-27, 2007), Vol. 8 (2007) No. 2 (july 15-25, 2008), Vol. 9 (2008) No. 2 (july 14-24, 2009), Vol. 10 (2009) No. 2 (july 12-22, 2010), Vol. 11 (2010) No. 2 (july 11-21, 2011), Vol. 12 (2011) No. 2 (july 9-19, 2012), Vol. 13 (2012) No. 2 (july 14-21, 2013), Vol. 14 (2013) No. 2 (july 20-28, 2014), Vol. 15 (2014) No. 2 (july 11-19, 2015), Vol. 16 (2015) No. 2 (july 11-19, 2016), Vol. 17 (2016) No. 2 (july 9-17, 2017), Vol. 18 (2017) No. 2 (june 23-july 1, 2018), Vol. 19 (2018) No. 2 (july 6-14, 2019), Vol. 20 (2018) No. 2 > Hadrons as Solitons (july 6-17,1999) > Few-Quark Problems (july 8-15, 2000), Vol. 1 (2000) No. 1 > Selected Few-Body Problems in Hadronic and Atomic Physics (july 7-14, 2001), Vol. 2 (2001) No. 1 > Quarks and Hadrons (july 6-13, 2002), Vol. 3 (2002) No. 3 > Effective Quark-Quark Interaction (july 7-14, 2003), Vol. 4 (2003) No. 1 > Quark Dynamics (july 12-19, 2004), Vol. 5 (2004) No. 1 > Exciting Hadrons (july 11-18, 2005), Vol. 6 (2005) No. 1 > Progress in Quark Models (july 10-17, 2006), Vol. 7 (2006) No. 1 > Hadron Structure and Lattice QCD (july 9-16, 2007), Vol. 8 (2007) No. 1 > Few-Quark States and the Continuum (September 15-22, 2008), Vol. 9 (2008) No. 1 > Problems in Multi-Quark States (june 29-july 6, 2009), Vol. 10 (2009) No. 1 > Dressing Hadrons (july 4-11, 2010), Vol. 11 (2010) No. 1 > Understanding hadronic spectra (july 3-10, 2011), Vol. 12 (2011) No. 1 > Hadronic Resonances (july 1-8, 2012), Vol. 13 (2012) No. 1 > Looking into Hadrons (july 7-14, 2013), Vol. 14 (2013) No. 1 > Quark Masses and Hadron Spectra (july 6-13, 2014), Vol. 15 (2014) No. 1 > Exploring Hadron Resonances (july 5-11, 2015), Vol. 16 (2015) No. 1 > Quarks, Hadrons, Matter (july 3-10, 2016), Vol. 17 (2016) No. 1 > Advances in Hadronic Resonances (july 2-9, 2017), Vol. 18 (2017) No. 1 > Double-charm Baryons and Dimesons (june 17-23, 2018), Vol. 19 (2018) No. 1 IV > Electroweak Processes ofHadrons (July 15-19, 2019), Vol. 20 (2019) No. 1 > o Statistical Mechanics of Complex Systems (August 27-September 2, 2000) o Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry (August 25-28, 2001) Contents Preface in English and Slovenian Language............................VII Talk Section........................................................ 1 1 Corollary Analyses After the Recent Model-Independent Results of DAMA/LIBRA-Phase2 R. Bernabei etal...................................................... 1 2 Conspiracy of BSM Physics and Cosmology M.Yu. Khlopov...................................................... 21 3 New Way of Second Quantized Theory of Fermions With Either Clifford or Grassmann Coordinates and Spin-Charge-Family Theory N.S. Mankoc Borštnik and H.B.F. Nielsen............................... 36 4 Understanding the Second Quantization of Fermions in Clifford and in Grassmann Space — New Way of Second Quantization of Fermions — Part I N.S. Mankoc Borštnik and H.B.F. Nielsen...............................109 5 Understanding the Second Quantization of Fermions in Clifford and in Grassmann Space — New Way of Second Quantization of Fermions — Part II N.S. Mankoc Borštnik and H.B.F. Nielsen...............................120 6 Deriving Locality From Diffeomorphism Symmetry in a Fiber Bundle Formalism H.B. Nielsen and A. Kleppe...........................................135 7 How Compact Stars Challenge Our View About Dark Matter G. Panotopoulos.....................................................151 8 Phenomenological Studies of Models With a Pseudo Nambu Goldstone Boson T. Shindou..........................................................168 Discussion Section..................................................175 VI Contents 9 Analysis of Programming Tools in Framework of Dark Matter Physics and Concept of New MC-generator K.M. Belotsky, A.H. Kamaletdinov and E.S. Shlepkina....................179 10 Tessellation Approach in Modeling Properties of Physical Vacuum and Fundamental Particles E.G. Dmitrieff.......................................................190 11 Mass Matrix Parametrization for Pseudo-Dirac Neutrinos a. Gorin............................................................204 12 Relations Between Clifford Algebra and Dirac Matrices D. Lukman, M. Komendyak and N.S. Mankoc Borštnik...................211 13 Second Quantization as Cross Product N.S. Mankoc Borštnik and H.B.F. Nielsen...............................223 14 Novel String Field Theory and Remaining Problems H.B.F. Nielsen and M. Ninomiya.......................................232 15 Local Temperature Distribution in the Vicinity of Gravitationally Bound Objects in the Expanding Universe P.M. Petriakova and S. G.Rubin........................................237 Virtual Institute of Astroparticle Physics Presentation...................247 16 The Platform of Virtual Institute of Astroparticle Physics for Studies of BSM Physics and Cosmology M.Yu. Khlopov......................................................249 Preface The series of annual workshops on "What Comes Beyond the Standard Models?" started in 1998 with the idea of Norma and Holger for organizing a real workshop, in which participants would spend most of the time in discussions, confronting different approaches and ideas. Workshops take place in the picturesque town of Bled by the lake of the same name, surrounded by beautiful mountains and offering pleasant walks and mountaineering. In our very open minded, friendly, cooperative, long, tough and demanding discussions several physicists and even some mathematicians have contributed. Most of topics presented and discussed in our Bled workshops concern the proposals how to explain physics beyond the so far accepted and experimentally confirmed both standard models — in elementary particle physics and cosmology — in order to understand the origin of assumptions of both standard models and be consequently able to make predictions for future experiments. Although most of participants are theoretical physicists, many of them with their own suggestions how to make the next step beyond the accepted models and theories, experts from experimental laboratories were and are very appreciated, helping a lot to understand what do measurements really tell and which kinds of predictions can best be tested. The (long) presentations (with breaks and continuations over several days), followed by very detailed discussions, have been extremely useful, at least for the organizers. We hope and believe, however, that this is the case also for most of participants, including students. Many a time, namely, talks turned into very pedagogical presentations in order to clarify the assumptions and the detailed steps, analyzing the ideas, statements, proofs of statements and possible predictions, confronting participants' proposals with the proposals in the literature or with proposals of the other participants, so that all possible weak points of the proposals, those from the literature as well as our own, showed up very clearly. The ideas therefore seem to develop in these years considerably faster than they would without our workshops. This year neither the cosmological nor the particle physics experiments offered much new, as also has not happened in the last two years, which would offer new insight into the elementary particles and fields and also into cosmological events, although a lot of work and effort have been put in, and although there are some indications for the existence of the fourth family to the observed three, due to the fact that the existence of the fourth family might explain the existing experimental data better, what is mentioned in this proceedings. It looks like, that "nature" does not "like" to help us to better understand the assumptions, put into the standard models, as it is written in one of contributions to this proceedings. There were talks accompanied by very lively discussions about the way which could lead to next step beyond both standard models, some of them appear in this proceedings, the others might contribute to the next year proceedings. Understanding the universe through the cosmological theories and theories of the elementary fermion and boson fields, have, namely, so far never been so dependent on common knowledge and experiments in both fields. On both fields there appear proposals which should explain assumptions of these models. The competition, who will have right, is open and exciting. We are keeping expecting that new cosmological experiments will help to resolve the origin of the dark matter. Since the results of the DAMA/LIBRA experiments, presented in this year proceedings, can hardly be explained in some other way than with the signal of the dark matter, it is expected that sooner or latter other laboratories will confirm the DAMA/LIBRA results. This has not yet happened and our discussions clarified the reasons for that. Several contributions in this proceedings discuss proposals for the origin of the dark matter, suggesting that they might belong to the stable neutrons of the second group of four families, decoupled from the observed three, to the dark atoms made of dark baryons and ordinary baryons, and to the new scalar fields, new bosons, which manifest inside stars as a Bose-Einstein condensate. These contributions discuss also the possibilities that some of these kinds of the dark matter candidates were already observed by DAMA/LIBRA scattering events or if dark matter objects decay or annihilate too strongly they discuss reasons why experiments do not observe the corresponding gamma rays. The experiments on the LHC and other laboratories around the world do not so far offer the accurately enough mixing matrices for quarks and leptons, so that it will become clear whether there is the fourth family to the observed three and whether there are several scalar fields, which determine the higgs and the Yukawa couplings, predicted by the spin-charge-family theory. The symmetry in all orders of corrections of the 4 x 4 mass matrices, determined by the scalars of this theory, studied in the previous proceedings, limits the number of free parameters of mass matrices, and would for accurately enough measured matrix elements of the 3 x 3 sub-matrices of the 4 x 4 mixing matrices predict properties of the fourth family of quarks and leptons. The fourth family with the masses close to 1 TeV for leptons and above 1 TeV for quarks is weaker coupled with the rest three families than it is the third u-quark coupled to the rest of quarks. Calculations show that the larger the masses of the fourth family - up to 6 TeV seems to be allowed by experiments - the smaller the unwanted mixing elements which could cause the flavour-changing neutral currents and the better is agreement with the experimental data, which require, that there should be the fourth family due to the nonunitarity of the 3 x 3 so far measured mixing matrix for quarks. The new data might answer the question, whether laws of nature are elegant (as predicted by the spin-charge-family theory and also — up to the families — other Kaluza-Klein-like theories and the string theories) or "she is just using gauge groups when needed" (what many models assume, also some presented in this proceedings). Can the higgs scalars be guessed by smaller steps from the standard model case, appearing as pseudo Nambu Goldstone bosons and in many other possibilities, or they originate in gravity in higher dimensions as also the gauge fields do? Is there only gravity as the interacting field, which manifests in the low energy regime all the vector gauge fields as well as the scalar fields? Should correspondingly gravity be quantizable? Is masslessness of all the bosons and fermions essentail, while masses appear at low energy region due to interactions and break of symmetries? Do fermions charges manifest spins in higher dimensions? What is then the dimension of space-time? Infinite, or it emerges from zero? Is the law of nature emerging from random mathematical structure, which then de-velope to differentiability, diffeomorphism symmetry, locality, Lorentz invariance, so that fermions spin in higher dimension manifests as charges at low energies? Why and how? The evidences obviously tell that fermion fields have half integer spin and the charges in the fundamental representations of the so far observed groups. The Grassmann space offer on the other side the possibility that fermions would carry the integer spin and the charges in adjoint representations. Shall the study of Grassmann space in confrontation with Clifford space for the description of the internal degrees of freedom for fermions, discussed in this proceedings, offering explanation for the second quantization of fermions, help to better understand the "elegance of the laws of nature"? If "nature would make a choice" of the Grassmann instead of the Clifford algebra, all the atoms, molecules and correspondingly all the world would look completely different, but yet might be still possible. Why "she made a choice" of the Clifford algebra? Is the working hypotheses that "all the mathematics is a part of nature" acceptable and must be taken seriously? We need and correspondingly use so many mathematical concepts in order to derive a consistent theory, but in most cases still several questions remain open. Since, as every year also this year there has been not enough time to mature the very discerning and innovative discussions, for which we have spent a lot of time, into the written contributions, only two months, authors can not really polish their contributions. Organizers hope that this is well compensated with fresh contents. Questions and answers as well as lectures enabled by M.Yu. Khlopov via Virtual Institute of Astroparticle Physics (viavca.in2p3.fr/site.html) of APC have in ample discussions helped to resolve many dilemmas. Google Analytics, showing more than 240 thousand visits to this site from 153 countries, indicates world wide interest to the problems of physics beyond the Standard models, discussed at Bled Workshop. At XXII Bled Workshop VIA streamning made possible to webcast practically all the talks. The reader can find the records of all the talks delivered by cosmovia since Bled 2009 on viavca.in2p3.fr/site.html in Previous - Conferences. The three talks delivered by E. Kiritsis (Emergent gravity (from hidden sector)), Maxim Yu. Khlopov (Conspiracy of BSM Physics and BSM Cosmology) and Norma Mankoc Borstnik (Experimental consequences of spin-charge family theory) as well as students' scientific debuts talk by Valery Nikulin (Inflationary limits on the size of compact extra space) can be accessed directly at http://viavca.in2p3.fr/what_comes_beyond_the_standard_model_XXII.html Most of the talks can be found on the workshop homepage http://bsm.fmf.uni-lj.si/. Bled Workshops owe their success to participants who have at Bled in the heart of Slovene Julian Alps enabled friendly and active sharing of information and ideas, yet their success was boosted by vidoeconferences. Let us conclude this preface by thanking cordially and warmly to all the participants, present personally or through the teleconferences at the Bled workshop, for their excellent presentations and in particular for really fruitful discussions and the good and friendly working atmosphere. We express our gratitude to MDPI journals "Symmetry" and "Particles" for travel support for young and Senior participants and our hope that this tradition will be continued and extended. Norma Mankoi BorStnik, Holger Bech Nielsen, Maxim Y. Khlopov, (the Organizing comittee) Norma MankoC BorStnik, Holger Bech Nielsen, Dragan Lukman, (the Editors) Ljubljana, December 2019 1 Predgovor (Preface in Slovenian Language) Vsakoletne delavnice z naslovom ,,Kako preseči oba standardna modela, koz-moloskega in elektrosibkega" ("What Comes Beyond the Standard Models?") sta postavila leta 1998 Norma in Holger z namenom, da bi udeleZenci v izčrpnih diskusijah kritično soočali različne ideje in teorije. Delavnice domujejo v Plemljevi hisi na Bledu ob slikovitem jezeru, kjer prijetni sprehodi in pohodi na čudovite gore, ki kipijo nad mestom, ponujajo priložnosti in vzpodbudo za diskusije. K našim zelo odprtim, prijateljskim, dolgim in zahtevnim diskusijam, polnim iskrivega sodelovanja, je prispevalo veliko fizikov in čelo nekaj matematikov. Večina tem in vprasanjpredstavljenih in diskutiranih na nasih Blejskih delavničah, zadeva predloge za razlago pojavov onkraj obeh standadnih modelov — v fiziki osnovnih delčev in kozmologiji — z namenom razumeti izvor predpostavk obeh standardnih modelov, kar bi omogočilo naovedi za nove poskuse. (Čeprav je večina udelezenčev teoretičnih fizikov, mnogi z lastnimi idejami kako narediti naslednji korak onkraj sprejetih modelov in teorij, so še posebej dobrodošli predstavniki eksperimentalnih laboratorijev, ki nam pomagajo v odprtih diskusijah razjasniti resničšno sporočšilo meritev in nam pomagajo razumeti kaksšne napovedi so potrebne, da jih lahko s poskusi dovoljzanesljivo preverijo. Organizatorji moramo priznati, da smo se na blejskih delavničah v (dolgih) predstavitvah (z odmori in nadaljevanji preko več dni), ki so jim sledile zelo podrobne diskusije, naučili veliko, morda več kot večina udelezenčev. Upamo in verjamemo, da so veliko odnesli tudi študentje in večina udelezenčev. Velikokrat so se predavanja spremenila v zelo pedagoške predstavitve, ki so pojasnile predpostavke in podrobne korake, soočšile predstavljene predloge s predlogi v literaturi ali s predlogi ostalih udelezenčev ter jasno pokazale, kje utegnejo tičati šibke točke predlogov. Zdi se, da so se ideje v teh letih razvijale bistveno hitreje, zahvaljujoč prav tem delavničam. Tako kot v preteklih dveh letih tudi to leto niso eksperimenti v kozmologiji in fiziki osnovih fermionskih in bozonskih polj ponudili rezultatov, ki bi omogočili nov vpogled v fiziko osnovnih delčev in polj, čeprav je bilo vanje vloženega veliko truda in četudi razberemo iz eksperimentov, da četrta druzina k ze izmerjenim trem mora biti, saj lahko s štirimi druzinami lazje pojasnimo izmerjene podatke, kar je omenjeno tudi v tem zborniku. Zdi se, kot omenja eden od prispevkov v tem zborniku, da nam "narava no če pomagati", da bi bolje razumeli predpostavke v obeh standardnih modelih. Nekatera predavanja so spremljale zelo zšivahne diskusije o predlogih, ki nam lahko pomagajo razumeti privzetke obeh standardnih modelov. Nekatere od teh razprav so v tem zborniku, druge so bodo mordna pojavile v zborniku prihodnje delavniče. Kozmoloska spoznanja in spoznanja v teoriji osnovnih fermionskih in bozonskih poljse nikoli doslejniso bila tako zelo povezana in soodvisna. Na obeh področjih "rastejo" novi predlogi, ki najpojasnijo privzetke teh modelov. Tekma, kdo bo imel prav, je odprta in razburljiva. Priakujemo, da bodo novi kozmološki poskusi razkrili izvor temne snovi. Ker rezultate poskusov DAMA/LIBRA, predstavljene v tem zborniku, tezko pojasnimo drugače kot da gre za temno snov, je pričakovati, da bodo sčasoma tudi poskusi v drugih laboratorijih potrdili rezultate poskusa DAMA/LIBRA. To se se ni zgodilo, nase razprave so razjasnile razloge za to. Kar nekaj je prispevkov v zborniku, ki obravnavajo izvor temne snovi: za delce temne snovi predlagajo stabilne nevtrone druge skupine sštirih druzšin, ki niso sklo-pljene z ze izmerjenimi tremi in pričakovano četrto, temne atome, ki jih sestavljajo temni in običajni barioni ali nova skalarna polja, nove bozone, ki se znotrajzvezd zgostijo v Bose-Einsteinov kondenzat. Avtorji v teh prispevkih obravnavajo tudi moznost, da so v poskusu DAMA/LIBRA nekatere od teh delčev ze opazili. En prispevek obravnava mozšnost, da temna snov morda razpada ali se anihilira dovlj hitro, da bi morali opaziti pri tem nastale zarke gama, pa jih zaradi absorpčije ne opazimo. Poskusom na pospesevalniku LHC in v drugih laboratorijih doslejni uspelo izmeriti mesšalnih matrik za leptone in kvarke dovolj natančšno, da bi lahko ugotovili, ali poleg izmerjenih treh druzin obstaja tudi četrta družina in ali obstaja tudi več skalarnih polj, ki določajo higgsov skalar in Yukawine sklopitve. Teorija spinov-nabojev-druzin napoveduje obstoj četrte druzine in obstoj več skalarnih polj. Simetrija masnih matrik 4 x 4 v vseh redih popravkov, obravavana v prispevkih v prejšnjih zbornikih, omeji stevilo prostih parametrov masnih matrik. Za dovolj natančno izmerjene matrične elemente podmatrik 3 x 3 v mesalnih matrikah 4 x 4 bi ta teorija lahko napovedala lastnosti četrte druzine kvarkov in leptonov. Cetrta druzina, ki ima mase leptonov blizu 1 TeV, mase kvarkov pa nad 1 TeV, je šibkeje sklopljena s preostalimi tremi druzinami, kot je tretji kvark u (top) sklopljen s preostalimi kvarki. Izračuni pokažejo, da se z večanjem mas četrte druzine — poskusi dopusčajo mase do 6 TeV — zmansujejo matrični elementi, ki povzročšajo nevtralne tokove in spremembo druzšinskega kvantnega sštevila, Hkrati se izboljsša ujemanje z eksperimentalnimi podatki, ki zahtevajo čšetrto druzšino zaradi neunitarnosti dosedajizmerjene mesalne matrike 3 x 3 za kvarke. Nove meritve bodo morda odgovorile na vprasanje, ali so zakoni narave elegantni (kot to napove teorija spinov-nabojev-druzšin in, z izjemo druzšin, ostale teorije Kaluza-Kleinovega tipa in teorije strun) ali pa samo "uporablja umeritvene grupe po potrebi" (kot to predpostavi veliko modelov, tudi nekateri v tem zborniku). Ali se da uganiti izvor higgsovega skalarja z metodo malih odmikov od standardnega modela, tako, denimo, da se pojavi kot psevdobozon Nambu-Goldstoneovega tipa (moznosti je se veliko več), ali pa izvirajo skalarji iz gravitačije v visjih dimenzijah, tako kot tudi umeritvena polja, in tudi naboji fermionov? Je gravitačija edino polje, s katerimi fermioni interagirajo, pri nizkih energijah pa se manifestira kot običajna gravitačija in tudi kot poznana vektorska ter skalarno Higgsovo polje? Se gravitačijo da kvantizirati? Ali je brezmasnost vseh bozonov in fermionov osnovna lastnost, masam pa so pri nizkih energijah vzrok interakčije in zlomitve simetrij? Ce so fermionskim nabojem vzrok spini fermionov v višjih dimenzijah, kolikšna je tedajdimenzija prostor-časa? Neskončna ali pa se pojavi iz nič? Se zakon narave rodi iz naključnih matematičnih struktur, ki nato v svojem razvoju porodijo odvedlijivost, difeomorfno simetrijo, lokalnost, Lorentzovo invarianco, zaradi česar se spin fermionov iz visjih dimenzijkaze v nizjih kot naboji? Zakajin kako? Podatki kazejo, da imajo polja fermionov polstevilski spin in naboje v fundamen-talni upodobitvi dosedajopazenih grup. Grassmannov prostor pa ponuja motnost, da bi imeli fermioni čelostevilski spin in naboje v adjungirani upodobitvi grupo. Lahko primerjava Grassmannovega in Cliffordovega prostora za opis notranjih prostostnih stopenj fermionov, obravnavana v tem zborniku, ponudi razlago za drugo kvantizačijo fermionov in pomaga bolje razumeti "elegančo zakonov narave"? Ce bi narava " izbrala" za opis notranjih prostostnih stopenjGrassmannovo algebro namesto Cliffordove, bi vsi atomi, molekule in posledično čel svet izgledali poplnoma drugače. Zakajje"izbrala" Cliffordovo algebro? Ali je domneva, da je "vsa matematika del narave" sprejemljiva in jo je potrebno vzeti resno? Za razvoj skladne teorije potrebujemo in zato uporabljamo veliko različnih matematičnih končeptov, vendar ostaja veliko vprasanjodprtih. Ker je vsako leto le malo časa od delavniče do zaključka redakčije, manj kot dva meseča, avtorji ne morejo izpiliti prispevkov, vendar upamo, da to nadomesti svezina prispevkov. Cetudi so k uspehu „Blejskih delavnič" največ prispevali udelezenči, ki so na Bledu omogočili prijateljsko in aktivno izmenjavo mnenj v osrčju slovenskih Julijčev, so k uspehu prispevale tudi videokonferenče, ki so povezale delavniče z laboratoriji po svetu. Vprasanja in odgovori ter tudi predavanja, ki jih je v zadnjih letih omogočil M.Yu. Khlopov preko Virtual Institute of Astropartičle Physičs (viavča.in2p3.fr/site.html, APC, Pariz), so v izčrpnih diskusijah pomagali razčistiti marsikatero vprasanje. Na letosnji delavniči je "pretočno predvajanje" omogočilo, da so vsa predavanja in diskusije tekle tudi preko spleta. Braleč najde zapise vseh predavanj, objavljenih preko "čosmovia" od leta 2009, na viavča.in2p3.fr/site.html v povezavi Previous - Conferenčes. Letosnja predavanja na čosmoviji so prispevali: E. Kiritsis (Emergent gravity (from hidden sečtor)), Maxim Yu. Khlopov (Conspiračy of BSM Physičs and BSM Cosmology), Norma Mankoč Borštnik (Experimental čonsequenčes of spin-čharge family theory). Cosmovia predstavlja prvikrat študentsko predavanje, imel ga je Valerij Nikulin (Inflationary limits on the size of čompačt extra spače). http://viavča.in2p3.fr/what_čomes_beyond _the_standard_modeLXXIEhtml Večino predavanjnajde braleč na spletni strani delavniče na http://bsm.fmf.uni-lj.si/. Najzaključimo ta predgovor s prisrčno in toplo zahvalo vsem udelezenčem, prisotnim na Bledu osebno ali preko videokonferenč, za njihova predavanja in se posebno za zelo plodne diskusije in odlično vzdusje. Zahvaljujemo se tudi revijam "Symmetry" in "Particles" založbe MDPI za podporo pri potovalnih stroških za mlade in starejše udeležence delavnice ter upamo, da se bo to sodelovanje lahko nadaljevalo in se razsirilo. Norma Mankoč Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (Organizacijski odbor) Norma MankoC Borštnik, Holger Bech Nielsen, Dragan Lukman, (uredniki) Ljubljana, grudna (decembra) 2019 Talk Section All talk contributions are arranged alphabetically with respect to the authors' names. Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 1) Bled, Slovenia, July 6-14, 2019 1 Corollary Analyses After the Recent Model-Independent Results of DAMA/LIBRA-Phase2 * R. Bernabeia'b, P. Bellia'b, F. Cappellac'd, V. Caraccioloe, R. Cerullia'b, C.J. Daif, A. d'Angeloc'd, A. Di Marcob, H.L. Hef, A. Incicchittic'd, X.H. Maf, V. Merloa'b, F. Montecchiab'9, X.D. Shengf, Z.P. Yef'h aDip. di Fisica, Universita di Roma "Tor Vergata", Rome, Italy bINFN, sez. Roma "Tor Vergata", Rome, Italy cDip. di Fisica, Universita di Roma "La Sapienza", Rome, Italy dINFN, sez. Roma, Rome, Italy eINFN Laboratori Nazionali del Gran Sasso, Assergi (AQ), Italy f Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, P.R. China gDip. Ingegneria Civile e Ingegneria Informatica, Universita di Roma "Tor Vergata", Rome, Italy hUniversity of Jinggangshan, Ji'an, Jiangxi, P.R. China Abstract. The first DAMA/LIBRA-phase2 model-independent results (exposure: 1.13 ton x yr, and software energy threshold at 1 keV) have recently been released. They further confirm — with high confidence level — the evidence already observed by DAMA/NaI and DAMA/LIBRA-phasel on the basis of the exploited model-independent Dark Matter (DM) annual modulation signature. The total exposure above 2 keV of the three experiments is 2.46 ton x yr. Here several DM candidate particles and related scenarios are analyzed including the latest results. These analyses permit to constrain the parameters' space of the considered candidates in the given scenarios, restricting their values with respect to previous analyses thanks to the increase of the exposure and to the lower energy threshold. Povzetek. Avtorji prispevka so pred nedavnim objavili zadnje analize svojih večletnih opazovanj temne snovi na eksperimentu DAMA/LIBRA, ki so jim dodali zadnjo fazo, Fazo 2. Ta zadnja faza ima skupno ekspozicijo 1.13 ton x let in energijski prag 1 keV. Avtorji zagotavljajo, da so rezultati poskusov, ki temeljijo na letni modulaciji signalov, neodvisni od modelov, ki poskusajo razloziti izvor temne snovi in njihovo detekcijo. Skupna ekspozicija obeh faz poskusov, Faza 1 je imela energijski prag 2 keV, je 2.46 ton x let. Zadnji rezultati potrjujejo z visoko zanesljivostjo rezultate prejsnje faze poskusa (DAMA/NaI and DAMA/LIBRA-Faza 1). Avtorji analizirajo ustreznost modelov glede na njihove meritve, ter omejijo prostor parametrov obravnavanih modelov. Keywords: Scintillation detectors, elementary particle processes, Dark Matter * Talk presented by F. Cappella 2 R. Bernabei et al. 1.1 Introduction Recently the model-independent results of the first six full annual cycles measured by DAMA/LIBRA-phase2 with a software energy threshold of 1 keV 1 [1,2] have been released [3-7]. The model-independent evidence for the presence of DM particles in the galactic halo is further confirmed on the basis of the exploited DM annual modulation signature after the previous DAMA/LIBRA-phasel [1,2,8-14] and the former DAMA/Nal [15,16] experiments. The cumulative Confidence Level (C.L.) is increased from the previous 9.3 ct (data from 14 independent annual cycles for an exposure of 1.33 ton x yr) to 12.9 ct (data from 20 independent annual cycles for an exposure of 2.46 ton x yr). We recall that the expected DM particles differential counting rate depends on the Earth's velocity in the galactic frame: vE(t) = vq + vecosYCos^(t — to), where the Sun velocity with respect to the galactic halo is vQ — v0 + 12 km/s with v0 local velocity), and ve — 30 km/s is the Earth's orbital velocity around the Sun on a plane with inclination y = 60o with respect to the galactic one. Moreover, 2n/T with T = 1 year and roughly t0 — June 2nd (when the Earth's speed in the galactic halo is at maximum). Thus, the expected counting rate averaged in a given energy interval can be conveniently worked out through a first order Taylor expansion: S (t) = S0 + Smcos^(t — t0), (1.1) with the contribution from the highest order terms being less than 0.1 %. The Sm and S0 are the modulation amplitude and the un-modulated part of the expected differential counting rate, respectively. In the DAMA experiments the experimental observable is the modulation amplitude, Sm, as a function of the energy, and the identification of the constant part of the signal, S0, is not required to point out the presence of a signal in the exploited model-independent annual modulation approach. It has several advantages; in particular, the only background of interest is the one able to mimic the signature, i.e. able to account for the whole observed modulation amplitude and to simultaneously satisfy all its many specific peculiarities (see e.g. Ref. [5]). No background of this sort has been found, see Refs. [2-13]. The modulation amplitudes, Sm, for the whole data sets: DAMA/Nal, DAMA/ LIBRA-phase1 and DAMA/LIBRA-phase2 (total exposure 2.46 tonxyr) are plotted in Fig. 1.1; the data below 2 keV refer only to the DAMA/LIBRA-phase2 exposure (1.13 tonxyr). It can be inferred that positive signal is present in the (1-6) keV energy interval, while Sm values compatible with zero are present just above [5]. In the following the implications on some models, we already investigated with lower exposure and higher software energy threshold in the past, are updated by including the data of DAMA/LIBRA-phase2 [17]. 1 Throughout this paper: i) keV means keV electron equivalent, where not otherwise specified; ii) ton means metric ton (1000 kg). 1 Corollary Analyses After the Recent Model-Independent... 0.05 > la 0.025 I u wg0.025 ITi -0.05 0 0 2 4 6 8 10 12 14 16 18 20 Energy (keV) 3 Fig. 1.1. Modulation amplitudes, Sm, for the whole data sets: DAMA/Nal, DAMA/LIBRA-phasel and DAMA/LIBRA-phase2 (total exposure 2.46 tonxyr) above 2 keV; below 2 keV only the DAMA/LIBRA-phase2 exposure (1.13 ton x yr) is available and used. The energy bin AE is 0.5 keV. A clear modulation is present in the lowest energy region, while Sm values compatible with zero are present just above. 1.2 Data analysis The corollary analyses presented in the following are model-dependent; thus, it is important to point out at least the main topics which enter in the determination of the results and the related uncertainties. These arguments have been already addressed at various extents in previous corollary model-dependent analyses. The DM candidates considered here have been previously discussed in the Ref. [17] and references therein. A specific phase-space distribution function (DF) in the galactic halo has to be adopted in order to derive the allowed regions of the parameter's space for the considered DM particles and scenarios. A large number of possibilities is available in literature; these models are continuously in evolution thanks to new simulations and astrophysical observations, as the recent GAIA ones (see e.g. Refs. [18,19] and references therein). Thus, large uncertainties in the predicted theoretical rate are present. Here, to account at some extent for the uncertainties in halo models and to allow direct comparisons, the same not-exhaustive set of halo models as in previous published analyses [15,16,20], is considered; they are illustrated in Table II of Ref. [20]. In particular, the considered classes of halo models correspond to: (1) spherically symmetric matter density with isotropic velocity dispersion (Class A); (2) spherically symmetric matter density with non-isotropic velocity dispersion (Class B); (3) axisymmetric models (Class C); (4) triaxial models (Class D); (5) moreover, in the case of axisymmetric models it is possible to include either an halo co-rotation or an halo counter-rotation. We also consider the physical ranges of the local velocity v0: from 170 km/s to 270 km/s, and of the local total DM density, p0. For p0, its minimal, pmin, and its maximal, pmax, values are estimated imposing essentially two astrophysical constraints: one on the amount of non-halo components and the other on the flatness of the rotational curve in the Galaxy. The values for pmin and pmax are related to the DF and the considered v0; they are reported in Table III of Ref. [20]. The halo density p0 ranges from 0.17 to 0.67 GeV/cm3 for v0 = 170 km/s, while p0 ranges from 0.29 to 1.11 GeV/cm3 for v0 = 220 km/s, and p0 ranges from 0.45 to 4 R. Bernabei et al. 1.68 GeV/cm3 for v0 = 270 km/s, depending on the halo model. Moreover, to take into account that the considered DM candidate can be just one of the components of the dark halo, the £ parameter is introduced; it is defined as the fractional amount of local density in terms of the considered DM candidate (£ < 1). Thus, the local density of the DM particles is pDM = £p0. Finally, the DM escape velocity, vesc, from the galactic gravitational potential is considered; actually, it is also affected by significant uncertainty. In the following vesc = 550 km/s is adopted as often considered in literature; however, no sizable differences are observed in the final results when vesc values ranging from 550 to 650 km/s are considered; in fact, for low-mass DM particles scattering off nuclei, the Na contribution is dominant and has a small dependence on the tail of the velocity distribution. We note that the possible presence of non-virialized components, as streams in the dark halo coming from external sources with respect to our Galaxy [21-23] or other scenarios as e.g. that of Ref. [24-26], are not included in the present analyses. In the interaction of DM particles in the NaI(Tl) detectors the detected energy, Edet, is a key quantity. It is connected with the energy released by the products of the interaction, Erel; two possibilities exist: 1) the products of the interaction have electromagnetic nature (mainly electrons); 2) a nuclear recoil with ER kinetic energy is produced by the DM particle scattering either off sodium or off iodine nucleus. Since, the detectors are calibrated by using y sources, in the first case Edet = Erel, while in the second case a quenching factor (q.f.) for each recoiling nucleus must be included: Edet = qNa,i x Erel. In literature there are available a lot of measurements on the Na and I q.f.'s, that show a wide spread, since they are a property of the specific detector and not general properties of any NaI(Tl), particularly in the very low energy range. The same procedures previously adopted in Refs. [27-30] are considered here, i.e. the following three instances are accounted for: • (QI) Na and I q.f.'s "constants" with respect to the recoil energy ER: the adopted values are qNa = 0.3 and qI = 0.09, measured with neutron source integrating the data over the 6.5 - 97 keV and the 22 - 330 keV recoil energy range, respectively [31]; • (QII) quenching factors depending on ER, evaluated as in Ref. [32]; • (Qm) quenching factors with the same behavior of Ref. [32], but normalized in order to have their mean values consistent with QI in the energy range considered there. Another important effect is the channeling of low energy ions along axes and planes of the NaI(Tl) DAMA crystals. This effect can lead to a further important deviation, in addition to the uncertainties discussed in section II of Ref. [27] and in Ref. [28]. In fact, the channeling effect in crystals implies that a fraction of nuclear recoils are channeled and experience much larger q.f.'s than those derived from neutron calibration (see Refs. [33,27] for a discussion of these aspects). Anyhow, the channeling effect in solid crystal detectors is not a well fixed issue and there could be several uncertainties in the modeling. Because of the difficulties of experimental measurements and of theoretical estimate of the channeling effect, in the following 1 Corollary Analyses After the Recent Model-Independent... 5 it will be either included using the procedure given in Ref. [33] or not in order to give idea on the related uncertainty. Finally, three discrete cases are considered in the following to cautiously account for possible uncertainties on the quenching factors measured by DAMA in its detectors and on the parameters used in the SI and SD nuclear form factors [17]: • Set A considers the mean values of the parameters of the used nuclear form factors [15] and of the quenching factors. • Set B adopts the same procedure as in Refs. [34,35,16], by varying (i) the mean values of the 23Na and 127I quenching factors as measured in Ref. [31] up to +2 times the errors; (ii) the nuclear radius, rA, and the nuclear surface thickness parameter, s, in the SI nuclear form factor from their central values down to -20%; (iii) the b parameter in the considered SD nuclear form factor from the given value down to -20%. • Set C where the iodine nucleus parameters are fixed at the values of set B, while for the sodium nucleus one considers [15]: (i) 23Na quenching factor at qNa = 0.25; (ii) the nuclear radius, rA, and the nuclear surface thickness parameter, s, in the SI nuclear form factor from their central values up to +20%; (iii) the b parameter in the considered SD nuclear form factor from the given value up to +20%. In conclusion, model-dependent analyses through a maximum likelihood procedure, which also takes into account the energy behavior of each detector, can be pursued. In particular, for each considered scenario, the allowed domains in the corresponding parameters' space will be obtained by marginalizing over the halo models, over halo parameters (v0 and po) and over the sets A, B, C2. This procedure shows the impact of the uncertainties in the astrophysical, nuclear and particle physics on the model-dependent analyses. However, for simplicity the allowed regions in the parameters' space of each considered scenario can also be derived by comparing - for each k-th energy bin of 1 keV - the measured DM annual modulation amplitude, S^XP ± 3, with the theoretical expectation in each considered framework, Smhk. Of course, the Smhk values depend on the free parameters of the model 0, such as the DM particle mass, the cross section, etc., on the uncertainties accounted for, on the proper accounting for the detector's features, and on priors. In particular, as mentioned in previous works (as e.g. recently in Refs. [28,29]), a cautious prior on S0,k - assuring safe and more realistic allowed regions/volumes - can be worked out from the measured counting rate in the cumulative energy spectrum; the latter is given by the sum of the un-modulated background contribution bk (whose existence is shown by the detailed analyses on residual radioactive contaminations in the detectors [8]) and of the constant part of the signal S0,k. By adopting a standard procedure, used in the past in several low background 2 In particular, each allowed domain encloses all the allowed regions obtained for each chosen configuration of model and parameters. 3 The distributions of the measured modulation amplitudes around their mean value show a perfect Gaussian behaviors, justifying the use of a symmetric uncertainty [9,2,11,3,5]. 6 R. Bernabei et al. fields, one can derive lower limits on bk and, thus, upper limits on S0,k (Smk1*)-In particular, in DAMA/LIBRA-phase2 is obtained: S0 <0.80 cpd/kg/keV in the (1-2) keV energy interval; S0 <0.24 cpd/kg/keV in (2-3) keV, and S0 <0.12 cpd/kg/keV in (3-4) keV. ~ ~ Thus, the following x2 can be calculated for each considered model: /<-.exp _ ^th. Cfll)2 /cmax _ oth. r0"l)2 X2(0) = £ lSm,k Sm,k(0)J + £ lS°,a ¿OJc'(0JJ 0 (sothk, (0) - sj^aX) k k' CT0,k' , , (1.2) where the second term encodes the experimental bounds about the un-modulated part of the signal; cr0,k/ ~ 10-3 cpd/kg/keV, 0 is the Heaviside function, and Sft.' is the average expected signal counting rate in the k' energy bin. The sum in the first term in eq. 1.2 runs here from 1 keV to 20 keV. The x2 defined in eq. (1.2) can be calculated in each considered framework and is function of the model parameters 0. Thus, we can define: ax2(0 )= x2 (0)- x2 (1.3) where x2 is the x2 for 0 values corresponding to absence of signal. The Ax2 is used to determine the allowed intervals of the model parameters 0 at 10 ct from the null signal hypothesis. We have verified that the Qui option for the quenching factors provides results similar to the case of the QI option; thus, to avoid the overloading of the figures in the following the QIII case is not considered. 1.3 Updated corollary model-dependent scenarios 1.3.1 DM particles elastically interacting with target nuclei A lot of candidates have been proposed in theory extending the Standard Model of particles that includes candidates for DM elastically scattering off target nuclei. In the DM particle-nucleus elastic scattering, the differential energy distribution of the recoil nuclei can be calculated by means of the differential cross section of the DM-nucleus elastic process [31,36,15,16,33]. The latter is given by the sum of two contributions: the SI and the SD one. In the purely SI case, the nuclear parameters can be decoupled from the particle parameters and the nuclear cross sections, which are derived quantities, are usually scaled to a defined point-like SI DM particle-nucleon cross section, oSI. In principle, this procedure could allow - within a framework of several other assumptions (that in turn introduce uncertainties in final evaluations) - a model-dependent comparison among different target nuclei, otherwise impossible. In the following, the usually considered coherent scaling law for the nuclear cross sections is adopted: osi(A,Z) « m2ed(A, DM) [fpZ + fn(A - Z)]2 , (1.4) 1 Corollary Analyses After the Recent Model-Independent... 7 where osi (A, Z) is the point-like cross section of DM particles scattering off nuclei of mass number A and atomic number Z, mred (A, DM) is the reduced mass of the system DM particle and nucleus, fp and fn are the effective DM particle couplings to protons and neutrons, respectively. The case of isospin violation fp _ fn will be discussed in Sect. 1.3.1; now we assume fp = fn and, thus, we can write: o (a z)_ mr2ed(A,DM) a2o (1 5) OSl(A,Z)= mr2ed(1,DM) A °SI- (1.5) As for nuclear SI form factors, the Helm form factor [37,38] has been adopted4 (for details on the used form factors see Ref. [15]). As described above, some uncertainties on the nuclear radius and on the nuclear surface thickness parameters in the Helm SI form factors have been included in the following analysis by considering three discrete cases, labeled as set A, B, and C in Sect. 1.2. The purely SD case is even more uncertain since the nuclear and particle physics degrees of freedom cannot be decoupled and a dependence on the assumed nuclear potential exists. Also in the purely SD case all the nuclear cross sections are usually scaled to a defined point-like SD DM particle-nucleon cross section, osd [34,15]. The adopted scaling law for this case profits of the proportionality of the SD nuclear cross section to the nuclear spin factor Ar J(J + 1) and to the squared reduced mass. To take into account the finiteness of the nucleus, a SD nuclear form factor is also used; for details of its parametrization used in the following see Ref. [15]. Moreover, a further parameter must be introduced; in fact, following the notations reported in Ref. [34]: tan0 _ ^, where ap,n are the effective DM-nucleon coupling strengths for SD interactions. The mixing angle 0 is defined in the [0, n) interval; in particular, 0 values in the second sector account for ap and an with different signs. Therefore, further significant uncertainties in the evaluation of the SD interaction rate also arise from the adopted spin factor for the single target-nucleus. In fact, the available calculated values are well different in different models (and differently vary for each nucleus) and, in addition, at fixed model they depend on 0 [34,15]. It is worth noting that for the SD part of the interaction not only the target nuclei should have spin different from zero (for example, this is not the case of Ar isotopes, and most of the Ca, Ge, Te, Xe, W isotopes) to be sensitive to DM particles with a SD component in the coupling, but also well different sensitivities can be expected among odd-nuclei having an unpaired proton (as e.g. r3Na and 127I, and 1H, 19F, 27Al,133Cs) and odd-nuclei having an unpaired neutron (as e.g. the odd Xe and Te isotopes and r9Si, 43Ca, 73Ge,183W). Spin-Independent interaction For the purely SI scenario in the considered model frameworks the allowed region in the plane mDM and £,osi have been calculated and are shown in Fig. 1.2. Of course, best fit values of cross section and DM mass span over a large range in the considered model frameworks. 4 It should be noted that the Helm form factor is the least favorable one e.g. for iodine and requires larger SI cross-sections for a given signal rate; in case other form factor profiles, 8 R. Bernabei et al. ■c a O 10 10 ' 10 ' 10 ' 10 ' 10 ' 10 mDM (GeV) 10 Fig. 1.2. Regions - allowed at 10 a from absence of signal - in the nucleon cross-section vs DM particle mass plane allowed by DAMA experiments in the case of a DM candidate elastically scattering off target nuclei and SI interaction. Three different instances for the Na and I quenching factors have been considered: (i) Qi case [(green on-line) vertically-hatched region], (ii) with channeling effect [(blue on-line) horizontally-hatched region)] and (iii) QII [(red on-line) cross-hatched region]. 1 The allowed domains in Fig. 1.2 are obtained by marginalizing all the models for each considered scenario (see Sect. 1.2); they represent the domains where the likelihood-function values differ more than 10 ct from absence of signal. The three different instances described above for the Na and I quenching factors have been considered: (i) Qi case, (ii) with channeling effect, and (iii) Qn. When comparing with the previous results obtained with DAMA/NaI [15] and DAMA/LIBRA-phase1 [11] data, one can derive that: 1) the C.L. associated to the allowed regions is improved; 2) the allowed regions are restricted (i.e. several configurations are no more supported by the cumulative data at the given C.L.); 3) in the QI and QII cases the low and high mass regions, driven by the Na and I nuclei, respectively, are disconnected; 4) including the channeling effect the lower available mass is 4 GeV, instead of 2 GeV as in the previous analysis [27,2]. In conclusion, the purely SI scenario is still supported by the data both for low and high mass candidates; the inclusion of channeling effect also offers stringent agreement in many considered SI scenarios. Candidates with isospin violating SI coupling To study the case of a DM candidate with SI isospin violating interaction, where fp = fn, a third parameter, namely the ratio fn/fp, must be considered together with £,o-SI and mDM. Obviously the previous case of isospin conserving is restored whenever the ratio fn/fp = 1. considered in the literature, would be used, the allowed parameters' space would extend [15]. 1 Corollary Analyses After the Recent Model-Independent... 10 11dm (GeV) 10 mau (GeV) 10 'dm (GeV) Fig. 1.3. Regions in the fn/fp vs mDM plane allowed by DAMA experiments in the case of a DM candidate having isospin violating SI interaction. The Na and I quenching factors are: Qi [left (green on-line)], Qn [center (red on-line)], and with channeling effect [right (blue on-line)]. The considered halo is A0 (isothermal sphere) with the vo and po in the range of Table III of Ref. [20]. The three possible sets of parameters A, B and C are considered (see Sect. 1.2). The color scales give the confidence level in units of a from the null hypothesis. 9 0 The results of the analysis for a single halo model hypothesis described later are reported in Fig. 1.3, where the allowed regions in the fn/fp vs mDM plane are shown after marginalizing on £,aSI. For simplicity the halo model A0 (isothermal sphere) with the v0 and p0 in the range of Table III of Ref. [20], and three choices of the Na and I quenching factors: QI, QII, and including the channeling effect are considered. Typically, few considerations can be done: • Two bands of mDM can be recognized, as expected: one at low mass and the other at higher mass. • The low mass DM candidates have a good fit in correspondence of fn/fp — -53/74 = -0.72, where the 127I contribution vanishes and the signal is mostly due to 23Na recoils. • Similarly, at larger mass fn/fp — -0.72 is instead disfavored. • The case of isospin-conserving fn/fp = 1 is well supported at different extent both at lower and larger mass. • When the channeling effect is included (panels on the right of Fig. 1.3), the case of fn/fp = 1 at low mass has even a stronger support, that is higher confidence level. • Contrary to what was stated in Ref. [39-41] where the low mass DM candidates were disfavored for fn/fp = 1 by DAMA data, the inclusion of the uncertainties related to halo models, v0 and p0, quenching factors, channeling effect, nuclear form factors, etc., and correctly accounting for other aspects, can also support low mass DM candidates either including or not the channeling effect. In conclusion, at present level of uncertainties the DAMA data, if interpreted in terms of DM particle inducing nuclear recoils through SI interaction, can account either for low and large DM particle mass and for a wide range of the ratio fn/fp, even including the "standard" case fn/fp = 1. 10 R. Bernabei et al. Spin-Dependent interaction The purely SD interaction, to which Na and I nuclei are fully sensitive, can also be considered. The complete results would be described by a 3-dimensional volume: (£,osd, mDM, 9). Thus, a very large number of possible configurations are available; here for simplicity we show, as examples, the results obtained only for 4 particular couplings, which correspond to the following values of the mixing angle 9: (i) 9 = 0 (an = 0 and ap = 0 or |ap| > |aj); (ii) 9 = n/4 (ap = au); (iii) 9 = n/2 (an = 0 and ap = 0 or |an| > |ap|); (iv) 9 = 2.435 rad (an/ap = -0.85, pure Z0 coupling). The case ap = —an is nearly similar to the case (iv). e = o e = n/4 e = n/2 e = 2.435 pQ a 10 10 1 Q -.1 u?5 10 D -2 JJL^ 10 10 10 1 10 1 10 1 10 mDM (GeV) 1 Fig. 1.4. Slices of the 3-dimensional volume (£,osd, mDM, 9) allowed at 10 o from absence of signal by the DAMA experiments in the case of a DM candidate elastically scattering off target nuclei and SD interaction. Three different instances for the Na and I quenching factors have been considered: (i) Qi case [(green on-line) vertically-hatched region], (ii) with channeling effect [(blue on-line) horizontally-hatched region)] and (iii) QII [(red on-line) cross-hatched region]. In Fig. 1.4 slices of the 3-dimensional allowed volume (£,osd, mDM, 9) at 10 o from absence of signal are shown. For each configuration three regions are depicted accounting for the quenching factors uncertainties. Finally, Fig. 1.5 shows the allowed regions in the tan9 vs mDM plane after marginalizing on £,osd. For simplicity the halo model A0 (isothermal sphere) with the v0 and p0 in the range of Table III of Ref. [20], and three choices of the Na and I quenching factors: QI, QII, and including the channeling effect are considered. In conclusion, the purely SD scenarios are in good agreement with the DAMA results and can explain the different capability of detection among targets with different unpaired nucleon. The large uncertainties e.g. in the spin factor also offer additional space for compatibility among different target nuclei. Mixed coupling framework The most general case is when both SI and SD couplings are considered. Details of related calculations can be found in Ref. [34,15]. In this scenario, both the uncertainties on the SI and SD frameworks have to be accounted. The complete result is given by a 4-dimensional allowed volume: (£,osi, £osd, mDM, 9). The isospin violating SI interaction is not included hereafter. 1 Corollary Analyses After the Recent Model-Independent... 11 mDM (GeV) mDM (GeV) mDM (GeV) Fig. 1.5. Regions in the ton.0 vs mDM plane allowed by DAMA experiments in the case of a DM candidate with SD interaction. The Na and I quenching factors are: Qi [left (green on-line)], QII [center (red on-line)], and with channeling effect [right (blue on-line)]. The considered halo is A0 (isothermal sphere) with the vo and po in the range of Table III of Ref. [20]. The three possible sets of parameters A, B and C are considered (see Sect. 1.2). The color scales give the confidence level in units of a from the null hypothesis. e = o e = n/4 e = n/2 e = 2.435 -Q 10-2 & _ ^aSD (pb) Fig. 1.6. Slices of the 4-dimensional volume (£,asI, £,osd, mdm, 0) allowed by all DAMA experiments in the case of a DM candidate with elastic scattering off target nuclei and mixed SI and SD interaction. Three different instances for the Na and I quenching factors have been considered: (i) QI case [(green on-line) vertically-hatched region], (ii) with channeling effect [(blue on-line) horizontally-hatched region] and (iii) QII [(red on-line) cross-hatched region]. Few examples of slices (£,ctsi, £,ctsd) at 10 a from the null hypothesis (absence of modulation) are shown in Fig. 1.6 for some values of 0 and mDM = 10 GeV. Obviously, the proper accounting for the complete 4-dimensional allowed volume and the existing uncertainties and complementarity largely extend the results and any comparison. Finally, let us now point out that configurations with £,ctsi (£,ctsd) even much lower than those shown in Fig. 1.2 (Fig. 1.4) would be possible if a small SD (SI) contribution would be present in the interaction. This possibility is clearly pointed out in Fig. 1.7 where some examples of regions in the plane £,ctsi vs mDM are reported. Similar plots can be obtained for the £,ctsd vs mDM case (see Ref. [17]). As it can be seen, these arguments clearly show that even a relatively small SD (SI) contribution can drastically change the allowed region in the (mDM, £,ctsi(sd]) plane; therefore, the typically shown model-dependent comparison plots between exclusion limits at a given C.L. and regions of allowed parameter space do not hold e.g. for mixed scenarios when comparing experiments with and without sensitivity to the SD component of the interaction. The same happens when comparing 12 R. Bernabei et al. -°SD = 0 pb °SD 0.02 pb °SD 0.04 pb °SD 0.05 pb °SD = 0.06 pb ...........°SD 0.08 pb - 1 -4 10 -7 10 / 10 102 10 102 10 102 mDM (GeV) mDM (GeV) mDM (GeV) Fig. 1.7. An example of the effect induced by the inclusion of a SD component different from zero on allowed regions given in the plane £,ffsi vs mDM. In this example the B1 halo model with vo = 170 km/s and po = 0.42 GeV/cm3, the set of parameters A and the particular case of 0 = 0 for the SD interaction have been considered. The used quenching factors are QI (left), QII (center) and with channeling effect (right). From top to bottom the contours refer to different SD contributions: ctsd = 0 pb (solid black line), 0.02 pb, 0.04 pb, 0.05 pb, 0.06 pb and 0.08 pb. Analogous situation is found for the other model frameworks. regions allowed by experiments whose target-nuclei have unpaired proton with exclusion plots quoted by experiments using target-nuclei with unpaired neutron when the SD component of the interaction would correspond either to 0 ~ 0 or e ~ n. 1.3.2 DM particles with preferred electron interaction Some extensions of the standard model provide DM candidate particles, which can have a dominant coupling with the lepton sector of the ordinary matter. Thus, such DM candidate particles can be directly detected only through their interaction with electrons in the detectors of a suitable experiment, while they cannot be studied in those experimental results where subtraction/rejection of the electromagnetic component of the experimental counting rate is applied5. These candidates can also offer a possible source of the 511 keV photons observed from the galactic bulge. This scenario was already investigated by DAMA with lower exposure [42]. The analyses updated by including the new data of the first six annual cycles of DAMA/LIBRA-phase2 with lower software energy threshold is reported in Ref. [17]. The lower energy threshold achieved by DAMA/LIBRA-phase2 at 1 keV prevents to find configurations for these DM candidates distant more than 10 ct from the null hypothesis. However, allowed regions can be found when lowering the number of required ct [17]. This is an example how to disentangle among some scenarios, improving the sensitivity of the set-up. 5 If the electron is assumed at rest, considering the DM particle velocity, the released energy would be of order of few eV, well below the detectable energy in any considered detector in the field. However, the electron is bound in the atom and, even if the atom is at rest, the electron can have non-negligible momentum, as shown in Ref. [42]. 1 Corollary Analyses After the Recent Model-Independent... 13 1.3.3 Inelastic Dark Matter Another scenario regards the inelastic DM: relic particles that cannot scatter elasti-cally off nuclei. Following an inelastic scattering off a nucleus, the kinetic energy of the recoiling nucleus is quenched and is the detected quantity. As discussed in Refs. [43-45,35], the inelastic DM could arise from a massive complex scalar split into two approximately degenerate real scalars or from a Dirac fermion split into two approximately degenerate Majorana fermions, namely x+ and X-, with a 6 mass splitting. In particular, a specific model featuring a real component of the sneutrino, in which the mass splitting naturally arises, has been given in Ref. [43]. JAJ1 8(keV) 8(keV) Fig. 1.8. Slices of the 3-dimensional volume (£,op, 6, mom) allowed by DAMA experiments in the case of a DM candidate with preferred inelastic interaction. Three different instances for the target nuclei quenching factors have been considered: (i) Qi case [(green on-line) vertically-hatched region], (ii) with channeling effect [(blue on-line) horizontally-hatched region] and (iii) QII [(red on-line) cross-hatched region]. In the right plots the inelastic scattering off thallium nuclei is also included; here the regions due to inelastic scattering only off Na and I nuclei, already shown on the left, are reported in (yellow on-line) light-filled. 1 10 1 io 10 10 10 10 1 10 10 10 10 10 1 10 10 10 4 = 10 10 1 3 TeV TeV 10 10 10 0 100 200 100 200 300 0 100 200 100 200 300 The discussion of the theoretical arguments on such inelastic DM can be found e.g. in Ref. [43], where it was shown that for the x- inelastic scattering off target nuclei a kinematic constraint exists which favors heavy nuclei (such as 127I) with respect to lighter ones (such as e.g. natGe) as target-detectors media. In fact, x-can only inelastically scatter by transitioning to x+ (slightly heavier state than x- ) and this process can occur only if the x- velocity, v, is larger than: Vthr = W-^-7, (1.6) V mred(A,x) 14 R. Bernabei et al. where mred(A,x) is the x—nucleus reduced mass. This kinematic constraint becomes increasingly severe as the nucleus mass, mN, is decreased [43]. For example, if 6 > 100 keV, a signal rate measured e.g. in Iodine will be a factor about 10 or more higher than that measured in Ge [43]. Moreover, this model scenario implies some characteristic features when exploiting the DM annual modulation signature since it gives rise to an enhanced modulated component, Sm, with respect to the un-modulated one, S0, and to largely different behaviors with energy for both S0 and Sm (both show a higher mean value) [43] with respect to elastic cases. Details of calculation procedures can be found in Ref. [35]. Accounting for the uncertainties mentioned above, in the inelastic DM scenario an allowed 3-dimensional volume in the space (£,ap, mDM, 6) is obtained. Here, following the notation of Ref. [35], ctp is a generalized SI point-like x—nucleon cross section and mDM is the x mass. For simplicity, Fig. 1.8 left shows slices of such an allowed volume at 10 ct from the null hypothesis for some values of mDM; the different cases of quenching factors are considered as well. It can be noted that when mDM ^ mN, the expected differential energy spectrum is trivially dependent on mDM and, in particular, it is proportional to the ratio between £,ctp and mDM. Thus, allowed regions for other mDM > mN can be obtained from the last panel of Fig. 1.8, straightforward. Significant enlargement of such regions should be expected when including complete effects of model (and related experimental and theoretical parameters) uncertainties. mDM (GeV) mDM (GeV) mDM (GeV) Fig. 1.9. Regions in the 6 vs mDM plane allowed by DAMA experiments in the case of a DM candidate with preferred inelastic interaction. The Na and I quenching factors are: Qi [left (green on-line)], QII [center (red on-line)], and with channeling effect [right (blue on-line)]. The considered halo is A0 (isothermal sphere) with the v0 and p0 in the range of Table III of Ref. [20]. The three possible sets of parameters A, B and C are considered (see Sect. 1.2). The color scales give the confidence level in units of ct from the null hypothesis. Fig. 1.9 shows the allowed regions in the 6 vs mDM plane after marginalizing on £,ctp . For simplicity the halo model A0 (isothermal sphere) with the v0 and p0 in the range of Table III of Ref. [20], and three choices of the Na and I quenching factors: QI, QII, and including the channeling effect are considered. It is worth noting that in the case of Inelastic DM the thallium dopant (stable isotopes with mass number 203 and 205, and natural abundances 29.5% and 70.5% respectively) can also play a role as it has been described in Ref. [46], where it has 1 Corollary Analyses After the Recent Model-Independent... 15 been shown how the DM interaction on thallium nuclei would give rise to a signal which cannot be detected with lower mass target-nuclei. This also can decouple theoretical and experimental aspects from different experiments. The slices of the 3-dimensional volume (£,ctp, 6, mDM), allowed by DAMA experiments when the inelastic scattering off thallium nuclei is also included, have been evaluated in Fig. 1.8 right marginalizing all the considered models (see Sect. 1.2). Two instances for the Tl quenching factor in NaI(Tl) are considered: (i) Qi case with qTl = 0.075, tentatively obtained by extrapolating the qNa and qI measured by DAMA with neutrons [31]; (ii) QII quenching factors varying as a function of ER evaluated as in Ref. [32]. Moreover, the thallium is assumed to be homogeneously distributed in each crystal and among the crystals at level of 0.1% in mass (corresponding to 2.95 x 1021 Tl atoms/kg). As shown in Fig. 1.8 right, new regions with £,ctp >1 pb and 6 >100 keV are allowed by DAMA after the inclusion of the inelastic scattering off thallium nuclei. Such regions are not fully accessible to detectors with target nuclei having mass lower than thallium. In conclusion, we point out that here the analysis of the inelastic DM particle has been limited only to SI coupling. Recently analyses of the inelastic DM candidate with SD coupling have been reported in Refs. [47,48]. They show that also this scenario can be compatible with the DAMA result. This conclusion can be further confirmed considering e.g. the effects of uncertainties in the models that in those papers have not been included. 1.3.4 Investigation on light dark matter Some extensions of the Standard Model provide DM candidate particles with sub-GeV mass; in the following these candidates will be indicated as Light Dark Matter (LDM). Several LDM candidates have been proposed in Warm DM scenarios, as keV-scale sterile neutrino, axino, gravitino, and MeV-scale particles (for details see Ref. [30]). In this section the direct detection of LDM candidate particles is investigated considering the possible inelastic scattering channels either off the electrons or off the nuclei of the target. Firstly we note that - since the kinetic energy for LDM particles in the galactic halo does not exceed hundreds eV - the elastic scattering of such LDM particles both off electrons and off nuclei yields energy releases hardly detectable by the detectors used in the field; this might prevent the exploitation of the elastic scattering as detection approach for these candidates. Thus, the inelastic process could be the only possible viable one for the direct detection of LDM [30]. The following process is, therefore, considered for detection: the LDM candidate (hereafter named vH with mass mH) interacts with the ordinary matter target, T, with mass mT. The target T can be either an atomic nucleus or an atomic electron depending on the nature of the vH particle interaction. As result of the interaction a lighter particle is produced (hereafter vL with mass mL < mH) and the target recoils with an energy ER, which can be detectable by suitable detectors. The lighter particle vL is neutral and it is required that it interacts very weakly with ordinary matter or not at all; thus, the vL particle escapes the detector. In particular, the vL particle can also be another DM halo component (dominant or 16 R. Bernabei et al. sub-dominant with respect to the vH one), or it can simply be a Standard Model particle (e.g. vL can be identified with an active neutrino) [30]. Since the sub-GeV LDM wavelength (A = h > 103 fm) is much larger than the nucleus size, the targets can be considered as point-like and the form factors of the targets can be approximated by one. The cross section of the processes, oT, is generally function of the LDM velocity, v, and can be written by adopting the approximation for the non-relativistic case [30]: oTv ~ a + bv2 , (1.7) where a and b are constant depending on the peculiarity of the particle interaction with the target T. In the analysis, the cross sections oT = ^ and o^ = bv0 are defined [30]; they are related to the a and b parameters rescaled with the DM local velocity, v0. In particular, the o^ is responsible for the annual modulation of the expected counting rate for LDM interactions, and in the following it will be used as free parameter, together with mH and the mass splitting A = mH — mL. Moreover, for the case of LDM interaction on nuclei, following the prescriptions given in Ref. [30], two different nuclear scaling laws are adopted: the coherent (omOh « or/ANa « o^/A2) and the incoherent (o^ 511 keV (dark area in Fig. 1.10) are instead of interest for the possible annihilation processes: vHVH —» e+e-, vHVL —} e+ e-, vLVH —} e+e- and and vLVL —» e+e- in the galactic center. Some slices of the 3-dimensional allowed volume for various mH values (including the mH = A case, that is a massless or a very light vL particle) in the (£,omm vs A) plane are reported in Ref. [17]. In conclusion, it is worthwhile to summarize that electron interacting LDM candidates in the few-tens-keV/sub-MeV range are allowed by DAMA experiments (see Fig. 1.10). This can be of interest, for example, in the models of Warm DM particles, such as e.g. weakly sterile neutrino. Moreover, configurations with 6 For values of mH greater than O(GeV), the definition of LDM is no longer appropriate. Moreover, the kinetic energy of the particle would be enough for the detection in DAMA experiments also through the elastic scattering process, as demonstrated in Ref. [42]. 1 Corollary Analyses After the Recent Model-Independent... 17 CT 10 3 0> w C 10 2 10 1 Fig. 1.10. Projection of the allowed 3-dimensional volume on the plane (mH, A) for electron interacting LDM. The dashed line (mH = A) marks the case where vL is a massless particle. The decay through the detection channel, vH —> vLe+e-, is energetically not allowed for the selected configurations. The configurations with mH >me (dark area) are interesting for the possible annihilation processes: vHVh —> e+e-, vHVl —> e+e-, vLVh —> e+e- and vLVl —> e+ e- in the galactic center. The three nearly vertical curves are the thresholds of these latter processes as mentioned in Ref. [30]. mH in the MeV/sub-GeV range are also allowed; similar LDM candidates can also be of interest for the production mechanism of the 511 keV gammas from the galactic bulge. Interaction with nuclei With regard to the interaction of LDM with target nuclei, an allowed volume can be obtained in the space (mH, A, ^am11^1615). The projections of such a region on the plane (mH, A) are reported in Fig. 1.11 for the two above-mentioned illustrative cases of coherent and incoherent nuclear scaling laws. They have been obtained by marginalizing all the models for each considered scenario (see Sect. 1.2) and they represent the domain where the likelihood-function values differ more than 10 a from the null hypothesis (absence of modulation). The allowed mH values and the splitting A are in the intervals 8 MeV 2me (dark area in Fig. 1.11), while the annihilation processes into e+e- pairs are energetically allowed for almost all the allowed configurations. It is worth noting that for nuclear interacting LDM the 3-dimensional allowed configurations are contained in two disconnected volumes, as seen e.g. in their projections in Fig. 1.11. The one at larger A at mH fixed is mostly due to interaction on Iodine target, while the other one is mostly due to interaction on Sodium target. !U mH (keV) 18 i? 154 R. Bernabei et al. ¡j < 10 5 10 10 10 mH (keV) 4 5 6 10 10 10 mH (keV) 10 10 mH (keV) 105 106 mH (keV) 10 10 mH (keV) s 105 106 mH (keV) Fig. 1.11. Case of nucleus interacting LDM. Projections of allowed 3-dimensional volumes on the plane (mH, A) for coherent (top) and incoherent (bottom) nuclear scaling law, considering for the quenching factors: (i) QI case (left), (ii) with channeling effect (center), and (iii) QII (right). The dashed lines (mH = A) mark the case where vL is a massless particle. The decays through the diagram involved in the detection channel are energetically forbidden. <1 10 <1 10 <1 10 10 10 10 10 10 10 10 10 10 <1 10 <1 10 10 10 10 10 10 10 10 10 10 10 10 Some slices of the 3-dimensional allowed volumes for various mH values (including the mH = A case, that is a massless or a very light vL particle) in the (£,omOh',1'nc vs A) plane are reported in Ref. [17]. Finally, it is worthwhile to summarize that LDM candidates in the MeV/sub-GeV range are allowed by DAMA experiments (see Fig. 1.11). Also these candidates, such as e.g. axino, sterile neutrino, can be of interest for the positron production in the galactic bulge. 1.3.5 Mirror Matter Well-motivated DM candidates are represented by the so called Mirror particles. The Mirror scenario can be introduced by considering a parallel gauge sector with particle physics exactly identical to that of ordinary particles, coined as mirror world. In this theory the Mirror particles belong to the hidden or shadow gauge sector and can constitute the DM particles of the Universe. A comprehensive discussion about Mirror Matter as DM component can be found in Refs. [28,29]. In these two papers the annual modulation effect measured by DAMA experiments with lower exposure has been analyzed in the framework of Asymmetric and Symmetric Mirror Matter scenarios. The analyses updated by including the new data of the first six annual cycles of DAMA/LIBRA-phase2 with lower software energy threshold is reported in Ref. [17]. This new analysis restricts a significant part of the parameters' space of the Mirror DM scenarios. 1 Corollary Analyses After the Recent Model-Independent... 19 I.4 Conclusions A high C.L. model-independent evidence for the presence of DM particles in the galactic halo has been achieved by DAMA/Nal, DAMA/LIBRA-phasel and by the first six full annual cycles of DAMA/LIBRA-phase2 on the basis of the exploited signature. The corollary investigation on the nature of the DM particles is an open problem; it always requires a large number of assumptions. In this paper several possible scenarios [17] for DM candidates are analyzed on the basis of the longstanding DAMA results exploiting the DM annual modulation signature. In particular, the DAMA/LIBRA-phase2 data, collected over the first six full annual cycles (1.13 ton x yr) with a software energy threshold down to 1 keV, are analyzed with the DAMA/NaI and DAMA/LIBRA-phase1 data for several scenarios, improving the confidence levels and restricting the allowed parameters' space of the considered DM candidate particles with respect to previous analyses. Several scenarios are compatible with the observed signal; other possibilities are open as well. For example other scenarios as e.g. Refs. [49,50] are planned to be analysed as well. 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Bled Workshops in Physics Vol. 20, No. 2 JLV Proceedings to the 22nd Workshop What Comes Beyond ... (p. 21) Bled, Slovenia, July 6-14, 2019 2 Conspiracy of BSM Physics and Cosmology M.Yu. Khlopov * 1 Institute of Physics, Southern Federal University Stachki 194 Rostov on Don 344090, Russia 2 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia 3 APC laboratory 10, rue Alice Domon et Leonie Duquet 75205 Paris Cedex 13, France Abstract. The lack of experimental evidence at the LHC for physics beyond the Standard model (BSM) of elementary particles together with necessity of its existence to provide solutions of internal problems of the Standard model (SM) as well as of physical nature of the basic elements of the modern cosmology demonstrates the conspiracy of BSM physics. Simultaneously the data of precision cosmology only tighten the constraints on the deviations from the now standard ACDM model and thus exhibit conspiracy of the nonstandard cosmological scenarios. We show that studying new physics in combination of its physical, astrophysical and cosmological probes, can not only unveil the conspiracy of BSM physics but will also inevitably reveal nonstandard features in the cosmological scenario. Povzetek. Poskusi na pospesevalniku LHC niso ponudili doslej ničesar, kar bi prispevalo k razumevanju izvora predpostavk standardnega modela osnovnih delcev. Tudi vedno bolj natancne kozmoloske meritve ne kaZejo odstopanja od standardnega modela. Zdi se, meni avtor, kot da se je narava "zarotila" in vstraja na standardnem modelu in "nam noce pokazati poti" k razumevanju privzetkov standardnega modela. Avtor v prispevku demonstrira, kako lahko teoreticne raziskave skupaj s fizikalnimi, astrofizikalnimi in kozmoloskimi meritvami pokazzejo nestandardne znacilnosti v kozmoloskem scenariju. Keywords: cosmology, particle physics, cosmoparticle physics, inflation, baryosyn-thesis, dark matter, primordial black holes, antimatter, dark atoms, composite dark matter, stable double charged particles 2.1 Introduction The now standard description of the structure and evolution of the Universe is based on inflationary models with baryosynthesis and dark matter/ energy. The interpretation of the data of precision cosmology ascribes about 95% of the modern cosmological energy density to the impact of physics beyond the Standard model (BSM) of elementary particles. BSM physics is involved in virtually all the * E-mail: khlopov@apc.univ-paris.fr 22 M.Yu. Khlopov mechanisms of inflation and baryosynthesis, explaining the initial conditions of the cosmological evolution. It makes the observed homogeneous and isotropic expanding Universe, origin and structure of its inhomogeneities with their observed baryon asymmetry an evident reflection of the BSM physics. The problem of experimental studies of BSM physics is generally related with necessity to address effects of a high energy scale F 1. At the energy release E > F it leads to appearance of new heavy particles with the mass M ~ F or new interactions that manifest their full strength at these energies. If the energy is much less, than F, only virtual effects of new physical scale are possible, which are suppressed by some power of E/F. Therefore we can either turn to rare low energy processes, in which new high energy physics phenomena can appear, like proton decay, or probe at the currently available energies E the extensions of the Standard model (SM), which involve new physics at scales F < E. Probes for supersymmetric (SUSY) models at the LHC corresponded to the latter case, but the lack of positive evidence for existence of SUSY particles at the energy of hundreds GeV probably moves the SUSY scale to higher energies, at which direct search of SUSY particle production at the LHC is not possible. The only experimentally proven evidence for new physics is the effect of neutrino oscillations, but the physical nature of neutrino mass is still unknown. Following [1] we characterize here the current situation as the conspiracy of the BSM physics: there is no doubt in its existence, but all its features are hidden, since the experimental data put only more and more stringent constraints on the new physics effects. We discuss the physical motivation for extension of SM model and their possible physical, astrophysical and cosmological signatures in Section 2.2. We draw attention in the Section 2.3 that BSM physics involved in the description of the now standard cosmological model (which we consider in Section 2.2 as the motivation for the SM extension) should inevitably add nonstandard model dependent features like a plethora of non-WIMP forms of dark matter, primordial black holes or antimatter domains in the baryon asymmetrical Universe. We express the hope in the Conclusion (Section 2.4) that revealing of specific model dependent signatures of BSM physics can not only unveil its conspiracy, but can also enrich the theory of structure and evolution of the Universe by nonstandard cosmological scenarios. 2.2 Motivations for the SM extension 2.2.1 Physics of neutrino mass The discovery of neutrino oscillations proves the existence of the nonzero mass of neutrino. It may be considered as a manifestation of BSM physics, since neutrinos are strictly massless in the Standard model. However, the very existence of neutrino mass doesn't shed light on its physical nature and the corresponding new physics. Neutrino mass term relates ordinary left-handed neutrino state to some right handed state. The latter can be ordinary right-handed antineutrino. It corresponds 1 Henceforth,if it is not otherwise specified, we use the units h _ c _ k _ 1 2 Conspiracy of BSM Physics and Cosmology 23 to Majorana mass term, in which lepton number L conservation is violated and L changes as AL = 2. In the SM lepton number is conserved at the tree level and Majorana mass term is the example of BSM physics. Smallness of ordinary neutrino Majorana mass mv relative to the Dirac mass mo of the corresponding charged lepton is explained by "see-saw" mechanism, involving right handed neutrino with large Majorana mass M, so that ordinary neutrino mass is given by 2 mD mo , . m = M" = Mmo < mo- (Z1) Majorana mass term of electron neutrino leads to neutrinoless double beta decay. In the nonrelativistic limit interaction of Majorana neutrino with nuclei is proportional to spin operator acting on nuclear wave function. It leads to spin dependent interaction of nonrelativistic Majorana neutrino with nuclei. Another possibility is a Dirac neutrino mass term. It corresponds to transition to a new state of sterile right handed neutrino. Such neutrino doesn't participate in the ordinary weak interactions, being another possible example of BSM physics, related to the mechanism of neutrino mass generation. In the nonrelativistic limit Dirac neutrino interaction with nuclei is spin independent and leads to coherent scattering of low energy neutrinos in the matter. V.Shwatsman has noted in his diploma work in late 1960s that neutrino with mass m and velocity v can scatter coherently on the piece of matter with size I ~ ft/(mv) and cause its acceleration. This idea, published in [2,3] was probably the first step towards direct detection of cosmological dark matter. It is the stable prediction of the Big Bang theory that primordial thermal neutrino background should exist with number density 3 nvv = lyny, (2.2) where Uy « 400cm-3 is the the number density of CMB photons. Multiplied by neutrino mass it gives the predicted contribution of relic massive neutrinos to the cosmological density. Experimental constraints on the mass of electron neutrinos (see [4] for the latest results) together with the data on the neutrino oscillations exclude explanation of the measured dark matter density by this contribution. However, while ordinary massive neutrinos cannot play dominant dynamical role in the Universe, BSM physics of neutrino mass can lead to important cosmological effects, like sterile neutrino dark matter [5]. 2.2.2 Supersymmetry and the SM problems SUSY models provide natural solution for the internal SM problems, if the SUSY scale is in the range of several hundred GeV. Then contribution of SUSY partners in loop diagrams of radiative effects in the Higgs boson mass cancel the quadratic divergent contribution of the corresponding SM particles. Renormalization group analysis of evolution of scalar field potential 24 M.Yu. Khlopov from superhigh energy scale leads to the Higgs form of this potential at lower energy, explaining the nature of the electroweak symmetry breaking. R-parity or some continuous symmetry provides stability of the lightest SUSY particle. Such particle with mass of several hundred GeV has interaction cross section at the level of weak interaction and can play the role of Weakly Interacting Massive Particle (WIMP) candidate for dark matter. The lack of experimental signatures for SUSY particles at the LHC as well as of positive result of underground WIMP searches 2 implies nontrivial ways of search for SUSY (see [11] for the latest review). In the extreme case SUSY scale may be close to the scale of Grand Unfication (GUT). This case implies non-SUSY solution for the problem of divergence of the Higgs mass and origin of the electroweak symmetry (see the next subsection), but has the advantage to unify all the four fundamental natural forces, including gravity, in the framework of Supergravity. Starobinsky supergravity can provide simultaneous BSM solution for dark matter in the form of superheavy gravitino [12-14] and Starobinsky inflation [15]. This solution can be hardly probed by any direct experimental mean and makes cosmological consequences the unique way for its indirect test. 2.2.3 Nonsupersymmetric solutions. Composite Higgs. Multiple charged particles Nonsupersymmetric solution for the problem of Higgs mass divergence may be related to the composite nature of Higgs boson [16-21]. Then this divergence is cut at the scale, at which Higgs constituents are bound. In parallel such constituents can form bound states with exotic charges. Such situation can take place in the model of composite Higgs based on Walking Technicolor (WTC) [22-27]. The minimal walking technicolor model (WTC) involves two techniquarks,U and D. They transform under the adjoint representation of a SU(2) technicolor gauge group. Neutral techniquark-antitechniquark state is associated with the Higgs boson. Six bosons UU, UD, DD, and their corresponding antiparticles carry a technibaryon number. If the technibaryon number is conserved, the lightest technibaryon should be stable. Electric charges of UU, UD and DD are given in general by q + 1, q, and q — 1, respectively, where q is an arbitrary real number [28-30]. To compensate the anomalies the model includes in addition technileptons v' and Z that are technicolor singlets. Their electric charges are in terms of q, respectively, (1 — 3q)/2 and (—1 — 3q)/2. Fractional value of q would correspond to stable fractionally charged tech-niparticles. Their creation in the early Unvierse would lead to their presence in the terrestrial matter that is severely constrained by the experimental data. On the same reason, stable techniparticles should not have odd charge 2n + 1. Positively 2 Though interpretation of positive result of DAMA/NaI and DAMA/LIBRA experiments in the terms of WIMPs is not excluded [6,7], theoretical analysis [8], proving such a possibility indicates its contradiction with the results of XENON1T [9] and PICO [10] experiments. 2 Conspiracy of BSM Physics and Cosmology 25 charged +(2n +1) stable particles are bound with electrons in anomalous isotopes of elements with Z = 2n + 1. Negatively charged particles with charge — (2n + 1), created in the early Unvierse, bind with n + 1 nuclei of primordial helium, produced in the Big Bang Nucleosynthesis, and form a +1 charged ion that binds with electrons in atoms of anomalous hydrogen. The experimental data put severe constraints on such anomalous isotopes. The case of stable multiple charged particles with even value of negative charge —2n avoids these troubles, since it forms with n nuclei of primordial helium neutral dark atom. Their bound states with primordial helium can play the role of dark matter and can even solve the puzzles of dark matter searches (see [1,31-33] for the latest review). 2.2.4 Axion and axion-like models The popular solution for the problem of strong CP violation in QCD involves the additional U(1 )pq symmetry which provides automatic suppression of the CP-violating 9-term [34]. Breaking of this Peccei-Quinn symmetry spontaneously at the scale f, followed by its manifest breaking at the scale A C f results in appearance of a pseudo-Nambu-Goldstone (PNG) particle, axion, a. In the axion models the second step of breaking is generated by instanton transitions. The mass of axion is given by [35] ma = Cm„f„/f, (2.3) where mn and fn « mn are the pion mass and constant, respectively. The constant C ~ 1 depends on the choice of the axion model. The relationship (2.3) of axion to neutral pion makes possible to estimate the cross section of axion interactions from the corresponding cross section of pion processes multiplied by the factor (fn/f)2. The existence of ayy vertex leads to a two-photon decay of axion, as well as to effects of ay conversion [38] like axion-photon conversion in magnetic field (see e.g. [39] for review and references). The principles of experimental search for axion by "light shining through walls" effects are based on such a conversion [40]. Axion couplings to nondiagonal quark and lepton transition can lead to rare processes like K —» na or ^ —» ea. In the gauge model of family symmetry breaking [41] the PNG particle called archion shares properties of axion with the ones of singlet Majoron and familon, being related to the mechanism of neutrino mass generation. In the axion-like models the condition of Eq. (2.3) is absent and the mass of the PNG particle may be very small. In cosmology, in spite of a very small mass (2.3) primordial axions appear in the ground state of Bose-Einstein condensate and, being created initially nonrela-tivistic, represents a specific form of Cold Dark Matter. 2.2.5 BSM physics of the standard cosmology The now Standard cosmological model involves inflation to explain the homogeneity and isotropy of the Universe as well as initial impulse for Big Bang expansion. 26 M.Yu. Khlopov Observed absence of antimatter objects is explained by baryosynthesis, in which baryon asymmetry was generated in the intially baryon symmetric Universe. Formation and evolution of Large Scale Structure is described in the framework of the standard ACDM model, assuming dominance in the modern total cosmological density of dark energy with vacuum-like equation of state (cosmological constant A in the simplest case) and dark matter dominating in the matter content of the Universe. All these elements of the Standard Cosmological model imply BSM physics, making the observational confirmation of this model an evidence for existence of BSM physics. On the other hand, the data of precision cosmology (planck15,planck18) analysed in the terms of parameters of this standard cosmological model continuously tighten the constraints on deviations of the measured parameters from the model predictions. These measured parameters involve dark matter density nDMh2 = 0.120 ± 0.001, baryon density nbh2 = 0.0224 ± 0.0001 (where the dimensionless constant h is the modern Hubble constant H0 in the units of 100 km/s/Mpc), scalar spectral index ns = 0.965 ± 0.004, and optical depth t = 0.054 ± 0.007 [43]. These results are only weakly dependent on the cosmological model and remain stable, with somewhat increased errors, in many commonly considered extensions. Assuming the ACDM cosmology, the inferred late-Universe parameters were determined: the Hubble constant H0 = (67.4 ± 0.5) km/s/Mpc; matter density parameter Hm = 0.315 ± 0.007; and matter fluctuation amplitude dg = 0.811 ± 0.006. Combining with the results of studies of baryon acoustic oscillations (BAO) by measurement of large scale distribution of galaxies [44]3 Planck collaboration has constrained the effective extra relativistic degrees of freedom to be Neff = 2.99 ± 0.17, and the sum of neutrino mass was tightly constrained to Y. mv < 0.12. These results prove the basic ideas of inflationary model with baryosynthesis and dark matter/energy, but cannot provide definite choice for the corresponding BSM physics. PLANCK collaboration has found no compelling evidence for extensions to the ACDM model, but has mentioned the 3d difference with the results of local determination of H0 [46]. Such a discrepancy may be a hint to a necessity to extend the standard cosmological model. Indeed, the conspiracy of Beyond the Standard model (BSM) Cosmology [1] is puzzling taking into account the plethora of nontrivial cosmological consequences of BSM particle models. Some of these nonstandard features which have probably found their experimental evidence are discussed in the next Section 2.3. 2.3 Features of BSM cosmology 2.3.1 Plethora of dark matter candidates Well motivated BSM models offer a plethora of dark matter candidates. In the essence such candidates follow from the extension of the SM symmetry. If the additional symmetry acting on new sets of particles is strict or nearly strict, the 3 Such oscillations were first discussed by A.D. Sakharov [45] and are also called Sakharov oscillations 2 Conspiracy of BSM Physics and Cosmology 27 lightest particles that possess this symmetry are stable or sufficiently long living to play the role of dark matter. In addition to massive sterile neutrinos, superheavy gravitino or invisible axion that follow respectively from solutions of the origin of neutrino mass, Starobinsky supergravity or solution of the problem of strong CP violation in QCD there are mirror or shadow particles, whose existence is related to restoration of equivalence of left- and right-handed coordinate systems. Grand Unification, string phenomenology or phenomenology of extra dimensions extend this list by many other nontrivial candidates accompanied by a very extensive hidden sector of new particles and fields. Such extensions naturally lead to multi-component dark matter that can include unstable or decaying components, like it takes place in the model of broken family gauge symmetry [41] (see e.g. [28,37,35] for review and references). In this large list of possibilities the model of dark atoms, in which stable —2n charged particles are bound with n nuclei of primordial helium, is of special interest not only owing to the minimal set of the involved new physics parameters (their number is reduced to the mass of a hypothetical negatively charged stable particles only), but also since it may provide a solution for controversial results of direct dark matter searches. The idea of this solution is that nuclear interacting dark atoms are slowed down in the terrestrial matter and thus cannot cause significant nuclear recoil in the underground detector. However, in the matter of these detectors dark atoms can bind with intermediate mass nuclei with the binding energy of few keV (see [1,31,33] for recent review and references). Since the concentration of dark atoms in the matter of underground detectors is adjusted to their incoming cosmic flux, energy release in such binding should experience annual modulations. It explains positive results of DAMA/NaI and DAMA/LIBRA experiments. In a simple rectangular wall and well approximation it was shown in [47] that a level of about 3 keV can exist in binding of dark atoms with intermediate mass nuclei and doesn't exist for heavy nuclei, like xenon, explaining absence of positive results in the corresponding detectors. If such level exists, transition to it is determined by isospin violating electric dipole operator and its rate is proportional to the temperature, being suppressed in cryogenic detectors [1,31,33]. The open problem of this explanation is a selfconsistent treatment of Coulomb and nuclear interactions of dark atoms. Such treatment needs special study in the lack of all the usual approximations of atomic physics: there are no small parameters like small ratio of sizes of nucleus and atom and the electroweak interaction of electronic shell. Dark atoms has strongly interacting nuclear shell with the radius of the order or equal to the nuclear radius. Dark atom cosmology contains such notrivial features as Warmer than Cold Dark Matter scenario and can explain the observed excess of radiation in positron annihilation line from the center of Galaxy as indirect effect of dark atoms (see [1,31-33] for recent review and references). This explanation assuming electron-positron pair production in de-excitation of dark atoms excited in collisions in the center of Galaxy is possible only for a narrow range around 1.25 TeV of the mass of dark atom, which is determined by its constituent with multiple negative 28 M.Yu. Khlopov charge [1,35]. It challenges search for multiple charged stable particles at the LHC that provides complete test of such an explanation [48]. In a two-component dark atom model, a possibility to explain the observed excess of high energy positrons by decays of +2 charged dark atom constituents was proposed in [49]. However, any source of positrons is simultaneously the source of gamma radiation and to avoid contradiction with the observed gamma background the mass of the decaying +2 consituent of dark atom should be less, than 1 TeV. Moreover, in view of the difference of propagation in the Galaxy by gamma radiation and positrons the condition not to exceed the observed gamma background may cause troubles for any explanation for the high energy positron excess, involving indirect effects of dark matter [50]. In any case, the results of searches for stable double charged particles in the ATLAS experiment at the LHC put lower limit on the mass of such particles [51], excluding explanation of high energy positron anomaly by decaying +2 charged constituents of dark atoms [1,35]. 2.3.2 Primordial Black holes Strong primordial inhomogeneities are a prominent tracer of BSM physics of very early Universe and Primordial Black Holes (PBH) are the most popular example of this kind (see e.g. [12,52] for review and references). To form a black hole in the homogeneously expanding Universe the expansion should stop in some region and it corresponds to a very strong inhomogeneity [53-55]. In the universe with equation of state the probability of forming a black hole from fluctuations within the cosmological horizon is given by [56] where (S2) C 1 is the amplitude of density fluctuations. For relativistic equation of state (y = 1/3) the probability (2.6) is exponentially small. It can increase, if the amplitude of density fluctuations in the early Universe was much larger, than in the period of galaxy formation, or the equation of state was much softer, corresponding to matter dominated stage with y = 0. Therefore PBH origin may be related with early matter dominated stages, phase transitions in the early Universe or nonflat features in the spectrum of primordial density fluctuations. All these phenomena are not only originated from BSM physics, but also represent strong deviation from the Standard cosmological scenario. PBHs with mass M < 1015 g evaporate by the mechanism of Hawking [57,58]. This process is the universal process of production of any type of particles with P = Ye with numerical factor y being in the range 0 < y < 1 (2.4) (2.5) (2.6) 2 Conspiracy of BSM Physics and Cosmology 29 mass m < TeVap « 1013 GeVM. It can be the source of superweakly interacting particles, like gravitino [59] as well as of fluxes of particles with energy much larger, than the thermal energy of particles in the surrounding medium. It causes non equilibrium processes in the hot Big Bang Universe, nonequilibrium cosmological nucleosynthesis [60], in particular. PBHs with mass M > 1015 g should survive to the present time and represent a specific form of dark matter. It was noticed in [61] that taking into account PBH formation in clusters the constraints on PBH contribution into the total density [62] can be relaxed and even the possibility of PBH dominant dark matter is not excluded. It would make primordial nonhomogeneities in the form of PBHs the dominant matter content of the modern nonhomogeneities. Mechanism of PBH cluster formation can be illustrated with the use of the axion-like model, discussed in subsection 2.2.4, in which the first step of symmetry breaking at scale f takes place on the inflationary stage [35,52]. Then at the second stage of the symmetry breaking at T ~ A closed massive walls are formed so that the larger wall is accompanied by a set of smaller walls. Their collapse form a PBH cluster, in which the range of PBH masses M is determined by the model parameters f and A [35,36] f(^)r < M < fmp!(^)r (2.7) Here the minimal mass is determined by the condition that the width of wall doesn't exceed its gravitational radius, while the upper limit comes from the condition that the wall enters horizon, before it starts to dominate within it [36]. At A < 100 MeV(mpl/f)1/r the maximal mass exceeds 100Modot. Collapse of massive walls to such black holes takes place at t > mpi mpi "AT" • (2.8) At A < 1 GeV and f _ 1014 GeV it happens at t > 0.1 s, what can lead to interesting observable consequences. Closed wall collapse leads to primordial gravitational wave (GW) spectrum, estimated as peaked at [35] v0 _ 3 x 1011 (A/f)Hz. (2.9) Their estimated contribution to the total density can reach Qgw « 10-4(f/mpi), (2.10) being at f ~ 1014 GeV HGW « 10-9. For 1 < A < 108 GeV the maximum of the spectrum corresponds to 3 x 10-3 (13 + 1) dimenzij, uporabi eno vrsto spina za opis spina in nabojev karkov in leptonov in antikvarkov in antileptonov, drugo vrsto spina pa * Talk presented by N.S. Mankoc Borstnik 3 New Way of Second Quantized Theory of Fermions... 37 za opis družin. Avtorica teorije spinov-nabojev-družžin je dokazala, da vektorji, ki so lastni vektorji Car-tanove podalgebre Lorentzove algebre in so produkt lihega stevila Cliffordovih operatorjev, izpolnjujejo vse lastnosti fermionov v drugi kvantizaciji. To pomeni, da opis fermionov v Cliffordovi algebri razlozi Diracove postulate za drugo kvantizacijo fermionov. Kreacijski in anihilacijski operatorji, ki določajo v tej drugi kvantizaciji 1 -fermionska stanja, zadostijo antikomutacijskim relacijam za drugo kvantizacijo fermionov, ce jih zapisemo kot produkt niloptentov in projektorjev lihega stevila Cliffordovih operatorjev. Kreacijski operatorji za n fermionska stanja so v tej drugi kvantizaciji produkti enofermionskih kreacijskih operatorjev, ki delujejo na praznem vakuumskem stanju. V tej teoriji ni potrebe po negativnih energijskih stanjih zapolnjenih s fermioni. Avtorja postavita zahtevo, ki ohrani le enega od obeh vektorskih prostorov, druga vrsta operatorjev pa poveze neodvisne nerazcepne upodobitve Lorentzove algebre v tem prostoru in jim " podeli" kvantno stevilo "druzin ". Tako omogoci Cliffordova algebra opis ne le spinov in nabojev kvarkov in leptonov in antikavarkov in antileptonov, ampak tudi njihovih druzin. (Članek pokaze, da tudi Grassmannova algebra ponudi kreacijske in anihilacijske operatorje, ki zadoscajo antikomutacijskim relacijam za 1 fermionska stanja. Vendar so spini teh vektorjev celostevilski. Grassmannov prostor ne ponudi druzin. Akcija in enecbe gibanja, ko so v Cliffordovi algebri poznani, za Grassmannov algebro pa clanek predlaga akcijo in diskretne operatorje. Za obe algebri ponudi resitve ustrezne "Weylove" enacbe za proste "fermione" brez mase in jih komentira. Avtorja ponudita tudi kratek pregled dosezkov teorije spinov-nabojev-druzin in njenih odprtih problemov. Primerjava Grassmannovega in Cliffordovega primera osvetli mnoga odprta vprasanja fizike osnovnih fermionov in bozonov ter kozmologije. Keywords: Second quantization of fermion fields in Clifford and in Grassmann space, Spinor representations in Clifford and in Grassmannspace, Kaluza-Klein-like theories, Discrete symmetries, Higher dimensional spaces, Beyond the standard model 3.1 Introduction More than 50 years ago the standard model offered an elegant new step in understanding elementary fermion and boson fields by postulating: i. Massless family members of coloured quarks and colouress leptons, the left handed members as the weak charged doublets and the weak chargeless right hand members, the left handed quarks distinguishing in the hyper charge from the left handed leptons, each right handed member having a different hyper charge. All fermion charges are in the fundamental representation of the corresponding groups. Antifermions carry the corresponding anticharges and opposite handedness. The existence of massless families to each family member is as well postulated. There is no right handed neutrino, since it would carry none of the so far observed charges, and correspondingly there is also no left handed antineutrino. ii. The existence of the massless vector gauge fields to the observed charges of quarks and leptons, carrying charges in the corresponding adjoint representations. 38 N.S. Mankoc Borstnik and H.B.F. Nielsen iii. The existence of a massive scalar Higgs, gaining at some step of the expanding universe the nonzero vacuum expectation value, causing masses of fermions and heavy bosons and the Yukawa couplings. The Higgs carry a half integer weak and hyper charge. iv. Fermion and boson fields can be (second) quantized. The standard model assumptions have in the literature several explanations, mostly with many new not explained assumptions. The most successful seem to be the grand unifying theories [12-28], if postulating in addition the family group and the corresponding gauge scalar fields. The spin-charge-family theory, the project of N.S.M.B. [1-7,9,8,10], is offering the explanation for all the assumptions of the standard model, unifying not only charges, but also charges and spins and families, explaining the appearance of families, of the vector gauge fields, of the scalar field and the Yukawa couplings, offering the explanation for the matter-antimatter asymmetry, making several predictions. This theory also offers the explanation for the appearance of creation and annihilation operators, fulfilling the anticommutation relations for fermions,which in the Dirac theory [67] is just assumed. The spin-charge-family theory is a kind of the Kaluza-Klein like theories [2936,8] due to the assumption that in d > 5 (in the spin-charge-family theory d > (13+1)) fermions interact with the gravity only. Correspondingly this theory shares with the Kaluza-Klein like theories their weak points, at least: a. Not yet solved the quantization problem of the gravitational field. b. Breaking spontaneously the starting symmetry, which would at low energies manifest the observed almost massless fermions [30]. Concerning this second point we proved on the toy model of d = (5+1) that the break of symmetry can lead to (almost) massless fermions [6870]. It remains to study how does appear the spontaneous breaking of the starting symmetry in d = (13 + 1), first with the appearance of the condensate of two right handed neutrinos, Table 3.3, Ref. [4], and then when scalar fields with space index (7,8) obtain nonzero vacuum expectation values. (This second point is common to all the unifying theories.) Since in d = (3 + 1)-dimensional space — at low energies — the gauge gravitational fields manifest as the observed vector gauge fields [5], which can be quantized in the usual way, quantization procedure of gravity can wait to be made. The author is in mean time trying to find out (together with the collaborators) how far can the spin-charge-family theory — starting in d = (13 + 1)-dimensional space with a simple and "elegant" action, Eq. (3.1) — reproduce in d = (3 + 1) the observed properties of quarks and leptons [3-7,9,8,10], the observed gauge fields, the assumed scalar field, the appearance of the dark matter and of the matter-antimatter asymmetry, as well as the other open questions, connecting elementary fermion and boson fields and cosmology. The work done so far seems promising. Let us in what follows and in Subsect. 3.1.1 overview shortly the starting assumptions and so far achievements of the spin-charge-family theory, and discuss as well open problems. 3 New Way of Second Quantized Theory of Fermions... 39 The recognition that there are in Grassmann space two kinds of the Clifford algebra objects [2] (ya and Ya) enables that the spin-charge-family theory is explaining the origin of families [47-49,1,2], Table 3.1. The assumption made in the spin-charge-family theory that the dimension of space is > (13 + 1) enables the explanation for by the standard model assumed spins and charges of quarks and leptons [71,72], explaining as well the miraculous cancellation of triangle anomalies [8,9,4] in the standard model, however, without relating handedness and charges "by hand" as needed in SO (10) [37-39]. Since there are in SO(13 + 1) additional quantum numbers to those assumed by the standard model, the theory predicts that right handed neutrinos and left handed antineutrinos, carrying nonzero additional quantum numbers — t23 and t4 instead of Y in the standard model (Y = (t23 + t4) in the spin-charge-family theory as presented in Table 3.6 and in Eqs. (3.111, 3.112, 3.113, 3.114)) — are regular members of families of quarks and leptons [71,72,3,9]. This prediction is common also to S0(10) [37-39]. In the spin-charge-family theory spins and charges are described by the superposition of Sab (= 4 (YaYb — YbYa), Eq. (3.2)), with Ya belonging to the first kind of the Clifford algebra objects and with Smn, (m, n) = (0,1,2,3), describing spins and handedness of quarks and leptons (Eq. (3.111)), and Sst, (s, t) = (5,6, • • • , 14), describing their charges, Table 3.6, Eqs. (3.112, 3.113) and Refs. [2,47,49,72]. Family quantum numbers are determined by the second kind of the Clifford algebra objects, by the superposition of Sab (= 4 YaYb — YbYa)), Eq. (3.2), Table 3.1 [2,48]. The vector gauge fields, assumed in the standard model as the gauge fields of the corresponding fermion charges, are in the spin-charge-family theory explainable as the superposition of the gauge fields of the generators of the Lorentz transformations Sst (Sst custm, (s,t) = (5,6, •• • ,14), Eqs. (3.1, 3.9, 3.111)), with the vector index m = (0,1,2,3), Eq. (3.10), Ref. [5]. In the standard model the scalar fields appear as the Higgs scalar and the Yukawa couplings by the assumption. In the spin-charge-family theory both kinds of the gauge fields, Y.s' t' cs t ds t s, which are the gauge fields of Sst with (s', t') = (5,6,7,8), and Y_ab cab d>abs, which are the gauge fields of Sab, with (a, b) = (0,1, • • • ,8), both with the scalar index s = (7,8), manifesting properties of the Higss scalar (by carrying weak and hyper charges in the "fundamental representation"), define masses of quarks and leptons and of heavy bosons, Eq. (3.10), Refs. [72,9,3]. These scalar fields determine in the spin-charge-family theory masses of the two groups of four families [51,53-56,3,9]. The lower group predicts the existence of the fourth family of quarks and leptons, coupled to the observed three families [51,53,56,54,70]. From the symmetry of the mass matrices predicted the 4 x 4 mixing matrix of quarks [56] appear to be in better agreement with the experiments than if only three families are assumed [40]. The lowest family of the upper four families offers the explanation for the existence of the dark matter [54,61]. There are additional scalar fields in the spin-charge-family theory [4], having the scalar space index t G (9,10,..., 14). They carry colour charges in the "fun- 40 N.S. Mankoc Borstnik and H.B.F. Nielsen damental" representations, cause transitions of antileptons and antiquarks into quarks and back, enabling the decay of baryons. These scalar fields are offering in the presence of the right handed neutrino condensate, Table 3.3, Ref. [4], which breaks the CP symmetry, the answer to the question about the matter-antimatter asymmetry in the universe [4]. Authors of this paper proved on the toy model of d = (5 + 1) that breaking the symmetry in Kaluza-Klein theories can lead to massless fermions [68-70]. The authors determine as well the discrete symmetries operators in observable dimensions d = (3 + 1) for any d, Eqs. (3.94), Ref. [65]. The breaking of the starting symmetry SO(13 + 1) is in the spin-charge-family theory triggered by the appearance of the condensate (Table 3.3) of the right handed neutrinos [4] and, like in the standard model, by the nonzero vacuum expectation values of the scalar fields with the space index s = (7,8). In this paper it is demonstrated that the odd products of nilpotents and projectors, which are the "egienfunctions" of the Cartan subalgebra of the Lorentz algebra in Clifford space, and which solve the Weyl equations for free massless fermions, fulfill together with the corresponding Hermitian conjugated annihilation operators the anti-commutation relations as needed in the second quantized fermion fields [50]. No assumption of the Dirac kind about the creation and annihilation operators is needed. The spin-charge-family theory has many common points with other unifying theories ([12-17,29-36] and other references), and because of that and because of the fact that by starting with the very simple action, Eq. (3.1), the theory is able to offer explanations for so many observed phenomena, built into assumptions of the standard model(s) of the elementary boson and fermion fields and also of cosmology, and also in other unifying theories, it might be that it is the right next step beyond the standard models. The achievements of the spin-charge-family theory are discussed in more details in Subsect. 3.1.1. There also problems waiting to be solved are presented. Let us present a very simple starting action of the spin-charge-family theory of N.S.M.B., in which massless fermions in d = (13 + 1)-dimensional space interact with massless bosons, that is only with gravity — the vielbeins faa (the gauge fields of moments pa) and the two kinds of the spin connections (daba and daba, the gauge fields of the two kinds of the Clifford algebra objects Ya and ya, respectively). A = ddxE 1 Yapca^)+ h.c. + ddxE (aR + aR), (3.1) with P0a = faaP0a + 21E {Pa, Ef%}-, P0a = Pa — "Sabdaba — 2Sabdaba and R = J {fa[afPb] (Wab«)P — Wcaa d%p)} + h.C., R = 2 {f^f^1 (daba,p — dcaa dcbp)} + h.c.. Here 1 fa[afPb] = faafPb — fabfPa. 1 faa are inverted vielbeins to eaa with the properties eaafab = Sab, eaafpa = SjJ, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while 3 New Way of Second Quantized Theory of Fermions... 41 The two kinds of the Clifford algebra objects, Ya and Ya, Eq. (3.2), anticom-mute and determine the infinitesimal generators of the Lorentz transformations in the internal space of fermions — Sab for SO(13,1), arranging states into representations (Table 3.6), and Sab for SO (13,1), arranging states into families (Table 3.1). Eq. (3.69) relates these two internal dgrees of freedom, keeping the relations of Eq. (3.2) unchanged. {Ya,Yb}+ = 2nab = {Y a,Y b}+ , {Ya,Y b}+ = 0, Sab = 4(Ya Yb - Yb Ya), Sab = 4(Ya Yb - Yb Ya). (3.2) The generators Sab are used in the spin-charge-family theory to determine spins and charges of spinors of any family, Table 3.6, another kind, Sab, determines the family quantum numbers, Table 3.1. These two degrees of freedom are connected by the requirement, presented in Eq. (3.69). The scalar curvatures R and R determine dynamics of the gauge fields — the spin connections and the vielbeins — manifesting in d = (3 + 1) as all the known vector gauge fields, as well as the scalar fields [5], which offer the explanation for the appearance of the Higgs and the Yukawa couplings, of the ordinary matterantimatter asymmetry [4] and the dark matter [54], provided that the symmetry breaks from the starting SO(13,1) to SO(3,1) x SU(3) x U(1). In this paper we start to study the possibility that fermions are described in Grassmann space, in order to better understand how far can the simple starting action, Eq. (3.1), of the spin-charge-family theory agree with the at low energies observed properties of fermions and bosons. We demonstrate in this paper that besides Clifford space also Grassmann space offers the description of the internal degrees of freedom of fermions in the second quantized procedure. In both cases there exist the creation and annihilation operators, which fulfill the anticommutation relations required for fermions, Eqs. (3.54, 3.81). But while the internal spins determined by the generators of the Lorentz group of the Clifford objects Sab and Sab — we repeat here that in the spin-charge-family theory Sab determine the spin degrees of freedom and S ab the family degrees of freedom — are half integer, the internal spin determined by Sab (expressible with Sab + Sab) is integer. Correspondingly Clifford space offers according to the spin-charge family theory the description of spins, charges and families, all in the fundamental representations of the subgroups of the Lorentz group SO(d — 1,1), while Grassmann space offers spins and charges in the adjoint representations of the subgroups Greek indices a, .., |i, v, ..ff, T,.. denote an Einstein index (a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c,.. and a, Y,.. ), from the middle of both the alphabets the observed dimensions 0,1,2,3 (m, n,.. and v,..), indexes from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and ff, T,..). We assume the signature r|ab = diag{1, —1, —1, • • • , -1}. 42 N.S. Mankoc Borstnik and H.B.F. Nielsen of the Lorentz group SO(d — 1,1) and no family degrees of freedom. Fermions with integer spins would lead to completely different nucleons, nuclei, atoms, molecules, matter than the so far observed ones. Let us make a short introduction into the Grassmann space as well. In Grassmann space the infinitesimal generators of the Lorentz transformations Sab are expressible with anticommuting coordinates 0a and their conjugate momenta p0a = i[2], {ea,eb}+ = 0, {p0a,peb}+ = o, {p0a,eb}+ = inab, sab = eapeb — ebpea. (3.3) Taking into account that Ya and ya, expressible in terms of ea and their conjugate momenta pea, anticommute [2], Ya = (ea — ipea), Ya = i (ea + ipea), (3.4) one recognizes Sab = Sab + Sab , (3.5) from where one concludes, if taking into account Eq. (3.1), that in the Grassmann case the covariant momenta p0a are poa = pa — 1 SabHaba , (3.6) with naba as the only kind of the connection fields (instead of the two kinds in the Clifford case — daba, which is the gauge fields of Sab and d aba, which is the gauge fields of S ab). Let us point out that Eq. (3.69) relates the two anticommuting degrees of freedom, {ya, Yb}+ = 0, making a choice of Ya to detremine the internal degrees of freedom in Clifford space, while keeping all the relation of Eq. (3.2) unchanged. It follows for Sab {sab, Scd}_ = i{Sadnbc + Sbcnad — Sacnbd — Sbdnac}, Sabt = naanbbSab. (3.7) The same relations are true also if Sab is replaced with either Sab or Sab. These infinitesimal generators of the Lorentz group — the two kinds of the Clifford operators and the Grassmann operators — operate as follows {Sab, Ye}_ = —i (nae Yb — nbe Ya), {Sab, ye}_ = —i (naeyyb — nbeYYa), {Sab, Scd}- = o, {Sab, ee}_ = —i (nae eb — nbe ea), {Sab, pee}_ = —i (nae peb — nbe pea), {Mab, Ad...£...9}_ = —i(nae Ad...b...g — nbe Ad...a...g), (3.8) 3 New Way of Second Quantized Theory of Fermions... 43 where Mab are defined in the Clifford case by the sum of Lab plus either Sab (if Ya's are chosen to describe the basis, otherwise Sab replace Sab), while in the Grassmann case Mab is Lab + Sab (which is, Eq. (3.5), Mab = Lab + Sab + Sab). In Sect. 3.2 the actions and norms for free massless fermions, with the internal degrees of freedom described in Clifford and in Grassmann space in d-dimensional spaces are presented. The discrete symmetry operators in d-dimensional space — Clifford and Grassmann — and their manifestation in d = (3 + 1)-dimensional space are presented in Subsect. 3.3.3 of Sect. 3.3. While the action and the discrete symmetry operators in Clifford space are known from before [9,65], the action in Grassmann space as well as the discrete symmetry operators are here assumed by The new way of second quantization of fermion fields in both spaces is discussed in Sect. 3.3. We treat in both spaces only massless free particles. Sect. 3.4 presents what we learn from this work. This work is a part of the project of both authors, which includes the fermion-ization procedure of boson fields (or the bosonization procedure of fermion fields), discussed in Refs. [42,43,45] for any dimension d (by the authors of this contribution, while one of them, H.B.F.N. [44], has succeeded with another author to do the fermionization for d = (1 +1)), and which would hopefully also help to understand a little better the content and dynamics of our universe. 3.1.1 Comments on the achievements of the spin-charge-family theory so far and the open questions to be solved Let us illustrate the achievements of the spin-charge-family theory, presented in the introduction, by adding some comments. I. In the action, Eq. (3.1), fermions carry in d = (13 + 1) two kinds of spins — no charges and interact with gravity only — with the vielbeins faa and the two kinds of the spin connection fields, the gauge fields of Sab — daba — and the gauge fields of Sab — daba. One can formally rewrite the fermion part of the action so that it manifests in d = (3 + 1) the free massless fermion part (first line in Eq. (3.9)), the interaction of fermions with the vector gauge fields (the second line in Eq. (3.9)), the interaction of fermions with the scalar fields (the third line in Eq. (3.9)), and the rest. N.S.M.B.. Lf = iYmPm^ m -Y_ if YmTAiAmi i+ A,i + f YSPos f (3.9) t=5,6,9,...,14 with xAl = Y.st cstAlSst, (s,t) = (5,6, • •• , 13,14), which are generators of the subgroups of SO(13,1), determining charges of fermions, Eq. (3.112, 3.113,3.114), 44 N.S. Mankoc Borstnik and H.B.F. Nielsen with Am1, which are the corresponding superposition of wstm ([4,9] and the references therein), pos = Ps - 1 Ss vws/S"S - 2Sabd)abs and pot = pt - 2St't"t -2SabcDabt, while m G (0,1,2,3), s G (7,8), (s',s") G (5,6,7,8), (a,b) (appearing in S ab)run within (0,1,2,3) and (5,6,7,8), t G (5,6,9,..., 13,14), (t ',t") G (5,6,7,8) and G (9,10,..., 14). I.i The spinor function ^ represents all the family members, 2d -1 = 64 for d = 13 + 1, of all the 2z++1-1 = 8 families, including fermions and antifermions. Tables 3.6 and 3.1 represent the creation operators for the states of one family and the creation operators for the eight families, respectively. The rest of families are assumed to have very large masses as discussed and proved for a toy model in Ref. [68-70,73]. The creation operators operate on a vacuum state, Eq. (3.79). I. A. The Clifford object Ya are in the spin-charge-family theory used to determine from the point of view of d = (3 + 1) spins and all the charges of fermions. I. A.i. d = (13 + 1)-dimensional space offers 2d -1 = 64 members of SO(13,1). In Table 3.6 the properties of quarks and leptons and antiquarks and antileptons, forming 64 members, are presented from the point of view of subgroups of SO(13,1) breaking first into SO(7,1) x SU(3) x U(1), keeping connection between handedness and the two SU(2)^n charges, and further to — SU(2)r x SU(2)l xSU(2)i xSU(2)„ x SU(3) x U(1)' — representing in d = (3 +1) the spin and handedness, the weak charge t13 of SU(2)I, the second t23 of SU(2)II, the colour charge t33 and t38 of SU(3) and t4 of U(1) for quarks and leptons and for antiquarks and antileptons. Cartan subalgebra has f = 7 members, the standard model assumes one commuting operator less. I. A.ii. Due to the additional commuting operator (the member of the Cartan subalgebra of Sab) in the spin-charge-family theory, the neutrinos become a regular members of quarks and leptons, with masses determined by the interaction with the scalar fields as all the rest of family members [51,53-56,3,9] (in Eq. (3.9) the interaction of fermions with the scalar fields is contained in the third line). This is the case also in S0(10) theories [12-15]. The difference in the spin-charge-family theory is, that spin and handedness are correlated with charges, while in SO (10) this is not the case (and must be correlated by "hand"). This fact is discussed in details in Ref. [8]. Let us point out that colour chargeless leptons and quarks of any of the three colours have completely the same SO(7,1) part. Quarks and leptons distinguish only in the SU(3) x U(1) part. I. B. The second Clifford object ya offers the explanation for the existence of families. I. B.i. There are twice four families of quarks and leptons in the spin-charge-family theory ([3] and the references therein) after the appearance of the condensate of the two right handed neutrinos, presented in Table 3.3, Ref. [4]. Since we have not really shown yet how this dynamically happens (we did this so far only for the toy model [68-70]), this remains as an open problem. All eight families obtain masses when the scalar gauge fields with the space index (7,8) — third line in Eq. (3.9) — gain nonzero vacuum expectation values at the 3 New Way of Second Quantized Theory of Fermions... 45 electroweak phase transition. Table 3.1 represents in the left column eight families of creation operators of UR1 ^ — the first member in Table 3.6 — and of chargeless VR — the 25th member in Table 3.6. (S11 12, for example, transforms UR^ into VR and opposite). I. B.ii. The eight-plets separate into two groups of four families: One group contains doublets with respect to N R and T2, these families are singlets with respect to N L and f1. Another group of families contains doublets with respect to N L and T1, these families are singlets with respect to N R and T2. Mass matrices of both groups manifest correspondingly, when the scalar fields — the gauge fields of (NR, T2, U(1)) and (Nj., f1, U(1)) — obtain nonzero vacuum expectation values. Correspondingly both groups manifest SU(2) x SU(2) x U(1) symmetry, with the same U(1) and two different SU(2)(L,R) x SU(2)(I,II) symmetries, Ref. [57]. To the lower four families the observed three families of quarks and leptons contribute [51-53,55,56,58]. By the spin-charge-family theory predicted SU(2) x SU(2) x U(1) symmetry of mass matrices, which limits the number of free parameters of mass matrices, the properties of the fourth family could be predicted by fitting free parameters to the experimental data. However, the accuracy of the so far measured 3 x 3 mixing (sub)matrices are even for quarks far from the required precision, which would enable prediction of masses of the fourth family members [55,56]. We predict for the assumed masses of the fourth family of quarks the corresponding matrix elements. Calculations show [56] that the larger the masses of the fourth family — up to 6 TeV seems to be allowed by experiments [40] — the smaller the unwanted mixing elements which could cause the flavour-changing neutral currents and the better is agreement with the experimental data, which require, due to the observations in Refs. [40,41], that there should be the fourth family due to the nonunitarity of the 3 x 3 so far measured mixing matrix for quarks and that the 4 x 4 mixing matrix elements should have the properties: VUl d4 > VUl d3, Vu2 d4 < VUl d4, and V^ d4 < VUl d4. Here ut, di, i = 1,2,3,4 represent u, c, t, u4 and d, s, b, d4 quarks. The lowest of the upper four families is, as evaluated in Refs. [54,61], the candidate, which can explain (or at least can contribute to) the appearance of the dark matter in the universe. Comparing the results from following the fifth family members in the expanding universe with the astrophysical observations of dark matter and the direct measurements of the dark matter, the predicted masses of the fifth family quarks would be 102 TeV < mq5 c2 <4 • 102 TeV, and the scattering cross section ct for the fifth family neutron at least 10-6 x smaller than the cross section for the first family neutron. These values change if the fifth family neutron is not the only source of the dark matter. The fifth family would correspondingly manifest completely different "nuclear force" than the members of the lower four families [54], leading to different atoms and molecules, if they would success to form a matter in the expanding universe. II. The gauge fields — the vielbeins, faa, and the two kinds of the spin connection fields, daba and daba, of Eq. (3.1), appearing in the 2nd,3rd and 4th lines in Eq. (3.9) — manifest in d = (3 + 1) as the vector gauge fields of t3, f13 t23 Nt N3 T Ctc1t R 1 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 11 12 13 14 I (+i) [+] W ( + ) II (+) (- -) (-) (+i) [+] W (+) ( + ) [+] [+] 1 2 0 i 2 0 1 2 -ftc1t R 2 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 11 12 13 14 I [+i] (+) W ( + ) II (+) (- -) (-) [+i] (+) W (+) ( + ) [+] [+] 1 2 0 1 2 0 1 2 -ftc1t R 3 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 11 12 13 14 I (+i) [+] (+) W II (+) (- -) (-) (+i) [+] ( + ) W ( + ) [+] [+] 1 2 0 1 2 0 1 2 Ctc1t R 4 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 11 12 13 14 I [+i] (+) (+) W II (+) (- -) (-) [+i] (+) ( + ) W ( + ) [+] [+] 1 2 0 1 2 0 1 2 -ftc1t R 5 03 12 56 78 9 10 11 12 13 14 ♦ts 03 12 56 78 9 10 1112 13 14 II [+i] [+] [+] W II (+) (- -) (-) [+i] [+] [+] [+] (+) [+] [+] 0 1 2 0 1 2 1 2 Ctc1t R 6 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 11 12 13 14 II (+i) (+) 1 [+] [+] II (+) (- -) (-) (+i) (+) 1 [+] [+] ( + ) [+] [+] 0 1 2 0 1 2 1 2 -ftc1t R 7 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 11 12 13 14 II [+i] [+] 1 (+) ( + ) II (+) (- -) (-) [+i] [+] 1 (+) (+) ( + ) [+] [+] 0 1 2 0 1 2 1 2 •ftc1t R 8 03 12 56 78 9 10 1 12 13 14 03 12 56 78 9 10 11 12 13 14 II (+i) (+) 1 (+) (+) II (+) ( -) (-) (+i) (+) 1 (+) (+) II (+) [+] [+] 0 1 2 0 1 2 1 3 Table 3.1. Eight families of creation operators of Ct^1'' — the right handed u-quark with spin \ and the colour charge (t33 = 1 /2, t38 = 1 /(2\/3>)), appearing in the first line of Table 3.6 — and of the colourless right handed neutrino ^ — of spin appearing in the 25th line of Table 3.6 — are presented in the left and in the right column, respectively. Table is taken from [9], Families belong to two groups of four families, one (I) is a doublet with respect to (Nl and t'1') and a singlet with respect to (NR and t'2'), the other (II) is a singlet with respect to (Nl and t'1') and a doublet with with respect to (Nr and i2'), Eq. (3.111). All the families follow from the starting one by the application of the operators (NR L, t'2,1 Eq. (3.129). The generators (NR L, t'2,1 (Eq. (3.129)) transform itjR to all the members of one family of the same colour. The same generators transform equivalently the right handed neutrino j R to all the colourless members of the same family. 3 New Way of Second Quantized Theory of Fermions... 47 Eq .(3.113), t4, Eq. (3.113), t1 , Eq. (3.112), and t2, Eq. (3.112), if the space index is m = (0,1,2,3) (2nd line in Eq. (3.9)), as well as the scalar gauge fields, if the space index is s > 5 (3rd and 4th line in Eq. (3.9)), of the same operators as in the vector gauge fields case, Ref. [5]. Only if there are no fermion present, then both, and daba, are uniquely expressed by vielbeins, Ref. ([9], Eq. (C9)). dab« = daba = -^ |ee«eby 9P (Ef^f^a]) + ee«eay 3p(EfY[bfpe]) - eeaeeT 3P(EfY[afPb])} - ¿{eaaEeVp (Efy[dfpb]) - ebaEedy3p (EfY[dfpa])j . (3.10) II. A. It is proven in Ref. [5] that the vector (as well as the scalar gauge fields) can indeed be expressed with the spin connections (rather than with the vielbeins), A^1 = CAist dstm, (3.11) s,t demonstrating the symmetry of space with (s, t) > 5, making the spin-charge-family theory transparent and correspondingly "elegant", so that it is easier to recognize that the origin of charges of the observed fermions, vector gauge fields, Higgs's scalar and Yukawa couplings might really be in (d - 4) space. In the presence of the condensate, Table 3.3, of the right handed neutrinos, all the vector gauge fields and the scalar gauge fields, which interact with the condensate, gain masses. Only the weak (SU(2^), the colour (SU(3)) and the hyper (U(1), Y = t4 + t23) gauge fields, which do not interact with the condensate, remain massless. II. A.i. The weak vector gauge fields Am , the gauge field of SU(2)^ and Am, the gauge fields of SU(2)n, are the superposition of gauge fields ds/t/s (Ref. [9], Eqs. (8,9,10)), Am = (d58m - d67m, d57m + d68m, d56m - d78m) , Am = (d58m + d67m, d57m - d68m, d56m + d78m) . (3.12) Taking into account Eq. (3.113) one easily finds the colour vector gauge field expressed with dstm. Am get masses by interaction with the condensate. In Ref. [5], Eqs. (24-25), the reader can find Lagrange density for the R(d-4) part of the gravity field R, Eq.(3.1), expressed by the vector gauge fields Am. II. B. The scalar gauge fields are the superposition of either ds t/s, with (s ',t ',s) = (5,6, ••• ,14), Ref. [5], or d abs, with (a,b) = (0,1, ••• ,8) and (s) = (5,6,7,8), Refs. [4,7,9], the fourth line in Eq. (3.9). 48 N.S. Mankoc Borstnik and H.B.F. Nielsen Both kinds of scalar fields with s = (7,8) contribute to the masses of the two groups of four families. Scalar fields ds't's, with (s', t') = (5,6, • • • ,14), s = (9,10, • • • ,14) contribute to matter-antimatter asymmetry and to proton decay [4]. II. B.i. In the spin-charge-family theory the scalar fields with the space index s = (7,8) carry with respect to this space index the weak charge and the hyper charge (t 2, ± 1), respectively, independent of whether they are superposition of ds't's or of d>abs, s = (7,8), Refs. [9,3,4]. There are twice two triplets, the superposition of dabs, Eqs. (3.111,3.112) with Sab replaced by Sab, the gauge scalar fields of either the group sU(2)jO(3 1 )L x SU(2)i or of the group sU(2)gg(3 x SU(2)n, the first two triplets interacting with one group of four families, the second two triplets interacting with another group of four families, both groups presented in Table 3.1. There are also three singlets, the gauge scalar fields of = (Q,Q , Y ), Eq. (3.114), which are the superposition of d s't's and interact with members of all the eight families of Table 3.1 [7,9,3,4]. Let us use a common notation AA1 for all the scalar fields, independently of whether they originate in dabs or dabs, s = (7,8), aa1 e (a? ,a?' ,ay' ,A 1 ,ANL ,A2,ANR), TA1 d (Q, Q', Y', T1, NL, T2, Nr) . (3.13) Here tai represent the operators of the groups the gauge scalar fields of which are As A1 Let us rewrite the third line in Eq. (3.9) as follows, Ref. ([9], Eqs. (18-19)). £ f Ys (-TA1 AA1) f = s=(7,8),A,1 { (+) ta1 (AA1 - iAA1)+(-) (TA1 (AA1 + iAA1) }f , A,1 78 1 (±)= 1 (Y7 ± iY8 ), AA1 := (AA1 T iAA1), (3.14) 2 (±) with the summation over A, i performed, since AA1 represent the scalar fields (A?, A? , Aj', A^, A], A2, Anr and A). In the low energy regime the momentum ps, s = (7,8) can be neglected. Taking into account that t13 = ± (S56-S78), Y = (t23 +t4), t23 = 1 (S56+S78), while t4 = -3(S910 + S11 12 + S1314),and SabAc = i(Aa5b - A^), one finds 1 2 1 2 Q (AA1 T iAA1) = 0. (3.15) T13 (AA1 T iAA1) = ± 1 (AA1 t iAA1), Y (AA1 T iAA1) = T 1 (AA1 T iAA1), This are quantum numbers of the by the standard model assumed Higgs. These scalar gauge fields with the space index (7,8), gaining nonzero vacuum expectation values (by assumption as in the standard model so far), cause the electroweak 3 New Way of Second Quantized Theory of Fermions... 49 break, breaking the weak and the hyper charge, explaining the appearance of in the standard model assumed Higgs and the Yukawa couplings, predicting the existence of several scalars — two triplets and three singlets, which couple to the lower four families, making them massive and giving masses to weak bosons. These scalar fields manifest the SU(2) x SU(2) xU(1) symmetry, which reduces the number of free parameters in mass matrices of quarks and leptons, enabling predictions of properties of the four families [55-57]. II. B.ii. The scalar fields with the space index s = (9,10, • • • ,14), presented in Table 3.2, carry triplet or antitriplet colour charges and the "spinor" charge equal to twice the quark or antiquark "spinor" charge, and the fractional hyper and electromagnetic charge. They carry in addition the quantum numbers of the adjoint representations originating in Sab or in S ab. (Although carrying the colour charge of one of the triplet or antitriplet quantum numbers, these fields can not be interpreted as superpartners of the quarks, since they do not have quantum numbers as required by, let say, the N = 1 supersymmetry. The hyper charges and the electromagnetic charges are namely not those required by the supersymmetric partners to the family members.) Let us have a look what do the scalar fields, appearing in the fourth line of Eq. (3.9) and in the seventh line of Table 3.2, do when applying on the left handed members of the Weyl representation presented on Table 3.6, containing quarks and leptons and antiquarks and antileptons [71,72,65]. Fig. 3.1 presents the creation of proton due to the interaction of quarks and leptons with these scalar fields. One can read on this Fig. 3.1 all the quantum numbers of a positron (57th line of Table 3.6), an antiquark (43rd line of Table 3.6), and a quark (9th line of Table 3.6), as well as of the scalar field A^, seventh line of Table 3.2, involved in the proton birth. The opposite transition at low energies would make the proton decay. After the appearance of the condensate of the two right handed neutrinos, Table 3.3, the discrete symmetry CNPN is obviously broken. In the expanding universe, fulfilling the Sakharov request for appropriate non-thermal equilibrium, the triplet scalars from Table 3.2 have a chance to explain the matter-antimatter asymmetry in the universe [4]. III. The spin-charge-family theory suggests two kinds of phase transitions — two kinds of breaking symmetries: The appearance of the condensate and the nonzero vacuum expectation values of the scalar fields with the space index s = (7,8). III. A. Table 3.3 represents the properties of the condensate of the two right handed neutrinos VR8 — Table 3.1 — of spin up and spin down, breaking the discrete CNPN symmetry Subsect. 3.3.3, [4,65]. Due to the interaction with the condensate of Table 3.3 the gauge vector fields of t2 and t4 become massive. The colour vector gauge fields of t3 , the weak vector gauge fields of t1 and the hyper vector gauge field of Y do not interact with the condensate (the corresponding quantum numbers of the condensate are zero) and correspondingly remain massless, the gravity in d = (3 + 1), which is the gauge field of Smn and pm, remains massless as well. 50 N.S. Mankoc Borstnik and H.B.F. Nielsen a 1Q a9 10 (© a 13 a9 10 (© a10 a11 12 (© a 13 a 11 12 (© a10 a13 14 (© a 13 a13 14 _ (Q 2, © © 3 © 3 (Q 2, © (© 2 (© 2 273 2^73 ) -J—) 273 ) -J—) 273 ) © 3 + © 1 © 3 © 3 © © 3 + m 1 © 3 © 33 + © 1 © 3 3 (0, © (0, © 73' 9 10 (© ,23 9 10 (© © 3 + © © 2 73 © 3 + m 1 ©© 910 (© 910 (© © 3 © 3 273' 273' © 3 © 3 © 3 © 3 , 2Q 910 910 (© 273' 273' ©© ©© 910 (© 910 (© © © 2 73 2 73 ©© ©© 910 (© . nR 910 (© © © 2 73 2 73 ©© ©© 9 10 (© 4 9 10 (© + © , ± 273) ©© ©© © 1, © —l) © 2 , © 273) ©© ©© 4 3 23 (t33,T38) 4 3 23 3 field y q r © 3 © 3 0 0 0 © 3 © 3 0 0 0 0 0 0 0 0 0 0 0 © 3 © 3 ) 0 0 0 © 3 © 3 0 0 0 0 © 3 2, © 2773) © 3 1 2 © 3 0 2 0 2 © 3 1 2 © 3 © 3 1 2 © 3 0 n l © 3 1 2 © 3 © 3 1 2 © 3 0 n R © 3 1 2 © 3 © 3 1 2 © 3 0 © 3 1 2 © 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 Table 3.2. Quantum numbers of the scalar gauge fields carrying the space index t = (9,10, • • • , 14), appearing in the fourth line of Eq. (3.9), are presented. To the colour charge of all these scalar fields the space degrees of freedom — the space index — contribute one of the triplets or antitriplet values. These scalars are with respect to the two SU(2) charges, (f1 and T2), and the two SU(2) charges, (f1 and T2), triplets (that is in the adjoint representations of the corresponding groups), and they all carry twice the "spinor" number (t4) of the quarks or antiquarks. The quantum numbers of the two vector gauge fields, the colour and the U(1 )n ones, are added. These Table is taken from Ref. [4], Table I. We invite the reader to visit Ref. [4] for more details. 3 New Way of Second Quantized Theory of Fermions... 51 = 2 ,T 1 ^33 _38\ (t 33 ,t 38) = (0,0) y=1,0=1 (T 33,T 38) = ( 2 , ^ ) Y=-3 ,Q=- 3 -dR 4_1 _13 23 t 23 1 t 2 + l ; A2B (+) 4=2x (— 6 ),t 13=0,t 23= (t 33,t38) = ( 2 , ) 0,t 23_ 8) — ( 1__L_ ) ) = ( 2 , 2V3) -2 0=-2 3,0= 3 t 4 = 1 ,t 13 = 0,t 2 = 2 (t 33,t 38)=(0,—-13) VS' y = 3,0=3 t 3,0 4 3 u r l 4 1 T13 6 t u r 0,t 23 = (t .4= 1 t13= = 6 ,t =0,' = 2 33 t38) = / 1 ) t ) = ( 2 , 2—3) y=3,0=3 u r Fig. 3.1. The birth of a "right handed proton" out of a positron e+, antiquark ur2 and quark (spectator) ur2. The family quantum number can be any. state S03 S12 T13 T23 T4 Y Q T13 t23 T4 Y Q INI l IN r (|v1R >1 |v2R >2J 0 0 0 1 -1 0 0 0 1 -1 0 0 0 1 il ,viii . 1 viii . 1 l|v1 r >1 |evr >2j iloviii . , viii . i (|e1r >1 |evr >2 j 0 0 0 0 -1 -1 -1 0 0 0 -1 -1 -2 -2 0 1 -1 0 0 0 1 0 1 -1 0 0 0 1 Table 3.3. The condensate of the two right handed neutrinos vR, with the quantum numbers of the VIIItH family, coupled to spin zero and belonging to a triplet with respect to the generators T2i, is presented, together with its two partners. The condensate carries T1 = 0, t23 = 1, t4 = -1 and Q = 0 = Y. The triplet carries T4 = -1, T23 = 1 and INIr = 1, N r = 0, Y = 0, Q = 0. The family quantum numbers of quarks and leptons are presented in Table 3.1. Due to nonzero family quantum numbers of the condensate the corresponding scalar gauge fields become massive. The condensate gives masses to all the scalars from Table 3.2, either because they couple to the condensate due to t4 or t4 or t23 or T23 quantum numbers. It gives masses also to all the scalar fields with s € (5,6,7,8), since they couple to the condensate due to the nonzero t23. The scalar fields with the quantum numbers of the upper four families couple in addition through their family quantum numbers. III. B. The electroweak phase transition is caused by the nonzero vacuum expectation values of twice two triplets and three singlet scalars, giving masses to the lower fourth families — two of twice two triplets and three singlets — and to the upper four families — another two triplets and the same three singlets. 52 N.S. Mankoc Borstnik and H.B.F. Nielsen IV. Predictions of the spin-charge-family theory so far. IV. A. The spin-charge-family theory predicts the fourth family to the observed three to be observed at the LHC [53]. By predicting symmetry of mass matrices (in all orders of loop corrections [57]) the theory enables for accurate enough measured mixing matrices of the 3 x 3 submatrices (the sensitivity of the fitting procedure on masses of the so far measured quarks and leptons is much smaller [55,56]), and due to other measured properties of quarks and lep-tons [40], to predict the properties of the 4 x 4 mixing matrices and to explain correspondingly the origin of Higgs and Yukawa couplings. The 4 x 4 mixing matrix elements for quarks are predicted to have the properties: VU1 d4 > VU1 d3, Vu2 d4 < VUl d4, and V^ d4 < VUl d4, here ut, di, i = 1,2,3 represent u, c, t, U4 and d, s, b, d4 quarks. The theory explains [58] why the fourth family has not yet been observed, which is the main argument against the existence of four families [59,60] among experts in high energy physics. IV. B. The theory predicts the existence of several scalar fields — there are two triplets and three singlets which determine masses of the lower four families [9,7,3,6] — some of which will be observed in the near future measurements. IV. C. The theory predicts the second group of four families, the stable one of these four families contributing to the dark matter [54]. The nuclear force among these baryons differs a lot from the so far observed nuclear force [54,61]. IV. D. The masses of quarks and leptons are, according to these two groups of four families, spread from 10-3 eV to few TeV — at least 12 orders of magnitude for the first four families — and from 100 TeV to 1013 TeV — at least 11 orders of magnitude for the second four families, offering the explanation for the hierarchy problem. (The mass matrices of the two groups of mass matrices are very closed to the democratic ones [55,56]). IV. E. The spin-charge-family theory predicts the masses of the dark matter baryons [54]. IV. F. The spin-charge-family theory predicts the scalar fields which contribute to the matter-antimatter asymmetry in the universe [4] and correspondingly also to the proton decay. V. The spin-charge-family theory has (so far) several open problems, although it is also true that the more work is done, the more solutions of the open problems follow. V. A. In the spin-charge-family the vector and scalar gauge fields originate in gravity as the two kinds of the spin connection fields and the vielbeins. In the low energy region these vector and scalar gauge fields can be quantized in the usual way [5]. Yet the quantization of gravity remains as an open problem when the energies rise up to 1016 GeV and above. V. B. The dimension of space time — 13 + 1 — remains as an open problem: Why d = (13 + 1), why not 00? (Only 0 and 00 need no explanation.) How has the universe come to d =(13 + 1) [77]? V. C. Breaking the symmetry with the appearance of the condensate [4], which lead to observable properties of fermion and boson fields, explaining all the 3 New Way of Second Quantized Theory of Fermions... 53 assumptions of the standard models, needs to be studied as a dynamical appearance of the condensate in the expanding universe. V. D. It should be demonstrated dynamically how do the scalar fields gain nonzero vacuum expectation values, leading to the effective fields as assumed for the Higgs. The demonstrations, made in Refs. [68-70] for the toy model in d = (5 + 1) must be done also for d = (13 + 1). V. F. The coupling constants of the gauge and scalar fields in the low energy regime should be evaluated when starting with the simple action of Eq. (3.1) in d = (13 + 1), with only one (or already with two) coupling constants. V. G. There are additional open problems which we already see and either solve, like the one treated in this paper about the internal degrees of freedom of fermions in Clifford and Grassmann space and the new way of second quantization procedure, which explains the usual way of second quantization, or they wait to be solved, like the lepton number non conservation in the spin-charge-family theory. And there are open problems which we do not see yet or which we could better understand if learning more from all the trials to understand the evolution of the universe and the creation of hadrons of all kinds in the literature. 3.2 Fermions in Grassmann and in Clifford space In the literature the Clifford algebra is frequently discussed as an useful tool to describe internal degrees of freedom of fermions [62-64]. In the spin-charge-family theory Clifford space is used to describe all the internal degrees of fermions — quarks and leptons with their families included [1,2,9]. In this paper we demonstrate that the Clifford algebra offers an elegant and transparent way to better understand fermions properties: In even dimensional spaces — we make a choice of d = 2(2n + 1), n = 3 — the creation operators of an odd Clifford character can be defined (they are superposition of odd numbers of the Clifford algebra objects (ya's or Ya's, Eq. (3.2)), each of them is a product of f nilpotents and projectors, Eq. (3.27,3.70) [47,48], so that they are the eigenvectors of twice all the f members of the two kinds of the Cartan subalgebras of the Lorentz algebra — Sab and Sab — with the half integer eigenvalues, Eq. (3.72). These creation operators, Eq. (3.76), and their Hermitian conjugated partners — the annihilation operators, Eq. (3.77) — fulfill on the vacuum state, Eq. (3.79), the anti commutation relations required for fermions, Eq. 3.81. The superposition of these creation operators solve for a particular momentum pa the equation of motions for free massless fermions, Eq. (3.36), determining in d = (3 + 1) spins, handedness, charges and family quantum numbers. Again they fulfill on the vacuum state, Eq. (3.79), together with their Hermitian conjugated annihilation operators, the anti commutation relations required for fermions, Eq. (3.83). Correspondingly the creation and annihilation operators are indeed defined with the first quantized fermion fields already. We demonstrate in this paper that there exist also in Grassmann space of anticommuting coordinates, Eq. (3.3), the eigenvectors of the Cartan commuting subalgebra of the Lorentz algebra Sab, Eq. (3.3, 3.21), the f products of which form creation operators, Eq. (3.51), and which fulfill together with their Hermitian 54 N.S. Mankoc Borstnik and H.B.F. Nielsen conjugated partners the annihilation operators, Eq. (3.18), as well the anticommutation relations required for fermions, Eq. (3.54). However, the eigenvalues of the Cartan subalgebra are in this case integer. Also in the Grassmann case the superposition of these creation operators solve for a particular momentum pa the equation of motions for free massless fermions, presented in Eq. (3.43), determining in d = (3 + 1) spins, handedness and charges. There are no families in this case. For both cases, Clifford and Grassmann, we present the proofs for the above statements and illustrate the properties of fermions of both kinds on a few examples. 3.2.1 Actions and equation of motion in Clifford and in Grassmann space We define in d = ((d — 1) + 1)-dimensional space states with integer spin — in Grassmann space — and states with half integer spin — in Clifford space — proving that norms in both spaces can be determined by the integral in Grassmann space, Eqs. (3.32,3.33), since the Clifford algebra objects are expressible with the Grassmann algebra objects, Eq. (3.4) 2. When reformulating the vacuum in the Clifford case, Eq. (3.79), half integer spinors presentation in Clifford space become more elegant, that is easier to recognize properties of fermions. We present as well actions in both cases, Grassmann, Eq. (3.41), and Clifford, Eq. (3.36), leading to the equations of motion (in the Clifford case the Weyl equation is known for a long time, in the Grassmann case it is present for the first time by N.S.M.B.). We compare Euler-Lagrange equations in both cases to compare properties of Grassmann "fermions" with the Clifford fermions. a. Fields with the integer spin in Grassmann space A point in d-dimensional Grassmann space of anticommuting coordinates ea, (a = 0,1,2,3,5,..., d), is determined by a vector {ea} = (e0, e1, e2, e3, e5,..., ed). A linear vector space over the coordinate Grassmann space has correspondingly the dimension 2d, due to the fact that (eai )2 = 0 for any at e (0,1,2,3,5,..., d). Correspondingly are fields in Grassmann space expressible in terms of the Grassmann algebra objects d B = ^ aa,a2...ak eai ea2 ...eak |^og >, at < at+1 , (3.16) k=0 where |^og > is the vacuum state, here assumed to be |^og >= |1 >, so that "ae?|^0g >= 0 for any ea. The Kalb-Ramond boson fields aai a2 ...ak are antisymmetric with respect to the permutation of indexes, since the Grassmann coordinates anticommute {ea, eb}+ = 0, Eq. (3.3). 2 Observations in this paper might help also when fermionizing boson fields or bosonizing fermion fields [42]. 3 New Way of Second Quantized Theory of Fermions... 55 The left derivative -g|p on vectors of the space of monomials B(9) is defined as follows -A. B(0) = dBM, 3ea ( ) 39a ' deea'ae^l + B = foraUB• (3.17) The commutation relations are for p0a = idefined in Eq. (3.3), where the metric tensor nab (= diag(1, -1, — 1, • • •, -1)) lowers the indexes of a vector {ea}: ea = nab eb (the same metric tensor lowers the indexes of the ordinary vector xa of commuting coordinates). Defining3 (ea)f = ^naa = —ip0anaa, (3.18) it follows sea (— )t = naa ea, (p0a )t = — inaaea • (3.19) oea Making a choice for the complex properties of ea, and correspondingly of , as follows {ea}* = (e0 ,e\ —e2,e3, — e5,eV„, —ed-1,ed), r = (A —A — A —) (3 20) xae/ (ae0, ae1, ae2, ae3, ae5, ae6,•••, aed-1, aed), v' ' it follows for the two Clifford algebra objects Ya = (ea + ), and Ya = i(ea — g|p), Eq. (3.4), that Ya is real if ea is real, and Ya is imaginary if ea is imaginary, while ya is imaginary when ea is real and Ya is real if ea is imaginary, just as it is required in Eq. (3.26). Applying the operator Sab of Eq. (3.3) on the "states" ^(ea + nkeb), a = b, and (1 + kkeaeb), a = b, it follows 1 naa 1 naa Sab -= (ea + ^eb) = k — (ea + ^eb), v2 ik V2 ik Sab --= (1 + -Veb) = °, (3.21) 2k k2 =naanbb. We define here the commuting objects yG, which will be helpful when looking for the appropriate action for Grassmann fermions, Eq. (3.41). These operators will be needed also when looking for the definition of appropriate discrete symmetry operators in the Grassmann case. Following the definition of the discrete symmetry 3 In Ref. [2] the definition of was differently chosen. Correspondingly also the scalar product needed a (slightly) different weight function in Eq. (3.32). 56 N.S. Mankoc Borstnik and H.B.F. Nielsen operators in the Clifford algebra case [65] in ((d — 1) + 1) space-time and in (3 + 1) space-time, the discrete symmetry operators (Cg, Tg, Vg) in ((d — 1) + 1) and (Cng, Tng, Vng) in (3 + 1) will be defined in Subsct. 3.3.3, respectively. yG = (1 — 29anaa^- )=—inaa YaYa , {YG,YG}- = 0. (3.22) 39 a Index a is not the Lorentz index in the usual sense. yG are commuting operators for all (a, b). They are real and Hermitian. yG1 = yG , (yGr = yG . (3.23) Correspondingly it follows: yG^Yg = I, YgYg = I.I represents the unit operator. By introducing [2] the generators of the infinitesimal Lorentz transformations in Grassmann space, as presented in Eq. (3.3), and making use of the Cartan subalgebra of commuting operators, Eq. (3.110), the basic states in Grassmann space can be arranged into representations of the eigenstates of the Cartan subalgebra operators, Eq. (3.21), Ref. [2,46]. All these states have integer spins (k is ±i or ±1). The starting state in d-dimensional space, for example, with the eigenvalues of the Cartan subalgebra equal to either i or 1 is: (90 — 93)(91 + i92)(95 + i96) • • • (9d-1 + i9d)|^og >, with |^og >= |1 >, Eq. (3.21). All the states of the representation, which starts with this state, follow by the application of those Sab, which do not belong to the Cartan subalgebra of the Lorentz algebra. S01 , for example, transforms this starting state into (9093 +i91i92)(95 + i96) • • • (9d-1 +i9d)|^og >, while (S01 — iS02) transforms this state into (90 + 93)(91 — i92)(95 + i96) • • • (9d-1 + i9d)|^og >. b. Fields with the half integer spin in Clifford space Let us present as well the properties of the fermion fields with the half integer spin, expressed by the Clifford algebra objects Ya's ([1,2,9,3,5,4,47] and the references therein) F =Y- aa,a2...ak Ya Ya2 ... Yak|^oc >, a. < ai+1 , (3.24) k=0 where |^oc > is the vacuum state. The Kalb-Ramond fields aai a2...ak are again in general boson fields, which are antisymmetric with respect to the permutation of indexes, since the Clifford objects have the anticommutation relations, Eq. (3.2), {Ya, Yb}+ = 2nab. The linear vector space over the Clifford coordinate space has, as in the Grassmann case, the dimension 2d, due to the fact that (Yai)2 = naiai for any at G (0,1,2,3,5,..., d). As written in Eq. (3.4), Ya are expressible in terms of the Grassmann coordinates and their conjugate momenta, as Ya = (9a — ip0a), and Ya = i (9a + ip0a), with the anticommutation relation of Eq. (3.2), {Y,Yb}+ = 2nab = {Ya,Yb}+, {Ya, Yb}+ = 0. Taking into account Eqs. (3.18, 3.19, 3.4) one finds (Ya)f = Yanaa, (Ya )f = Y anaa, YaYa = naa , Ya(Ya )f = I, YaYa = naa , Y a(Y a)f = I, (3.25) 3 New Way of Second Quantized Theory of Fermions... 57 where I represents the unit operator. Making a choice for the 0a properties as presented in Eq. (3.20), it follows for the Clifford objects {yT = (Y0,Y1, -Y2,Y3, -Y5,Y6,..., -Yd-1,Yd), {YT = (-Y0, -•Y1 ,Y2, -•Y3,Y5, -•Y6,...,^d-1, -•Yd), (3.26) Applying the operators Sab and Sab, Eq. (3.2), on 2(Ya + ntTYb) and on 2 (1 + kYaYb), and taking into account the relation of Eq. (3.69), one obtains 1 naa k 1 naa Sab 1 (Ya + ¥Yb) = (Ya + ^Yb), 1 naa k 1 naa S ab 1 (Ya + V Yb) = ^ (Ya + V Yb), Sab 1 (1 + kYaYb) = 2} (1 + kYaYb) SabI0 + kYaYb) = -2 }(1 + £yV). (3.27) One could make a choice of Ya instead of Ya and change correspondingly the relations in Eqs. (3.69, 3.27). All the three choices for the linear vector space — spanned over either the Grassmann ea's, or over the vector space of Ya's, or over the vector space of Ya's — have the dimension 2d. More about the meaning of these degrees of freedom in any of theses cases can be found in Ref. [11]. Let us point out here that 0a's and dp's (each of them has 2d degrees of freedom) are expressible with Ya's and Ya's (with 2d degrees of freedom each) and opposite. Since {Ya, Yb}+ = 0, Ya's and Ya's form independent degrees of freedom. We should therefore allow also Ya's to form the vector space. We can express Grassmann coordinates 0a and momenta p0a = idp in terms of Ya and Ya as well4 0a = 1 (Ya - iYa), 7\ 1 ae; = 2 (Ya + iYa), (3.28) with dipea11 >= nab|1 >. Requiring that the application of Ya's on Ya's are determined by Eq. (3.69), the Ya's part is sacrificed [11]. The two possibilities are no longer acceptable: Ya's are chosen to span the basis, while Ya's become operators which determine the family quantum numbers. From Eqs. (3.28, 3.69) follows that ea = 0 and 0a = y. All the relations of Eq. (3.2) remain valid, while the space of Ya's is sacrificed and the Grassmann space has lost , the Hermitian conjugated partner of 0a. (Of course, we can still replace Ya by Ya, if we change correspondingly the vacuum state |^oc > and relation in Eq. (3.69)). 4 In Ref. [76] the author suggested in Eq. (47) a choice of superposition of ya and Ya, which resembles the choice of one of the authors (N.S.M.B.) in Ref. [2] and both authors in Ref. [47,48] and in present article. 58 N.S. Mankoc Borstnik and H.B.F. Nielsen The vacuum state |^og >= 11 > must after Eq. (3.69) be transformed into l^oc > with the property [2,7,9] <^oclYa|^oc > = 0, ya|^oc >= iYa|^oc >, YaYb|^oc >=-iYbYa|^oc >, YV|^oc > |a=b = -YaYb|^oc > , YaYb|^oc > |a=b = nab|^oc > . (3.29) This is in agreement with the requirement Ya F(y) |^oc >:= ( ac Ya + aa, Ya Yai + aa,a2 Ya Yai Ya2 + ••• + aa, ...ad Ya Yai ••• Yad ) |^oc >, Ya F(y) |^oc >: = (i acYa - i aa, Yai Ya + i aa, a2Yai Ya2 Ya + • • • + i (-1)d aa, -ad Yai ••• Yad Ya )|^oc > . (3.30) The basic states in Clifford space can be arranged in representations, in which any state is the eigenstate of the Cartan subalgebra operators of Eq. (3.110). The state, for example, in d-dimensional space with the eigenvalues of Sc3, S12, S56,...,Sd-1 d and of Sc3, S12, S56,..., Sd-1 d equal to 1 (i, 1,1,..., 1) is (yc - Y3)(Y1 + iY2)(Y5 + iY6) • • • (Yd-1 + iYd). The states of one representation follow from the starting state by the application of Sab, which do not belong to the Cartan subalgebra operators, while S ab, which operate on family quantum numbers, cause jumps from the starting family to the new one. Norms of vectors in Grassmann and Clifford space Let us look for the norm of vectors in Grassmann space, B = Y.d=c aa, a2...ak 0a 0a2... 0ak|^og >, and in Clifford space, F = Y_£=c aa, a2... ak Yal Ya2 ... Yak|^oc >, where |^og > and |^oc > are the vacuum states in the Grassmann and Clifford case, respectively. In what follows we refer to Ref. [2]. a. Norms of Grassmann vectors Let us define the integral over the Grassmann space [2] of two functions of the Grassmann coordinates < B|9 >< C|0 >, < B|9 >=< 0|B >t, by requiring d0a = 0, d0a0a = 1 , {d0a,0b}+ = 0, 3 dd0 0c01•••0d = 1 , dd0 = d0d ... d0c , d = n£=c (-^p + 0k), (3.31) with 0c = nac. We shall use the weight function d = njLc(-d^ + 0k) to define the scalar product < B|C > < B|C > = dd-1xdd0a d < B|0 > < 0|C >= k=C J dd 1xbb,...bkcb,...bk , (3.32) where, according to Eq. (3.18), it follows: d < B|0 >= ^(-i)p aa, ...app0ap napap • • • peal naa • p=c 3 New Way of Second Quantized Theory of Fermions... 59 The vacuum state is chosen to be |^og >= |1 >, as assumed in Eq. (3.16). The norm < B|B > is correspondingly always nonnegative. Let us notice that the choice of the Hermitian conjugated value of 0a is gf^ ((0a)t = naa dp, Eq. (3.18)) makes that we easily evaluate in any d the scalar product < foSl(didagd-r^ • • • wwje0«' • • 0d-20d-'ed)|^og >=1 for |^og >= |1 > (without integration over coordinate space of 0a's). b. Norms of Clifford vectors To evaluate norms in the Clifford space for vectors of Eq. (3.24) we can use as well Eqs. (3.31, 3.32), if expressing ya in terms of 9a and p0a: < (9a — ip0a)|F >. In this case |^oc >= |^og >= |1 >. It follows < F|G > = dd-1 xdd9a œ < F|y > < y|G >= ^ k=0 dd 1xaa,...akbbi ...bk . (3.33) To simplify the evaluation we use instead [3,47] in the Clifford case the vacuum state |^oc >, Eq. (3.79), which is the product of projectors, Eq. (3.70). It takes care of the orthogonality of states (like if we would evaluate the integration in Grassmann space). Correspondingly we can write dd9a w(aai a2...ak Yai Ya2 ... Yak)t(aai a2...ak Yai Ya2 ... Yak) = aa,a2...ak aaiü2...ak . (3.34) The norm of each scalar term in the sum of F is nonnegative. Actions in Grassmann and Clifford space We construct an action for free mass-less fermion in which the internal degrees of freedom is described: i. in Grassmann space, ii. in Clifford space. In the first case the internal degrees of freedom manifest integer spins, in the second case the half integer spin. While the action in Clifford space is well known since long [67], the action in Grassmann space will be defined here (by N.S.M.B.). In both cases we present an action for free massless fermions in ((d — 1 ) + 1 ) space5. States in Grassmann space as well as states in Clifford space will be arranged to be the eigenstates of 5 In d = (3 + 1 ) space masses of fermions are in the spin-charge-family theory in the Clifford case caused by the interaction of fermions with scalar gauge fields with the space index (7, 8), that is the vielbeins and the spin connections of two kinds — the gauge scalar fields of Sab and of S ab. We expect that masses of "fermions" appear also in the Grassmann case due to the interaction of fermions with scalar gauge fields with the space index (7,8), but in this case due to the vielbeins and the spin connection of one kind only — the gauge field of Sab 60 N.S. Mankoc Borstnik and H.B.F. Nielsen the Cartan subalgebra — with respect to Sab in Grassmann space and with respect to Sab and Sab in Clifford space, Eq. (3.110), and orthogonal and normalized with respect to Eq. (3.31)6. In both spaces the requirement that states are obtained by the application of creation operators on the vacuum state — in the Grassmann case 6® t on 11 >, Eq. (3.58), obeying together with the k the anti commutation relations of Eq. (3.54) on the vacuum state |^og >= |1 >, and in the Clifford case Eq.(3.76), obeying together with the ttf the equivalent anticommutation relations of Eq. (3.81) on the vacuum states |^oc >, Eq. (3.79) — reduces the number of states, in Clifford space more than in Grassmann space. But while in Clifford space all physically applicable states are reachable by either Sab (defining family members quantum numbers) or by S ab (defining family quantum numbers), the states in Grassmann space, belonging to different representations with respect to the Lorentz generators, seem not to be connected. a. Action in Clifford space In Clifford space the action for a free massless fermion must be Lorentz invariant A = ddx 1 (^Y0 YaPa^) + h.c., (3.35) pa = i dda, leading to the equations of motion YaPal^> = 0, (3.36) which fulfill also the Klein-Gordon equation YaPaYbPbl^ > = papal^ >= 0, (3.37) for each of the basic states l^oc > = l^f >. Y0 appears in the action since we pay attention that Sabt y0 = Y0 Sab , SV = Y0S-1 , i /cab , t ab \ S = e-2œab(s +L ). (3.38) The Lagrange density, Eq. (3.35), LC = 2(V Y0 Ya Pa^ - Pa^t Y0 Ya ^}, (3.39) leads to = 0 = Y0Ya Pa = 0 = pa Ya . (3.40) 3LC P 9£C a^t Pa 3Pa^t 9LC P 9£c Pa 3(pa^) 6 In the Clifford case the states can be orthogonalized also with respect to Eq. (3.79), while taking into account Eq. (3.71). 3 New Way of Second Quantized Theory of Fermions... 61 All the states, belonging to different values of the Cartan subalgebra — they differ at least in one value of either the set of Sab or the set of Sab, Eq. (3.110) — are orthogonal according to the scalar product, defined as the integral over the Grassmann coordinates, Eq. (3.31), for a chosen vacuum state |1 >. Correspondingly the states generated by the creation operators, Eq. (3.76), on the vacuum state, Eq. (3.79), are orthogonal as well. b. Action in Grassmann space We define here the action in Grassmann space, for which we require — similarly as in the Clifford case — that the action for a free massless fermion is Lorentz invariant Ag = 1 ddxdd0œ{*(1 - 29°^) ¿0aPa*} + h.c.. (3.41) We use the integral over 0a coordinates with the weight function w from Eq. (3.31, 3.32). Requiring the Lorentz invariance we add after * the operator yG (yG = (1 — 20a dda)), which takes care of the Lorentz invariance. Namely Sabt 0 — 20° ^ ) = 0 — 20° ^ ) Sab, St (1 — 29° d0°) = (1 — 20° d0°) S-1 , S = e-2— (LQb+Sqb), (3.42) while 0a, ddr and pa transform as Lorentz vectors. The equations of motion follow from the action, Eq. (3.41), 1 2 ,G 39c yG = (1 - 20°^) , (3.43) 2 yG (0a — ^ ) Pal*> = 0, 3 30° as well as the Klein-Gordon equation, yG (9a — dda) Pa yG (9b — afk) Pb I* >= 0, leading to {0aPa, Pb}+ = papa = 0. (3.44) 30b From the Lagrange density, presented in Eq. (3.41), using Eqs. (3.18,3.19,3.28) _ ^aa-.aj-.a fûa d ' G = —y y , (0 — ae7- 1 (yG = —inaaYaYa, (0a — adh) = —i""a) it follows, up to the surface term, Lg =— "G Ya (Pa*) = —4{* yG Ya Pa* — Pa*t yG Ya *}. (3.45) One correspondingly finds "3*r — pa dp^ = 0 = T"G Y pa *, 3Lg ^ 3Lg „ i d* — Pa d7dLG*) = 0 = -pa *t"G Ya , (3.46) 3* 3 (p a *) 2 62 N.S. Mankoc Borstnik and H.B.F. Nielsen The solutions of these equations are presented in Eq. (3.98). We shall see that, if one identifies the creation operators in both spaces with the products of odd numbers of either 9a — in the Grassmann case — or ya — in the Clifford case — and the annihilation operators as the Hermitian conjugated operators of the creation operators, the creation and annihilation operators fulfill the anticommutation relations, required for fermions. The internal parts of states are then defined by the application of the creation operators on the vacuum state. But while the Clifford subalgebra defines states with the half integer "eigenvalues" of the Cartan subalgebra operators of the corresponding Lorentz algebra, the Grassmann algebra defines states with the integer "eigenvalues" of the Cartan subalgebra operators of the corresponding Lorentz algebra. 3.3 Second quantization of Grassmann and Clifford vectors It is proven in this section that solutions of the Weyl equations — following from the Hermitian and Lorentz invariant actions for free massless fermions, using to describe their internal degrees of freedom either Clifford space, Eqs. (3.35, 3.36), or Grassmann space, Eq. (3.41,3.43), — can be represented as creation operators, operating on the appropriate vacuum state. The corresponding Hermitian conjugated operators, taken as their annihilation partners, fulfill together with the creation operators, if both are of an odd either Clifford or Grassmann character, the anticommutation relations required for fermions. Correspondingly there is no need to assume the anticommutation relations as done in the Dirac theory [67,74,75], since the creation and annihilation operators of an odd either Clifford or Grasmmann character by themselves fulfill the anticommutation relations for fermions without postulating them. Creation operators in both spaces determine the Hilbert space of n fermions for any integer n and have all the properties of the corresponding Slater determinants, if we recognize that a product of two creation operators of two different moments in the ordinary space (pk, pi) — 60pk • , applying on the vacuum state |^oc >, are zero if and only if i = j, a = | and pk = pl. In the Grassmann case 6™pk • bj, is replaced by • and the vacuum |^oc > by |^og >. Let us point out that fermions with the internal degrees of freedom described in Clifford space manifests half integer spins, while "fermions" with the internal degrees of freedom described in Grassmann space demonstrate integer spins. We pay attention in this paper on d = 2(2n+1)-dimensional spaces, arranging all the vectors to be "eigenvectors" of the Cartan subalgebra operators of Sab and Sab in the Clifford case and of Sab in the Grassmann case, Eqs. (3.110, 3.2, 3.3). In d-dimensional spaces the linear vector space, spanned over either the Clifford coordinates ya's or the Grassmann coordinates 0a's, has the dimension 2d. One can in both cases represent the vector space as 2d operators, which — when applied on the vacuum state — create 2d vectors. Half of these operators have an odd and half an even either Clifford (with respect to odd or even products of Ya's) or Grassmann (with respect to odd or even products of 0s's) character. In the Clifford case there are in the group of an odd Clifford character two groups of operators: each member of one group has its Hermitian conjugated 3 New Way of Second Quantized Theory of Fermions... 63 partner in another group. One of the two groups can be therefore chosen to represent the creation operators, the other to represent the corresponding annihilation operators. Each of the two groups has 2d-2 members. Each of the two Clifford odd groups, one with 2d-2 creation the other with 2d-2 annihilation operators, further divides into 2d-1 subgroups with 2d-1 members. All the 2 d -1 members of one particular subgroup are related by the operators Sab, while S ab transform each member of this subgroup of particular family into the same member of one of 2 d -1 families. In the group of the Clifford even operators there are again two groups, each with 2 d -1 • 2d -1 operators related by either Sab or by Sab. Within each of the group there are 2d-1 subgroups with 2d-1 members, related by the application of Sab, while S ab transform each member of a particular subgroup into the same member — with respect to the operators Sab — of another subgroup with again 2 d -1 members. These two groups are not related by the Hermitian conjugation as in the case of odd Clifford objects. In each of the two groups of an even Clifford character there are 2d-1 self adjoint operators. The rest of 2d-1 • (2d-1 — 1) Clifford even operators have the Hermitian conjugated partners within the same group. YaYa transform 2d-1 self adjoint operators of one Clifford even group into 2d-1 self adjoint operators of another Clifford even group, while yaYa transform the rest of this group — that is 2 d -1 • (2d -1 — 1) operators, having the Hermitian conjugated partners within the same subgroup — into 2d-1 x (2d-1 — 1) operators of another Clifford even group, having again the Hermitian conjugated partners within the same subgroup. Any odd Clifford member of the assumed (chosen to be) creation operators gives, when applied on one (only one) of the even self adjoint operators of only one of the two groups with (2d-1 )2 members, a nonzero contribution, which is the same creation operator back. It gives nonzero contribution also on one (only one) of the rest 2d-1 • (2d-1 — 1) operators of the same group to which also the self adjoint operator belong, transforming it to one of creation operators, belonging to another family of the creation operators. On all the others Clifford even objects this creation operator gives zero. The annihilation operators manifest, when applied on the Clifford even objects, equivalent properties as creation operators. Let bTat be the creation operator of an odd Clifford character, a denoting the subgroup with a particular value of the Cartan subalgebra of Sab (family) and with i denoting a particular member of a family a. To all the members of particular a one and only one of the selfadjoint operators of an even Clifford character corresponds, which, when any of these members applies on it, gives the same creation operator back. (bft)t = bf, denoting the corresponding annihilation operator of an odd Clifford character, gives zero when applied on the selfadjoint operators on which gives nonzero contribution. We choose the superposition of these selfadjoint operators to determine the vacuum state in the Clifford case, Eq. (3.79). 64 N.S. Mankoc Borstnik and H.B.F. Nielsen All the members of the odd Clifford character, half of them creation operators and half of them annihilation operators, fulfill the anticommutation relations, required for fermions. Correspondingly there are only 2t-1 • 2t-1 creation operators, determining 2 t -1 families with 2 t -1 family members each, which when applied on the superposition of selfadjoint operators of one group of Clifford even operators, create fermion states. These creation operators determine n fermions Hilbert space. In the Grassmann case there are two kinds of operators, 0a and g^r, Hermi-tian conjugated to each other, Eqs. (3.18,3.19). If 0a represent the creation operators, then gd^ are the corresponding annihilation operators. Not having the Hermitian conjugated partner with the property that when applying on |fog >= | ! > gives zero, the identity (I) can not belong either to creation or to annihilation operators. In d = 2(2n + 1 )-dimensional Grassmann spaces there are correspondingly 2d — 1 creation operators. The largest two representations have together -¿dry 2 ' 2 ' creation operators and the same number of annihilation operators of an odd Grassmann character, Eq. (3.59), chosen to be eigenstates of the Cartan subalgebra, Eq. (3.110), of Sab. All the irreducible representations of the Grassmann case are decoupled. The application of the creation operators, which are products of f 0a's, on the identity (I) gives them back, while the annihilation operators applied on I give zero. ÏTTT The ¿ddr creation operators split into two by the generators of the Lorentz transformations Sab unconnected groups, each with 2 -ddrr members. 2 2 ' 2 ' We introduce common notation for the Clifford and Grassmann case to simplify the discussion: Let be the creation operator of an odd Grassmann character with a = (1,2) denoting one of the two (by Sab unconnected) the largest subgroups and let i denotes one of the -2 ¿fa' members related among them- 2 2 ' 2 ' selves by Sab. We make a choice of the vacuum state in the Grassmann case to be Ifoa >= I 1 >. All members of two groups of 1 number of creation operators of an 2 ' 2 ' odd Grassmann character, and their Hermitian conjugated partners, fulfill the anticommutation relations, required for fermions. The number of vectors in the Hilbert space of n-fermions depends for a chosen momentum p£ on the number of the creation operators, creating a particular fermion in the Clifford case or a particular "fermion" in the Grassmann case. There are for each p£ in the odd Clifford case 22-1 -22-1 and in the odd Grassmann case (for the two the largest representations) ¿dU- creation operators b"P 2 ' 2 ' Pk of an odd character — either Clifford odd character, with a = (1, ••• ,2 t -1), i = (1, • • • ,2t-1), or Grassmann odd character, with a = (1,2),i = (1, • • • , -¿drr)), 2 ' 2 ' creating the corresponding single particle states, when applied on the vacuum states o > — in the Clifford case is the vacuum state oc >, the superposition of all selfadjoint operators, on which an odd gives a nonzero contribution, and in the Grassmann case the vacuum state is og >= | 1 >. Let the zero fermion state for any p£ in either Clifford or Grassmann space, be written as T >: = O^V A^V ,...,0S=1p1 v-^CPr,..., 00==1p2, 3 New Way of Second Quantized Theory of Fermions... 65 0ta=21p2 > ^ >... > 0ar«1p2 >... > 0CPr > ■ ■ ■ > ■ ■ ■ ^ >, with ^ >= o^c >, i1 > ), in the Clifford and the Grassmann case, respectively, and amax = (22-1,2) and imax = (2 2 -1, 2 xfrr), again in the Clifford and the Grassmann case, respectively. 2 ' 2 ' Then the vector space with n fermions in the Clifford case or n "fermions" in the Grassmann case, for any n looks like brpk i ^o > = i oi=iip1, oGV ,...,cxTr,..., ifpk ■ ■ ■ > i ^o > there are amax ■ imax such 1 — fermion states for each pk , n«=1,Omax ni=1,imax bipk ^o >, , na=1,Omax ni=1,imax ^o > , , there are 2amax lmax Slater determinants of fermions for each pk , • • • (3.47) «max = (22-1, 2) and imax = (22-1, 2 drrT) in the Clifford and Grassmann case, 2 2 '2 ' respectively. One sees that bat ipk b fpti ioa= a= 1 a a= 1 1 Pi ' 0i=2 p i : 1 'Pk ,..., 0?,,= 1Pk VPki b Pt b at |0a=1 0a=1 1a "jpi "tPk |0i=1 Pi '0i=2pi >...> 'i 1 'Pk' . ' 1 P' pt, '...' |^o > = Pkiv 0a 1 a if'p,,...,I^o >, (3.48) 7. 7 Each single particle state caries its own internal space, described by a creation operator with a superposition of an odd number of yf's, and its own coordinate space, described by xf's (or pf). The creation operators of any two pairs of particles therefore anti-commute. Correspondingly the two states of two particles must distinguish in either internal space or in the coordinate space, as it follows from Eq. (3.86). The property of the creation operators bopii>a'p'j applying on the n-particle state |1?p1, I^p ,2,1?p"3,... i,... i0?^ j,..., >,presented in Eq. (3.86), can be as well described by (superposition of) Slater determinants of single particle states. Let us add that the vacuum state, having the sum of the spins of both kinds of operators, Sab and S ab, equal to zero and therefore neutral, remains neutral also when filled with fermions of all the spins, Sab and S ab. 66 N.S. Mankoc Borstnik and H.B.F. Nielsen One fermion states are either in Clifford or in Grassmann space already second quantized, since in both cases they fulfill the anticommutation relations required for fermions, Eqs. (3.66, 3.87). All together there are 22 Slater determinants for a chosen pk in the Clif- d! ford case and 2 t! t ! Slater determinants for a chosen pk in the Grassmann case (if only the two largest group of odd irreducible representations are taken into d_1 account, if we take all odd representations into account we have 22 Slater determinants), pk has a continuously changing value, p0 = (0, oo), —od < pl < oo, I =(1,2,3,5, -,d). It can be concluded that there are only second quantized states, since the anticommuting creation and annihilation operators, creating a Clifford fermion or Grassmann "fermion" states, determine all the properties of the n-particle Hilbert space for any n. We shall as well recognize that no Dirac sea is needed either in the Clifford or in the Grassmann case, since the same Lorentz representation includes in both cases fermions and antifermions. We discuss in the subsections the second quantization procedure in both spaces, Clifford and Grassmann, when dimension of the space-time is larger than four. We demonstrate that if the dynamics manifests only in d = (3 + 1), that is when momentum is different from zero only in d = (3 + 1), pa = (p0,p",p2,p3,0,0, ••• ,0) — what happens at low energies after the break of Lorentz symmetries in d > 5 — spins in d > 5 manifest as charges in d = (3 + 1). While the Clifford case offers the explanation for all the properties of observable fermions (after sacrificing the space of Ya's), the Grassmann case, having difficulties in describing energy within the usual second quantized procedure, as long as the Lorentz invariance in internal space is unbroken, leads to unobserved "fermions" with integer spins. Let us point out that states in Grassmann space as well as states in Clifford space are organized to be — within each of the two spaces — orthogonal and normalized with respect to Eq. (3.31, 3.32, 3.33). All the states in each of spaces are chosen to be eigenstates of the Cartan subalgebra — with respect to Sab in Grassmann space, Eqs. (3.3,3.5, 3.110), and with respect to Sab and Sab, Eq. (3.2), in Clifford space, Eq. (3.110). We pay attention in this paper almost only to spaces with d = 2(2n + 1)8. 3.3.1 Second quantization in Grassmann space There are 2d states in Grassmann space, orthogonal to each other with respect to Eqs. (3.31, 3.32). To any coordinate there exists the conjugate momentum. We pay attentionin what follows mostly to spaces with d = 2(2n + 1). The states, which 8 The main reason that we treat here mostly d = 2(2n + 1) spaces is that one Weyl representation, expressed by the product of the Clifford algebra objects, manifests in d = (1 + 3) all the observed properties of quarks and leptons, if d > 2(2n + 1 ),n = 3, and that the breaks of the starting symmetry down to d = (3 + 1) can lead to massless fermions [68,69]. 3 New Way of Second Quantized Theory of Fermions... 67 contribute in the second quantization procedure and manifest anticommutation relations required for fermions, are Grassmann odd products of eigenstates of the Cartan subalgebra, Eq. (3.110), of the Lorentz algebra. In d = 2(2n + 1 ) spaces there are two Grassmann odd irreducible representations of the Lorentz algebra with the largest number of members, divided into two separated groups of 1 d! d, d, 2 ! 2 ! members, Eq. (3.59). (All states of one group are reachable from a starting state by the application of Sab.) Any Grassmann odd state can be written as a creation operator, operating on the vacuum state, while the Hermitian conjugated creation operator is the corresponding annihilation operator. Creation and annihilation operators of an odd Grassmann character fulfill the anticommutation relations of Eq. (3.50, 3.54). Let us see how it works. if 6et is a creation operator, which creates a state in the Grassmann space when operating on a vacuum state |^og > and 6? = (6^ )t is the corresponding annihilation operator, then for a set of creation operators and the corresponding annihilation operators it must be 6f|$og > = 0, S^^g > = 0. (3.49) We first pay attention on only the internal degrees of freedom of the Grassmann "fermions": the spin in any dimension d = 2(2n + 1), n is a positive integer. Choosing = ea, then it follows that (SI^ = gfa, Eqs. (3.18, 3.19). One correspondingly finds 6at = ea, 6a = 9 aea' {6ea,6bt} + l^og > = ^ab^og > , {b0a,6b}+|^og > = 0, {6eat,6bt}+^og > = 0, bfae |^og > = ea|^og >, ba^og > = 0. (3.50) The vacuum state |^og > will in this case be chosen as |^og >= | 1 >. The number operator NN a = Sa^S has the property, due to thefirst line in Eq. (3.49) and the second line in Eq. (3.50), that (NNa)2 = NNa, with the eigenvalue 0 or 1 . The identity I (It = I) can not be taken as a creation operator, since its annihilation partner does not fulfill Eq. (3.49). The identity is obviously selfadjoint operator determining the vacuum state |^og >= | 1 >. We can use the superposition of products of ea's as creation operators and the corresponding superposition of products of -g^-'s as the corresponding annihilation operators, provided that they fulfill the requirements for the creation and annihilation operators, Eq. (3.54), with the vacuum state |^og >= |1 >. In general they would not. Only an odd number of ea in any superposition would have the required anticommutation properties. 1^1 > = (-p)2 (e0-e3)(01 + ie2)(e5 + ie6)••• (ed-1 + ied)b®T11 >, 68 N.S. Mankoc Borstnik and H.B.F. Nielsen To construct creation operators it is convenient to take products of such superposition of vectors 0a and 0b that each factor is the "eigenstate" of one of the Cartan subalgebra members of the Lorentz algebra (3.110). Let us start in d = 2(2n + 1) with the creation operator, which is a product of f "eigenstates" of an odd Grassmann character of the Cartan subalgebra Sab ^ (0a + ^or0b) = kvf (ea + "[""eb),Eq. (3.21). Then the corresponding annihilation is a product of f of the corresponding factors (-g®" + -ik~d®r), In both cases (a, b) belong to (0,3), (1,2), (5,6), ••• , (d - 1,d). ' Let us in d = 2(2n + 1), n is a positive integer, start with the state = b^n >, with b0Tt: = (—=)d (e0 -e3)(eT + ie2)(e5 + ie6) ••• (ed-1 + ied). (3.51) v2 One finds for the eigenvalues of the Cartan subalgebra operators, Eq. (3.110), the values (+i, +1, +1,----h 1). The rest of states, belonging to the same Lorentz irreducible representation, follow from the starting state by the application of the operators Scf, which do not belong to the Cartan subalgebra operators. One can find creation and annihilation operators for d = 4n in App. 3.5. i. We proposed in Eq. (3.51) the starting creation operator b01 t, the upper index indicates one of the two groups, the lower index indicates the starting member. By taking into account Eqs. (3.18,3.19) the starting creation operator and its annihilation partner are for d = 2(2n + 1) equal to b01t = (—)d (e0-e3)(eT + ie2)(e5 + ie6)••• (ed-1 + ied), v2 £01 c 1 , d • 9 . d 9 bl —2)2 (_9ed-T-i_9ed)••• ("9e°-le^, for d = 2(2n + 1). (3.52) The rest of creation operators belonging to this group (group 1) in d = 2(2n + 1) follow by the application of operators Sef. The corresponding annihilation operators are the Hermitian conjugated partners of the corresponding of creation operators. For d = 2(2n + 1) one finds by the application of S01 another creation operator and the corresponding annihilation operator as follows b0it = (-_)d-1 (e0e3 + ieTe2)(e5 + ie6) ••• (ed-1 + ied), 22 b0i =( —)d-1 ( 9 - ) (- ) 2 (-2) ( 9ed-T 9ed ( 9e3 9e0 9e2 9eT), in general : b01t « Sab••• Sefb01t, b01 =(b01t)t. (3.53) 3 New Way of Second Quantized Theory of Fermions... 69 It was taken into account in the above equation that any Sac (a = c), which does not belong to the Cartan subalgebra, Eq.(3.110), transforms (^)2(ea + ieb)(ec + ied) (a = c and a = d, b = c and b = d, naa = nbb) into ^(eaeb + eced). The states are normalized and the simplest phases are assumed. One evaluates that either Sab or Scd, applied on (eaeb ± eced), gives zero. The vacuum state is in all these cases | 1 >. All the creation operators of an odd Grassmann character — the Grassmann even S ac does not change the oddness of the creation operators and neither do the Hermitian conjugation — fulfill the anticommutation relations {6tek,6®li}+^0g > = Sij Ski |^og >, {6fk,6j'l}+^0g > = 0|^og >, {6ekt,6eit}+^o9 >=oi^o9 >, 6ekti*o9 > = ^ >, 6|k|^og > = 0|^og >, (k,l) = (1,2) . (3.54) Since there is another group of states, presented in Eq. (3.56), not reachable from the starting state by Sab, we denote, to generalize the notation, creation operator with 6ekt and the annihilationoperator with 6|k. It is not difficult to see that states included into one representation, which started with 6e1111 > as presented in Eq. (3.52) for d = (2n +1 )2 have the properties, required by Eq. (3.54) for k = 1 : i.a. In any d-dimensional space the product -gear • • • "aeW, with all different at (if all or some of them are equal, then this is trivially true since ( gl^- )2 = 0), if applied on the vacuum 11 >, is equal to zero. Correspondingly the second equation and the fifth equation of Eq. (3.54) are fulfilled. i.b. In any d-dimensional space the product of different 0as — 0ai 0a2 • • • 0ak with all different 0a's (at = aj for all at and aj) — applied on the vacuum | 1 >, is different from zero. Since all the 0's, appearing in Eqs. (3.52,3.53), are different, forming orthogonal and normalized states, the fourth equation of Eq. (3.54) is fulfilled. i.c. The third equation of Eq. (3.54) is fulfilled provided that there is an odd number of 0 s in the expression for a creation operator. Then, when in the anticommutation relation different 0a's appear (like in the case of d = 6 {000305,0102 06}+), Such a contribution gives zero. When two or several equal 0's appear in the anticommutation relation, the contribution is zero (since (0a)2 = 0). i.d. Also for the first equation in Eq. (3.54) it is not difficult to show that it is fulfilled only for a particular creation operator and its Hermitian conjugated partner: Let us show this for d = (3 + 1) and the creation operator -Jj (0° — 03) 0102 and its Hermitian conjugate (annihilation) operator: {-g|T -ger ("alô — W), Ti(0° - 03) 01 02}+. Applying (^ — ) on (0° — 03) gives two, while "a!7 "alT applied on 01 02 gives one. 70 N.S. Mankoc Borstnik and H.B.F. Nielsen i.e. If we define the number operator Nek as follows n ek=eektbek, (3.55) it follows, taking into account the third equation of Eq. (3.54), that (Nek)2 = , requiring that the eigenvalue of this operator N ek on the state bfkt > is 0 or 1. ii. There is one additional irreducible representation of creation and annihilation operators in d = 2(2n + 1), which follows from the starting state >=be2tn >, 1 for d = 2(2n + 1). (3.56) be2t : = (—) , (0c + 03 )(01 + i02)(05 + i06) ••• (0d-3 + i0d-2)(0d-1 + i0d), This state can not be obtained from the previous group of states, presented in Eqs. (3.52, 3.53) by the application of Sef, since each Sef changes an even number of factors, never an odd one. All the other states of this new group of states follow from the starting one by the application of Sef. The corresponding creation and annihilation operators are ee2t = (—)d (0c+03)(01 + i02 )(05 +i06) ••• (0d-1 + i0d), 12 %/2 — ( 1 )a ( d • d ) ( 9 9 ) — ( rn) 2 ( ^nd-1 i and ) • • • ( ^nc + ^n3 ) , b (-2) ( 90d-1 1 90d) •••( 90c + 903' for d — 2(2n + 1). (3.57) The corresponding annihilation operators follow by the Hermitian conjugation of the creation operators. be2t K Sab • • • Sef6e2t, tif2 — (6e2t)t. (3.58) Also all these creation and annihilation operators fulfill the requirements for the creation and annihilation operators, presented in Eq. (3.54), due to the same reasons as in the first case. It is true also in this case, as stated below Eq. (3.55), that Nek applied on the state |$k > gives 0 or 1, due to the fact that (Nek)2 — Nek. Thus the basic states, determined by the application of creation operators of Eqs. (3.53, 3.58) on the vacuum state | 1 > have the properties required for fermions. Let us now count the number of states in each of the two groups presented in Eqs. (3.53, 3.58). There are in (d — 2) two creation ((0c t 01, for nab — diag(1, —1)) and correspondingly two annihilation operators (-gd^ T -geT), each belonging to its own group with respect to the Lorentz transformation operators, both fulfilling Eq. (3.54). 3 New Way of Second Quantized Theory of Fermions... 71 It is not difficult to see that the number of all creation operators of an odd Grassmann character in d = 2(2n + 1 )-dimensional space, with all Ya's included is equal to -3727. 2 '2 ' We namely ask: In how many ways can one put on j places d different 9a's. And the answer is — the central binomial coefficient for x2 1 2 — with all x different. This is just ¿drr. But we have counted all the states with an odd 2 ' 2 ' Grassmann character, while we know that these states belong to two different groups of representations with respect to the Lorentz group. Correspondingly one concludes: There are two groups of states in d = 2(2n + 1) with an odd Grassmann character with all 9a's included, each of these two groups has 1 dd (3.59) 2 2 ' 2 ' members. In d = 2 we have two groups with one state, which have an odd Grassmann character, in d = 6 we have two groups of 10 states, in d = 10 we have two groups of 126 states with an odd Grassmann characters. And so on. All together there are 2d-1 the states of an odd Grassmann character. Correspondingly we have in d = 2(2n + 1 )-dimensional spaces two groups of creation operators of the kind presented in Eqs. (3.53,3.58), each kind with i dirr 2 ' 2 ' members, creating states with an odd Grassmann character and the same number of annihilation operators. Creation and annihilation operators fulfill anticommutation relations presented in Eq. (3.54). The rest of creation operators [and the corresponding annihilation operators] with the opposite Grassmann character than the ones studied so far — like 9091 [afr a§0] in d = (1 + 1) (90 T 93)(91 ± i92) [(¿fr T ia§r)(a§0 T afr], 909391 92 [ afr air adF ddk] in d = (3 + 1), do not fulfill the anticommutation relations required for fermions in Eq. (3.54), with bf1 and bf1 f replaced by b§k and b§kt, k = (1,2) and correspondingly with {b§k, b§lt}|^og >= 5kl Sij|^og >, (k, 1) = (1,2), (i,j) running from (1,..., 2 ¿727). 2 2 ' 2 ' All the states >, k = (1,2), generated by the creation operators, Eqs. (3.54, 3.58), on the vacuum state |^og > (= |1 >) are the eigenstates of the Cartan subalgebra operators and are orthogonal and normalized with respect to the norm of Eq. (3.31) < ' > = Sij 6kk' , 1 d! (k,k') = (1,2), (i, j) = (1,2,...,-). (3.60) 2 2 ' 2 ' All these basic states describing the internal degrees of freedom can be used to solve Eq. (3.43) for free massless "fermions", with the part in ordinary space proportional to e-ipa*a. The eigenstates of Eq. (3.43) are superposition of the basic 72 N.S. Mankoc Borstnik and H.B.F. Nielsen states > with coefficients depending on momentum pa, a = (0,1,2,3,5,..., d) i |^p > = l^og > , > = L ckspil^k >, (3.61) s represents different solutions of the equations of motion, and, since they are orthogonalized, they fulfill the relation < ^kp|^k,p, >= 5kk/ 5SS/ 5pp , where we assumed the discretization of momenta. The corresponding creation operators, creating the basic states describing free massless "fermions" — 60pJ — are superposition of creation operators 60kJ, 60k = i ckspi60kt and fulfill together with the corresponding annihilation operators 60p = (60pJ)J the relations {^^p J}+|^og > = ¿¡J' §ss'6pp ' |^og >, {60pk,60kp,}+|^g > = 0|^og >, {bfpkt,6®kp'J}+|^g > = 0|^og >, S^og > = 0 |^og >, 60k^og > = |^p >, |^og > = |1 > . (3.62) Again index k =(1,2) in (60p, 60pj) (602, 60J2) denotes creation and annihilation operators of one of the two groups of states describing the internal space of "fermions", reachable by Sab, and 60pJ creates the state for a particular momentum in ordinary space pa, solving Eq. (3.43). The number operator for a "fermion" state |^kp > is now NN 0k = b0ktfiek sp sp sp (NN 0pk )2 = NN fpk, (3.63) with the eigenvalues 0 or 1, since the states of a chosen discretized pa are orthogonal. Correspondingly each state can be occupied or empty. If 11 ®kp i ,1 02p2, 1 0kp3,..., 0®kpk,..., 0®ikpi,..., > is a n particle state of "fermions" (and "antifermions"), where 1 denotes the occupied state and 0 the unoccupied state, then it follows, for example, due to the third line in Eq. (3.62), that bekJ b0kt |1 0k 10k 10k 00k 00k > = 6 S ipi 6 S j pj | 1 S 1 p 1 , 1 S 2p2 , 1 s 3p 3 , . . . , 0 S ipi , . . . , 0 s jpj , . . . , > -bSkpt 60ikpJi |10kpi ,102p2 ,103kp3 , . . . ,00kpi ,...,00jkpj ,...,>. (3.64) Any n "fermion" state is therefore a product of n creation operators 60pJ as presented in Eq. (3.47). The number operator for "fermions" in the n-particle state of Eq. (3.64) is correspondingly NN0 = X NN 0ip (3.65) k,Sipi 3 New Way of Second Quantized Theory of Fermions... 73 When coefficients ckspi depend also on coordinates xa (for free "fermions" ckspi (x) ckspi • e-ipa*a), it follows for papa = 0 r jd-1,. bfkV, X)= ckspi(x) bf-t. {b®k(x0,x), b®k V,y)}+|Toc > = 6kk' 6ss'6d-1(X- y) |Toc >, {b®kt(x0,x), b®kV,y)}+|Toc > = 0, {b®k(x0,x), b®k V,y)}+|Toc >= 0. (3.66) It is discussed in the subsection 3.3.3 how do discrete symmetry operators in the Grassmann case take care of "fermion" and "antifermion" states. Let us now take into account Eq. (3.45) with lg = 4{$tT°GTa(Pa^) - (Pa^)Y°GT>}. The Euler-Lagrange equations lead to —i2Y°Yapa$ = 0 and i2pa$tY°Ya = 0. Let us find the Hamilton function for a second quantized field: $(x0,y), generated by one of the creation operators lb®t on the vacuum state |$og >, rr dLG 1 if 0-0 rr dL° 1 0 _0 i ^ = deM) = 4$ tY°Y, t = ^^ = - 4Y°Y$ , hg = n^ (p0$) + (p0$ ^ t — lg , i r S W0 Y^ift. $ ) — (P TT t)Y0 Y^rh = 4 [$ Vgy1«^) - (PiÛ>VGy^] , HG = dd-1xHg . (3.67) A vector $ depends on k = (I, II) and on spins (what in d = (3 + 1) manifests as spins and charges). Hamilton function is obviously an odd Grassmann object and does not define the energy of the system. However, if assuming the relation: 2Y0p0 $k(x0,x) = {$k(x0, x), Hg}— , one still ends up with the equations of motion, Eq. (3.45). One namely obtains Y°Po$k(t,x) = { $k(t,x), H°}_ = -YgY'Mk(t,x), (3.68) what might help to find the procedure to define the energy for the interacting "Grassmann fermions". One must at this point either give up with the Grassmann "fermions" with the integer spins or find a consistent unconventional way to define the energy, like the one suggested in Eq. (3.68). 3.3.2 Second quantization in Clifford space In Grassmann space the requirement that products of "eigenstates" of the Cartan subalgebra operators form the creation and annihilation operators, obeying the relations of Eq. (3.54), reduces the number of creation operators and correspondingly 74 N.S. Mankoc Borstnik and H.B.F. Nielsen the number of states from 2d (allowed for "eigenstates" of the Cartan subalgebra operators) to two isolated groups of 2 ¿ddr creation operators. (There are no gener- 2 ' 2 ' ators of the Lorentz transformations Sab that would connect both groups of states and correspondingly there are no families.) Let us study what happens, when, let say, Ya's are used to create the basis and correspondingly also to create the creation and annihilation operators. Here we briefly follow Ref. [50]. Let us point out that Ya is expressible with 9a and its derivative (Ya = (0a + )), Eq. (3.4), and that we again require that creation (annihilation) operators create (annihilate) states, which are "eigenstates" (Eq. (3.72)) of the Cartan subalgebra operators, Eq. (3.110). Then the application of Ya on any Clifford algebra object A(Ya), (determined by Ya's), can be evaluated as follows, Eq. (3.29, 3.30), (YaA = iR(A)AYa)|^oc >, (3.69) where (-)(A) = — 1, if A is an odd Clifford algebra object and (-)(A) = 1, if A is an even Clifford algebra object, while |^oc > is the vacuum state, replacing the vacuum state in the Grassmann case |^og >= |1 > with the one of Eq. (3.79), in accordance with the relation of Eqs. (3.4, 3.32, 3.31), Refs. [50,10]. We could as well make a choice of Ya = i(0a — ) instead of Ya's to create the basic states, exchanging correspondingly the role of Ya and Ya 9). Making a choice of the Cartan subalgebra "eigenstates" of Sab, Eq. (3.27), one ab ab defines nilpotents (k) and projectors [k] ab 1 naa ab (k): = 2 (Ya + n^Yb), (k) 2 = 0, ab 1 i ab ab [k]: = 2(1 + kYaYb), [k] 2 =[k], (3.70) ab where k2 = naanbb. Recognizing that the Hermitian conjugate values of (k) and ab [k] are ab ^ ab ab ^ ab (k) = naa (—k), [k] =[k], (3.71) while the corresponding "eigenvalues" of Sab and S ab on nilpotents and projectors, Eq. (3.27), are ab k ab ab k ab Sab (k) = ^ (k), Sab [k]= ^ [k], ab k ab ab k ab Sab (k) = ^ (k), Sab [k]= — ^ [k], (3.72) 9 In the case that we would choose Ya's instead of ya's, Eq.(3.4), the role of Ya and ya should be then correspondingly exchanged in Eq. (3.69). 3 New Way of Second Quantized Theory of Fermions... 75 we find for d = 2(2n + 1) that from the starting state made as a product of an odd number of only nilpotents |'1 > = b^oc >, 03 12 35 d-3 d-2 d-1 d b1t :=(+i)(+)(+) ••• (+) (+) , d-1 d d-3 d-2 35 12 01 b1 =(b1t)t = (—) (—) ••• (—)(—)(—i), (3.73) having correspondingly an odd Clifford character, all other states of the same Lorentz representation, there are 2d-1 members, follow by the application of Scd (which do not belong to the Cartan subalgebra) on the starting state 10, Eq. (3.110), (Scd |'1 >= ' >). b1t«Sab...Sefb1t, ' >= Sab..SefiM1 >, b = > are normalized [50,10]. The operators Scd, which belong to the Cartan subalgebra of Sab, Eq. (3.110), generate "eigenstates" of the Cartan subalgebra operators (S03, S12, S56, • • • , S d-1 d), with the eigenvalues which determine the "family" quantum numbers. There are 2d-1 families. From the starting new member with a different "family" quantum number the whole Lorentz representation of family members with this "family" quantum number follows by the application of Sef: Sab • • • Sef Scd|' >= |'f >. All states of one Lorentz representation of any particular "family" quantum number have an odd Clifford character, since neither Scd nor Scd — both of an even Clifford character — can change the odd character of the starting state. Any vector > follows from the starting vector, Eqs. (3.73), by the application of either Sef, which change the family quantum number, or Sgh, which change the family member quantum number of a particular family or with the corresponding product of Sef and Sef ' > . (3.75) Again, a denotes "family" quantum numbers, i denotes family member quantum number. Correspondingly we define b^ (up to a constant) to be bat < Sab ••• S efSmn ••• Sprb1t < Smn ••• Sprb1tSab ••• Sef . (3.76) This last expression follows due to the property of the Clifford object ya and correspondingly of Sab, presented in Eqs. (3.69, 3.120). We accordingly have for an annihilation operator £•*(= (6ixt)t) b« = (bat)t < Sef • • • Sabb1 Spr • • • Smn. (3.77) 10 The smallest number of all the generators Sac, which do not belong to the Cartan subalgebra, Eq. (3.110), needed to create from the starting state all the other members, is 2d-1 — 1. This is true for both even dimensional spaces - 2(2n + 1) and 4n. 76 N.S. Mankoc Borstnik and H.B.F. Nielsen The proportionality factor ought to be chosen so that the corresponding states >= > are normalized when the vacuum state |^oc > is normalized, < ^oc|^oc >= 1, while all the states belonging to the physically acceptable 03 12 56 78 d-3 d-2 d-1 d states, like [+i] [+] [—] [—] • • • (+) (+) |^oc >, must not give zero for either d = 2(2n + 1) or for d = 4n. We also want that states, obtained by the application of ether Scd or Scd or both, are orthogonal. To make a choice of the vacuum it is needed to know the relations of Eq. (3.116). It must be ab 1 ab < ^oc| • • • (k) • • • | • • • (k') • • • |^oc > = §kk' , ab1 ab < ^oc| • • • [k] • • • | • • • [k'] • • • |^oc > = §kk' , ab1 ab <^oc| ••• [k] ••• | ••• (k') ••• |^oc > = 0. (3.78) We must choose the vacuum state in a way that fulfills the above requirements as well as the requirements bf1|^oc >= 0 and bf |"^oc >= 0 for all members i ef gh ef gh ab ab of any family p. Since any Seg changes (+) (+) into [+] [+] and [+] 1 =[+], while ab ab ab (+) 1 (+)=[—], the vacuum state |^oc > must be |^oc > = 03 12 56 d-1 d 03 12 56 d-1 d 03 12 56 d-1 d [—i][—][—] ••• [—] +[+i][+][—] ••• [—] +[+i][—][+] ••• [—] + "-|0>, for d = 2(2n + 1), (3.79) n is a positive integer. There are 2d -1 summands, since we can start with the vac- 03 12 56 d-1 d uum state [—i] [—] [—] • • • [—] 11 >, which fulfills the requirement for ^1 |"^oc >= 0 and 61|^oc >= 0, and then we must step by step replace all possible pairs of ab ef 03 12 35 d-1 d ab ef [—] • • • [—] in the starting part [—i] [—] [—] • • • [—] into [+] • • • [+] and include new terms into the vacuum state so that the last (2n + 1) summands have for d = 2(2n + 1) case, n is a positive integer, only one factor [—] and all the rest [+], each [—] at different position 11. This vacuum has all the spins, either with respect to Sab or with respect to Sab, equal to zero. The vacuum state has then the normalization factor 1/\/2d/2-1, while there is 2 d-1 2 d-1 (3.80) 11 The choice of Eq. (3.79) for the vacuum state is not unique. If one would multiply any of summands by a number P a, where a represents the a-th family, and then multiply each of 2 d-1 members of creation operators belonging to this family b^ by —pa and the corresponding annihilation operator ba by —pa, Pa is the complex conjugated value of p a, it would still be true that ba 6ft = Saf Sij times the corresponding summand of the vacuum back. 3 New Way of Second Quantized Theory of Fermions... 77 number of creation operators, defining the orthonormalized states when applying on the vacuum state of Eqs. (3.79) and the same number of annihilation operators, which are Hermitian conjugated to creation operators. Again, operators Sab connect members of different families, operators S ab generate all the members of one family. Paying attention on only internal degrees of freedom, that is on the spin, the creation and annihilation operators must fulfill the relations {6f,tj't}+l^oc > = ' ôijl^oc >, {6a,tj'}+l^oc > = 0 l^oc >, {b^ttf 't}+l^oc > = 0l^oc >, bfl^oc > = 0 l^oc >, 61atl^oc > = >, (3.81) with (i, j) determining family members quantum numbers and (a, a') denoting "family" quantum numbers. Only Clifford odd objects fulfill the relations of Eq. (3.81), since the odd Clifford objects anti-commute (like: {(y0 — Y3), (y1 + iY2)}+ = 0), while the Clifford even objects commute (like: {(1 — Y0Y3), (1 — iY1Y2)}+ = 2 (1 — Y0Y3)(1 — iY1Y2))-The reader can find the detailed proofs for the above statements, for either d = 2(2n + 1 ) or d = 4n, in Refs. [50,10]. Let us again, like in the Grassmann case, Eq. (3.62), look for the creation (and their annihilation operators) which, when applied on the vacuum state, Eq. (3.79), solve the equation of motion, Eq. (3.36). The solution for each momentum p£, a = (1,..., d), for discretized values of momenta, is a superposition of 6"t, tapk = L casi(Pk) , (3.82) i applied on the vacuum state, Eq. (3.79). Since 6at and j fulfill the relations of Eq. (3.81) and, if the states for two different momenta are orthogonalized, it follows {6sV, tta/p}+l^oc > = 6aa' 5SS, 6pk Pl l^oc > , {6saPk, }+|^oc > = 0 |^oc >, {6apk ,bsa;pi }+i^oc > = 0 i^oc >, 6aPk i^oc > = 0 i^oc >, bfikl^oc > HCP >, (3.83) with the vacuum state |^oc > defined in Eq. (3.79), with s denoting the corresponding solution of equations of motion and for a discretized momentum space. The number operator of a particular solution s, a particular momentum pk and a particular "family" a, Nfpk = , (Nfpk)2 = ^, (3.84) 78 N.S. Mankoc Borstnik and H.B.F. Nielsen has the eigenvalues 1 or 0. The number of fermions in the n-particle state of Eq. (3.86) is correspondingly N = X NfPk . (3.85) a,s,pk For a n fermion and antifermion state, Eqs. (3.47, 3.48) in the introduction to Sect. 3.3,11 orlp,, 10=2pi, 1 O^p,,..., 0SV,..., 0«^ ,...,> it follows, for example, due to the third line in Eq. (3.83), that ft«'t b« t |1«=1 1 a=1 1 a=1 0a 0a > = bs 'Pi sp;1 1 s=1 pi , 1 s=2 p i , 1 s=3 p i ,...,0spi ,...,0sivpj ,...,> = - bapjO -iV, 1^V, 1 aslpi,..., 0api,... ,..., >, (3.86) where 1 denotes the occupied state and 0 the unoccupied state, and |1J=11pi >= ba=iV+i l^oc >. Eq. (3.86, 3.47) demonstrates properties of Slater determinants. One fermion state is obviously second quantized by construction. Two states with n1 and n2 fermions each, defined by Aat as n1 products of b^pi (which distinguish among themselves in at least one of the properties (a, s, pi)) and by Abt as n2 products of . (which distinguish among themselves in at least one of the properties (a', s',pj)), applying on |^oc >, must distinguish in either internal space or in the coordinate space, as it follows from Eq. (3.86), that the product of /Aat and /Abt applying on |^oc > would give a state with (n1 + n2) fermions. Let us add that the vacuum state, having the sum of the spins of both kinds of operators, Sab and S ab, equal to zero and therefore neutral, remains neutral also when filled with fermions of all the spins, Sab and S ab. When coefficients casi(pk) depend also on coordinates xa (for free fermions casi(pk,x) = casi(pk) • e-ipa*a), it follows for papa = 0, r d d—1p bxx) = (L-—1casi(Pk,x)ba. {ba(x0,x), bsa tlx0,y)}+^oc > = öaa' öss' 6d-1 (X- y) l^oc > , {bsat(x0,X), bsaV,y)}+|^oc > = 0, {bsa(x0,x), b$(x0,y)}+|^oc >= 0. (3.87) Let us now take into account Eq. (3.35) with Lc = 1 $ W(Pa^) - (PaV)YV^}. The Euler-Lagrange equations lead to y0yapa^ = 0 and —pa^ty0ya = 0. Let us look for the Hamilton function for fermions determined by one of the creation operators, like i|>a(x0, x) = bat(x0,x)]|"^oc >, which is already the second quantized state. 3 New Way of Second Quantized Theory of Fermions... 79 For a vector $ and ij>t it therefore follows 3£e 1 Tt n 3£c 1 T ^ = ^ = , ^ = ^ = 2 $ , He = n^ (p°$) + (pc^t)n^t - Le , = -1 [^ Y° Y1 (Pi$) - (MVY^ ] , He = dd-1xHe , (3.88) Correspondingly one finds for a component $0(x°, X) [74], X is a vector in (d — 1)-dimensional coordinate space, Pbo^a (x°,X) = { (x°,x), He}_ = { $sa(x°,X), dd-1x' ^ $sa't (x°,x") y°Y1 (p'^f/(x°,x"/))} a ',s' dd-1x' {^a(x°,X) ,$saV,xc') }+yV (pb,i^sa' (x°,x"')) a's' = -Y°Yi (Pi^sa(x°,X)). (3.89) (We took into account that y°y1 transforms (x°,x') into^Is'' ca s's'' (x°,x') which anticommute with ^f(x°,X) (Eq. (3.87)), we also assumed that states, obtained when operators operate on a vacuum state, do not contribute to the surface term. Integrating per partes and dropping the surface term simplifies HC into - J H dd-1x't (x°,x') Y°Yi (p^' (x°,x')).) The obtained equations of motion agree with the ones from Eqs. (3.39, 3.40). Correspondingly the energy of the n-fermion state of free massless fermions created by bft on the vacuum state |$oc > all with zero momentum po (solving the Weyl equation Eqs. (3.36,3.40)) is equal to E = Y as NN fp°. The current is correspondingly ja = $atY°Ya$a. The observed fermions — quarks and leptons — manifest their properties obviously in d = (3 + 1). The internal space in d = (3 + 1) can therefore be used to describe the spin and handedness of massless fermions, in the spin-charge-family theory also families, while the internal space in d > 5 can be used to describe charges of fermions, contributing in the spin-charge-family theory as well to families. One family representation contains in d = 2(2n + 1),n = 3, 2d-1 = 64 members, described by the creation and annihilation operators fulfilling the anticommutation relations of Eq. (3.81), explaining from the point of view of d = (3 + 1) spins, handedness and charges of the observed quarks and leptons and antiquarks and antileptons. Correspondingly there is no need for the negative energy "Dirac sea". We discuss below discrete symmetry operators for both cases, the Clifford one and the Grassmann one, in d and in observable dimension d = (3 + 1). In Subsect. 3.3.4 we present a few examples. 80 N.S. Mankoc Borstnik and H.B.F. Nielsen 3.3.3 Discrete symmetries in Grassmann space and in Clifford space in d and in d = (3 + 1 ) part of the space We have treated so far free massless fermions in Grassmann and in Clifford space. The fermion "nature" of states are in both spaces demonstrated by the fact that the corresponding creation and annihilation operators fulfill the anticommutation relations of Eq. (3.62) in Grassmann case and of Eq. (3.83) in Clifford space. Fermions — in both spaces — are in superposition of eigenstates of the Cartan subalgebra operators of Sab in the Grassmann case, in the Clifford case they are in superposition of the Cartan subalgebra operators of Sab as well as of S ab. We distinguish in d-dimensional space two kinds of discrete symmetry C, P and T operators with respect to the internal space in which the fermion properties are described. In the Clifford case we have [65] Ch = n Ya K, yaea Th _ Y° n Ya KIx0 , yaen ' H _ Y Ix , Ix xa _—xa , Ixo xa _ (—x°, x) , Ixx _—X, Ix3xa _ (x°, —x1, —x2, —x3,x5,x6,...,xd). (3.90) The product ^ Ya is meant in the ascending order in Ya, K stands for complex conjugation. In the Grassmann case we correspondingly define Cg _ n yGK, yge3ya TG _ yG n YG KIx0, yg £Mya PGd-1) _ YG Ix, (3.91) yG is defined in Eq. (3.22) as yG _ (1 — 20anaaâfb), while Ixxa _ —xa , Ixoxa _ (—x°,x), Ixx _ —x, Ix3xa _ (x°, —x1, —x2, —x3,xf ax6,... ,xd), like in the Clifford case. Let be noticed, that since yG (_ —inaa YaYa) is always real as we see in Eq. (3.28)12. Since Ya is either real or imaginary, Eq. (3.22), we use in Eq. (3.91) Ya to make a choice of appropriate yG. In what follows we shall use the notation as in Eq. (3.91). Let us define in the Clifford case and in the Grassmann case the operator "emptying" 13. The operation "emptyingNH" after the charge conjugation CH in 12 If we choose a real 0a, then ya is real and Ya imaginary, if 0a is imaginary, then ya is imaginary and Ya real, as is demonstrated in Eq. (3.28). 13 The operator "emptying" empties the "Dirac sea" of negative energies [65], although in the spin-charge-family theory is no need for the "Dirac sea" of negative energies, as 3 New Way of Second Quantized Theory of Fermions... 81 the Clifford case [65,7,9] (arxiv:1312.1541) and "emptyingNG" after the charge conjugation CG in the Grassmann case, namely transforms the positive energy fermions into positive energy antifermions in both cases, solving Eq. (3.36) in the Clifford case, and Eq. (3.43) in the Grassmann case. "emptyingNH" = ya K in Clifford space, "emptyingNG" = yG K in Grassmann space. (3.92) Then the anti-particle state creation operator to the corresponding particle state creation operator can be obtained by the application of CH,g = WptymgN H,NG" • CH,G (3.93) CH and CG, with indexes H and NH denoting the Clifford case and with g and NG denoting the Grassman case, on the creation operator for a particle state, or opposite. Let us remind the reader that in the spin-charge-family theory, using the Clifford algebra, the family members of each family include fermions and antifermions — quarks and leptons and antiquarks and antileptons. This is the case also for Grassmann fermions and antifermions, but in this casethere are instead of families two by Sab unconnected representations. Ref. [65] proposes in the Clifford case the following discrete symmetry operators, manifesting dynamics in d = (3 + 1) 3 Cn = n Ym r(3+1) KIx6,x8,...,xa , 3ym,m=0 3 TN = n Ym r(3+1 ' K Ix0 Ix5,x7,...,xd-1 , Mym,m=1 PNd-1) = y0 r(3+1) r(d) ix3, CN = Y0Y5Y7 ••• Yd-1Ix3 Ix«,x8,...,xd CnPNd-1) = Y0 Y2 Ix3 KIx6,x8,...,xd , CnPNd-1) = Y0Y5 ••• Yd-1 Ix3 Ix6,x8,...,xd . (3.94) In the Grassmann case we use the Grassmann even, Hermitian and real operators yG, Eq. (3.22), to determine discrete symmetries in ((d — 1) + 1) space (as presented in Eq. (3.91)) and in d = (3 + 1) space. In (3 + 1) space we proceed — we discussed already in the introduction of Sect. 3.3, for either Clifford or Grassmann fermions. The operation of "emptyingnh" after the charge conjugation Ch in the Clifford case, which transforms the state put on the top of the Clifford "Dirac sea" into the corresponding negative energy state, namely creates the anti-particle state to the starting particle state, each anti-particle state, put on the top of the "Dirac sea", solving the Weyl equation in the Clifford case, Eq. (3.36). 82 N.S. Mankoc Borstnik and H.B.F. Nielsen in analogy with the operators in the Clifford case [65] — as follows CNG = Yg KIx6x8...xd , yjjeKy™ TNG = YG n YG KIXoI x0 Ix5x7...xd-1 PNdG-1) = YGn ySGiX , s = 5 CNG = n YG ,I x6x8 ...xd Yg £Mys CNG PNg 1) = Yg Yg KIx3 Ix6x8...xd , d CNGP(dG-1) = YG H YSG % w...xd , Yge3Ys,s=5 CngTngP(dG1) = n YG IxK. (3.95) rge3Ya 3.3.4 Examples of massless fermion and antifermion states in Clifford and in Grassmann space Let us illustrate solutions for free fermion states, represented by the creation operators applied on the vacuum states for the Clifford and the Grassmann case in ((d — 1) + 1)-dimensional space, representing indeed the contribution of a one fermion second quantized state in the Fock space of any number of fermions. We analyze states in both cases from the point of view d = (3 + 1 )-dimensional space, with the momentum in ordinary space pa = (p0,p1,p2,p3,0, ••• , 0), so that the charges "seen" in d = (3 + 1) are determined by the generators of the Lorentz transformations in the internal space — Sst, (s, t) = (5,6,7, • • • , d) in the Clifford case and Sst, (s, t) = (5,6,7, • • • , d) in the Grassmann case. In the Clifford case we discuss one family in details (let be reminded that the generators Sab connect all the members belonging to one family, while S ab transform a particular member of one family into the same member of another family), commenting also on the appearance of families (all the families are reachable by S ab) and present them briefly. In the Grassmann case different representations can not be reached by the generators of the Lorentz representations Sab. The discrete symmetry operators are in the Clifford case presented in Eq. (3.94), and in the Grassmann case in Eq. (3.95). We start with examples in d = (5 + 1)-dimensional space, with charges determined by Sst, (s, t) = (5,6) in the Clifford case and Sst, (s, t) = (5,6) in the Grassmann case. The dimension (13+1), used in the spin-charge-family theory to describe quarks and lepton as well the gauge fields and scalar fields, offers to free fermions at low energies additional charges, what explains observable properties of quarks and leptons. We present the creation operators creating all the states of one family Y"leJym 3 New Way of Second Quantized Theory of Fermions... 83 members in Clifford space. The family members creation operators are reachable by Sab. All the families are reachable from the starting family by S ab in the case of Clifford odd representations. In the case of the Clifford even representations there are Sab and YaYa, which take care of all irreducible representations. In Ref. [50,68-70] (d = 5 + 1)-dimensional space is studied as a toy model to manifest that the break of symmetry from the higher dimensional space to the (3 + 1 )-dimensional space can lead to massless fermions. Fermions were described in Clifford space. Here we briefly follow these references, and Refs. [65,66], adding new observations. The first study of Grassmann case can be found in Ref. [46]. Clifford fermions and antifermions Let us start with the examples in the Clifford case. To make discussions transparent let us first treat the d = (5 + 1) case. The d = (13 + 1) case is not so easy to present in particular when also families are treated. Clifford case in d = (5 + 1): In Table 3.4 the basic creation operators bO=(cK s) and their annihilation partners b°L(chs) in d = (5 + 1) are presented for all four (2d-1) families a = (I, II, III, IV). Index i is devided into s, determining spin and into ch to point out that S56 represents the charge from the point of view of d = (3 + 1), having two values, +1 and — 1. The vacuum state, Eq. (3.79), is the sum of selfajoint 03 12 56 03 12 56 03 12 56 03 12 56 operators ([—i] [—] | [—], [+i] [+] | [—], [+i] [—] | [+], and [—i] [+] | [+]), needed that the first, second, third and fourth family creation operators, respectively, applying on the vacuum state, give nonzero value. There are superposition of the basic creation operators — bi=(ch s) — which solve, applied on the vacuum state, the Weyl equation Eq. (3.36). Let us make the choice of pa = (p0,p1 ,p2,p3,0, • • • ,0) to see how the spin in d = (5,6) manifest charges in d = (3 + 1). pa = (p0,p1,p2 ,p3,0, ••• ,0), Sri,sol)(p)l^c > = X cai=(ch'sWoi) (p) 6ri(ch,s)e-ipaX>oc >,(3.96) s where index (ch,sol), represents charges and different solutions, respectively, of the Weyl equation for massless free fermions. We present in Eq. (3.97) the creation operators, the superposition of the first family members, presented in Table 3.4, which solve the Weyl equation, Eq. (3.36), for pa = (p0,p1 ,p2,p3,0,0). The corresponding annihilation operators follow by the Hermitian conjugation of the creation operators. There are two fermion solutions with the charge 1 and two antifermion solutions with the charge — 2, both having the positive energy. The first two creation operators are related by the time reversal operator (Tn = Y1 Y3 K Ixo Ix5)X7,... ,xd-i), while the second two follow from the first two by the application of CnPj/-1) = Y0 Y5 ••• Yd-1 Ix3 Ix6,x8 ,...,xd, both are presented inEq. (3.94). 84 N.S. Mankoc Borstnik and H.B.F. Nielsen family a (ch, s « at S03 « ch,s 03 12 56 56 12 03 (+i) ( + ) ( + ) (-) (-) |(-) (-) -i) 2 03 12 56 56 12 03 [-i] [-] ( + ) (-) (-) [-] -i] 2 03 12 56 56 12 03 [-i] ( + ) I [-] [-] |(-) (-) -i] 2 03 12 56 56 12 03 (+i) [-] I [-] [-] [-] -i) 2 03 12 56 56 12 03 [+i] [+] ( + ) (-) (-) [+] +i] 2 03 12 56 56 12 03 (-i) (-) ( + ) (-) (-) |(-) (+) +i) - 2 03 12 56 56 12 03 (-i) [+] I [-] [-] [+] +i) 2 03 12 56 56 12 03 [+i] (-) I [-] [-] I(-) (+) +i] 2 03 12 56 56 12 03 [+i] ( + ) I [+] [+] |(-) (-) +i] 2 03 12 56 56 12 03 (-i) [-] I [+] [+] [-] +i) 2 03 12 56 56 12 03 (-i) ( + ) (-) (-) ( + ) |(-) (-) +i) - 2 03 12 56 56 12 03 [+i] [-] (-) (-) (+) [-] +i] 2 03 12 56 56 12 03 (+i) [+] I [+] [+] [+] -i) 2 03 12 56 56 12 03 [-i] (-) I [+] [ + I(- ) ( + [-] 2 03 12 56 56 12 03 [-i] [+] (-) (-) (+) [+] -i] 2 03 12 56 56 12 03 (+i) (-) (-) (-) ( + ) I(-) ( + ) -i) 2 ( 1 1 ( 2 , 2 ( 1 -1 ( 2 , 2 (__l 1 ( 2 , 2 ( - 2,- ( 2, (- ; — 1 _ (__l__l ( 2 , 2 ( 1__l ( 2 , 2 (-1 1 ( 2 , 2 (__1__1 ( 2 , 2 Table 3.4. The basic creation operators — 6ft , ch (charge) and s (spin) explain the index i — and their annihilation partners — 6a=(ch.,s) — are presented for the d — (5 + 1)-dimensional case. The basic creation operators are the products of nilpotents and projectors, which are the "eigenstates" of the Cartan subalgebra generators, (Sc3, S12, S56), (Sc3, S12, S56), presented in Eq. (3.110). Operators s and 6ch,s fulfill the commutaion relations of Eq. (3.81). , Ê:4| 2 Creation operators for the fermion states in Clifford space for d = (5 + 1) „0 «v t1 (p) 2 , 2 «j^ _± (p) 2 , 2 «I3| , (P) -2 , 22 ( 03 1 2 56 p1 + ip P (+i) ( + ) I ( + ) +—;- V Ip0 I + Ip 03 12 [ —i] [-] 56 \ (+) e_ -i(Ip0I*0-p-X) p 03 12 56 -i] [-] I ( + ) ^ r" , p0 -^ (+3) (+) | (+)) e-i(IP°l*°+P-x) Ip0 I + lp3l j Creation operators for the antifermion states in Clifford space for d = (5 + 1) 0 |P 1 , 03 12 -i] (+) 56 p1 +ip2 [-] +—0-r Ip01 + I p3 I 03 12 (+i) [-] 56 i ( I p0 I x0 +p • X ) /03 12 1 (p) = -P* (+i) [-] I [-] - Ip0 I + Ip3 I 03 12 56 [-i] ( + ) I [-] I e 56] 1 e i(Ip0Ix0 p • X ) Index i=(1,2,3,4) counts the solutions, while |3*|3 = |p2|p0p 1 takes care that the corresponding states are normalized. All the states are correspondingly or-thogonalized. The coefficients cai=(ch,s) (ch,sol) (p) can be read from the solutions. The solutions have the definite handedness and orientation of the spin with respect to the momentum: S1! ^± defines the state with r(3+1 ) = 1 and the spin and 2 , 2 momentum both up, ± defines the state with r(3+1 ) = 1 and with spin and 2 , 2 momentum both down, ± defines the state with r(3+1 ) = —1 and the spin up 2 56 03 2 56 s s 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 , 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 2 ( 2 1 2 1 2 1 2 1 2 1 2 2 ( 2 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 2 2, 2 2 2 2 ( 1 1 ( 2 , 2 1 2 1 2 1 2 1 2 2 2 2 2 2 2 0 p 2 3I 0 p P 2 56 p ip (3.97) 3 New Way of Second Quantized Theory of Fermions... 85 and the momentum down, b-l _± defines the state with r(3+1' = —1, the spin 2 ' 2 down and the momentum up. The same indexes — cai=(ch's) (ch,sol) (p) — define the solution of the Weyl equation also for for the rest three families presented in Table 3.4. The phases of creation operators are in agreement with the application of discrete symmetry operators Cn • Pn, and Tn. Let us point out that the scalar fields, interacting with fermions (in the spin-charge-family theory [[4,3] and the references cited therein] the scalar fields origin in the spin connection fields — dabc, the gauge fields of Sab, and dabc, the gauge fields of Sab, appearing in Eq. (3.1) — with the space indexes c > 5) can make massless fermions massive [68,69,73,66]. In this case the creation operators (and correspondingly the annihilation operators) start to be superposition of basic operators of different charges ch as well: (bol' (p)= Z Ca'ch'S°lch,sol' (p) 6ctsoie_iPaXQ moc >. ch,sol In this case the solutions of the corresponding equations of motion, presented in Eq. (3.97) for massless states, become superposition of different charges and different families. For pm = (0,0,0),m = (1,2,3) and one massive family only [66] the creation operators for the basic states (usually used in text books [74,75] for massive states) are presented at Table 3.5. The creation operators, presented in Table 3.5, define family a Ba+ bs,m S12 03 12 56 03 12 56 1 i V2 [-!](+)[-]) i 2 03 12 56 + 03 12 56 2 1 V2 m-](+) + (+i)[-][-]) 1 2 Table 3.5. The basic creation operators — b^m — for massive states, the first with spin up and the second with spin down, are presented. i^m e_imx , s = ± 1, solve the equations 56 56 0 of motion (po + Y0((+) m++ (—) m_)}b^m e_imx = 0, for the two positive energy states, (1,2), (one with spin up and the other with spin down). m2 = m+m_, m+ = —m_, (p0)2 = m2, p0a = — 1 Scddcda is assumed to be real [66]. orthonormal states when applied on the vacuum state and fulfill, together with the annihilation operators, the anticommutation relations presented in Eq. (3.83). Clifford case in d =(13 + 1 ): There are 2 J2-1 = 64 creation operators for family members of one family, all reachable from the starting one by Sab. They are presented in Table 3.6, analyzed so that the internal degrees of freedom manifest in d = (3 + 1 ) quantum numbers of the observed quarks and leptons. Applied on the vacuum state |^oc > they form in the spin-charge-family theory 64 basic states for quarks and leptons and antiquarks and anti-leptons for each family. In the spin-charge-family theory there are 86 N.S. Mankoc Borstnik and H.B.F. Nielsen two times four families — 2 2-1 — getting masses after the two triplet scalar fields, the superposition of cDabc, (a, b) = (0,1, • • • ,8) and three singlet scalar fields, the superposition of cabc, (a, b) = (5,6) or (7,8) or (9, • • • 14), while c = (5,6,7,8) for all these scalar fields, get nonzero vacuum expectation values at low energies [9,3,4,6,7]. Table 3.1 represents the creation operators creating 8 families of ûR1 ^ and of VR. All the family members of each of these families follow by the application of SRb. All the rest of families not included in these eight families get in the spin-charge-family theory masses by the interaction with the condensate [9,3,4,6,7]. To the lower four families the three so far observed families of quarks and leptons belong. i aÉt r (3+1 ) S12 t13 t23 t33 t38 t4 Y Q (Anti)octet, r (7 + 1 ) = ( — 1 ) 1 , r(6) = (1 ) — 1 of (anti)quarks and (anti)leptons 1 ac1 t 03 12 56 78 910 1112 1314 (+i) [+] I [+] ( + ) || ( + ) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 a/3 1 6 2 3 2 3 2 ac1 t 03 12 56 78 910 1112 1314 [ —i] ( —) I [+] ( + ) II ( + ) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 a/3 1 6 2 3 2 3 3 ac11 03 12 56 78 910 1112 1314 (+i) [+] I ( —) [ —] II ( + ) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 A3 1 6 1 3 1 3 4 ac1t 03 12 56 78 910 1112 1314 [ —i] ( —) I ( —) [ —] II ( + ) [ —] [ —] 1 1 — 2 0 1 — 2 1 2 1 2 A/3 1 6 1 — 3 1 — 3 5 aft 03 12 56 78 910 1112 1314 [ —i] [+] I ( —) ( + ) II ( + ) [ —] [ —] -1 1 2 1 — 2 0 1 2 1 2 a/3 1 6 1 6 1 — 3 6 aft 03 12 56 78 910 1112 1314 (+i) ( —) I ( —) ( + ) II (+) [ —] [ —] -1 1 — 2 1 — 2 0 1 2 1 2 13 1 6 1 6 1 — 3 7 aft 03 12 56 78 910 1112 1314 [ —i] [+] I [+] [ —] II (+) [ —] [ —] -1 1 2 1 2 0 1 2 1 2 13 1 6 1 6 2 3 8 aft 03 12 5678 910 1112 1314 (+i) ( —) I [+] [ —] II ( + ) [ —] [ —] -1 1 2 1 2 0 1 2 1 2 13 1 6 1 6 2 3 9 ûc2t 03 12 56 78 910 1112 1314 (+i) [+] I [+] ( + ) I I [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 1 2 13 1 6 2 3 2 3 10 a c 2t 03 12 56 78 910 1112 1314 [ —i] ( —) I [+] ( + ) I I [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 1 2 13 1 6 2 3 2 3 11 a c 2t 03 12 56 78 910 1112 1314 (+i) [+] I ( —) [ —] I I [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 1 2 13 1 6 1 3 1 3 12 a c2t 03 12 56 78 910 1112 1314 [ —i] ( —) I ( —) [ —] I I [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 1 2 13 1 6 1 3 1 3 13 aft 03 12 56 78 910 1112 1314 [ —i] [+] I ( —) ( + ) I I [ —] ( + ) [ —] -1 1 2 1 2 0 1 2 1 2 13 1 6 1 6 1 3 14 aft 03 12 56 78 910 1112 1314 (+i) ( —) I ( —) ( + ) 11 [ —] (+) [ —] -1 1 2 1 2 0 1 2 1 2 13 1 6 1 6 1 3 15 aft 03 12 56 78 910 1112 1314 [ —i] [+] I [+] [ —] 11 [ —] (+) [ —] -1 1 2 1 2 0 1 — 2 1 2 -13 1 6 1 6 2 3 16 aft 03 12 5678 910 1112 1314 (+i) ( —) I [+] [ —] I I [ —] ( + ) [ —] -1 1 — 2 1 2 0 1 — 2 1 2 a/3 1 6 1 6 2 3 17 aR3t 03 12 56 78 910 1112 1314 (+i) [+] I [+] ( + ) I I [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 13 1 6 2 3 2 3 18 aR3t 03 12 56 78 910 1112 1314 [ —i] ( —) I [+] ( + ) I I [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 13 1 6 2 3 2 3 19 a R3t 03 12 56 78 910 1112 1314 (+i) [+] I ( —) [ —] I I [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 13 1 6 1 3 1 3 20 a c31 03 12 56 78 910 1112 1314 [ —i] ( —) I ( —) [ —] I I [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 13 1 6 1 3 1 3 21 aft 03 12 56 78 910 1112 1314 [ —i] [+] I ( —) ( + ) I I [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 13 1 6 1 6 1 3 22 aft 03 12 56 78 910 1112 1314 (+i) ( —) I ( —) ( + ) 11 [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 13 1 6 1 6 1 3 23 aft 03 12 56 78 910 1112 1314 [ —i] [+] I [+] [ —] 11 [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 13 1 6 1 6 2 3 24 aft 03 12 5678 910 1112 1314 (+i) ( —) I [+] [ —] I I [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 13 1 6 1 6 2 3 25 aR 03 12 56 78 910 1112 1314 (+i) [+] I [+] ( + ) II ( + ) ( + ) ( + ) 1 1 2 0 1 2 0 0 1 2 0 0 Continued on next page 3 New Way of Second Quantized Theory of Fermions... 87 (Anti)octet, F (7+ of (anti) quarks and ( anti ) leptons mtt (-1)1 , r 16" -(3 + 1 ) S12 t13 t23 t33 1 2 0 1 2 0 1 2 0 1 2 0 1 - 2 0 1 2 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 1 2 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 A. 03 12 [-i] ( 56 78 9 10 11 12 13 14 [+ (+) II (+) (+) (+) 56 78 9 10 11 12 13 14 ( - -] II (+) (+) (+) 56 78 9 10 11 12 13 14 ( - -] II (+) (+) (+) 56 78 9 10 11 12 13 14 - (+) II (+) (+) (+) 56 78 9 10 11 12 13 14 ( - (+) II (+) (+) (+) 56 78 9 10 11 12 13 14 [+ -] II (+) (+) (+) 56 78 9 10 11 12 13 14 [+] -] II (+) (+) (+) 03 12 (+i) [ + 03 12 [ -i] ( 03 12 [ -i] [ + 03 12 (+i) ( - V 03 12 [ i] [+ V 03 12 (+i) ( - i c»1t L 03 12 [-i] [ + 78 + ) II 10 1112 13 14 -] ( + ) ( + ) 2 v3 âl_1t 03 12 (+i) ( 78 +) ii 10 11 12 13 14 -] (+) (+) 2 v3 - c-1 t uL T 03 12 [-i] [ + 78 -] II 10 11 12 13 14 -] ( + ) ( + ) 2 -jï llnt 03 12 (+i) ( 10 11 12 13 14 ] (+) (+) 2 v3 âc1t dR 03 12 (+i) [ + 78 -] II 10 11 12 13 14 -] ( + ) ( + ) 2 v3 » c»1 t R 03 12 [-i] ( - 78 -] II 10 11 12 13 14 -] ( + ) ( + ) 2 v3 sc»1t lr 03 12 (+i) [+ 78 +) II 10 11 12 13 14 -] (+) ( + ) 2 v3 âc»1 t âR t 03 12 [-i] (- 78 +) II 10 11 12 13 14 -] (+) (+) 2 v3 âl"2t 03 12 [-i] [+ 78 +) II 10 11 12 13 14 +) [-] (+) 2 v3 i c"2t L 03 12 (+i) (- 78 +) II 10 11 12 13 14 + ) [-] ( + ) 2 v3 li2t 03 12 [-i] [+] 10 11 12 13 14 +) [-] (+) 2 v3 li2t 03 12 (+i) (- 78 -] II 10 11 12 13 14 +) [-] (+) 2 v3 â^t dR 03 12 (+i) [ + 10 1112 13 14 + ) [-] ( + ) 2 v3 âc2t dR 03 12 [-i] (- 78 -] II 10 1112 13 14 + ) [-] ( + ) 2 -,/j ,r2t 03 12 (+i) [+ 78 +) II 10 1112 13 14 + ) [-] ( + ) 2 v3 ,r2t 03 12 [-i] (- 78 +) II 10 1112 13 14 + ) [-] ( + ) 2 v3 » ¿3t L 03 12 [-i] [+ 78 +) II 10 1112 13 14 + ) ( + ) [-] » clSt L 03 12 (+i) ( 78 +) II 10 1112 13 14 + ) (+) [-] âc3t âL t 03 12 [-i] [+] 78 -] II 10 1112 13 14 + ) ( + ) [-] }c3t L 03 12 (+i) ( 10 11 12 13 14 + ) (+) [-] i c»3 t R 03 12 (+i) [ + 78 -] II 10 11 12 13 14 + ) ( + ) [-] âc»3t dR 03 12 [-i] ( 10 11 12 13 14 +) (+) [-] }c3t R 03 12 (+i) [+ 78 +) II 10 11 12 13 14 + ) (+) [-] >c"3t R 03 12 [-i] ( 78 +) II 10 11 12 13 14 + ) (+) [-] v3 03 12 [-i] [+ 78 +) II 10 11 12 13 14 03 12 (+i) (- 78 +) II 10 11 12 13 14 03 12 [-i] [+ 10 11 12 13 14 03 12 (+i) (- 10 11 12 13 14 03 12 (+i) [+ 78 +) II 10 11 12 13 14 03 12 [-i] ( 78 +) II 10 11 12 13 14 03 12 (+i) [ + 10 11 12 13 14 03 12 [-i] ( 10 11 12 13 14 Table 3.6. The left handed (F 13,1) = - 1 ), multiplet of creation operators of spinors — the members of the fundamental representation of the SO (13, 1 ) group, manifesting the subgroup SO (7, 1 ) of the colour charged quarks and anti-quarks and the colourless leptons and anti-leptons — is presented in the massless basis using the technique presented in App. 3.7. It represent the left handed (F (3 + 1) = - charged (t ± 2 ,(t r(s58 - sb + S68,S56 - S )) and SU(2)jj chargeless (t2 1, App. 3.7) weak (SU(2 )j) t2 = 1 (S58 + 38 4 y q 26 0 0 0 0 27 0 28 0 29 30 0 -1 2 31 0 0 2 32 0 0 2 33 6 3 1 6 1 3 34 3 5 6 1 6 2 j 2 3 35 8 1 6 2 j 2 3 36 I 37 6 3 38 6 3 56 1 6 2 3 39 56 40 6 3 1 6 1 3 41 1 6 1 3 42 3 56 78 1 6 2 3 2 3 43 I 2 3 44 6 3 78 1 6 1 6 1 3 45 I 1 6 1 6 1 3 46 56 1 6 1 6 2 3 47 5 6 48 6 6 3 49 1 6 1 3 1 3 50 56 1 6 2 3 2 3 51 78 1 6 2 3 2 3 I 52 53 6 6 3 78 1 6 1 6 1 3 54 I 56 1 6 1 6 2 3 55 56 1 6 1 6 2 3 56 57 0 1 2 58 0 56 78 ] II 0 0 0 59 78 60 I 0 0 0 56 61 0 0 56 62 0 0 78 63 I 0 8 1 2 1 2 64 I 0 3 57 S 88 N.S. Mankoc Borstnik and H.B.F. Nielsen S67, S57 — S68,S56 + S78 )) quarks and leptons and the righthanded (r (3 + 1) = 1), weak (SU (2 ) j) chaigeleas and SU(2)jj charged (t23 = ± 1) quarks and leptons, both with the spin S12 upanddown (± ij, respectively). The creation operators of quarks distinguish from those of leptons only in the SU(3) X U( 1 ) part: Quarks are triplets of three colours ( = (t33,t38)= [(1, —L= ), (— 1 , —1 ), (0,--L )], 2 2\/3 2 2\/3 v3 (T3 = 1 (S9 12 — S10 11 , S9 11 + S10 12 , S9 10 — S11 12 , S9 14 — S1013,S913 + S1014,S1114 — S12 13, 5 11 13 + S12 14 , -L (S9 10 + S11 12 — 2S13 14 )) , carrying the "fermion charge" (t4 = -6, = — 1 (S9 10 + S 12 + S ) The colourless leptons carry the "fermion charge" (t ' chargeless and SU (2 ) jj charged anti-quarks and anti-leptons and the right handed weak (SU (2 ) j) charged and SU (2 ) jj chargeless anti-quarks and anti-leptons. Anti-quarks distinguish from anti-leptons again only in the SU (3 ) X U(1) part: Anti-quarks are anti-triplets, carrying the "fermion charge" (t4 = — 6). The anti-colourless anti-leptons carry the "fermion charge" (t4 = 2). Y = (t23 + t4 ) is the hyper charge, the electromagnetic charge is Q = (t13 + Y). The creation operators of opposite charges (anti-particle creation operators) are reachable from the particle ones besides by Sab also by the application of the discrete symmetry operator C^f Pj\f, presented in Refs. [65,66]. The reader can find this Weyl representation also in Refs. [4,71,72,9] and in the references therein. Table 3.6 represents in the spin-charge-family theory the basic creation operators for observed quarks and leptons and anti-quarks and anti-leptons for a particular family. Hermitian conjugation of the creation operators of Table 3.6 generates the corresponding annihilation operators, fulfilling together with the creation operators anticommutation relations for fermions of Eq. (3.81). In observable dimension d = (3 + 1) the d = (13 + 1) case differs from d = (5 + 1) case, Table 3.5, in a much reacher offer of charges. The kinematics of the fermion states in d = (13 +1), Table 3.6, in d = (3 +1) is, however, very similar to the one of Table 3.97. The coefficients of the superposition of the basic creation operators — — which solve, applied on the vacuum state, the Weyl equation, Eq. (3.36), for the choice of pa = (p0,p1 ,p2,p3,0, • • • ,0), can be taken from Eq. (3.97). For the 03 12 56 positive energy solution of spin 2 one only has to replace (+i) (+) (+) by UR j/2 03 12 56 with spin 2 and [—i][-](+) by UR -1/2 with spin — 1. The coefficients, |3 and 12 lP0+ip3|, remain the one of the case with d = (5 + 1). The operator Tp = Y1 Y3 K Ix0 I*5,*7 .. ,xd—1 transforms this superposition of creation operators, p (<,1/2 + -pr+f^UR!-1/2) • e-i(p0*0-px), into p* (UR,-1/2 — p1 -jp2 ttc1t ) . e-i(p°x0+p.x) |p0 | + |p3| UR,1/2) e . The operator CpPj^ -1) = y0 Y5 Y7 • • • Yd-1 I*3 • • • I*6,*8,... ,*d transforms the positive energy solution creation operator for u quark, p (UR, 1t/2+ |_po i+Ti^P31 itR,-1/2) • e-i(p0*0-p x), into the positive energy solution of anti-u quark, —p (tuR^1t/2 + p1 +iP2 uc_11 ) • e-i(P0x0+P *) |p0 + |p3| UL,-1/2) e . One can proceed in the same way also for the itL11, dR11, and all the other quarks ci, as well as for leptons. Spins in higher dimensional space manifest charges in d = (3 + 1), Table 3.6, provided that the angular momentum in ordinary space at higher dimensions do not contribute, which is supposed to be the case at low energies. All the creation operators of any family and any family member, or the orthogonal superposition of them, together with their Hermitian conjugate annihilation operators fulfill the anticommutation relations of Eqs. (3.81, 3.82,3.83). The commuting part of the operators of Sab, Eq. (3.110), determine in d = (3 + 1) the handedness (r(3i1)) = —4i• S03S12)), the spin (S12), the third component of the weak SU(2) charge (t13), the third component of the second SU(2) charge 3 New Way of Second Quantized Theory of Fermions... 89 (t23), the two components of the SU(3) colour charge (t33, t38) and the "fermion charge" (t4, originating in U(1) from SO(6), which includes SU(3) x U(1)). The hypercharge Y, which is in the standard model "guessed" from the experimental data, is in the spin-charge-family theory equal to (t4 + t23), while electromagnetic charge Q is, like in the standard model, equal to (Y + t13). One representation of creation operators with 2d -1 members includes all the left and the right handed coloured quarks and colourless leptons and left and right handed (anti coloured) antiquarks and (anti colourless) antileptons. The right handed neutrinos and the left handed antineutrinos, like all the other members of one Lorentz representation, carry the additional hypercharge (the additional superposition of t4 and t23) and are correspondingly not chargeless like in the standard model. The sum of the charges, the sum of the spins and the sum of the handedness —properties defined with respect to d = (3 + 1) — over all the members of one representation are equal to zero in any d, as it is the case of d = (5 + 1). However, in the d = (13 + 1) case this is true even within quarks and leptons separately and within antiquarks and antileptons separately. Let be repeated that this is so since the right handed neutrinos and the left handed antineutrinos are the regular members of one representation, as it is true for quarks and charged leptons. This can be checked in Table 3.6. Exclusion of the right handed neutrinos and left handed antineutrinos makes nonzero the sum of (r(3+1)), t23 and t4 over the spinor part separately and correspondingly also over the antispinor part. The whole representation has even in this case sums over all the quantum numbers of spins and charges equal to zero. Grassmann "fermions" and "antifermions" Let us represent creation and annihilation operators in Grassmann space, like we did in the Clifford case. In the Grassmann case the representations in d = (13 + 1) space start to be very large and correspondingly almost uncontrollable, Eq. (3.59). We learn in the Clifford case that at the low energy regime, when we treat the equations of motion for free massless fermions with nonzero momentum only in d = (3 +1), the higher dimensional space contributes charges, which are reacher the larger is space, but kinematics in d = (3 + 1) are in all such cases the same. We treat therefore only the d = (5 + 1) case. In Table 3.7 the basic creation operators for d = (5 + 1) case, with Grassmann space used to describe internal degrees of freedom of "fermions" and "antifermions", are presented. "Fermions" carry in Grassmann space integer spins and charges in the adjoint representations. There are two independent decuplets (unconnected by Sab). Both decuplets [46] of creation operators are of an odd Grassmann character, representing the second quantized n = 1 "fermion" states, Eq. (3.54), which belong in general to n (any n) "fermion" states. There are, from the point of view of d = (3 + 1) space, two triplets, one doublet and two singlets in each of the two decouplets. In Subsect. 3.3.3 the discrete symmetry operators in Grassmann space are discussed, with the discrete symmetry operators for the case that "fermions" 90 N.S. Mankoc Borstnik and H.B.F. Nielsen I i decuplet of creation operators S03 S12 S56 1 (0° —03)(01 +i02)(05 +i06) i 1 1 2 (0°03 + Í01 02 )(05 + i06) 0 0 1 3 (0° + 03)(0' — i02)(05 + i06) —i 1 4 (0° — 03)(0' — i02)(05 — i06) i 5 (0°03 — Í01 02 )(05 — i06) 0 6 (0° + 03)(0' + i02)(05 — i06) —i 1 7 (0° — 03 )(0 1 02 + 05 06) i 0 8 (0° + 93)(0'e2 — 05 06) —i 0 9 (0°03 + i0506 )(0 1 + i02 ) 0 1 0 10 (0°03 — i0506 )(0 1 — i02) 0 0 II i decuplet of creation operators S03 S12 S56 1 (0° + 03 )(01 +i02)(05 +i06) —i 1 1 2 (0°03 — Í01 02 )(05 + i06) 0 1 3 (0° — 03)(0' — i02 )(05 + i06) i 1 4 (0° + 03 )(01 —i02)(05 —i06) —i 5 (0°03 + Í01 02 )(05 — i06) 0 6 (0° — 03)(0' + i02 )(05 — i06) i 1 7 (0° + 93)(0'e2 + 05 06) —i 0 8 (0° — 03 )(0 1 02 — 05 06) i 0 9 (0°03 — i0506 )(0 1 + i02 ) 0 1 0 10 (0°03 + i0506 )(0! — i02 ) 0 0 Table 3.7. Two decuplets of the basic creation operators 62kt, k = (I, II), i = (1,..., 10), of the orthogonal group SO(5,1) in Grassmann space are presented. The creation operators form "eigenstates" of the Cartan subalgebra, Eq. (3.110), (S03, S12, S56 for SO(5,1)) with integer spins and charges, defining "fermions" and "antifermions". The creation operators within each decuplet are reachable from any member by (a product of) Sab's (which do not belong to the Cartan subalgebra). Creation operators 62kt and their Hermitian conjugated annihilation operators 62k fulfill the anticommutation relations for fermions, Eq. (3.62). The product of the discrete symmetry operators CNG and png-1 ', Eq. (3.95), (CngPng-1 ' = YgYg Ix3 Ix6 in d = (5 + 1)) transforms, for example, 6®:t into 6®:t, 626It 656It and 636It into 62it , transforming "fermions" with the charge 1 into "antifermions" with the charge -1. manifest kinematics only in d = (3 + 1)-dimensional space, while the higher dimensions contribute charges, included. Let us notice that the Grassmann even operator CngpNG—1', Eq. (3.95), transforms the creation operator creating the positive energy particle state (pa = (|p0|, 0,0, |p3|, 0,0)) with the charge 1, b^, into the creation operator of the anti-particle state, b^1", with the positive energy |p0| and with — |p3| and with the charge — 1 , for example. Correspondingly CNGPI(jGr1', Eq. (3.95), transforms the particle state b2If with the positive energy into the anti-particle state with the positive energy. All these states belong to the same representation, the same decuplet. In Eq. (3.98) the superposition of the creation operators of the two triplets of the first decuplet of creation operators — ) —which solve Eq. (3.43) for free massless "fermions" in Grassmann space, with the space function e- ipaX , pa = (p0,p1 ,p2,p3,0,0), Eq. (3.66), is presented. Two indexes — (ch, s) — replace the index i, ch represents the charge, defined by S56, and s represents the spin, S12. 3 New Way of Second Quantized Theory of Fermions... 91 Creation operators for "fermion" states in Grassmann space ford = (5 + 1) P0 = ip0I , «TÍ (pp) = ß {(—= )3 (e° - e3 )(e1 + ie2) — 2(lp01 — lp3|) (-L )2(e0e3 + ieW) 1,1 —2 p1 — ip2 —2 -( p1 + ip2 )2 (-L)3 (e° + e3)(e1 — ie2)} (e5 + ie6)e—i(|p0|x°—p-x) , |p°| + |p3 | —2 + * 1 3 0 3 1 2 2(|p0| — |p3|) 1 2 0 3 1 2 «e2Í 1 (p) = ß* {(—= )3 (e0 + e3 )(e1 — ie2)-----— (— )2 (e0e3 + ie1 e2) —1 —2 p1 + ip2 —2 1 )2 (— )3 (e0 — e3 )(e1 + ie2)} (e5 + ie6) e—i(|p0|x0+p'x) , |p0| + |p3 | V2 Creation operators for "anti — fermion" states in Grassmann space ford = (5 + 1) Êe3,t 1 (p) = ß {(—)3(e0 + e3)(0i + te2) - 2(|p 1 ~ |p 11 (-L)2 (e0e3 _ieie2, _1 —2 p1 _ ip2 —2 _( p + tp2 )2 ( — )3 (e0 _ e3)(e1 _ ie2)} (e5 _ ie6) e_i(|p0|*0+p-x) , |p0| + |p3 |' " —2 ' !! _ 1 1 (p) = ß* {( —- )3 (e0 _ e3)(e1 _ ie2)_ 2(lp01 _ lp3|) ( — )2(e0e3 _ ie1e2) _ _ 1 -,fl p1 + ip2 -,fi _( p1 _ip2 )2(—)3 (e0 + e3)(e1 + ie2 )} (e5 _ ie6) e_i(|p0|*0_p-x) |p0 | + |p3 | —2 p0 p0| , (3.98) Here p* p = ^i^op-^3)^ • All the corresponding states are orthonormal. The corresponding annihilation operators follow from the creation ones by taking into account Eq. (3.18). Let us write down, as an example, the annihilation operator partner to the creation operator b®1] (p) from Eq. (3.98). Taking into account Eq. (3.18) (saying that 9a f = naa af]' = afa), it follows b®]1 (p) = (-]= )3p*(305 -i906^(301 -i302)(30c-30()-2(pP1l+-PP,31 1 V2(303 30c -Í30230l )- (Jpc—p()2 (30, +1302)(30o + 303)}ei( lpCl xc-pX). The creation and annihilation operators fulfill the anti-commutation relations of Eq. (3.62). Creation operators b0h *s(p) e-l(Pm*m), m = (0, • • • ,3), while p5 = 0 = p6, generate states, which solve the equation of motion (9 a - dp) p a s (x°, x) = 0, Eq. (3.43),14. 1 ' Let be noticed that the second creation operator b02-1 follows from the first one — b01] — by the application of the operator Tng = Yg Yg K Ix Eq. (3.95). When applying on the first two creation operators of positive charge (b01/, 1), defining the "fermion" states of positive energy, the operator Cng •P^ 11 (= YgYgYg • • • Yg-1 Ix3 IX6'X8'- - ,xd), the third and the fourth creation operators follow, defining the "antifermion" states of negative charge and positive energy (b-1,1, b041|-1). Solutions of the equation of motion of the second decouplet, and correspondingly the creation and annihilation operators, can be obtained in equivalent way. 14 The equation (0a — ^f-) pa ^(0, x) = 0 can be rewritten into —iya pa ^ = 0, from where the equation (F(3+1) p0 = 2(f>23 p1 + S31 p2 + S12 p3)} $(0,x) follows, leading to the same solutions as presented in Eq. (3.98). Similar relation appears also in the Clifford case. 92 N.S. Mankoc Borstnik and H.B.F. Nielsen We learned that states transform under the application of the discrete symmetry operators (defined in the Clifford case in Eq. (3.90) and Eq. (17) in Ref. [65], or Eq. (10) in Ref. [66], and in the Grassmann case in Eqs. (3.91, 3.95)), equivalently in the Clifford and in the Grassmann case. 3.3.5 What do we learn from the second quantization procedure in Grassmann and in Clifford space? We proved that in both spaces, in Clifford space and in Grassmann space, the corresponding creation operators and their Hermitian conjugated annihilation operators of an odd (either Clifford or Grassmann) character fulfill the anticommutation relations as required for fermions, Eqs (3.83, 3.62), if operating on an appropriate vacuum state, representing in both spaces a n = 1 fermion space out of n, any n, fermion Hilbert space. No postulated creation operators are needed as in ordinary second quantization procedure. In Clifford space the creation operators are (after the requirement of Eq. (3.69)) products of odd numbers of Ya's, arranged into nilpotents and projectors, Eq. (3.70), which are the "eigenstates" of the Cartan subalgebras of Sab, Eq. (3.72), generating spins and charges, and of Sab, generating families, Eqs. (3.2, 3.4). In Grassmann space they are products of 0a, arranged in "eigenstates" of the Cartan subalgebra of Sab, Eq. (3.5, 3.52)). While in the Grassmann case the vacuum state is simple, |^og >= |1 >, in the Clifford case the vacuum state is a sum of 22-1 products of projectors, Eq. (3.79). In 2(2n + 1)-dimensional spaces there are in the Clifford case in one representation 22-1 creation operators. The whole representation is reachable from the (any) starting operator by products of Sab, while products of S ab transform each of these creation operators into the creation operator of the same family member, but belonging to another family, Eq. (3.76). There are correspondingly 22-1 • 22-1 creation operators, and correspondingly the same number of states, reachable by products of Sab's or Sab's or of both, Sab's and Sab's. Each state follows by the corresponding creation operator on the vacuum state and it is annihilated by its Hermitian conjugated operator, Eq.(3.71). In 2(2n + 1)-dimensional spaces there are in the Grassmann case (before the requirement of Eq. (3.69)) two decoupled representations with all the 0a's included into the representations, each with 2 -jfrr creation operators, and correspondingly 2 2' 2' with the same number of states. Each state can be obtained by the corresponding creation operator operating on the vacuum state and any state is annihilated by the corresponding Hermitian conjugated creation operator. While all of 2 2 -1 • 2 2 -1 states in Clifford space of an odd character are reachable from any of Clifford odd states by either products of Sab's or by products of S ab's or by products of both, and states of an even Clifford character by either products of Sab's or by products of S ab's or YaYa or all of them, in Grassmann space all the irreducible representations are decoupled — no products of Sab's transform states of one group into states of another groups. 3 New Way of Second Quantized Theory of Fermions... 93 The creation (annihilation) operators — which are superposition of the creation (annihilation) operators defining the eigenstates of the Cartan subalgebra in the internal space, fulfilling the relations of Eqs. (3.62,3.83), respectively — form the eigenstates of the equations of motion for free massless "fermions" with integer spins and no families in the Grassmann case, Eqs. (3.43,3.61), and for free massless fermions with half integer spins and families in the Clifford case, Eqs. (3.36, 3.82). The number operators for the odd part of either Clifford or Grassmann case have the eigenvalues 0 or 1, Eqs. (3.55, 3.84). One can as well define in both cases the Hamilton functions, which lead to the equations of motion in the Grassmann case, Eqs. (3.67, 3.68), and in the Clifford case, Eqs. (3.88,3.89). While in the Clifford case the procedure to find the Hamilton function is the usual one, that is the known one, in the Grassmann case is not. It remains to understand better the Hamilton function in the Grassmann case. Comparing solutions for free massless states in a toy model with d = (5 + 1) from the point of view of d = (3 +1) (assuming that pa = (p0,p1 ,p2,p3,0, • • • ,0)) for the Clifford case and for the Grassmann case, one observes several similarities. The main differences are: i. that spins and charges are in the Clifford case half integer while in the Grassmann case are integer, ii. that Clifford space offers, after the assumption of Eq. (3.69), the existence of families, while Grassmann space, before the assumption of Eq. (3.69), does not, and iii. that the requirement that the action is Lorentz invariant leads in Clifford space to well defined Hamilton function, while in the Grassmann case this point needs further study. We can conclude: a. The — odd part of the — Clifford algebra presentation of the internal degrees of freedom of fermions offers the n = 1 second quantized fermion part of the n second quantized Hilbert space, offering the fermion creation and annihilation operators, fulfilling the required relations, explaining therefore the assumption of Dirac about introducing creation and annihilation operators in the second quantized fields. b. The spin-charge-family theory of N.S.M.B., assuming d > (13+1 )-dimensional space and the Clifford algebra to explain internal degrees of freedom of fermions, enables to justify the assumption of the usual second quantized procedure. The group theory alone, without connecting the internal degrees of freedom with the Clifford objects for explaining spins, charges, and families, can not do that. c. Table 3.6 demonstrates that any family contains all the fermions and an-tifermions, what in the spin-charge-family theory means all the quarks and the antiquarks and leptons and anileptons, left and right handed. No Dirac sea of negative energy states is needed to explain the existence of antifermions. Correspondingly the vacuum state is simple, of an even Clifford character, with the sum of all the quantum numbers over the family members equal to zero. d. The sum of all the quantum numbers within one family representation, but also separately within fermions and separately within antifermions within the same representation, is zero. Also the sum over family quantum numbers is zero. e. In the Clifford case the operator CnPIf-1', Eq. (3.94), transforms the fermion state into the anti-fermion state. In the Grassmann case it is the operator CngPNG"1 ', which transforms the Grassmann "fermion" into the "antifermion". 94 N.S. Mankoc Borstnik and H.B.F. Nielsen 3.4 Conclusions We have learned in the present study that both Clifford and Grassmann space offer 1 -fermion second quantized part of vector space, with creation and annihilation operators — defined as an odd products of either Clifford or Grassmann eigenstates of the corresponding Cartan subalgebra operators in even dimensional space, Eq. (3.110) — fulfilling the desired anticommutation relations for fermions, Eqs. (3.62, 3.83). The corresponding number operators have the eigenvalues 0 or 1 in both cases. The fact that states, solving equations of motions, fulfill the desired anticommutation relations for second quantized fermions explains the second quantization postulates of Dirac. Grassmann coordinates and Clifford coordinates offer the same degrees of freedom: Two times 2d each. 9a's and their Hermitian conjugated partners are expressible with the two kinds of Clifford coordinates, Ya's and Ya's — defining two independent spaces — and opposite. The vacuum states ought to be changed from 11 >in the Grassmann case to the one presented in Eq. (3.79) for either Ya's or Ya's. The Grassmann states carry integer spins, while Clifford states carry in both spaces half integer spins. The requirement of Eq. (3.69) breaks the equivalence of both kinds of the Clifford coordinates and opens the possibility for the appearance of families. Clifford space, defined by the two kinds of objects, narrow now to only one of the two, determined by Ya's, while Ya's take care of families. Correspondingly also in Grassmann space there remain only 9a's, becoming Ya's, while their Hermitian conjugated partners d® a no longer exist. Consequently, after the requirement of Eq. (3.69), the possibility of having integer spins "fermions" no longer exists. The 1-fermion second quantized vector space has for a chosen momentum p£ in the Clifford case (after the requirement of Eq. (3.69)) 2d-1 • 2d-1 members (that is 2 d -1 families, each family having 2 d -1 members), and in the Grassmann case (before the requirement of Eq. (3.69)), when all 0a's contribute in forming a state, drr members in two decoupled representations. 2 ' 2 ' In both spaces the members of one representation include fermions and antifermions and correspondingly there is no need for the Dirac sea of negative energies filled by fermions. In both cases the creation and annihilation operators of different momentum pa and the same internal part represent different creation operators. The n (any n) second quantized vector space of fermions (or "fermions" in the Grassmann case) follows in both cases as products of n creation operators defining each one fermion states when applying on the corresponding vacuum state (in the Clifford case on |Toc >, Eq. (3.79), in the Grassmann case |Tog >= | 1 >), if the creation operators distinguish at least either in one of the quantum numbers of the corresponding Cartan subalgebra or in momentum p£. But while in the Clifford case states carry spin and charges from the point of view of d = (3 + 1) in the fundamental representations of the Lorentz group carrying therefore half integer spins, states in the Grassmann case are in adjoint representations of the Lorentz group, carrying therefore integer spins. 3 New Way of Second Quantized Theory of Fermions... 95 We present in this paper as well the action (Eq. (3.41, 3.42)), describing free massless "fermions" with the internal degrees of freedom describable in Grassmann space. The action leads to the equations of motion (Eq. (3.43)), analogous to the Weyl equation in Clifford space (Eq. (3.36)), fulfilling as well the KleinGordon equation (Eq. (3.44)). We also present the discrete symmetry operators in the Grassmann case. Since the Clifford objects Ya and Ya are expressible with the Grassmann coordinates ea and their conjugate moments gfr — Ya = (ea + gjr;), Ya = i(ea - deh), Eq. (3.4) — either basic states in Grassmann space, Eq. (3.16), or basic states in Clifford space, Eq. (3.73), can be normalized with the same integral, Eq. (3.31, 3.32, 3.33). To understand better the difference in the description of the fermion internal degrees of freedom either with Clifford algebra (after the requirement of Eq. (3.69)) or with Grassmann algebra (before the requirement of Eq. (3.69)), let us replace in the starting action of the spin-charge-family theory, Eq. (3.1), using the Clifford algebra (after the requirement of Eq. (3.69)) to describe fermion degrees of freedom, the covariant momentum p°a = faa P°a, P°a = Pa - jSabdaba - jSabdaba, with p°a = Pa - 1 SabHaba, where Sab = Sab + Sab, Eq. (3.5), and Qaba are the spin connection gauge fields of Sab (which are the generators of the Lorentz transformations in Grassmann space), while faa p°a replaces the ordinary momentum when massless objects start to interact with the gravitational field through the vielbeins and the spin connections. Let us add that it follows, if varying the action with respect to either daba or d> aba when no fermions are present, that both spin connections are uniquely determined by the vielbeins ([9,3,5] and references therein) and correspondingly in this particular case Haba = daba = daba. The present study was stimulated by one of the author in order to better understand whether and to which extend the spin-charge-family theory offers the next step to both standard models — the one of the fermion and boson fields and the cosmological one. Correspondingly we present in Subsect. 3.1.1 of the introductory Sect. 3.1, the achievements so far of the spin-charge-family theory as well as the open problems of this theory, both suggested by the referees. In shortly, the spin-charge-family theory (using Clifford objects to describe the internal space of fermions) offers, while starting with the simple action in d > (13 + 1) with fermions interacting with gravity only (the vielbeins and the two kinds of the spin connection fields, the gauge fields of moments and the generators of the Lorentz transformations Sab and Sab, respectively), Eq. (3.1), the explanation for all the assumptions of the standard model — for quarks and leptons, antiquarks and antileptons, for fermion families, for the vector gauge fields, for the scalar Higgs and Yukawa couplings — explaining also the phenomena like the existence of the dark matter [54], of the matter-antimatter asymmetry [4], offering correspondingly the next step beyond both standard models — cosmological one and the one of the elementary fields, Sect. 3.1.1. This theory predicts the fourth family to the observed three, Sect. 3.1.1, and the new scalar fields, some of those which explains the properties of the observed Higgs and Yukawa couplings, Sect. 3.1.1, and which will be observed at the LHC and other experiments in the future. This theory predicts also the existence of the stable fifth family, manifesting 96 N.S. Mankoc Borstnik and H.B.F. Nielsen the dark matter and with the "new nuclear" force among the hadrons of these much heavier families, Sect. 3.1.1. To these achievements the present study adds the recognition that the creation operators for one fermion states are in Clifford space already second quantized, and that the creation operators for any n fermion second quantized vectors are products of one fermion creation operators, operating on the empty vacuum state. The spin-charge-family theory namely describes all the internal degrees of freedom of fermions in Clifford space — spins and charges. There is in this theory no need for the existence of the negative energy states filled with fermions. The most severe among the open problems of the spin-charge-family theory is the quantization of gravity gauge fields, although the spin-charge-family theory is explaining the phenomena in the low energy regime where all the vector and scalar gauge fields can be quantized in the known procedure. There are also other open problems, some of them needing only time to be solved, presented in Sect. 3.1.1. The second quantization of "fermions" with the internal degrees of freedom described in Grassmann space might help to understand better the properties of scalars and vectors in the spin-charge-family theory. Let us conclude with a question: Could "fermions" with integer spins and charges in adjoint representations be an acceptable possibility and no requirement of Eq. (3.69) needed? 3.5 APPENDIX: Creation and annihilation operators in Grassmann and Clifford space for d = 4n We discuss in Subsect. 3.3 mainly cases with d = 2(2n + 1), since if assuming no conserved charges in the fundamental theory with fermions, which carry only spins and interact with only the gravity — as the spin-charge-family theory assumes — the dimensions 4n, n is positive integer, as well as all odd dimensions, are excluded under the requirement of mass protection [77]. Let us nevertheless add in this appendix comments on the second quantization procedure in d = 4n spaces. i. Grassmann space In Eq. (3.51) we define in Grassmann space a possible starting creation operator for d = 2(2n + 1) spaces. In d = 4n we correspondingly start with the state 1*1 > = b?1t|1 >, b®1f = (—)d-1 (e° — e3)(e1 + ie2)(e5 + ie6) ••• (ed-3 + ied-2)ed-1ed, 12 (3.99) generated by the creation operator b®1 which is, as it ought to be — like in the d = 2(2n +1) case — of an odd Grassmann character to fulfill the anticommutation relations for fermions, Eq. (3.62). Again the rest of states, belonging to the same 3 New Way of Second Quantized Theory of Fermions... 97 Lorentz representation, follow from the starting state by the application of the operators Scf, which do not belong to the Cartan subalgebra operators. Their annihilation partners follow by Hermitian conjugation. One finds therefore for the (chosen) starting creation and the corresponding annihilation operator b®1f = (—)d-1 (e° — e3)(e1 + ie2 )(e5 + ie6) ••• (ed-3 + ied-2)ed-1 ed, v2 fi01 =( _L) 2-1 9 9 ( 9 — i 9 ) (J___) b1 (-2) aed aed-1 ( aed-3 L aed-2) ( ae0 ae3), d = 4n. (3.100) The application of S01 , for example, generates = (— )f-2(e0e3 + ie1e2)(e5 + ie6)••• (ed-3 + ied-2) ed-1 ed, v2 £21 / 1 —2 9 9 f9 - 9^f"9 9 -9 9^ 2 -r2 "39^ aed-1 ( aed-3 — i aed-2)••• (_ae3 "ae0 — i_ae2 "deT). (3.101) There is the additional group of creation and annihilation operators in d = 4n, which follows from the starting creation operator b22t = (—)2-1 (e0 + e3)(e1 + ie2)(e5 + ie6) • • • (ed-3 + ied-2) ed-1 ed, v2 b2- = (b22t)t = (_L)2-1 ____i 9 v.. + ) b1 (b ) (-2) sed aed-1 ( aed-3 1 aed-2) (ae0 + ae3), for d = 4n. (3.102) All the rest of creation operators follow from the starting creation operator of each of the two groups by the (left) application of products of Sab b2kt , the anticommutation relations required for fermions, Eq. (3.54). i. Clifford space In Eq. (3.73) we define in Clifford space a possible starting creation operator for d = 2(2n + 1) spaces. In d = 4n we correspondingly start with the state with an odd number of nilpotents and with one projector > = b1t|foc >, 03 12 35 d-3 d-2 d-1 d b1t : = (+i)(+)(+) ••• (+) [+] , d-1 d d-3 d-2 35 12 01 b =(b1t)t = [+] (—) ••• (—)(—)(—i) (3.104) 98 N.S. Mankoc Borstnik and H.B.F. Nielsen All the other creation operators, creating all the members of the representation of this particular family, are obtainable by the application of products of Sab on this creation operator from the left hand side. There are 2d—1 members of each family. All the other families follows from the starting one by the application of products of Sab. There are 2d—1 families with 2d—1 members each. A general creation operator in d = 4n follows by the application of Sab and Sab on the starting creation operator of Eq. (3.104) and the corresponding annihilation operator is its Hermitian conjugated value. Correspondingly we define b^ (up to a constant) to be bat K s ab ••• S efSmn ••• Sprb1t = [-i][-][-] ••• [-] [+] +[+i][+][-] ••• [-] [+] + •••|1>, d = 4n, (3.106) n is a positive integer. There are 2 d—1 summands, since we step by step replace ab ef 03 12 35 d—3 d—2d—1 d all possible pairs of [-] • • • [-] in the starting part [-i] [-] [-] • • • [-] [+] into ab ef [+] • • • [+] and include new terms into the vacuum state so that the last 2n + 1 d—1 d 03 12 35 summand has for d = 4n also the factor [+] in the starting term [-i] [-] [-] d—3 d—2d—1 d d—1 d • • • [-] [+] changed into [-] . The vacuum state has then the normalization factor 1/V2d/2—1. 3.6 APPENDIX: Lorentz algebra and representations in Grassmann and Clifford space The Lorentz transformations of vector components 0a, Ya, or Ya — usable for the description of the internal degrees of freedom of fermion fields obeying in the second quantization the anticommutation relations for fermions — and of vector components xa, which are real (ordinary) commuting coordinates, 9 'a = Aab eb, Y'a = Aab Yb, Y/a = Aab Yb and xa = Aab xb, leave forms aa,a2...al 9ai 9a2 . . . 9a, aa,a2...al Ya Ya2 . . . Ya, aa,a2...ai Ya Ya2 ...Ya and baia2...ai xai xa2 ...xai, i =(1,...,d), invariant. While ba,a2...ai (= na,b, na2 b2 ...ilai bi bblb2---bi ) is a symmetric tensor field, aa,a2...ai (= na,b, na2b2 ...naibi ablb2-"bi ) are antisymmetric Kalb-Ramond fields. 3 New Way of Second Quantized Theory of Fermions... 99 The requirements: x a x bnab = xc xdncd, 0/a0/b£ab = 0c0d£cd, Y/aY/b£ab = ycyd£cd and Y/aY/b£ab = YcYd£cd lead to Aab Acdnac = nbd.Here nab (in our case nab = diag (1, — 1, — 1,..., — 1)) is the metric tensor lowering the indexes of vectors ({xa} = nabxb, {0a} = nab 0b, {Ya} = nab Yb and {Ya} = nab Yb) and £ab is the antisymmetric tensor. An infinitesimal Lorentz transformation for the case with det A = 1, A00 > 0 can be written as Aab = 5£ + wab, where wab + wb a = 0. In Eqs. (3.4, 3.8) the commutation relations among the above objects are presented. 3.6.1 Lorentz properties of basic vectors What follows is taken from Ref. [2] and Ref. [9], Appendix B. Let us first repeat some properties of the anticommuting Grassmann and Clifford coordinates, taking into account Eqs. (3.3,3.4). An infinitesimal Lorentz transformation of the proper ortochronous Lorentz group is then i 2C ab c Ö0C = --Cab Sab0c = Wca0a , 6YC = -2^abSabYC = WCaYa , ÖYc = --WabSabYC = CCaYa , öxc = --cabLabxc = ccaxa , (3.107) where cab are parameters of a transformation and Ya and Ya are expressible by 0a and gd^ in Eqs. (3.3,3.4). Let us write the operator of finite Lorentz transformations as follows S = e-2-ab(sab+Lab), (3.io8) Sab have to be replaced by Sab and Sab in the Clifford case. We see that the Grassmann 0a and the ordinary xa coordinates and the Clifford objects Ya and Ya transform as vectors 0 'c = e-2 œab(Sab + Lab) 0c e 2 œab(Sab + Lab) = 0C - 2wab{Sab, 0C}- + • • • = 0C + Wca0a + • • • = Aca0a , x'c = Acaxa , Y'c = ACaYa , Y'c = ACaYa . (3.109) Correspondingly one finds that compositions like YaPa and Yap a, here pa are Pa (= -af^), transform as scalars (remaining invariants), while Sab cabc and Sab cDabc transform as vectors. Objects like R = 1 f«[afßb] (caba,ß — ccaa ccbß) and R = 1 f«[afßb] (cDaba,ß — cDcaacDcbß) from Eq. (3.1) transform with respect to the Lorentz transformations as scalars. Making a choice of the Cartan subalgebra set of the algebra Sab, Sab and Sab, Eqs. (3.2, 3.5, 3.7), S03 S12 S56 • • • Sd-1 d S03 s12 s56 ^ ^ ^ sd-1 d S03, S12, S56, ••• ,S d-1d, (3.110) 100 N.S. Mankoc Borstnik and H.B.F. Nielsen one can arrange the basic vectors so that they are eigenstates of the Cartan subalgebra, belonging to representations of Sab, or of Sab and Sab, with ab from Eq (3.110). 3.7 APPENDIX: Technique to generate spinor representations in terms of Clifford algebra objects Here we briefly repeat the main points of the technique for generating spinor representations from Clifford algebra objects, following Ref. [2,47]. We advise the reader to look for details and proofs in these references. No requirements for the second quantization is taken into account. We assume the objects Ya, Eq. (3.4), which fulfill the Clifford algebra relations of Eq. (3.2), {Ya, Yb}+ = I • 2nab , for a,b G {0,1,2,3,5, • • • , d}, for any d, even or odd. I is the unit element in the Clifford algebra, while {Ya, Yb}± = YaYb ± YbYa. The "Hermiticity" property for Ya's and Ya's, Eq. (3.25), follows from Eq. (3.18), Yat = naaYa Yat = naaYa, leading to YatYa = I, YatYa = I. The Clifford algebra objects Sab close the Lie algebra of the Lorentz group {Sab, Scd}_ = i(nadSbc + nbcSad — nacSbd — nbdSac), Eq. (3.7). One finds from Eq.(3.25) that (Sab)t = naanbbSab and that {Sab, Sac}+ = 2naanbc. Recognizing that two Clifford algebra objects (Sab,Scd) with all indexes different commute, we select (out of many possibilities) the Cartan subalgebra set of the algebra of the Lorentz group of Eq. (3.110) Let us present the operators of subgroups of the SO (13 + 1) group N±(= N(L,R)): = 1 (S23 ± iS01, S31 ± iS02,S12 ± iS03), (3.111) t1 := 1 (S58 — S67, S57 + S68, S56 — S78), t2 := 2(S58 + S67, S57 — S68, S56 + S78), (3.112) t3 := 1 {S912 — S1011 ,S911 + S1012, S910 — S11 12, S9 14 — S10 13 S913 + s10 14 Sn 14 — §12 13 S11 13 + S1214, -L(S910 + S11 12 — 2S1314)}, V3 t4 : =—1 (S910 + S11 12 + S1314). (3.113) Y := t4 + t23 , Y' :=—t4 tan2 32 + t23 , Q := t13 + Y, Q' := —Y tan2 + t13 . (3.114) The equivalent expressions for the group SO (13,1) follows from the above one, if replacing Sab by S ab. 3 New Way of Second Quantized Theory of Fermions... 101 To make the technique simple, we introduce the graphic representation, [47], Eq. (3.70), ab 1 r«a (k): = ^(Ya + , ab 1 i ab]: = -(1 + kYaYb), where k2 = naanbb. One can easily check by taking into account the Clifford algebra relation (Eqs. (3.4, 3.18)) and the definition of Sab (Eq. (3.2)) that if one ab ab multiplies from the left hand side by Sab the Clifford algebra objects (k) and [k], it ab ab ab ab ab follows that, Eq. (3.72), Sab (k)= 2k (k), Sab [k]= 1 k [k]. This means that (k) and ab 2 2 [k] acting from the left hand side on the vacuum state |^oc)), Eqs. (3.79, 3.106) for d = 2(2n + 1) and d = 4n respectively, are eigenvectors of Sab. We further find ab cd ab Ya (k) = ab naa [-k], ab Yb (k) = ab -ik [-k], ab Ya [k] = ab (-k), ab Yb [k] = ab -iknaa (-k) (3.115) i^aa^cc 2n n ab cd [-k] [-k], Sac ab cd [k][k]= 2 ab (-k)( cd ab cd -k), Sac (k) [k] ab cd ab cd ab cd -2naa [-k](-k), Sac [k](k)= 2ncc (-k)[-k]. It is useful to deduce the following relations ab ab (kb) (kb) = 0, ab ab (k)(-k)= naa ab [k], ab ab (-k)(k)= :^aa [ ab -k], ab ab (-k)(-k)= 0, ab ab [k] [k] ab = [k], ab ab [k][-k]= 0, ab ab [-k][k] = 0, a b a b [-k][-k]= ab [-k], ab ab (k) [k] = 0, ab ab [k](k)= ab (akb) , ab ab (-k)[k] = ab (-k), ab ab (-k)[-k]= 0, ab ab (k) [-k] ab = (k), ab ab [k](-k): = 0, ab ab [-k](k)= 0, ab ab [-k](-k)=( ab -k) (3.116) We recognize in the first equation of the first row and the first equation of the second row the demonstration of the nilpotent and the projector character of the ab ab Clifford algebra objects (k) and [k], respectively. Whenever the Clifford algebra objects apply from the left hand side, they always ab ab ab ab ab ab transform (k) to [-k], never to [k], and similarly [k] to (-k), never to (k). 102 N.S. Mankoc Borstnik and H.B.F. Nielsen We define in Eq. (3.79, 3.106) the vacuum state |^oc > so that one finds ab ^ ab < (k) (k) >= 1 , ab ^ ab < [k] [k] >= 1 . (3.117) Taking the above equations into account it is easy to find a Weyl spinor irreducible representation for d-dimensional space, with d even or odd. (We advise the reader to see Ref. [2,47] in particular for d odd.) For d even, we simply set the starting state as a product of d/2, let us say, ab only nilpotents (k) for d = 2(2n + 1), Eq. (3.73), or nilpotents and one projector, Eq. (3.104), for d = 4n, one for each Sab of the Cartan subalgebra elements (Eq. (3.110)), applying it on the vacuum state, Eqs. (3.79,3.106). Then the generators Sab, which do not belong to the Cartan subalgebra, applied to the starting state from the left hand side, generate all the members of one Weyl spinor. 0d 12 35 d-1 d-2 (kod)(kl2)(k35) • • • (kd-1 d-2) |^oc > , 0d 12 35 d-1 d-2 [-kod][-k12](k35) ••• (kd-1 d-2) |^oc >, 0d 12 35 d-1 d-2 [-kod](k12)[-k35] • • • (kd-1 d-2) |^oc > , od 12 35 d-1 d-2 (kod)[-k12][-k35] • • • [-kd-1 d-2] |^oc > , for d = 2(2n + 1), n = positive integer. (3.118) 0d 12 35 (kod)(k12)(k35) Od 12 35 -kod][-k12](k35) od 12 35 -kod](k12)[-k35] d-1 d-2 [kd-1 d-2] |^oc > , d-1 d-2 [kd-1 d-2] |^oc > , d-1 d-2 [kd-1 d-2] |^oc > , od 12 35 d-1 d-2 (kod)[-k12][-k35] ••• [kd-1 d-2] |^oc >, for d = 4n, n = positive integer. (3.119) 3.7.1 Technique to generate "families" of spinor representations in terms of Clifford algebra objects We found in this paper that for d even there are 2d/2-1 "family members" and 2d/2-1 "families" of spinors, which can be second quantized. (The reader is advised to see also Refs. [2,71,47,48,72,9].) We shall here pay attention on only even d. One Weyl representation forms a left ideal with respect to the multiplication with the Clifford algebra objects. We proved in Refs. ([9,48], and the references 3 New Way of Second Quantized Theory of Fermions... 103 therein) that there is the application of the Clifford algebra object from the right hand side, which generates "families" of spinors. Right multiplication with the Clifford algebra objects namely transforms the state with the quantum numbers of one "family member" belonging to one "family" into the state of the same "family member" (into the same state with respect to the generators Sab when the multiplication from the left hand side is performed) of another "family". We defined in Ref.[2,48] the Clifford algebra objects Ya's as operations which operate formally from the left hand side (as Ya's do) on any Clifford algebra object A as follows, Eq. (3.69), YaA = i(-)(A)Aya , (3.120) with (-)(A) = — 1, if A is an odd Clifford algebra object and (-)(A) = 1, if A is an even Clifford algebra object. Then it follows, in accordance with Eq. (3.4), that Ya obey the same Clifford algebra relation as Ya. (YaYb + YbYa)A = -ii((-)(A))2A(YaYb + yV) = I • 2.nabA (3.121) and that Ya and Ya anticommute (Y aYb + YbY a)A = i(-)(A)(-YbAYa + YbAYa) = 0. (3.122) We may write {Ya, Yb}+ = 0, while {Ya,Yb}+ = I • 2nab . (3.123) One accordingly finds Y a II a k( ab -i (k) Ya = ab -inaa [k], Y b ab (k): = ab -i (k) Yb = ab -k [k] , y a k][: ab II ab i [k] Ya = ab i (k), ab ab ab Yb [k]: = i [k] Yb = -knaa (k) . (3.124) If we define, Eq. (3.2), Sab = 4 [-Ya,Yb] = 4 {ya,Yb}- = 4(YV - YbYa) , (3.125) it follows SabA = a4(YbYa - YaYb), (3.126) manifesting accordingly that Sab fulfill the Lorentz algebra relation as Sab do. Taking into account Eq. (3.69), we further find {S ab,Sab}_ = 0, {Sab ,Yc}- = 0, {Sab,Yc}- = 0. (3.127) 104 N.S. Mankoc Borštnik and H.B.F. Nielsen One also finds {Sab,r}- = 0, (Ya,r}_ = 0, {Sab,r}- = 0, for d even, r(d) :=(i)d/2 ^ (VnaaYa), if d = 2n, a r(d) :=(i)d/2 ^ (VnaaYa), if d = 2n, (3.128) a where handedness r ({r, Sab}_ = 0) is a Casimir of the Lorentz group, which means that in d even transformation of one "family" into another with either S ab or Ya leaves handedness r unchanged. We advise the reader to read [2] where the two kinds of Clifford algebra objects follow as two different superpositions of a Grassmann coordinate and its conjugate momentum. Below some useful relations [71,72] are presented 03 12 03 12 N± = N+ ± iN+ = - (Ti)(±), N± = N- ± iN- =(±i)(±), 03 12 03 12 N± =-(Ti)(±), N± =(±i)(±), 56 78 56 78 T1± = (T)(±)(T) , T2^ = = (T) (T)(T), 56 78 T1± = (T)(±)(T) , t2^ = 56 78 = (T) (T)(T) . (3.129) ab Sab (k) k ab = k (k), ab Sab [k] k ab = - k[k], ab cd Sac (k)(k) i ab cd = 2^aancc [k][k], ab cd Sac [k][k] i ab cd = - 2 (k) (k.), ab cd Sac (k)[k] i ab cd = -2^aa [k] (k), ab cd Sac [k](k) i ab cd = 2ncc (k) M . (3.130) We transform the state of one "family" to the state of another "family" by the application of Sac (formally from the left hand side) on a state of the first "family" for a chosen a, c. To transform all the states of one "family" into states of another "family", we apply Sac to each state of the starting "family". It is, of course, sufficient to apply Sac to only one state of a "family" and then use generators of the Lorentz group (Sab) to generate all the states of one Dirac spinor d-dimensional space. ab ab One must notice that nilpotents (k) and projectors [k] are "eigenvectors"not only of the Cartan subalgebra Sab but also of Sab. 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Lukman, DMFA ZaloZnistvo, Ljubljana December 2013, p. 31-51, http://arxiv.org/abs/1212.4055. 56. G. Bregar, N.S. Mankoc Borstnik, "The new experimental data for the quarks mixing matrix are in better agreement with the spin-charge-family theory predictions", Proceedings to the 17th Workshop "What comes beyond the standard models", Bled, 20-28 of July, 2014, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA ZaloZnistvo, Ljubljana December 2014, p.20-45 [ arXiv:1502.06786v1] [arxiv:1412.5866]. 57. A. HernandeZ-Galeana and N.S. Mankoc Borstnik, ""The symmetry of 4 x 4 mass matrices predicted by the spin-charge-family theory — SU(2) x SU(2) x U(1) — remains in all loop corrections", Proceedings to the 21 st Workshop "What comes beyond the standard models", 23 of June -1 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA ZaloZnistvo, Ljubljana, December 2018 [arXiv:1902.02691, arXiv:1902.10628]. 58. N.S. Mankoc Borstnik, H.B.F. Nielsen, "Do the present experiments exclude the existence of the fourth family members?", Proceedings to the 19th Workshop "What comes 108 N.S. Mankoc Borstnik and H.B.F. Nielsen beyond the standard models", Bled, 11-19 of July, 2016, Ed. N.S. Mankoc Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2016, p.128-146 [arXiv:1703.09699]. 59. A. Ali in discussions and in private communication at the Singapore Conference on New Physics at the Large Hadron Collider, 29 February - 4 March 2016. 60. M. Neubert, in duscussions at the Singapore Conference on New Physics at the Large Hadron Collider, 29 February - 4 March 2016. 61. N.S. Mankoc Borstnik, M. Rosina, "Are superheavy stable quark clusters viable candidates for the dark matter?", International Journal of Modern Physics D (IJMPD) 24 (No. 13) (2015) 1545003. 62. D. Hestenes, G. Sobcyk, "Clifford algebra to geometric calculus", Reidel 1984. 63. P. Lounesto, P. Clifford algebras and spinors, Cambridge Univ. Press.2001. 64. M. Pavsic,"Quantized fields a la Clifford and unfication", [arXiv:1707.05695] 65. N.S. Mankoc Borstnik and H.B.F. Nielsen, "Discrete symmetries in the Kaluza-Klein theories", JHEP 04:165, 2014 [arXiv:1212.2362]. 66. T.Troha, D. Lukman, N.S. Mankoc Borstnik, "Massless and massive representations in the spinor technique", .Int.] Mod. Phys. A 29,1450124 (2014). 67. P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610. 68. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "An effective two dimensionality cases bring a new hope to the Kaluza-Klein-like theories", New ]. Phys. 13:103027, 2011. 69. D. Lukman and N.S. Mankoc Borštnik, "Spinor states on a curved infinite disc with nonzero spin-connection fields", ]. Phys. A: Math. Theor. 45:465401, 2012 [arxiv:1205.1714, arxiv:1312.541, arXiv:hep-ph/0412208 p.64-84]. 70. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "Families of spinors in d = (1 + 5) with a zweibein and two kinds of spin connection fields on an almost S2 ", Proceedings to the 15th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2012,157-166, [arXiv:1302.4305]. 71. A.Borstnik Bracic, N. Mankoc Borstnik,"The approach Unifying Spins and Charges and Its Predictions", Proceedings to the Euroconference on Symmetries Beyond the Standard Model", Portorož, July 12-17, 2003, Ed. by Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Založnistvo, Ljubljana December 2003, p. 31-57, [arXiv:hep-ph/0401043, arXiv:hep-ph/0401055]. 72. A. Borstnik Bracic, N. S. Mankoc Borstnik, "On the origin of families of fermions and their mass matrices", hep-ph/0512062, Phys Rev. D 74 073013-28 (2006). 73. N.S. Mankoc Borstnik, H.B. Nielsen, "Particular boundary condition ensures that a fermion in d=1+5, compactified on a finite disk, manifests in d=1+3 as massless spinor with a charge 1/2, mass protected and chirally coupled to the gauge field", hep-th/0612126, arxiv:0710.1956, Phys. Lett. B 663, Issue 3, 22 May 2008, Pages 265-269. 74. H.A. Bethe, "Intermediate quantum mechanics", W.A. Benjamin, 1964 (New York, Amsterdam). 75. C. Itzykson, J.B. Zuber, "Quantum field theory", McGraw-Hill, 1980 (New York). 76. M. Pavsic, "Quantized fields a la Clifford and unification" [arXiv:1707.05695]. 77. N.S. Mankoc Borstnik, H. B. Nielsen, "Fermions with no fundamental charges call for extra dimensions", Phys. Lett. B 644 (2007) 198-202 [arXiv:hep-th/0608006]. Bled Workshops in Physics Vol. 20, No. 2 JLV Proceedings to the 22nd Workshop What Comes Beyond ... (p. 109) Bled, Slovenia, July 6-14, 2019 4 Understanding the Second Quantization of Fermions in Clifford and in Grassmann Space — New Way of Second Quantization of Fermions — Part I * N.S. Mankoc Borštnik1 and H.B.F. Nielsen2 1 Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen 0, Denmark Abstract. Both algebras, Clifford and Grassmann, offer the second quantized fermions [13] without postulating the second quantization conditions of Dirac [13]. But while fermions with the internal degrees of freedom described by the Clifford algebras manifest the half integer spins — in agreement with the observed properties of quarks and leptons and antiquarks and antileptons — the Grassmann "fermions" manifest integer spins. In Part I properties of the second quantized integer spins "fermions" in Grassmann space are presented. In Part II the conditions are discussed under which the Clifford algebra offers the appearance of families of the second quantized fermions. Povzetek. Avtorja sta v clanku [3] pokazala, da ponudita obe algebri — Cliffordova in Grassmannova — razlago za Diracove postulate druge kvantizacije fermionov [1-3], saj imajo vektorji v obeh prostorih vse lastnosti, ki jih zahteva Diracov pogoj za drugo kvanti-zacijo [13]. Clanek razlozi v prvem delu tega prispevka drugo kvantizacijo v Grassman-novem prostoru. Pri tem opisu nosijo"fermioni" celostevilcni spin in naboje, kadar je prostor sest razsezen ali vec, v adjungirani upodobitvi. Avtorja demonstrirata lastnosti teh "fermionov" na primeru sest razsezznega prostora. Spin v peti in sesti dimenziji (se po zlomitvi simetrije) "vidi" v (3 + 1 )-razseznem prostoru kot naboj "fermiona". V drugem delu obravnavata lastnosti fermionov s polstevilcnimi spini v Cliffordovi algebri. Keywords: Second quantization of fermion fields in Clifford and in Grassmann space, Spinor representations in Clifford and in Grassmann space, Kaluza-Klein-like theories, Higher dimensional spaces, Beyond the standard model 4.1 Introduction It is demonstrated in this paper how does the Grassmann algebra — in Part I — and the two kinds of the Clifford algebras — in Part II — take care of the second quantization of fermions without postulating anticommutation relations [13]. * Talk presented by N.S. Mankoc Borstnik 110 N.S. Mankoc Borstnik and H.B.F. Nielsen In d-dimensional Grassmann space of anticommuting coordinates 0a's, i = (0,1,2,3,5, • • • , d), there are 2d operators ("vectors"), which are superposition of products of 0a. One can arrange them into irreducible representations with respect to the Lorentz group. There are as well derivatives with respect to 0a's, g^- 's, which are Hermitian conjugated to 0a's [3], (0at = naadr~, nab = diag{1, -1, -1, • • •, -1}, which again form 2d operators ("vectors"). Grassmann space offers correspondingly 2 • 2d degrees of freedom. There are two kinds of the Clifford operators ("vectors"), which are expressible with 0a and — Ya = (0a + ), ya = i (0a - ) [2,4,5]. Each of these two kinds of the Clifford algebra objects has 2d operators ("vectors"), together again 2 • 2d degrees of freedom. The Grassmann and each of the two Clifford algebras split into odd and even part with respect to the odd and even number of 0a's, dp-'s, Ya's, ya's. There is the odd algebra in all three cases which fulfills the second quantized anticommutation relations without postulating them [13]. We present in Sect. 4.2 properties of the Grassmann odd anticommuting algebra and even commuting algebra of the corresponding creation and annihilation operators representing the second quantized "fermion" fields, manifesting in the Grassmann case an integer spin, and offering in d-dimensional space, d > (3 + 1), the description of the corresponding charges in adjoint representations. We follow in this paper to some extent the Ref. [3]. In Part II we present in equivalent section properties of the two kinds of the Clifford algebras and discuss conditions under which operators of the two Clifford algebras demonstrate the anticommutation relations required for the second quantized fermion fields, this way with the half integer spin, offering in d-dimensional space, d > (3 + 1), the description of charges, as well as the appearance of families of fermions [3], both needed to describe the properties of the observed quarks and leptons and antiquarks and antileptons, explaining the appearance of families. In Sect. 4.3 we comment what we have learned from the second quantized "fermion" fields with integer spin when internal degrees of freedom is described in Grassmann space and compare these recognitions with the recognitions, which the Clifford algebra is offering, discussions on which appears in Part II. We discuss as well a possible action for such an integer spin "fermions" and the corresponding equations of motion, both taken from [3], which are needed that the theory would have any prediction power. The Clifford algebra offers in even d-dimensional spaces, d > (13 + 1) indeed, the description of the internal degrees of freedom for the second quantized fermions with the half integer spins, explaining all the assumptions of the standard model: The appearance of charges of the observed quarks and leptons and their families, as well as the appearance of the dark matter, of the matter/antimatter asymmetry, offering several predictions [1,2,6-12]. 4.2 Second quantized "fermions" in Grassmann space In Grassmann d-dimensional space there are d anticommuting operators 0ai, {0a, 0b}+ = 0, a = (0,1,2,3,5,.., d), and d anticommuting derivatives with respect 4 Understanding the Second Quantization of Fermions... Part I 111 to ®a, alb", {adL' }+ = 0, offering together 2 • 2d operators, the half of which are superposition of products of 0a and another half corresponding superposition all. {ea ,eb}+ = o, {—,—}+ = o, 1 ' + ' l9ea' 9eb + 3 Defining [3] }+ = 5ab , (a,b) = (0,1,2,3,5, • • • , d). 90b (4.1) (0a)t = naa^, 90a it follows 9 ( ^ )t = naa0a, 90a (4.2) The signature nab = diag{1, -1, -1, • • • , -1} is assumed. One can arrange products of ea into 2d irreducible representations with respect to the Lorentz group with the generators [2] S ab 7\ 7\ = i (0a^ - 0^ ), (Sab)t = naanabS 00b 00a ab ab (4.3) Half of the representations have an odd Grassmann character, those which are superposition of odd products of ea and half have an even Grassmann character, those which are superposition of even products of ea. Since Sab do not change the character of operators ("vectors"), that is the odd-ness and evenness of operators, all the members of one irreducible representation have the same Grassmann character. Different representations, either even or odd, are not reachable by Sab. The Hermitian conjugated 2d representations are reachable, due to Eq. (4.2), from the 2d representations of ea's. It is useful to make a choice of the Cartan subalgebra of the commuting operators of the Lorentz algebra. We make the ordinary choice S03, S12, S56, ••• , S d-1 d (4.4) and choose the irreducible representations of the Lorentz group to be the "eigenvectors" of the Cartan subalgebra. naa Sab —= (0a + 0b) = k — (0a + 0b), naa ja , n ab> V2 ik V2 ik Sab -= (1 + ^0a0b) = 0. a/2 k J (4.5) Let us point out that the Grassmann "vectors" have an integer spin. Making a choice of naa = 1, —1, —1,..., —1, the eigenvectors of S03, ^ (9° T 93), have k = ±i, respectively, all the others have k = ±1. 112 N.S. Mankoc Borstnik and H.B.F. Nielsen "Vectors" are normalized, up to a phase, in accordance with Eq. (4.21) of App. 4.4. Lorentz transformations change the Cartan subalgebra, correspondingly also the "eigenvectors" of the Cartan subalgebra change, since the choice of the Cartan subalgebra depends on the Lorentz frame. The Hermitian conjugated representations of (odd and even) products of 0a are obtainable according to Eq. (4.2). 1 naa 1 d naa d ' (0a + n_0b)t = naa (+ _H__) -2( + ik J n V2(30a + i(—k) 30b J, -Ln + k0a0b)t = -L (1 + AJL JL, (46) 4.2.1 Properties of Grassmann "vectors" 2d-1 odd and 2d-1 even Grassmann operators, which are superposition of odd and even products of 0a's are well separated from their 2d-1 odd and 2d-1 even Hermitian conjugated operators, which are superposition of odd and even products of dp's, Eq. (4.6)1. To make discussions concrete let us start with illustrating properties of the representations in Grassmann space in d = (5 + 1 )-dimensional space. Table 4.1 represents two decuplets, which are "egenvectors" of the Cartan subalgbra (S03, S12, S5'6), Eq. (4.4), of the Lorentz algebra Sab. The two decouplets represent two Grassmann odd irreducible representations of SO(5,1 ). One can read on the same table, from the first to the third and from the fourth to the sixth line of both decuplets, two Grassmann even triplet representations of SO (3,1 ), if paying attention on the "eigenvectors" of S03 and S12 alone, while the "eigenvactor" of S56 has, as a "spectator", the "eigenvalue" either +1 (the first triplet in both decouplets) or —1 (the second triplet in both decuplets). Each of the two decuplets contains also one fourplet ((7th, 8th, 9th, 10th) lines in each of the two decuplets (Table II in Ref. [2])). Paying attention on the eigenvectors of S03 alone one recognizes as well even and odd representations of SO(1,1 ): 0°03 (Table II in Ref. [2] includes instead 1 ± 9°e3) and 0° ± 03, respectively. The Hermitian conjugated "vectors" follow by using Eq. (4.6) and is for the first "vector" of Table 4.1 equal to (—)2( ^ )3( afj — i â§6 )( â§7 — i âfr )( â§0 + â§3 )• One correspondingly findsthat when ( ^)3( âfj — i dp)( a§7 — i afj)( al^ + all) applies on (d^ )3(0° — 03)(01 + i02)(05 + i06) the result is identity. Application of (^)3( afj — iâi) (â§7 — ia§2) (â§0 + â§3) on all the rest of "vectors" of the decuplet I as well as on all the "vectors" of the decuplet II gives zero. "Vectors" are orthonormalized with respect to Eq. (4.21). Let us notice that dp on a "state" 1 Relations among operators and their Hermitian conjugated partners in both kinds of the Clifford algebra objects are more complicated than in the Grassmann case. In the Grassmann case Hermitian conjugated operators follow by taking into account Eq. (4.2). In the Clifford case 1 (ya + pk^)1 is proportional to 1 (ya + p-kjYb), while -d(1 + kYaYb ) are self adjoint. This is the case also for representations in the sector of ya's. 4 Understanding the Second Quantization of Fermions... Part I 113 I i decuplet of "eigenvectors" S°3 S12 S56 r (5 + 1 ) r ( 3 +1 ) 1 —_ (e° — e3)(e' + ie2)(e5 + ie6) V 2 i 1 1 1 1 2 —= (e°e3 + ie1e2)(e5 + ie6) V2 ° 1 1 1 3 —_ (e° + e3)(e' — ie2)(e5 + ie6) V 2 —i 1 1 1 4 —_ (e° — e3)(e' — ie2)(e5 — ie6) V 2 i 1 5 —= (e°e3 — ie1e2)(e5 — ie6) V2 ° 1 6 —_ (e° + e3)(e' + ie2)(e5 — ie6) V 2 —i 1 1 7 —_ (e° — e3 )(e'e2 + e5e6) V 2 i ° 1 ° 8 —_ (e° + e3 )(e'e2 — e5e6) V2 —i ° 1 ° 9 —= (e°e3 + ie5e6)(e1 + ie2) V2 ° 1 ° 1 ° 10 —_ (e°e3 — ie5e6)(e' — ie2) V2 ° ° 1 ° II i decuplet of "eigenvectors" S°3 S '2 S56 Y ( 5 +1 ) Y (3 +1 ) 1 —_ (e° + e3)(e' + ie2)(e5 + ie6) V 2 —i 1 1 2 —_ (e°e3 — ie'e2)(e5 + ie6) v2 ° 1 3 —_ (e° — e3)(e' — ie2)(e5 + ie6) V 2 i 1 4 —= (e° + e3)(e1 — ie2)(e5 — ie6) V2 —i 1 5 —_ (e°e3 + ie'e2)(e5 — ie6) v2 ° 1 6 —_ (e° — e3)(e' + ie2)(e5 — ie6) V 2 i 1 1 7 —= (e° + e3 )(e1e2 + e5e6) V2 —i ° ° 8 —_ (e° — e3 )(e'e2 — e5e6) V 2 i ° ° 9 —_ (e°e3 — ie5e6)(e' + ie2) v2 ° 1 ° ° 10 —_ (e°e3 + ie5e6)(e' — ie2) V2 ° ° ° Table 4.1. The two decouplets, the largest odd "eigenvectors" of the Cartan subalgebra, Eq. (4.4), (S03, S12, S56, for SO(5,1)) of the Lorentz algebra in Grassmann (5 + 1)-dimensional space, forming two irreducible representations, are presented. Table is partly taken from Ref. [3]. "Vectors" within each decuplet are reachable from any member by Sab's and are decoupled from another decouplet. The two operators of handedness, F(d-1)+1 for d = (5,4) are invariants of the Lorentz algebra, Eq. (4.23). which is just an identity, | I >, gives zero, I I >= 0, while 0a | I >, or any superposition of products of 0a's applied on | I >, gives the "vector" back. The two by Sab decoupled Grassmann decouplets of Table 4.1 are the largest two irreducible representations of odd products of 0a's. There are 12 additional Grassmann odd "vectors", arranged into irreducible representation, (2 (00 T 03)(1 ± 01020506), 2 (01 ± i02)(1 ± 0O030506),1 (05 ± i06)(1 ± 0O0301 02)). And there are 32 Grassmann "vectors" arranged into irreducible representations, which are superposition of even products of 0a's. 4.2.2 Second quantized "Grassmann fermions" and bosons It is not difficult to see that Grassmann "vectors" of an odd Grassmann character — odd products of superposition of 0a's — anticommute among themselves and so do odd products of superposition of gf^ 's, while equivalent even products commute. Defining the vacuum state in the Grassmann case as 11 > [3] 2, one easily sees that application of products of superposition of 0a's on 11 > gives nonzero 2 We shall see in Part II that the vacuum states are for both kinds of the Clifford algebra objects, ya's and Ya's, the sums of products of projectors. 114 N.S. Mankoc Borstnik and H.B.F. Nielsen contribution, while application of products of superposition of gf^ 's on 11 > gives zero. Application of products of superposition of dda's on the corresponding Her-mitian conjugated partners, which are products of superposition of 0a's, leads to identity for either even or odd Grassmann character3. All these algebras of an odd character, the Grassmann one and the Clifford two, offer the description of the anticommuting second quantized fields, as postulated by Dirac. But the Grassmann "fermions" carry the integer spins, while the observed fermions — quarks and leptons — carry half integer spin. a. Grassmann anticommuting "vectors" with integer spins Let us first study properties of Grassmann odd "vectors". Let us use in d = 2(2n + 1 ), n is a positive integer, for the starting Grassmann odd "vector" — in d = (5 + 1 ) this is the first "vector" on Table 4.1 — the notation 6?11" : = (—)d (e0±e3)(e1 + ie2)(e5 + ie6) ••• (ed-1 + ied), v2 (gen)t = b?1 = (_L)d __i_L)...(—___— ) (4 7) ) b1 (-2) (aed-1 iaed) (aeo ae3). () b?1 is the Hermitian conjugate (6?1t)t. In the case of d = 4n, n is a positive integer, the starting Grassmann odd "vectors" of one Lorentz irreducible representation, and correspondingly the creation operator must be of the kind b?1t : = ( — )2-1 (e0 - e3)(e1 + ie2)(e5 + ie6) • • • (ed-3 + ied-2)ed-1 ed(4.8) 12 All the rest of "vectors" belonging to the same irreducible representation follow by the application of Sab. We denote them by 6?kt and their Hermitian conjugated partners by b?k. Let those "vectors" belonging to different irreducible representations be denoted by 6?kt and their Hermitian conjugated partners by 6?k = (b?kt)t. From Sect. 4.2.1 we derive {b?k,b?1t }+ii> = 5lJ 6kMi>, gek £eii {b?k,bfl }+ii> = o 11 >, {blekt,beit}+ii> = o11 >, b?k | 1 > = 0 | 1 > . (4.9) 3 The Clifford case requires more detailed analyses, as we shall see in Part II: Clifford odd "vectors" of each of the two Clifford algebras anticommute with all the members of the same irreducible representation and so do anticommute among themselves their Hermitian conjugated partners. One must, however, introduce the family quantum numbers in order that anticommutator of a "vector" only with its Hermitian conjugated parter gives a nonzero contribution. 4 Understanding the Second Quantization of Fermions... Part I 115 These anticommutation relations are just the relations among creation and annihilation operators required by Dirac [13] for fermions. Fermion states correspondingly follow by the application off creation operators on the vacuum state | 1 >. I^kb > = bt0kt I 1 > (4.10) But Grassmann "fermions" have an integer spin — this follows from Eq. (4.5), and is demonstrated on Table 4.1 — and not half integer spin as it is the case for the so far observed fermions. b. Grassmann commuting "vectors" with integer spins Grassmann even "vectors" commute, and not anticommute as it is the case for the Grassmann odd "vectors". Let us use in the Grassmann even case, that is in the case of even number of 0a's, and correspondingly of the commuting "vectors", in d = 2(2n + 1) the notation a®1t = () 2-1 (e0 - e3)(e1 + ie2)(e5 + ie6) ••• (ed-3 + ied-2)ed-1ed4.11) Again the rest of "vectors", belonging to the same Lorentz irreducible representation, follow by the application of Sab. The Hermitian conjugated partner of ft®11 is a?1 =(ft®1t)t a?1 =(-L)2__( 9 -i 9 )...(_?___^) (412) (v2) aed aed-1(aed-3 1 aed-2) ( ae0 ae3). ( ) Let us noticed, that the "vector" identity, 1, is not allowed, since the Hermitian conjugated "vector" of the identity is the identity back. Then the last requirement of Eq.(4.9) for the commutation relations in the case of Grassmann even "vectors", instead of the anticommutation relations in the case of Grassmann odd "vectors", presented in Eq. (4.9), could not be fulfilled. If ft®k represents a Grassmann even operator, then one obtains, with the index j denoting different irreducible representations and the index k denoting a particular member of the jth irreducible representations, taking into account Sect. 4.2.1, the relations { aek, afl}_ {a^j V aek aekt 1 > = öl, öki 11 >, 1 > = 0 | 1 >, 1 > = 0 11 >, 1 > = 0 | 1 >, 1 > = l^ka > . (4.13) c. Action for free massless Grassmann "fermions" with integer spin [3] To obtain the equations of motion for at least noninteracting Grassmann massless "fermions" the corresponding Lorentz invariant action for a free massless "fermions" must be proposed. We follow here the suggestion from Ref. [3]. Ag = 7\ 1 ddx dde œ{^(1 - 29°^) ^Pa+ h.c.. (4.14) 116 N.S. Mankoc Borstnik and H.B.F. Nielsen We use the integral over 9a coordinates with the weight function w from Eq. (4.21, 4.22). Requiring the Lorentz invariance we add after * the operator yG (yG = (1 — 29a gd^ )), which takes care of the Lorentz invariance. Namely Sabt (1 — 290^) = (1 — 290^) Sab , 30 9 > _ n ~>a0 9 ï c-1 St (1 — 290 ^ ) = (1 — 290 ^) S-1 , S = e-2 — (L-+S-), (4.15) while 9a, g|p and pa transform as Lorentz vectors. The equations of motion follow from the action, Eq. (4.14), 1 2 yg (9a — ^) Pal*> = 0, (4.16) as well as the Klein-Gordon equation, yG (9a — gfr) pa yg (9b — af^) Pb l* >= 0, leading to {9apa, Pb}+ = papa = 0. (4.17) 09b From the Lagrange density, presented in Eq. (4.14), using Eq. (4.2), and the 3 ^ „r-a _ : (aa d aerJ, Y = i - M: relations Ya = (9a + gf-), Ya = i (9a — air ), yG = —inaaYaYa, it follows, up to the surface term, lg = —yg Ya (Pa*) = —i4f{* yg Ya Pa* — Pa* yg Ya *}. (4.18) One correspondingly finds equations of motion 3£g ^ 3£g ~ —i 0 ~ a ^ j. — Pa dpL*t = 0 = TYG Y Pa *, dL* — Pa ^^^ 0 = 2pa *tYG Ya , (4.19) 3* 3(pa*) 2 The eigenstates of Eq. (4.16, 4.19) for free massless "fermions" are superposition of states * >, describing their internal degrees of freedom, with coefficients depending on momentum pa, a = (0,1,2,3,5,..., d) of the plane wave solution e-ipaxa l*kp > = Z ckspi bfkt 11 > e-ipaxa , (4.20) with s representing different solutions of the equations of motion, and, since they >kpl*k'p ' are orthogonalized, they fulfill the relation < *Î5p|*k/p/ >= / 5ss/ 5pp , where we assumed the discretization of momenta. One of the plane wave massless solutions of these equations, in d = (5 + 1 ), 1 2 for pa = (p0,p1,p2,p3,0,0), the positive energy p0 = |p0|, the spin 1 and the 4 Understanding the Second Quantization of Fermions... Part I 117 charge 2, from the point of view of d — (3 + 1), for example, is 6,t , (p) — 2 2 , 2 P{(72)3 (0c — 03)(01 + i02)— 2(ltp10_L-ipP23|) (^)2(0c03 + i0102) {(+)) e-i(|p0|x0-px), p is the normalization factor. The corresponding state follows by the application of the creation operator i6,1" , (p) on the vacuum state 11 >, , , >— d,1" , (P) 11 >. More solutions can 2 , 2 2 ' 2 2 " 2 be found in [3] and the references therein. 4.3 Conclusions We learn in this paper, in Part I, that products of superposition of 0a's, Eqs. (4.7, 4.5), exist, which together with their Hermitian conjugated partners, Eqs. (4.7, 4.6), fulfill all the requirements for the anticommutation relations for Dirac fermions. No postulation of anticommutation relations is needed. If using products of superposition of 0a's as creation operators to describe the internal degrees of freedom of "Grassmann fermions", these "fermions" carry the integer spin, and in spaces d > 5 the corresponding charges belong to adjoint representations. No families appear in this case, that means that there is no available operators, which would connect different irreducible representations of the Lorentz group (without breaking symmetries). The presented Lorentz invariant action leads to the equations of motion for free massless "Grassmann fermions" [3]. No elementary fermions with these properties have been observed. The interaction of such "Grassmann fermions" [3] with the corresponding gauge fields could tell more about the possibility whether or not these "Grassmann fermions" exist in nature, not yet observed. In Part II two kinds of operators are studied; There are namely two kinds of the Clifford algebra objects, Ya = (0a + ), Ya = i (0a — ), which anti-commute, |ya ,Y a}+ = 0, and correspondingly form two kinds of independent representations. Each of these two kinds of independent representations can be arranged into irreducible representations with respect to the two Lorentz generators — Sab = 4 (YaYb — YbYa), Sab = 4 (YaYb — YbYa). All the Clifford irreducible representations of any of the two kinds of algebras are independent and completely disconnected. The Dirac action in d-dimensional space for free massless fermions — A = J ddx 1 (^Y0 Ya Pa^)+ h.c. (or A = S ddx 2 (^fY0 YaPa^)+ h.c. ) —leads to equations of motions, which have the solutions in both kinds of algebras for either even or odd Clifford character, that is for an even or odd products of the superposition of Ya in one kind and Ya in another kind of the Clifford algebra objects. Although the "vectors" of one irreducible representation of an odd Clifford algebra character, anticommute among themselves and so do their Hermitian conjugated partners in each of the two kinds of the Clifford algebras, the anticommutation relations among creation and annihilation operators in each of the two 118 N.S. Mankoc Borstnik and H.B.F. Nielsen Clifford algebras separately, do not fulfill the requirement, that only the Hermitian conjugated partner of the creation operator gives nonzero contribution. The decision, the postulate, that only one kind of the Clifford algebra objects — let say Ya — is used to describe the internal space of fermions, while the second kind — Ya in this case — which does not contribute to description of the internal space of fermions, determines quantum numbers of the irreducible representations of the Sab, solves both problems: a. Different irreducible representations with respect to Sab carry now different "family" quantum numbers determined by d commuting operators among Sab. b. Creation operators and their Hermitian conjugated partners, which are odd products of superpositions of Ya, fulfill all the requirements which Dirac postulated for fermions. 4.4 APPENDIX: Norms in Grassmann space and Clifford space Let us define the integral over the Grassmann space [2] of two functions of the Grassmann coordinates < B|0 >< C|9 >, < B|0 >=< 0|B >, < b|9 >= Ld=o ba, ...ak 0ai • • • eak, by requiring {d9a,eb}+ = 0, dea = o, deaea = 1, dde e0e1 ••• ed = i, dde = ded...de0, w = nd=0(-^- + ek), 3 90k (4.21) with ec = nac. We shall use the weight function [2] w = + 0k) to define the scalar product in Grassmann space < B|C > < B|C > = ddea w< B|e >< e|cbb,...bkcbl...bk. (4.22) k=0 ' To define norms in Clifford space Eq. (4.21) can be used as well. 4.5 APPENDIX: Handedness in Grassmann and Clifford space The handedness r(d) is one of the invariants of the group SOd, with the infinitesimal generators of the LorentZ group Sab, defined as r(d) = a£a, a2...ad-, ad Sai a2 • Sa3a 4 ••• Sad-,ad , (4.23) with a, which is chosen so that r(d) = ±1. In the Grassmann case Sab is defind in Eq. (4.3), while in the Clifford case Eq. (4.23) simplifies, if we take into account that Sab|a=b = 2YaYb and Sab|a=b = 2Y b, as follows r(d) : = (i)d/2 n (VnaaYa), if d = 2n, a r(d) : = (i)(d_1)/2 ^ (V'naaYa), if d = 2n + 1. (4.24) 4 Understanding the Second Quantization of Fermions... Part I 119 Acknowledgement The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory, the author H.B.N. thanks the Niels Bohr Institute for being allowed to staying as emeritus, both authors thank DMFA and MatjaZ Breskvar of Beyond Semiconductor for donations, in particular for sponsoring the annual workshops entitled "What comes beyond the standard models" at Bled. References 1. N. Mankoc Borstnik, "Spin connection as a superpartner of a vielbein", Phys. Lett. B 292 (1992) 25-29. 2. N. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. of Math. Phys. 34 (1993) 3731-3745. 3. N.S. Mankoc Borstnik and H.B. Nielsen, "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA ZaloZnistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v4]. 4. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep-th/0111257]. 5. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 44 4817 (2003) [arXiv:hep-th/0303224]. 6. N.S. Mankoc Borstnik, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981, arXiv:1607.01618v2]. 7. N.S. Mankoc Borstnik, Phys. Rev. D 91 (2015) 065004 [arXiv:1409.7791]. 8. N.S. Mankoc Borstnik, D. Lukman, "Vector and scalar gauge fields with respect to d = (3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory", Eur. Phys. J. C 77 (2017) 231. 9. N.S. Mankoc Borstnik, [ arXiv:1502.06786v1] [arXiv:1409.4981]. 10. N.S. Mankoc Borstnik N S, J. of Modern Phys. 4 (2013) 823[arXiv:1312.1542]. 11. N.S. Mankoc Borstnik, J.ofMod. Physics 6 (2015) 2244 [arXiv:1409.4981]. 12. N.S. Mankoc Borstnik, H.B.F. Nielsen, Fortschritte der Physik, Progress of Physics (2017) 1700046. 13. P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610. Bled Workshops in Physics Vol. 20, No. 2 ^LV Proceedings to the 22nd Workshop What Comes Beyond ... (p. 120) Bled, Slovenia, July 6-14, 2019 5 Understanding the Second Quantization of Fermions in Clifford and in Grassmann Space — New Way of Second Quantization of Fermions — Part II * N.S. Mankoc Borstnik1 and H.B.F. Nielsen2 1Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen 0, Denmark Abstract. We discuss in Part I and Part II of this paper the possibility to present internal part of degrees of freedom of the second quantized fermions in Grassmann space — in Part I — and in Clifford space — Part II [1-3]. They both offer description for second quantized fermions [3]. It is no need in either of these algebras to postulate the second quantization relations as Dirac [13], since both algebras by themselves offer the appropriate anticommutation relations. But while fermions with the internal degrees of freedom described by the Clifford algebras manifest the half integer spins and charges in the fundamental representations — in agreement with the observed properties of quarks and leptons and antiquarks and antileptons — the "Grassmann fermions" manifest integer spins. In Part II we discuss properties of the two kinds of the Clifford algebra objects — both expressible with the Grassmann coordinates, ya = (6a + ) and ya = i (6ai - ) [2,4,5], {ya ,yb}+ = 0 — and conditions under which the members of the irreducible representation of the Lorentz algebra carry the family quantum numbers. Povzetek. Drugi del tega prispevka obravnava obstoj dveh neodvisnih vektorskih prostorov v Cliffordovi algebri, ki sta skupaj ekvivalentna prostoru, ki ga dolocša Grassmanova algebra. Vsak od vektorskih prostorov v Cliffordovi algebri ponudi kreacijske in anihi-lacijske operatorje, ki določajo na vakuumskem stanju, ki je vsota produktov anihilacijskih operatorjev na kreacijskih operatotorjih, stanja fermionov s spinom j in so resitve Weylove enacbe. Avtorja postavita zahtevo, da samo ena od obeh Cliffordovih algeber doloca vektorski prostor fermionov, druga pa opremi nerazcepne upodobitve Lorentzove grupe v prostoru prve s kantnim sštevilom druzšine. Zahteva zagotovi, da zadostijo kreacijski in anihilacijski operatorji Diracovim postulatom za fermione v drugi kvantizaciji. Keywords:Second quantization of fermion fields in Clifford and in Grassmann space, Spinor representations in Clifford and in Grassmann space, Kaluza-Klein-like theories, Higher dimensional spaces, Beyond the standard model * Talk presented by N.S. Mankoc Borstnik 4 Understanding the Second Quantization of Fermions... Part I 121 5.1 Introduction In Part I of this paper the properties of "Grassmann fermions" of integer spins are presented. Let us repeat: In d-dimensional Grassmann space of anticommuting coordinates 0a's, i = (1,.., d), there are 2d "vectors", which are superposition of products of 0a's. One can arrange them into irreducible representations with respect to the Lorentz group. There are as well derivatives with respect to 0a's, dp's, which again form 2d "vectors", representing Hermitian conjugated partners to the members of the irreducible representations of 0a's, Eq. (6) of Part I. Grassmann coordinates offer correspondingly 2 • 2d vectors. Taking superposition of products of 0a's as creation operators and their Hermitian conjugated partners as annihilation operators, the creation and annihilation operators fulfill, applied on a simple vacuum state 11 >, the anticommutation relations required for the second quantized fermions, if the unity is not included. The "Grassmann fermions" of an odd products of 0a's carry integer spins and the charges in adjoint representations. There are no elementary fermions with integer spin observed so far. In this Part II the properties of the two kinds of the Clifford algebras objects, Ya's and Ya's, both expressible with 0a's and g^'s (ya = (0a + g^), Ya = i (0a — ad") [2,4,5]), are presented and the conditions discussed, which limit the space of Clifford "vectors", so that the Clifford algebra "vectors" of each irreducible representation of the corresponding Lorentz algebra of this limited space are equipped by the family quantum numbers. This limited space of the Clifford algebra "vectors", when used to describe the internal degrees of freedom of (the second quantized) fermions, explain the anticommutation relations postulated by Dirac [13]. These Clifford second quantized fermions enable the descriptions for not only spins and all the charges of the observed quarks and leptons, but also for their families. We present in Sect. 5.2 properties of the Clifford algebra "vectors" in the space of d Ya's and d Ya's and discuss conditions, under which operators of these two kinds of the Clifford algebra objects demonstrate by themselves the anticommutation relations required for the second quantized "fermions", manifesting the half integer spins, offering the explanation for the spin and charges of the observed quarks and leptons and anti-quarks and anti-leptons and also for their families, Refs. [1,2,6-12,3]. In Sect. 5.3 we comment on what we have learned from the second quantized "Grassmann fermions", carrying the integer spins and (from the point of view of d = (3 + 1 )) the charges in the adjoint representations and compare these recognitions with the recognitions, which the Clifford algebra is offering for description of the fermions, appearing on families, with half integer spins and charges in the fundamental representations [1,2,6-11,3]. 122 N.S. Mankoc Borstnik and H.B.F. Nielsen 5.2 Second quantized fermions in Clifford space We learn in Part I that in d-dimensional space of anticommuting Grassmann coordinates (and of their Hermitian conjugated partners — derivatives), Eqs. (2,6) of Part I, there exist two kinds of the Clifford coordinates (operators) — Ya and Ya — which are expressible in terms of 0a and their conjugate momentum p0a = i g|-[2]. Q 7\ 7\ Ya = (0a + »971' Y" =1 (0a - »a1' 1 7\ 0a = _ (Ya - a), = _ (Ya + iYa), (5.1) 2 O0a 2 offering together 2 • 2d operators: 2d of those which are products of Ya and 2d of those which are products of Ya. Taking into account Eqs. (1, 2) of Part I, ({0a, 0b}+ = 0, {g^, g|^}+ = 0, {0a, al^}+ = 6ab, 0at = naaalQ and (g^)t = naa0a), one finds Q ' {Ya,Yb}+ = 2nab = {Ya,Y b}+ , {Ya,Yb}+ = 0, (a,b) = (0,1,2,3,5, ••• , d), (Ya1t = na Ya, (Y a1t = naa Ya, (5.2) with nab = diag{1, -1, -1, • • • , -1}. It follows for the generators of the Lorentz algebra of each of the two kinds of the Clifford algebra operators, Sab and Sab, that: Sab = 4(YaYb - YbYa), Sab = 4(YaYb - YbYa 1, Sab = Sab + Sab , {Sab, :§ab}_ = 0, {Sab,Yc}- = i(nbcYa -nacYb1, {Sab,Yc}- = i(nbcYa -nacYb1, {Sab,Yc}- = 0, {S ab,Yc}- = 0, (5.3) where Sab = i (0a g|^ - 0bg|^), Eq. (3) of Part I. Let us make a choice of the Cartan subalgebra of the commuting operators of the Lorentz algebra for each of the two kinds of the operators of the Clifford algebra, Sab and S ab, S03 s12 s56 Sd-1 d S03,S 12,S56, ••• ,S d-1 d. (5.4) The two kinds of the Lorentz algebras, the one generated by Ya and the other by Ya, are obviously completely independent. We make a choice of the irreducible representations of the two Lorentz groups to be the "eigenvectors" of 5 Understanding the Second Quantization of Fermions... Part II 123 the corresponding Cartan subalgebras of Eq. (5.4), and take into account Eq. (5.2), 1 raa k 1 naa 1(Ya + ¥Yb) = 21(Ya + ¥Yb) ' Sab 1 ° + kYaYb) = 22 ° + 2YaYb) 1 naa k 1 naa Sab 1 (Ya + ^Yb) = 21 (Ya + VYb), Sab2(1 + ¿YaYb] = 22° + kYaYb) • (5.5) The Clifford "vectors" of both kinds are normalized, up to a phase, with respect to Eq. (4.21) of App. 4.4. Both have half integer spin. The "eigenvalues" of the operator S03, for example, for the "vector" 1 (y0 ty3) are equal to ± 2, respectively, for the "vector" 1 (1 ± y0y3 ) are ± 2, respectively, while all the rest "vectors" have "eigenvalues" ± 2. One finds equivalently for the "eigenvectors" of the operator S03: for 1 (y0 T Y3) the "eigenvalues" ± 2, respectively, and for the "eigenvectors" 1 (1 ± Y0Y3) the "eigenvalues" k = ± 2, respectively, while all the rest "vectors" have k = ± 1. To make discussions easier let us introduce the notation for the "eigenvectors" of the two Cartan subalgebras, Eq. (5.4), Ref. [4,2]. ab (k): 1 naa = 1 (y' + Vy') ■ ab t (k) = = naa ( ab -k), ab ((k))2 = 0, ab [k]: = 1 (1 + ¿YaYb), ab ^ [k] = ab [k], ab ([k])2 = ab =fc], ab (k ): 1 naa = 1 (Ya + ¥ Yb), ab t (k) = = naa ( ab -k), ab ((k))2 = 0, ab [k]: = 1 (1 + kYaYb), abf [k] = ab [k], ab ([k])2 = ab z[k], (5.6) with k2 = naanbb- Let us point out that the eigenvectors of the Cartan subalgebras ab a_b ab ab are either the nilpotents — ((k))2 = 0 and ((k))2 = 0 — or projectors — ([k])2 = [k] ab ab and ([k])2 =[k]. Representations of Ya and representations of Ya are completely independent, each with 2d members in 2 • 2 • 2 d representations. 5.2.1 Properties of Clifford vectors 2d-1 odd and 2d-1 even Grassmann operators, which are superposition of odd and even products of 0a's, are well distinguishable from their 2d-1 odd and 2d-1 even Hermitian conjugated operators, which are superposition of odd and even products of 's, Eq. (6) in Part I. In the Clifford case (of either Ya's or Ya's) the "vectors", made of products ab ab ab ab of nilpotents ((k) or (k)) and projectors ([k] or [k]), Eq. (5.6), which each of them 124 N.S. Mankoc Borstnik and H.B.F. Nielsen are "eigenvectors" of one of the member of the Cartan subalgebra of one of the two kinds, Eq. (5.4), the relations among "vectors" and their Hermitian conjugated partners are less transparent (although easy to be evaluated). This can be noticed inEq. (5.6), since ^(ya + ^^V)1 isnaa (Ya + ^Yb),while ^(1 + ±yV) are self adjoint. This is the case also for representations in the sector of Ya's. Let us recognize the properties of the nilpotents and projectors. The relations are taken from Ref. [6]. ab ab (k)(k) = 0, ab ab (k)[k] = 0, ab ab ab ab ab ab (k)(-k)= naa [k], [k][k]=[k] : ab ab ab ab ab ab [k](k)=(k), (k)[-k]=(k) ab ab [k][-k]= 0, ab ab [k](-k)= 0. (5.7) ab ab ab The same relations are valid also if one replaces (k) with (k) and [k] with [k]. We illustrate properties of "vectors" of the Clifford algebra of Ya's on irreducible representations of the Lorentz group SO (5,1) and their subgroups SO (3,1) ab and SO(1,1), presented in Table 5.1, for the case of y~a's all (k)'s have to be replaced ab ab ab by (k)'s and all [k] by [k]'s. odd I i quadruplet a 03 12 56 quadruplet b 03 12 56 quadruplet c 03 12 56 quadruplet d 03 12 56 S03 S'2 S r(5+1 ) r (3 + 1 ) 1 2 3 4 03 12 56 (+i)( + )( + ) [ -i][ -]( + ) [ -i]( + )[-] (+i)[ -][ -] 03 12 56 [+i] [+] (+) (-i)(-)( + ) (-i)[+][-] [ + i](-)[-] 03 12 56 [+i]( + )[+] (-i)[-][+] (-i)( + )(-) [+i][-](-) 03 12 56 (+i)[+] [+] [-i](-)[+] [-i][+](-) (+i)(-)(-) i 2 i — 2 i 2 i 7 1 1 1 1 1 1 1 1 1 1 1 1 — 1 — 1 odd II i quadruplet a 03 12 56 quadruplet b 03 12 56 quadruplet c 03 12 56 quadruplet d 03 12 56 S03 S12 S5 6 r(5+1 ) r (3 + 1 ) 1 2 3 4 ( -i)( + )( + ) [+i][ -]( + ) [+i]( + )[-] ( -i)[ -][ -] [-i][+](+) (+i)(-)( + ) (+i)[+][-] [-i](-)[-] [-i]( + )[+] (+i)[-][+] (+i)( + )(-) [-i][-](-) (-i)[+][+] [+i](-)[+] [+i][+](-) (-i)(-)(-) i — 7 i 2 i 2 i 1 1 1 1 1 1 ? — 1 — 1 — 1 — 1 — 1 — 1 1 1 even I i quadruplet a 03 12 56 quadruplet b 03 12 56 quadruplet c 03 12 56 quadruplet d 03 12 56 S03 S12 S5 6 r(5+1 ) r (3 + 1 ) 1 2 3 4 [-i]( + )( + ) (+i)[-]( + ) (+i)( + )[-] [-i][-][-] (-i)[+]( + ) [+i](-)( + ) [+i][+][-] (-!)(-)[-] [-i][+][+] (+i)(-)[+] (+i)[+](-) [-!](-)(-) (-i)( + )[+] [+i][-][+] [+i]( + )(-) (-i)[-](-) i 2 i 2 i 2 i 1 1 1 1 7 — 1 — 1 — 1 — 1 — 1 — 1 1 1 even II i quadruplet a 03 12 56 quadruplet b 03 12 56 quadruplet c 03 12 56 quadruplet d 03 12 56 S03 S12 S5 6 r(5+1 ) r (3 + 1 ) 1 2 3 4 [+i]( + )( + ) (-i)[-]( + ) (-!)( + )[-] [+i][-][-] (+i)[+]( + ) [-i](-)( + ) [-i][+][-] (+i)(-)[-] [+i][+][+] (-i)(-)[+] (-i)[+](-) [+i](-)(-) (+i)( + )[+] [-i][-][+] [-!]( + )(-) (+i)[-](-) i 2 i — i i 2 i 7 1 "T —2 1 1 1 — 1 1 1 1 1 1 1 — 1 — 1 Table 5.1. 2d = 64 "eigenvectors" of the Cartan subalgebra, Eq. (5.4), of the Clifford Ya algebra in d = (5 + 1) are presented, divided into four groups of four irreducible representations. Two of four groups have an odd number of ya's. "Vectors" in the odd I part have Hermitian conjugated partners among "vectors" of the odd II part, and the opposite. The two groups with the even number of ya's, even I and even II, have their Hermitian conjugated partners within their own group each. Numbers — 03 12 56 — explain the indexes of the corresponding Cartan subalgebra. Equivalent table for Y a's follow ab a_b ab a„b by replacing all (k) by (k) and [k] by [k]. 4 Understanding the Second Quantization of Fermions... Part I 125 There are in the Ya part of the Clifford algebra "vectors" twice 2 2-1 = 4 odd irreducible representations, each representation with 2 6 -1 = 4 members and twice 4 even irreducible representations with 4 members, as presented in Table 5.1. ab The representations for the Ya sector follow from Table 5.1, if one replaces (k) with ab ab ab (k) and [k] with [k]. Hermitian conjugation transforms 2d -1 Clifford odd representations with 2 d -1 members, into 2 d -1 • 2 d -1 Hermitian conjugated partners for each kind of the two kinds of the Clifford algebra operators — Ya and Ya. Hermitian conjugated partners of one Lorentz irreducible representation with 2d-1 members, however, belong to 2d-1 Lorentz irreducible representations: The first column of the four representations in the odd I part has the corresponding Hermitian conjugated partners in the fourth line of the odd II, for example. In Table 5.2 only one quadruplet is presented, the quadruplet a from Table 5.1, together with the corresponding Hermitian conjugated partner. All the "vectors" of the quadruplet are orthogonal among themselves and so are also the "vectors" of the Hermitian conjugated partners. The product of each of the Hermitian 03 12 56 conjugated partner with its "vector" gives [-i][-1]([-1]. For the first "vector" one 03 12 56 03 12 56 03 12 56 finds: (-i)RR • (+i)(+)(+)=[-i][-1][-1]. This follows by taking into account Eq. (5.7). If we denote by 6m+, with f = 1 and m = (1,2,3,4), the first four "vectors" of Table 5.2, and their Hermitian conjugated partners by (6m+) + = 6m, with f = 1 and m = (1,2,3,4), we can write 03 12 56 • 6m+ = 5mm [—i] [—1 ]([—1 ], for f = 1 and all (m, m'). (5.8) One easily checks, taking into account Eq. (5.7), that quadruplets (a,b,c,d) of the irreducible representation odd I fulfill the equivalent relations, only the products of Hermitian conjugated partner m with its "vector" m change: It follows that 03 12 56 03 12 56 03 12 56 03 12 56 6m • 6m+ = smm ([-i] [-1 ][-1 ], [+i][+1 ][-1 ], [+i] [-1 ][+1 ], [-i] [+1 ][+1 ]) for f = (1,2,3,4), respectively. All these "vectors", which are products of 6m • 6m*, are products of selfadjoint projectors only, having an even Clifford character. One can check for d = (5 + 1), using Eq. (5.7), that it follows. 6m • 6?}' = 0, 6m • 6m'* = 0, 6m • 6m'+ = 5mm' |^oc >, for a chosen f, 6m+ i^oc >= i^m >, 6mi^oc >= 0, (5.9) 126 N.S. Mankoc Borstnik and H.B.F. Nielsen for all (f, f') and all (m, m') of Clifford odd Lorentz irreducible representations, 03 12 56 03 12 56 with the normalized vacuum state |^oc >= , 16 ([-i][-1][-1] +([+i][+1][-1] V 2 2 03 12 56 03 12 56 +([+i][-1][+1]+([-i][+1][+1]). The generalization of these recognitions to any even d, if d is either d = 2(2n + 1) or d = 4n, n is a positive integer, is straightforward. We shall do this in Subsect. 5.2.3). i quadruplet a Her. con. quadruplet a 03 12 56 03 12 56 1 (+i)(+)(+) (-i)HH 03 12 56 03 12 56 2 [-i][-](+) [-i][-](-) 03 12 56 03 12 56 3 -](+)[-] [-i](-)H 03 12 56 03 12 56 4 (+i)[-][-] (-i)HH Table 5.2. The quadruplet a of the irreducible representation odd I, from Table 5.1, d = (5 +1), together with the Hermitian conjugated partner is presented. Each member of the quadruplet a is a product of nilpotents and projectors, which are the "eigenvectors" of the Cartan subalgebra, Eq. (5.4), of the Clifford ya algebra. Let us noticed that all the vectors of the first column, odd I, when applied on the selfadjoint "vector" of the quadruplet a of even I, give the vectors of the first column, odd I, back, Eq. (5.7). The vectors of the second column, quadruplet b, odd I, when applied on the selfadjoint "vector" of the quadruplet b, even I, give the vectors of the second column back. This also happens to the third column, quadruplet c, odd I, when applied on the selfadjoint "vector" of the quadruplet c, even I, and to the fourth column, quadruplet d, odd I, when applied on the self adjoint vector of the quadruplet d even I. Similar properties follow when the columns of odd II apply on the corresponding selfadjoint operators of even II. Let us notice also that all the annihilation operators anticommute among themselves, ', }+ = 0, the same is true for creation operators, {6] *, 6]*}+ = 0, while {6], 6]ut}+|f '=f = 5mm'|^oc > is valid only for f' = f and not for the rest members of particular family to which ' belong 1. In any even dimensional space there is in any Clifford even irreducible representation of the corresponding Lorentz algebra of the two kinds of Clifford "vectors" (defined by either ya's or Ya's) one member, which is the product of selfadjoint projectors (1 + £yayb). Correspondingly the whole "vector" is self- 03 12 56 03 12 56 03 12 56 1 Anticommutator {(+i)(+)(+), [+i][+](—)}+ = — (+i)(+)[-], for example, and applied on 03 12 56 the first summand of |^oc > gives this Clifford even creation operator — (+i) (+) [—] back, which can be found in Table 5.1 among even I in the third line of the column quadruplet a, 03 12 56 03 12 56 while [+i] [+] (—) appears in the third line of quadruplet d in odd II and (+i) (+)(+) appears in the first line of quadruplet a in odd I of the same table. 4 Understanding the Second Quantization of Fermions... Part I 127 adjoint. In Table 5.1 there are in even I representations of Clifford even "vectors" four "vectors" (m = (4,3,1,2) of quadruplets (a,b,c,d), respectively), which can be obtained as well from the application of the annihilation operator ' (odd II) on its creation partner (odd I), for each irreducible representation f separately. The selfadjoint even "vectors" appear also in even II sector, belonging as well to different irreducible representations of the Lorentz group (in the quadruplets (a,b,c,d) they carry the family member number m = (4,3,1,2), respectively). All the Clifford even "vectors" of the same irreducible Lorentz representation, applied on their selfadjoint "vector", gives these "vectors" back. All the Clifford even representations follow from the products of the Clifford odd "vectors", Equivalent Clifford even representations as in the space of Ya's appear also in the space of Y a's. 5.2.2 Second quantized "Clifford fermions" We learned in Subsect. 5.2.1 that: a. The two vector spaces, the one spanned by Ya's and the second one spanned by Ya's, are completely independent vector spaces, each with 2d "vectors". The Clifford odd "vectors" (the superposition of products of odd numbers of Ya's or Ya's, respectively) can be arranged for each kind of the Clifford algebras as twice 2d-1 • 2d-1 irreducible representations of the Lorentz group. The Clifford even part (made of superposition of products of even numbers of Ya's and Ya's, respectively) splits again into twice 2d-1 • 2d-1 irreducible representations of the Lorentz group. b. The two groups of the Clifford odd parts (of each of the two kinds) of "vectors", each with 2d -1 irreducible representations of 2d-1 members, are Hermitian conjugated to each other. b.i. The members of one irreducible representation share all the quantum numbers (determined by the members of the Cartan sublagebra (of either Sab or Sab) with the corresponding members of another irreducible representations. The same is true also for their Hermitian conjugated partners. b.ii. The 2d -1 members of each of the 2d -1 irreducible representations are orthogonal and so are orthogonal their corresponding Hermitian conjugated partners. b.iii. Making a choice of "vectors" and denoting them by (where f denotes different irreducible representations and m a member in the representation f), and their Hermitian conjugate partners by = while choosing the vacuum state |^oc > as the sum of all the products of 6 m • for all f = (1,2, • • • ,2d-1 ), we end up with Eq. (5.9), valid for superposition of odd products of either Ya's or Ya's, each in its own "vector space". b.iv. The Clifford odd creation and annihilation operators of any irreducible representation f obey the anticommutation relations, postulated by Dirac for fermions. However (as we learn in Subsect. 5.2.1), there exist among annihilation operators 2 d -1 — 1 members of the same irreducible representation of annihilation operators, to which the particular Hermitian conjugated partner (of a particular creation operator belong (obviously obtainable by the generators of the 128 N.S. Mankoc Borstnik and H.B.F. Nielsen Lorentz transformations, Sab or Sab, respectively), the anticommutators of which with the creation operator gives one of the 2d -1 members (In Table 5.1 one gets quadruplets (a,b,c,d) of even I, if one chooses If from odd I — otherwise one would get one member of even II — which does not belong to self adjoint operators). c. There are the same number of the Clifford even irreducible representations — twice 2d -1, each with 2 t -1 number of members — as in the case of the odd irreducible representations. While in the case of the odd irreducible representations the two groups of 2 d -1 representations, each with 2 d -1 members, are Hermitian conjugated to each other, the Hermitian conjugated partners appear in the even case within each of the two groups separately. c.i. The members of one irreducible representation share all the quantum numbers (determined by the members of the Cartan sublagebra (of either Sab or Sab) with the corresponding members of another irreducible representations. c.ii. Only 2d -1 — 1 members of each of the 2d -1 irreducible representations of each of the two groups are orthogonal to each other, while their application on the member which is the product of the projectors only, gives the same member back. All the members of one irreducible representation are orthogonal to all the members of another representation and to all the members of all the representations of another group. c.iii. All the Clifford even "vectors" can be expressed as the products of the Clifford odd "vectors". The creation and annihilation operators of an odd Clifford algebras of both kinds, of either Ya's or Y a's, would obviously obey the anticommutation relations for the second quantized fermions, postulated by Dirac, provided that each of the irreducible representations would carry a different quantum number. But we know that a particular member m of all the irreducible representations have the same quantum numbers, that is the same "eigenvalues" of the Cartan subalgebra (for the vector space of either ya's or Ya's) Eq. (5.6). The only possibility to "dress" each irreducible representation of one kind of the two independent vector spaces with a new, let us say "family" quantum number, is that we "sacrifice" one of the two vector spaces, let us make a choice of Ya's, and use these operators to define the "family" quantum number for the irreducible representation of the vector space of Ya's, keeping the relations of Eq. (5.2) unchanged: {Ya,Yb}+ = 2nab = {Ya,Yb}+,{Ya,Yb}+ = 0, (Ya)f = naa Ya, (Y a)t = naa Ya, (a,b) = (0,1,2,3,5, ••• ,d). We therefore postulate: Let Ya's operate on Ya's as follows [5,2,10,11,5,3] YaB(Ya) = (—)B iBYa , (5.10) with (—)B = — 1, if B is an odd product of Ya's, otherwise (—)B = 1 [5]. The vector space of Ya's have correspondingly no meaning any longer, it is "frozen out". (No vector space of Ya's can be taken into account any longer). Taking into account Eq. (5.10) we can check that a. Relations of Eq. (5.2) remain unchanged. b. Relations of Eq. (5.6) remain unchanged. 4 Understanding the Second Quantization of Fermions... Part I 129 c. The eigenvalues besides of the operators Sab also of Sab on nilpotents and projectors of Ya's can be calculated, leading to ab k ab ab k ab Sab (k)= k (k), Sab (k)= k (k), ab k ab ab k ab Sab [k] = k [k], Sab [k]= —k [k], (5.11) demonstrating that the eigenvalues of Sab on nilpotents and projectors of Ya's differ from the eigenvalues of Sab, so that Sab can be used to denote irreducible representations of Sab with the "family" quantum number. ab ab ab d. We further recognize that Ya transform (k) into [—k], never to [k], while ab ab ab Ya transform (k) into [k], never to [—k] ab ab ab ab ab ab ab ab Ya (k)= naa [—k], Yb (k)= —ik [—k], Ya [k]=(—k), Yb [k]= —iknaa (—k), ab ab ab ab ab ab ab ab Y~a (k) = —inaa [k], Y~b (k) = —k [k], Y~a [k]= i (k), Y~b [k]= —knaa (k) (5.12) e. One finds, using Eq. (5.10), ab ab ab ab ab ab ab ab ab ab (k) (k) = 0, (—k) (k)= —inaa [k], (k) [k]= i (k), (k) [—k]= 0, ab ab ab ab ab ab ab ab ab ab [k] (k) = (k), [—k] (k)= 0, [k] [k]= 0, [—k] [k]= [k] . (5.13) f. From Eq. (5.12) it follows ab cd i ab cd ab cd i ab cd Sac (k)(k) = — 2naancc [—k][—k], Sac (k)(k)= 2naancc [k][k], ab cd i ab cd ab cd i ab cd Sac [k][k] = 2 (—k)(—k), Sac [k][k]= — 2 (k)(k), ab cd i ab cd ab cd i ab cd Sac (k)[k] = — 2^aa [—k](—k), Sac (k)[k]= — 2^aa [k](k), ab cd i ab cd ab cd i ab cd Sac [k](k) = 2ncc (—k)[—k], Sac [k](k)= 2ncc (k)[k] . (5.14) g. Each irreducible representation of the odd I has now the "family" quantum number, determined by Sab of the Cartan subalgebra of Eq. (5.4). Correspondingly the creation and annihilation operators fulfill the anticommutation relations of Dirac fermions, without postulating them. {6m, 6mm' f}+ i^oc > = 6mm ' Sff' i^oc >, {6m, 6mm'}+ i^oc > = 0 i^oc >, {^m1,6?^' f}+ i^oc >=0 i^oc >, 6mf i^oc > = i f >, 6? i^oc > = 0i ^oc >, (5.15) 130 N.S. Mankoc Borstnik and H.B.F. Nielsen with (m, m') denoting the "family" member and (f, f') denoting "families". h. The vacuum state for the vector space determined by Ya's remains unchanged |^oc >, Eq. (80) of Ref. [3]. 03 12 56 d-1 d 03 12 56 d-1 d 03 12 56 d-1 d |^oc > = [-i][-][-] • • • [-] + [+i][+][-] • • • [-] + [+i][-][+] • • • [-] + • • • |1 > , for d = 2(2n + 1), 03 12 35 d-3 d-2d-1 d 03 12 56 d-3 d-2 d-1 d |^oc > = [-i][-][-] • • • [-] [+] + [+i][+][-] • • • [-] [+] + • • • |1 > , for d = 4n, (5.16) n is a positive integer. i. Taking into account relation among 0a in Eq. (5.1) it follows from Eq. (5.10), since Ya • 1 = iYa 0a = Ya , ^ = 0. (5.17) 30a The Hermitian conjugated part of the space in the Grassmann case "freezed out" together with the "vector" space of Y a's. 5.2.3 Second quantization of "Clifford fermions" with families in any d Let us generalize what we learned in Subsect. 5.2.2 to any dimension d, with the vector space determined by Ya's, while Ya's define the family quantum numbers of each creation operator b^, which is the product of nilpotents and projectors, Eq. (5.6). Let us make a choice of the starting creation operator of an odd Clifford character and their Hermitian conjugated partner in d = 2(2n + 1) as follows 03 12 56 d-3 d-2 d-1 d 61t :=(+i)(+)(+) ••• (+) (-1) , d-1 d d-3 d-2 56 12 01 D1 =(b1t)t =(-) (-) ••• RRM) . (5.18) All the rest "vectors", belonging to the same Lorentz representation, follow by the application of the Lorentz generators Sab's. The representations with different "family" quantum numbers are reachable by Sab, since, according to Eq. (5.14), we recognize that Sac transforms two nilpo- ab cd abcd _ abcd tents (k)(k) into two projectors [k][k], without changing k (Sac transforms [k][k] ab cd ab cd ab cd into (k) (k), as well as [k](k) into (k) [k]). All the "family" members are reachable from one member of a new family also by the application of Sab's from any of the family members of a particular family. In this way, by starting with the creation operator 61t, Eq. (5.18), 2d-1 "families" each with 2d -1 "family" members follow. (In the odd I part of Table 5.1 we correspondingly recognize four representations with the "family" quantum numbers (S 03,s 12,s 56) = [(2,1,1), (-2, - 2,2), (-2,2, - 2), (2, - 2, - 2)], respectively, for d = (5 + 1).) 4 Understanding the Second Quantization of Fermions... Part I 131 The corresponding annihilation operators, that is the Hermitian conjugated partners of 2 d -1 "families", each with 2 d -1 "family" members, following from the starting creation operator 6^, can be obtained besides with the Hermitian conjugation also by the application of YaYa on any member of any "family" of the Clifford odd creation operators. (The application of Y0Y0 on b1' leads to b1), all the rest 2d -1 2 d -1 annihilation operators follow by the application of Sab and Sab on t^1). (Table 5.1 represents in the odd II part the annihilation operators to the creation partners of the odd I part.) The creation and annihilation operators of an odd Clifford character, expressed by nilpotents and projectors ofya's, obey the anticommutation relations of Eq. (5.15), without postulating the second quantized anticommutation relations. The even partners of the Clifford odd creation and annihilation operators follow by either the application of Ya on the creation operators, leading to 2d-1 "families", each with 2d-1 members, or with the application of Ya on the creation operators, leading to another group of the Clifford even operators, again with the 2d-1 "families", each with 2d-1 members. It is not difficult to recognize, that each of the Clifford even "families", obtained by the application of Ya on the creation operators contains one selfadjoint operator, which is the product of projectors only, determining the vacuum state, Eq. (5.16). (Table 5.1 represents in the even I part these four selfadjoint operators, together with the rest of (2-1 — 1 )-2-1 Clifford even operators.) The second Clifford even group of 2d -1 "families" with 2d -1 members, which follows by the application of Ya on the annihilation operators, has again 2d-1 selfadjoint operators, which would determine the vacuum state, if the annihilation and the creation operators would exchange their roles. (Table 5.1 represents in the even II part the second group of even operators, with 26-1 selfadjoint operators, together with the rest of (2d-1 — 1 )-2d-1 Clifford even operators.) 5.2.4 Action for free massless Clifford "fermions" with half integer spin The Lorentz invariant action for a free massless fermion in Clifford space is well known A = ddx 1 (^y0 YaPah.c., (5.19) pa = i gfr, leading to the equations of motion YaPal^> = 0, (5.20) which fulfill also the Klein-Gordon equation YaPaYbPbl^ > = papal^ >= 0, (5.21) for each of the basic states >. Y0 appears in the action to take care of the Lorentz invariance of the action. 132 N.S. Mankoc Borstnik and H.B.F. Nielsen Solutions of Eq. (5.20) are for free massless "fermions" superposition of b^1^, for a chosen "family" f, describing internal degrees of freedom, with coefficients depending on momentum pa, a = (0,1,2,3,5,..., d) of the plane wave solution g-ipaxa If > = L cmsfp f e-ipaxa |^oc >, m bp = L cmsfp bm^ e-ipaxa , (5.22) m s represents different solutions of the equations of motion, and, since they are orthonormalized, they fulfill the relation < ^SpI^s'p/ >= 5ss' 6ff / 5pp , where we assumed the discretization of momenta pa. 5.2.5 Solutions for n free massless Clifford "fermions" with half integer spin with the family quantum number The number of creation operators 6?p in d-dimensional space is 2 d• 2 d(5.23) for a chosen momentum pa, due to the number of families and number of members in each family, respectively. They all anticommute, fulfilling with the annihilation operators Eq. (5.15) ([3] andreferences therein). When we discus more then one "fermion", we must keep in mind that the number of creation operators for a particular momentum is d —1 d —1 22 2 2 2 , (5.24) since each state can be either fulfilled by a fermion or empty. Since the momentum can be any and the solutions of different momentum are, in the discretized case, orthogonal, the number of states is correspondingly infinite. Since the states are for different momentum orthogonal, the creation and annihilation operators fulfill the anticommutation relations of Eq. (5.15) for each momentum pa. {6sp,6s,p'}+ I^oc > = 6ss' Sff, 6pp, I^oc >, {6Sp,f p '}+ I^oc > = 0 |^oc >, {bsp,6?,'p,}+ I^oc > = o |^oc >, bspl^oc > = |f >, 6s fp In Ref. [3], Eqs. (47, 65,87), discuss properties of the n fermion states. 6?p I^oc > = 0 |^oc >. (5.25) 4 Understanding the Second Quantization of Fermions... Part I 133 5.3 Conclusions We learn in Part I of this paper, that odd products of superposition of 0a's, Eqs. (7,6) in Part I, exist, which together with their Hermitian conjugated partners, fulfill all the requirements for the anticommutation relations for the Dirac fermions. There is no need to postulate the anticommutation relations. However, these "fermions" carry the integer spin and the corresponding charges originating in d > 5 belong to adjoint representations. No families appear in this case, that means that there is no available operators, which would connect different irreducible representations of the Lorentz group. In Part II we learn that the Grassmann space offers two kinds of the Clifford operators — Ya's and Ya's. Both kinds of the Clifford objects define two kinds of independent Clifford spaces. "Vectors" of an odd products of Ya's or Ya's, respectively, carry the half integer spins and charges, originating in d > 5, in fundamental representations. Both kinds of odd Clifford "vectors" together offer two times 2d-1 •!d-1 creation operators and two times 2d-1 2d-1 annihilation operators. The Clifford odd creation and annihilation operators of both kinds of the Clifford spaces for each of the corresponding irreducible Lorentz representations separately fulfill the anticommutation relations for the Dirac fermions - without postulating them. To achieve that at least in one of the two groups of the Clifford odd creation and annihilation operators fulfill all the requirements for the Dirac fermions also when different irreducible representations are taken into account, the "family" quantum number must be introduced for any of the irreducible representation. To achieve this we "sacrifice" one of the two kinds of the Clifford vector spaces — the one determined by Ya's — and use the corresponding Sab's to define the "family" quantum number for each irreducible representation of Sab. The creation operators and the annihilation operators — (f f ') determine now family quantum numbers and (m, m') determine family members quantum numbers — fulfill the anticommutation relations of Eq. (5.15). The solutions of equations of motion for free massless fermions, Eq. (5.20), for a particular momentum pa fulfill correspondingly the anticommutation relations of Eq. (5.25). Solutions of equation of motion of different moments pa obviously anticom-mute, due to the fact that the creation and annihillation operators fulfil the anticommutation relations of of Eq. (5.15). There is no need to postulate anticommutation relations as Dirac did for the second quantized fermions. The Clifford algebra by itself, including "families", explains the Dirac assumption for second quantized fermions with the half integer spins and the charges in the fundamental representations, if charges origin in d > 5 . The reduction of the Clifford space, defined with two completely independent operators Ya's and Ya's, into the space spanned by Ya's only has as the conse-quencethat 9a"s become Ya's, while their Hermitian conjugated partners do not exist any longer. While in Grassmann space the Grassmann odd "vectors" fulfill the anticommutation relations for "fermions" with integer spins and charges in the adjoint representations (originating in d > 5), and the Grassmann even "vectors" com- 134 N.S. Mankoc Borstnik and H.B.F. Nielsen mute, with the vacuum state in both cases, which is just the identity, the Clifford even "vectors" are used to determine the (rather complicated) vacuum state. Acknowledgement The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory, the author H.B.N. thanks the Niels Bohr Institute for being allowed to staying as emeritus, both authors thank DMFA and Matjaz Breskvar of Beyond Semiconductor for donations, in particular for sponsoring the annual workshops entitled "What comes beyond the standard models" at Bled. References 1. N. Mankoc Borstnik, "Spin connection as a superpartner of a vielbein", Phys. Lett. B 292 (1992) 25-29. 2. N. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. of Math. Phys. 34 (1993) 3731-3745. 3. N.S. Mankoc Borstnik and H.B. Nielsen, "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v4]. 4. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep-th/0111257]. 5. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 44 4817 (2003) [arXiv:hep-th/0303224]. 6. N.S. Mankocs Borsstnik, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981, arXiv:1607.01618v2]. 7. N.S. Mankoc Borstnik, Phys. Rev. D 91 (2015) 065004 [arXiv:1409.7791]. 8. N.S. Mankoc Borstnik, D. Lukman, "Vector and scalar gauge fields with respect to d = (3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory", Eur. Phys. J. C 77 (2017) 231. 9. N.S. Mankoc Borstnik, [ arXiv:1502.06786v1] [arXiv:1409.4981]. 10. N.S. Mankoc Borstnik N S, J. of Modern Phys. 4 (2013) 823[arXiv:1312.1542]. 11. N.S. Mankoc Borstnik, J.ofMod. Physics 6 (2015) 2244 [arXiv:1409.4981]. 12. N.S. Mankoc Borstnik, H.B.F. Nielsen, Fortschritte der Physik, Progress of Physics (2017) 1700046. 13. P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610. Bled Workshops in Physics Vol. 20, No. 2 JLV Proceedings to the 22nd Workshop What Comes Beyond ... (p. 135) Bled, Slovenia, July 6-14, 2019 6 Deriving Locality From Diffeomorphism Symmetry in a Fiber Bundle Formalism * H.B. Nielsen1 ** and A. Kleppe2 *** 1 The Niels Bohr Institute, Copenhagen, Denmark 2SACT, Oslo, Norway Abstract. We normally assume that a quantum field theory should have an action of the form S = J L^gd4x, and we say that with this form the action is local. In the present work we however do not assume locality, but rather derive it. The point of departure for this derivation of locality, is a diffeomorphism symmetric, very general action S which is Taylor expandable as a functional. We are moreover only interested in long distance physics, compared to the fundamental scale. We already published - in a somewhat hidden way -such an argument in reference [2], but here we extract this derivation as the main point, and further formulate it in fiber bundle notation. Povzetek. Običajno privzamejo, da ima akcija v kvantni teoriji polja obliko S = J L^gd4x in recejo, da je akcija v tej obliki lokalna. Avtorja v tem prispevku ne predpostavita lokalnosti, ampak jo izpeljeta. Izhodisce je zelo splosna akcija S z difeomorfno simetrijo. Akcija se da razviti v Taylorjevo vrsto kot funkcional. Zanimajo ju lastnosti te akcije pri velikih razdaljah. To izpelijavo sta na kratko ze objavila v referenci [2], tukaj pa je osrednja tocka prispevka. Keywords: Deriving locality, fiber bundles 6.1 Introduction In a generic physical model, the property of locality is usually taken for granted. Its actual meaning is seldom discussed at great length, and this is even more true for nonlocality. The unreflected assumption that locality is fundamental, is reflected in the locality of the laws of nature, as well as in the continuity equation which tells that there are no jumps! We usually think of locality in terms of information being localized, propagating from one spacetime point to another by at most the speed of light. Another way of expressing it, is that all cause-and-effect relations are limited by the speed of light. Thus, an experiment in one place is not supposed to have an immediate influence on an experiment in an other place, this is also true for effects like the butterfly effect, because they take time. * Talk presented by A. Kleppe ** E-mail: hbech@nbi.ku.dk *** E-mail: astri.kleppe@gmail.com 136 H.B. Nielsen and A. Kleppe A theory is local when every degree of freedom is assigned a spatio-temporal site x^. That means that all interactions take place in one spacetime point, implying that there is a system for assigning one site x^ to each degree of freedom. In a local theory the action S can be factorized, S = S1 + S2 +..., such that each contribution only depends on the fields in limited regions of x^-spacetime. This locality concept is moreover coordinate choice dependent, S = L(x)d4x (6.1) In the present work we do not assume that locality is fundamental, on the contrary, our goal is to derive locality. Our point of departure is an analytic and diffeomorphic symmetric action, using fiber bundle formalism. The philosophical framework of this approach is Random Dynamics [1], which postulates that at a fundamental level, there is a "world machinery", meaning a very general, random mathematical structure, which merely contains non-identical elements and some set-theoretical notions. From this "world machinery", differentiability, space and time [2], diffeomorphism symmetry [3], locality, and eventually all other physical concepts, are to be derived. But even after locality has been derived, some smeared out left-over nonlocal effects remain, showing up in coupling constants (which feel an average over spacetime, and also depend on such averages). This remaining (mild) nonlocality is moreover supported by the Multiple Point Principle (MPP) [4]. The "locality" that we want to derive comes about by formulating the action as an integral over spacetime of a Lagrangian density £(*(x),3*(x)/3x) which only depends on the fields - such as * - and their (up to finite order) derivatives taken with values of the spacetime point x. Our starting point is a diffeomorphic symmetric action S[*], a fiber bundle of dimension > 4, and the idea that we can get genuine locality (not super locality) along p = 4-dimensional p-surfaces. When you have a field configuration on your fiber bundle (a compact fiber bundle that you can integrate over), it is namely in the spirit of fields that you can only make various local functions of them. 6.2 Diffeomorphisms A diffeomorphism is an isomorphism f on a smooth manifold M (thus preserving the structure of M); and the group of diffeomorphisms on M is the set of such mappings, D(M) = {f : M -^M} Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. To be diffeomorphic is a much stronger condition, since for a mapping f to be diffeomorphic, both f and its inverse need to be differentiable; while to be homeomorphic it suffices that f and its inverse are both continuous. 6 Deriving Locality From Diffeomorphism Symmetry 137 We can always define a coordinate system on the manifold, whereby we identify the points on the manifold in terms of the coordinates. Diffeomorphisms can be perceived as synonymous with reparametrizations, which in local coordinates x^ are analytic or at least smooth maps, ^ : x^ + — x(6.2) and in changing coordinates in M we establish the transformation rules between two different coordinate systems. We can introduce a new coordinate system and transform to it, or we can just keep our first coordinate system, and then intoduce a diffeomorphism ^ : M^M which is the same as "moving" the points on the manifold, after which we have to evaluate the coordinates of the new points. We then think of diffeomorphism invariance as reparametrization invariance, but the diffeomorphic symmetry exists whether we use coordinates or not. Our first challenge is to formulate what diffeomorphism symmetry means in the case of a functional over the fields on M. 6.3 Analyticity We do not assume that our initial action is local, so it may in principle comprehend nonlocal terms of the form f(x, y), which depend on more than one spacetime point. An analytic function is locally given by a convergent power series. There is however no demand that a power series must depend on a single variable, it can just as well be an infinite series of the form CO f(x,y, ••• ,z) — X (x — Cx)jx (y — Cy)jy ••• (y — Cz)jz (6.3) • ,jz=0 where jx, jy, • • • are natural numbers, ajx,jy,... and x,y, ..,z are variables, and Cj constitute the 'center'. The theory of such series is trickier than for single-variable series, with more complicated regions of convergence, but for example the power series Y.x^xj is absolutely convergent in the set {(x1, x2) : x1 x2 < 1} between two hyperbolas. This means that a nonlocal function can be analytic, implying that analyticity does not in itself guarantee locality. Our action, however, is not only analytic but also diffeomorphic symmetric, and we want to prove that this is enough to make it local. It remains to formulate analyticity in the case of a functional over the fields on M. 6.4 The action In our action S[^], the function ^ is defined over the manifold M. For a field on a manifold, a value in one single point has no signification (because it is of Lebesgue 138 H.B. Nielsen and A. Kleppe measure = 0), in the sense that in a continuous field a given point can always be replaced by some small integral piece, so these fields must be integrated over. The integration is taken over such small integral pieces, and the final, generic action is some complicated combination of these integrals. Moreover, if you have diffeomorphism symmetry, you cannot have boundaries on your integrals, so every integral must be taken over the whole space (which is assumed to be a connected manifold). Our action must thus be a function of integrals over the entire space. By loosely identifying a point on M with its coordinates xj, the summing over the various j=1,2,...,N gets replaced by integrals over the coordinate variable sets on the manifold, and the Taylor expansion of the functional for a single function ^ (x) is then defined as Sfo]= Sfo = 0] + 5S ,-d 1 6*(x) *Wddx + 2! 52S 5nS 5^(X1) • • • 6^(xn) wxwy) ^(y)ddxddy + •••- (6.4) - 0)^(X1) ••• ^(Xn)ddX1 ••• ddXn (6.5) Initially we only consider a subset of diffeomorphism transformations that leave the local d-volume invariant, i.e. a subset of diffeomorphism transformations x —» x '(x) such that de, (^) = 1, (6.6) which means that the d-volume of spacetime e^v . Tdx^dxv • • • dxT (6.7) remains invariant under this subset of diffeomorphism transformations. In the second step, we include "pseudoscalar" fields P(x) that have other transformation properties than mere scalars under diffeomorphism transformations, transforming as ( ) P(x) -> P(x') (^x^) (6.8) Imposing the diffeomorphism symmetry coordinate shift x x', preserving the integral ddx = ddx' on the Taylor expansion Sfo] ~ ^ ••• ^(xn)ddx1 ••• ddxn (6.9) n=0,1,.. leads to the requirement that the coefficients, i.e. the derivatives 5S 5^(X1) • • • 6^(xn) (6.10) must be invariant in a similar way under diffeomorphism transformations, except if some of the xj's are infinitely close or coinciding, which implies that we get terms like (6.9) multiplied with 5 functions 5(xj — xk). When including the terms 6 Deriving Locality From Diffeomorphism Symmetry 139 coming from 5 function forms at the coincidences, we however also get integrals over various products of the fields and their derivatives, though the integrals allowed by the diffeomorphism symmetry turn out to be such that the quantities getting integrated in the appearing integrals, are all time "pseudoscalars", like the P(x)'s themselves. This implies that the total Taylor expansion runs out to be a function of a lot of integrals over various pseudoscalars, which can be formed in terms of the various variables/fields in the theory. With the condition of diffeomorhism invariance, these integrals must thus be pseudoscalar (i.e. transform as ^g), resulting in an action which by definition is pseudoscalar, that is, a function (not a functional) of all the various pseudoscalar terms that we can construct, and we can make differentiation through all these pseudoscalar contributions. We can write the functional derivative of S as a partial derivative of S with respect to all the different integrals over the whole space summed over, multiplied by the functional derivatives of the latter, _5!SL = y- iliVl = (611) 5£,(x) 4" 9yk 5£(x) 5£,(x) ( . ) k where Seff — 7\ Ç MLdVk Pk(x)d4x, (6.12) and (m) « £k(x)y/g(x) = £k(m)y/g(m) are pseudoscalars, Lk(m) are La-grangian densities, and Vk are the integrals Vk = Pk(m)dm, (6.13) where m e M are spacetime points/events. In local coordinates x? on M, x?(m) are the coordinates of the event m, and d4x is a measure in the coordinates x?, such that d4x = dx1 dx2 • • • dxd = dm. In general relativity the transformation property of the metric tensor field g?v is g ^v(x)= dxv gnp(^(x)), (6.14) we infer that by having terms like a second order tensor field g=9"v »i? ^, (615) we can formulate a theory with effective locality (and no super-locality), there will however still be a certain nonlocality, because with the assumption of local diffeomorphism symmetry, the effective action comes out as Seff — L(pseudoscalar) dx1 A dx2 A • • • dxd (6.16) although with a very important detail: The coupling constants or coefficients become complicated integrals over the whole manifold/base space M, i.e. over all 140 H.B. Nielsen and A. Kleppe spacetime including both future and past. So in this restricted sense the resulting theory is still non-local, although the non-locality only comes in via the coupling constants. In our Taylor expansion philosophy, there is an interesting point: If we just take the usual inverse metric tensor field g as the field provididng the indices to be contracted with the ones from the derivative, on say the scalar fields $(x), we cannot obtain in a Taylor expansion that only provides polynomials, the needed pseudoscalar factor y g(x), where g(x) = det(g^v). The reason is that the square root is singular - at zero field - thus this conventional model of general relativity does not work for our philosophy, even in the case of a purely bosonic theory. So even for the purely bosonic theory - wherein one a priori expects that the gravity based on just the metric tensor would be o.k. - we cannot obtain in our Taylor expansion in the usual action form, because of the square root singularity is needed. In fact one can easily see that all polynomially constructed pseudoscalar-like field combinations based on the metric tensor alone, obtain transformation rules of the form of being multiplied with an even power of the det dXV factor. But for the construction of a diffeomorphism invariant integral we need an odd power, namely the "pseudoscalar" replacing \Jg (x) which transforms with this determinant of the partial derivative matrix to first power only. One could thus claim that we have a (kind of) prediction that there should be vier beins (or some replacement for vierbeins) rather than the simple metric tensor in the theory where locality is obtained in the spirit of our derivation. And this claim is of course only of much interest in the case where we ignore the fermions, because with fermions we need the vierbein anyway. As a consequence of this consideration we should say that the typical example of fields to be used for illustrating our model for deriving locality, should have a vierbein among the fundamental fields. 6.5 Fiber bundles A fiber bundle is often simply written as (E, B, F, n), where E, B and F are topological spaces, and n is a continuous map. E is known as the total space of the fiber bundle, B as the base space, F is the fiber, and in small regions of E, the fiber bundle n : E —» B behaves like a map from B x F to B. This is a local relation that is not necessarily globally valid. A simple example of a fiber bundle is the S1 x R surface of an infinitely long cylinder, which by definition is a differentiable manifold. Here the total space (E) is the entire cylinder, the base space (B) is the circle S1 running around the cylinder, and the fiber (F) is R. The fiber that runs through b G B is called a fiber over b, and is formally defined as the pre-image n-1 (b) G E, which is diffeomorphic to F. The fiber over c G B is a different fiber, but the fibers are all diffeomorphic to each other. It is this set that constitutes a fiber "bundle", in the sense that while there is only one total space E and one base space B, there is a whole set of fibers; we say that E is a fiber bundle of F over B. 6 Deriving Locality From Diffeomorphism Symmetry 141 In the case of the cylinder, the pre-image n-1(B) of B is trivial, i.e. it is the entire total space, n-1 (B) = B x F = E, so the topology of the total space E is the usual topology on a 2-dimensional manifold. In the general case, however, it is only the pre-image of an open set O in the base space that is (locally) trivial, i.e. n-1 (O) - B x F. Now, since E is locally diffeomorphic to a product space, a point p in E can be written as (b, f) where b G B and f G F, and n: (b, f) —» b. As we take b back to E under n-1, the pre-image n-1 (b) of b is not a point, but a subset of E that is diffeomorphic to F under n. Around any point in B, we can moreover find at least one neighbourhood Ot c B such that its pre-image n-1 (Ot) is trivial, i.e. diffeomorphic to Ot x F. There is thus a mapping n) : n-1 (Ot) —» Ot x F, where ^ : n-1 (Ot) —} F is a homeomorphism. A different open set Ok c B will have different pre-image in E, and projected on F by a different homeomorphism ^ : n-1 (Ok) —» F. Since n-1 (Ot) and n-1 (Ok) are connected to F by ^ and respectively, the intersection n-1 (Ot) n n-1 (Ok) is also connected to F by these two homeomorphisms, from which we can construct the diffeomorphisms ^ o and ^ o : F —> F. Such diffeomorphisms define the structure group G(F), which is a subset of the group of diffeomorphisms D(F) on F. Every transition function of the fiber bundle must belong to the structure group. 142 H.B. Nielsen and A. Kleppe E In the case of the cylinder, G(F) is the identity element, G(F) = I. On the Mobius band, most of the transition functions can be identified with the identity, but at least one of them must be negative, i.e. G(Mobius) - {I, -I}. 6.6 Analyticity in fiber bundle notation As a visualisation of the relation between fiber bundles and tangent spaces, consider all of the unit tangent vectors on the sphere. Over every point in S2, there is a circle of unit tangent vectors all of which constitute a principal bundle E on the sphere with the circle S1 as fiber, and every tangent vector projects to its base point in S2, giving the map n : E S2. For our purpose, we identify the 4-dimensional spacetime manifold as the basis space, i.e. B = M (spacetime), and the total space is then identified as the p-dimensional space E = B x F. We say that the functional S[^j, ^ : B E, is "analytic" provided we have a convergent Taylor expansion (6.17) 6 Deriving Locality From Diffeomorphism Symmetry 143 where m G B, and 5S 5S ———- ~ c , . .—-dA1 = dS G [cotangent space for F] (6.18) 5A(m) 5A1(m) Locally in B-space you have coordinates, so we define A^1(m) as AA1(m')l(at m) = (A1(m') - A0(m(6.19) which is a tangent vector, and 5S/5A1(m) is the basis of the tangent space. In terms of the coordinates x^ = (x1, x2, • • •), the basis in the tangent space is — = x^, (6.20) and the tangent vectors t^x^ G T, where T is the tangent vector space. The p = 4 surfaces should run through the d-dimensional E-space so as to have their 4-dimensional tangent vector space embedded (naturally) in the tangent space of B at the point m, in such a way that it is just the one that is spanned by the four tangent vectors Va = Va5/5x^. Now £S £S —AA1(m)|(at m) = K . W .d^1A^(m ')|(at m') = dS (m) A* (m) | (at m) 5 A1(m) 5 A1(m) (6.21) where 5 5 AMm) dA^TTTT::^ = 0 k (6.22) In order to express this in the language of functionals, we expand S around Ao S[A] = ¿. t!t5 A11 (mi)5A^O).^ Ain(mn) (miJ - ^ (miJ) x (6.23) n=0 x (a12 (mi) - AO2 (m2)) • • • (A1n (mn) - A0n (™n)) dmi • • • dmn (6.24) and then define a dual function to the tangent space vector, i.e. a covector, DA1(m), by using the tangent space basis vectors 5 (6.25) 5 Aj(m') to define the number 5Aj(m') One tests 5(m — m') by a test function K(m), 5 ^DA1(m)|-j—> = 515(m - m') (6.26) K(m)DA1(mL. J = 5 1K(m') (6.27) 5 A1(m') ' 144 H.B. Nielsen and A. Kleppe Is this the right definition of D^(m)? D^(m) should be in the dual space of the functional tangent space in which the basis vectors are 5/5^j (m'). So D^1 (m) is defined by defining its action W(m) 5 -> = 5i5(m - m') (m')' Inserting a product of n of these delta-functions in the action, we obtain 1 5nS[^ol (6.28) Sfo] = Y_ n=0 ' n! (mi )5^i2 (m2)..5^in (mn) W1 (mi)| 5 5^11 (mj ) > • • • (D^n(mn 5 5j (mn ) •(^ (mj (mj ))(^2 (m2 K )) •• • « « ))dmj Now define we get CO Sfo] = X n=0 ' A^ = (^j(m')- ^0(mj)Lj .-, dm' 5^1 (m ') dmn (6.29) (6.30) J__5nS[^o]_ n! (mi )5^i2 (m2)..5^in (mn) (mi ) ® • • • (mn) A^ <&•••<& A^ = = X 5®S(A^) (6.31) Actually the simple requirement that (6.29) should be constant when the arguments mi are all different, is only true if we restrict the diffeomorphisms by which we shuffle them around, to those transformations that keep the det( l^-) equal to unity. For more general diffeomorphisms we have to modify the functional derivatives for volume-non-preserving diffeomorphisms, by inserting density factors that are constant rather than simply derivatives. This is achieved by multiplying the functional partial derivatives by "pseudoscalar" correction factors, like (6.13). 6.7 Diffeomorphisms of the action Let us consider a symmetry under a group of bundle maps f: f : E —} E bijective; and f o n = n o f is a requirement for bundle map (6.32) This induces a transformation on M, f(M) —» M so that if n o f (e) = f(e) e G F(m) (6.33) f (m) = m', then if n o f (e) = m' ^ n(e) = m, (6.34) the symmetry transforms ^i —» where 6 Deriving Locality From Diffeomorphism Symmetry 145 • f : E -> E • f: M —» M (defined from f when bundle map). • n o f = f o n, i.e. f o n = n o f => the fiber F over say n-1 (m) is mapped onto/into itself, where F is a function F(V1, ....Vk, ^ Iscaiar,...). For e G n-1 (m),n o f(e) = f o n(e) is independent of e, except through m. So for each m there is a map fm(e) inside the fiber on m, te(e) = fn(e](^1(f-1(n(e)))) (6.35) A true diffeomorphism is defined by choosing an f rather than f, and then deduce a f according to semi-local rules like »" ^ ^ dX? <6-36> This is semi-local: it only depends on derivatives and values near or at x, and then going to x. Then we can almost choose f freely and still get 3xv/3xp, etc. Assuming: • that we have so much symmetry that all diffeomorphic maps f : M —» M are achievable. • that the full transformation f : E —» E as far as the moving around on the fiber is concerned, i.e. fm for all m G M, is determined by derivations of f in the neighbourhood of m, then we can prove that we can choose some subset of f's in the supposed symmetry group so that it follows that 5 S (6.37) S^1 (m1)5^i2 (m2) • • • (mn) must be the same even if one moves any of the mi's, except if this mi coincides with (up to infinitesimals) another m, say mj. This implies first that if we ignore any grouping of the mi, i.e. if they are all different, then (6.37) is independent of the mi's. We should and could (if we think of true diffeomorphisms with usual tensors) also assume that • we can arrange f in such a way that the subsequent f can locally "rotate" or "transform" indices (on the ^'s) so that the pseudoscalars are not transformed, so there is only a dependence on ^i(mi). With the assumption that we have a diffeomorphism invariant "expansion start field ^0" which is constant for scalars, and otherwise zero, the form F(^, • • • ) becomes constant over the entire base space product, and so the derivative 5 S (6.38) S^1 (m1 )5^i2 (m2) • • • S^n (mn) is only allowed in the form F(^ (m1 )|(scaiar), • • • (mn)I(scaiar)), where the scalars are the ^ -values corresponding to scalar components of the (m). Then 146 H.B. Nielsen and A. Kleppe only D^1 (m) with a scalar component will be relevant in 5nS 50nS = ö^11 (mi) • • • ö^n (mn) D^11 (mi) • • • (mn) dmi • • • dmn = = 60nS| (projected onto D^calar) (mi ) • • • D^(Scalar) (™n) dmi • • • dm^ = the "scalar" component) J = VnF(^101|sc ,^1o2lsc, ••• ^0nlsc) (6.39) The quantity (6.38) depends on a background field which a priori is a combination of all the fields in the theory, but for any fixed value of the fields it depends on n event-points m1 , m2,... mn. Now in (6.39) this dependence on the set of the n m-values (m1,..., mn) gets intergrated over the m's with a weighting by the duals of the partial derivatives called D(m1), i.e. by a product of n such D(m1 )'s. This contraction with such D's leads to (6.39). It is with the simplification to the case of a constant start field that we get that (6.38) can only be constant - as long as the arguments (m1, m2,..., mn) do not coincide. The background is that while we in the general case Taylor expand around start functions that are not necessarily diffeomorphism invariant, for the simplicity of the argument, we restrict ourselves to scalar or pseudoscalar fields Then by transformation with det{ dXV-} = 1, we can argue that merely with invariance under the restricted diffeomorphisms, in order to keep the total action S invariant it must not change as the m1's move around on the base manifold B, unless some of the m1 's are moved to a place with a different (m1) value. But inside a range with given value of the (^0(m1),..., ^0(mn)), the diffeo-morphism invariance implies that the expression (6.38) should be unchanged by such moving around, which in its turn implies that the expression (6.38) can only be of the form F(^0(m1),..., ^0(mn)). If we look for components with more complicated types of transformation, like vector components or tensor components, we could consider diffeomorphism transformations that are restricted in a different way, so that the components considered remain unchanged. The question is if the separate n m1's can still be moved around for such restricted diffeomorphisms, of course with the exception of coinciding points, since we naturally cannot move such points to different places with a continuous diffemorphism. Such an appropriate "moving around diffeomorphism" is in general easy to construct, because we do not require all vector components in a given direction to be unchanged under the diffeomorphism, but only certain components infinitesimally near the points m1 , m2 ,..., mn that we want to move. With the freedom to move as we please sufficiently far away from the n special points, it is easy to make the desired transformations, bringing the n m1's wherever we like, modulo coincidence. It means that for all these components of the fields, we can deduce the form F(^ (m1),..., (mn)) for the expression (6.38). Here the function F could of course be a complicated function of its n arguments, and its form depends on the original "fundamental" action functional S. We just derived the existence of such an F. To sum up, we argue that as a consequence of diffeomorphism invariance, the functional derivatives of the action S w.r.t. pseudoscalars, are constant over 6 Deriving Locality From Diffeomorphism Symmetry 147 the basis space. To come through this argument, we simplify by only considering the case where the function from which we expand ", is diffeomorphism invariant. That implies that all components which are not genuine scalars will be zero, because the field would otherwise tranform under the diffeomorphisms. But even the scalar fields have to be constant over the base space ( ~ spacetime), and the pseudoscalars must be zero in order to keep the reparametrization invariance of the "expansion start function ^0". This assumption is all right if we have assumed that S is "analytic" over the entire space of sections, so that we have Taylor expandability for all start functions, also for diffeomorphism invariant ones. Let us note that the A^ which of course are present in the Taylor expansion, have been left out in (6.39). Thus in (6.39) there is no dependence on the fields (=cross sections), apart for the dependence on the start field; but those are essentially taken to be zero. At least for a start field that is zero all over, (6.39) only depends on the action S, but not on any field configuration. The quantity 50nS in (6.39) thus is a tensor in function space in the sense that it is expanded on the D(m)'s, it is in fact a product of n such D(m), which in the last step in equation (6.39) is embedded in the definition of the V's. So for the separate points at which we differentiated, we only get non-zero functional differential quotients (6.38) for differentiation w.r.t. pseudoscalars, and then the differential quotients must be constant in the base space. We only took the true scalars in the start-function to be non-zero, and they are also constant over the base space. Now we get the integrals ^0(m))dm = V1 (6.40) M where i is "pseudoscalar". Our Taylor expansion then takes the form 1 n / \ ^ /n' CO 1 n , N = Z ^ Z p Wv 1)n-p(V2)p = F(V 1,V n=0 ' p=0 ^ "' (6.41) where Vk = jM Pkdm only depend on "pseudoscalar" components of ^T(m), and we shall think of = 0. There are also the cases where two or more of the mi's are infinitely close/coinciding. In such cases we however only get non-negligible contributions to S if we let the derivative ' S (6.42) 5^1 (m^S^12 (m2) • • • S^n (mn) have factors 5 (m1 — mj). Derivatives of 5-functions may also contribute, then the derivative (6.42) will have a series of terms classified by clusterings of the m1's. The number of ways of creating clusters corresponding to the partition n = Pi + P2 +-----F pi is " "' (6.43) Pi , P2, ••• ,PiJ Pi !P2! ••• ,Pi! For each cluster with a number of say p m1-values, we need a 5-function with p — 1 delta functions 5 (m1 — mj) to compensate for p — 1 of the dm1 integrations, so that 148 H.B. Nielsen and A. Kleppe only one integration remains and gives us an all-over the spacetime manifold B integral 5nS -dmt 5$^ (mi )5$i2 (m2)...6^in (mn) « n(p-1 of the i-values)5(mi — mj ) • • • n(p of the i-values) dmi • • • (6.44) When the $ik's are pseudoscalar, the integral J"A ddx integrated over a region A, will be the same as the integral over the image f(A) of this A, by a diffeo-morphism f of the base space M = B. This corresponds to the diffeomorphically transformed quantity i.e. 5S A ^ ddx = f(A) (with ddx transformed by the f). (6.45) Now, the integralA ddx actually gives the formal integral over the coordinates xl for the region A. By using an active diffeomorphism to push it around to f (A), and also correspondingly transforming the coordinates, we get the same number of coordinate in all points that are related by the diffeomorphism - i.e. we get the same integral, ddx = A (ddx transformed under f). (6.46) f(A) We indeed see that in order to have diffeomorphism invariance, small regions (with 5S/5$(i-k) removed) must transform in such a way that they are the same all over. Generalized this means that by taking the $jk as pseudosacalars we indeed get the constancy. 6.8 Locality The invariance under transformations that only transform f in the neighbourhood of one of the clusters, will only allow a non-zero contribution when the 5-functions of the cluster with the associated derivatives in the 5-functions eventually run out to extract a "pseudoscalar" component (of order p) from the product ($ik (mk) - $0" (mk))(^il (mk) - $0l (mk)) • • • that it is going to multiply. So apart from the S[$0]-term (though it best to just assume S[$0]=S[0]), the only non-zero cluster-contributions are total spacetime integrals over "pseudoscalar" combinations of the fields, such as yg(m)g^(m)^$(m)3v$(m)dm (6.47) Here we could think of ^g as just a (fundamental) pseudoscalar field transforming under diffeomorphism symmetry with a determinant of the transformation partial derivatives, Vg(x) -> det( — )Vg(x') (6.48) 6 Deriving Locality From Diffeomorphism Symmetry 149 arranged in such a way that J* ^gdm is diffeomorphism invariant. Everything in the Taylor expansion after choosing *0 = 0 (by field redefinition) becomes expressed by means of all integrals over M of the type Vk = jM Pkdm. For sufficiently high n, we can expect to get the same Pk out of several of the clusters into which we partition such "big enough" n. In that case we might evaluate the n ) (6.49) n! \pi•••pij and count the possibilities, but it is not really needed because the weight coefficients for the term combination can only be obtained if we somehow know the fundamental action functional S. We have already seen that we apriori shall get a series of terms in which all powers and all products of such powers of the integrals Vk = Jm Pkdm occur. That is to say, we get an expression of the form CO S[*]= ^ Cklk2...kqVklVk2 ••• Vkq (6.50) kl ,k2,---kq which in fact is the Taylor expansion for any function in the variables (V1, V2,..), provided one chooses the Cklk2...kq appropriately. So all we have derived is that S[*] is a function of these variables (V1, V2,..), but we do not know which function. The variables on which are all M-integrals of the "pseudoscalar" field combinations Vk. Now we shall however follow our earlier work where we derived an effective locality. The main use of the action is via the Euler-Lagrange equations. Suppose we have a field £ which can even be a component of some tensor field, or whatever; then the Euler-Lagrange equation for £ is SS[ Mpc), but encounters several problems at smaller (galactic) scales, like the core-cusp problem, the diversity problem, the missing satellites problem and the too-big-to-fail problem [9]. These problems may be tackled in the context of self-interacting dark matter [10,11], as any cuspy feature will be smoothed out by the dark matter collisions. In addition, if dark matter consists of ultralight scalar particles with a mass m < eV, and with a small repulsive quartic self-interaction a Bose-Einstein condensate (BEC) may be formed with a long range correlation. This scenario has been proposed as a possible solution to the aforementioned problems at galactic scales [12-14]. Boson stars are star-like, self-gravitating bosonic configurations, where bosons are exclusively trapped in their own gravitational potentials. Boson stars have been studied in [15-23], see also [24-27] for Newtonian self-gravitating Bose-Einstein condensates. The maximum mass of bosons stars in non-interacting systems was found in [15,16], while in [17,18] it was shown that self-interactions can cause significant changes. In [20,21] the authors constrained the boson star parameter space using data from galaxy and galaxy cluster sizes. Unlikely many other forms of matter, compact objects [28-30], which are formed at the end stages of stellar evolution, are unique probes to study the properties of matter under exceptionally extreme conditions. The matter inside such objects is characterized by ultra-high matter densities for which the usual classical description of stellar plasmas in terms of non-relativistic Newtonian fluids is inadequate. Therefore, such very dense compact objects are relativistic and as such, they are only properly described within the framework of Einstein's General Relativity (GR) [31]. Strange quark stars [32-37], at the moment hypothetical objects, can be viewed as ultra-compact NSs. Since quark matter is by assumption absolutely stable, it may be the true ground state of hadronic matter [38,39], and therefore this new class of relativistic compact objects has been proposed as an alternative to typical NSs. In fact strange quark stars may explain the observed super-luminous supernovae [40,41], which occur in about one out of every 1000 supernovae explosions, and which are more than 100 times more luminous than regular supernovae. One plausible explanation is that since quark stars are much more stable than NSs, they could explain the origin of the huge amount of energy released in super-luminous 7 How Compact Stars Challenge Our View About Dark Matter 153 supernovae. Many works have been recently proposed to validate its existence in different astrophysical scenarios [42,43]. It is well-known that the properties of stars, such as mass and radius, depend crucially on the equation-of-state. Furthermore,the presence of DM inside a star is expected to influence the structure, the evolution as well as certain properties of the object, such as mass-to-radius profiles and frequency oscillation modes. Even if dark matter does not interact directly with normal matter, it can have significant gravitational effects on stellar objects DM that can influence evolution and structure of compact objects [44-58]. Given the recent advances in Helioseismology and Asteroseismology in general, studying the oscillations of stars and computing the frequency modes offer us the opportunity to probe the interior of the stars and learn more about the equation-of-state, since the precise values of the frequency modes are very sensitive to the thermodynamics of the internal structure of the star [59]. For previous works on radial oscillations of stars see [60-68] and references therein. 7.2 Impact of DM on strange quark stars In the first part of the presentation we discuss the impact of bosonic self-interacting DM on properties of strange quark stars. 7.2.1 Mass-to-radius profiles -Structure equations: We briefly review relativistic stars in General Relativity (GR). The starting point is Einstein's field equations without a cosmological constant G^v = R^v Rg^v = (7.1) where we have set Newton's constant equal to unity, G = 1, and in the exterior problem the matter energy momentum tensor vanishes. For matter we assume a perfect fluid with pressure p, energy density p and an equation of state p(p), while the energy momentum trace is given by T = - p + 3p. For the metric in the case of static spherically symmetric spacetimes we consider the following ansatz ds2 = -f (r) dt2 + g (r) dr2 + r2 dn2 (7.2) with two unknown functions of the radial distance f(r), g(r). For the exterior problem one obtains the well-known Schwarzschild solution [69] f(r) = g(r)-1 = 1 - 2M (7.3) where M is the mass of the star. For the interior solution we introduce the function m(r) instead of the function g(r) defined as follows g(r)-1 = 1 - (7.4) 154 G. Panotopoulos so that upon matching the two solutions at the surface of the star we obtain m(R) = M, where R is the radius of the star. The Tolman-Oppenheimer-Volkoff (TOV) equations for the interior solution of a relativistic star with a vanishing cosmological constant read [70,71] m '(r) = 4nr2p(r) (7.5) m(r) + 4np(r)r3 r2(1 — ^) If \ f f \ i f w m(') + 4nP(')1 ,n P (r) = —(P(r) + P(r)) —77;-2m(Th (7.6) where the prime denotes differentiation with respect to r, and the equations are to be integrated with the initial conditions m(r = 0) = 0 and p(r = 0) = pc, where pc is the central pressure. The radius of the star is determined requiring that the pressure vanishes at the surface, p (R) = 0, and the mass of the star is then given by M = m(R). -Two-fluid formalism: Now let us assume that the star consists of two fluids, namely strange matter (de-confined quarks) and dark matter with only gravitational interaction between them, and equations of state ps (ps), px(px) respectively. The total pressure and the total energy density of the system are given by p = ps + px and p = ps + px respectively. Since the energy momentum tensor of each fluid is separately conserved, the TOV equations in the two-fluid formalism for the interior solution of a relativistic star with a vanishing cosmological constant read [72,73] m '(r) = 4nr2p(r) (7.7) r, ^ , , ^ , ^ m(r) + 4np(r)r3 pS (r) = —(ps(r) + Ps(r)) ^ Z^) (7.8) r, s , . . . .. m(r) + 4np(r)r3 pX (r) = —(px(r) + px(r)) (7.9) r2(1--T-1) In this case in order to integrate the TOV equations we need to specify the central values both for normal matter and for dark matter ps (0) and px(0) respectively. So in the following we show the mass-radius diagram for a certain value of the constant K = 2nl/mX and for fixed dark matter fraction e =_pxl°__(710) £ ps(0)+ px(0) (/.10) and we consider four cases, namely e = 0.02,0.035,0.05,0.09. We have chosen these values in agreement with the current dark matter constraints obtained from stars like the Sun. Actually, as shown by several authors, even smaller amounts of DM (as a percentage of the total mass of the star) can have a quite visible impact on the structure of these stars [74-76]. As we discuss in this work even such small amounts of DM can change the M — R relation of neutron stars. -Equation-of-states: For the condensed dark matter we shall consider the equation of state obtained in [77], namely Px = KpX, where the constant K = 2nl/mX is given in terms of the mass of the dark matter particles mx and the scattering length l. In a dilute and cold gas only the binary collisions at low energy 7 How Compact Stars Challenge Our View About Dark Matter 155 are relevant, and these collisions are characterized by the s-wave scattering length I independently of the form of the two-body potential [77]. Therefore we can consider a short range repulsive delta-potential of the form V(? -?2) = — S(3)(? -T2) (7.11) mx which implies a dark matter self interaction cross section of the form ctx = 4nl2 [52,77]. Following previous studies we fix the scattering length to be I = 1fm [52,77], and for 0 is the degree of angular momentum (or degree of the mode), and n = 0,1,2,... is the overtone number (or radial mode). The Sturm-Liouville boundary value problem at hand can be treated equiva-lently as a quantum mechanical problem by recasting the second order differential equation for Z into a Schrodinger-like equation [88,91,92] of the form dTf + [w2 - Ul(T)] ^ = 0. (7.43) Introducing the functions 2 2e ' (r) A(r) = 2 + 2Vr), (7.44) r e(r) ' ,1(1 + 1) Hi(r)= cs(r)2 -i-J-^, (7.45) and P(r)= A(r)cs(r) — cS (r). (7.46) The new variables t and ^ are defined as follows Z(r) ^(r) = UT) (7.47) where u satisfies the condition u'/u = —P/(2cS), and t is the acoustic time rr T = c-1(z)dz. (7.48) Finally, the effective potential is found to be U,(r)_ Hi(r) + () + , (7.49) and we thus obtain the effective potential as a function of the acoustic time in parametric form x(r),Ul(r). The acoustic potential with the first 7 eigenvalues, the corresponding eigen-functions as well as the large frequency separation in mHz for I = 2 are shown in the figures 7.9, 7.10 and 7.11, respectively. 7.4 Conclusions In this presentation we have presented results of our work on properties of self-interacting scalar field dark matter in two respects. In particular, in the first part we studied the impact of dark matter on the mass-to-radius profiles as well as on the radial oscillation modes of non-rotating, spherically symmetric strange quark stars in which dark matter is accumulated. Then, in the second part we studied radial and non-radial oscillations of self-gravitating bosonic (star-like) configurations. 7 How Compact Stars Challenge Our View About Dark Matter 163 t/t0 Fig. 7.9. Non radial oscillations: Acoustic potential vs acoustic time for 1 = 2. t/t0 Fig. 7.10. Non radial oscillations: Eigenfunctions vs acoustic time for 1 = 2. 1 .0n-'-'-'-1-'-'-'-1-'-'-'-1-'-'-'-1-'-'-'- 0.9- - 77 0.8- • - i • • • £ ______________•_ __• • ^ • ^ ^ • < 0.7- - 0.6- - 0.5 —1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1— 2 4 6 8 10 v [mHz] Fig. 7.11. Non radial oscillations: Large frequency separation in mHz for 1 = 2. Strange stars are hypothetical compact objects that are supposed to be much more stable than neutron stars, and thus could explain the super luminous supernovae. For the star interior problem we have solved numerically the Tolman- 164 G. Panotopoulos Oppenheimer-Volkoff equations in the two-fluid formalism. For strange matter we have assumed the simplest version of the MIT bag model (radiation plus the bag constant), while if dark matter is modelled inside the star as a BEC, it can be described by a polytropic equation of state with index n = 1. We have shown the mass-radius diagram assuming that strange stars are made of up of (5 — 10) % of dark matter. We conclude that if strange stars do exist, and if they accumulate dark matter, our findings limit in a certain way the radius and the mass of these compact objects. After that we studied the radial oscillations of dark matter admixed strange stars. Integrating numerically the equations for the perturbations we solved the corresponding boundary value problem to compute the first 11 frequency radial modes for three stars with the same mass and radius, but with different dark matter amounts. The large frequency separation were computed as well, and we showed them for all three models in the same plot for comparison reasons so that the impact of dark matter could be inferred. In the second part we studied radial oscillations of Dark BEC stars made of ultralight repulsive scalar particles in the Fermi-Thomas approximation. Using the known background solution to the Lane-Emden equation for a Newtonian polytropic star with index n = 1 we solved the Sturm-Liouville boundary value problem for the perturbation with the shooting method. We have computed the fundamental as well as several excited modes for two different star masses, and we have shown graphically i) several eigenfunctions corresponding to the first three and two highly excited oscillation modes, and ii) how the large frequency difference varies with the frequencies themselves. In addition, we have reformulated the boundary value problem equivalently by writing down a Schrodinder-like equation, and we have shown the effective potential together with the first five values of Finally, we have studied non-radial oscillations of bosonic configurations made of ultralight repulsive scalar particles in the Cowling approximation. For three different values of the angular degree 1 = 1,2,3 we have computed the lowest frequencies, several associated eigenfunctions, and the effective potential in the equivalent description in terms of a Schrodinder-like equation. The large frequency separations are shown as well. In all three cases, like in the radial oscillation case, for the higher excited modes the large separation tends to a constant determined entirely by the mass scale A = %/mF, where m is the mass of the scalar field and F is a high mass scale that determines the self-interaction coupling constant in the scalar potential. Acknowledgements I wish to thank the organizers of the Bled Workshop for their kind invitation to participate and present my work. 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The Astrophysical Journal 542,1071-1074. Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 168) Bled, Slovenia, July 6-14, 2019 8 Phenomenological Studies of Models With a Pseudo Nambu Goldstone Boson T. Shindou * Division of Liberal-Arts, Kogakuin University Nakano-machi 2665-1, Hachioji, Tokyo, 192-0015, Japan Abstract. We discuss how to probe a class of models where the Standard Model-like Higgs boson is identified with a pseudo Nambu Goldstone Boson (pNGB) associated with the spontaneous breaking of a global symmetry. We focus on the minimal version of such models. There SO(5) symmetry is broken to SO(4) so that four pNGBs appear which corresponds to the Higgs doublet in the SM. In order to probe such a model, double Higgs production process is found to be quite powerful. It is shown that the production cross section of this process has a model specific behavior so that we can distinguish different new physics scenarios. Povzetek. Avtor studira model, ki uporabi za opis Higgsovega skalarja psevdo Nambu-Goldstonove bozone, ki nastanejo pri spontani zlomitvi globalne simetrije SO(5) v SO(4). Stirje psevdo Nambu-Goldstonovi bozoni ustrezajo Higgsovemu dubletu standardnega modela. Avtor pokaze, da je presek za nastanek teh dveh Higgsovih bozonov odvisen od parametrov realizacije modela, s cimer se spremenijo tudi napovedi modela. Keywords: Hierarchy problem, composite Higgs model, double Higgs production processes 8.1 Introduction In 2012, the Higgs boson was discovered at the LHC experiments, and the Standard model (SM) is then experimentally established. However, the SM has several serious problems. There is a well-known theoretical problem known as a gauge hierarchy problem. If there is a unified theory including gravity, the unified theory is considered to be realized at around the Planck scale ~ 10i9 GeV. On the other hand, the typical energy scale of the SM (electroweak theory) is the scale of the Higgs vacuum expectation value, 175 GeV. There is a colossal hierarchy between these two energy scales. It is an origin of gauge hierarchy problem. The so-called gauge hierarchy problem is a mixture of two kinds of problems. One is how to set appropriate values for the model parameters of order of 100 GeV if the fundamental scale is of the order of 10i9GeV at the boundary. The naive expectation in such a case is that all the parameters with a mass dimension are set to be 0(10i9) GeV. * E-mail: shindou@cc.kogakuin.ac.jp 8 Phenomenological Studies of Models With a Pseudo Nambu... 169 Even if we can set the value of parameters to be 0(100) GeV by some mechanisms, the parameters can be affected by radiative corrections, and they may be as large as 0(1O19) GeV. In order to avoid it, we need a fine-tuning or some mechanisms to cancel such a huge radiative correction. It is the second type of problem. In order to address the hierarchy problem, several excellent mechanisms are proposed in the literature. For example, supersymmetry provides a cancellation between a bosonic loop and fermionic loop. Therefore quadratic divergences in scalar mass parameter disappear (The second type of problem is solved). The first type of problem in the SUSY model is known as ^-problem[1], and there are many attempts to solve it(e.g. [2]). There are other many ideas such as the gauge Higgs unification scenario, models with the classical conformal invariance, and so on. In a model where the Higgs boson is identified with a pNGB associated with some global symmetry breaking, the Higgs mass parameter is naturally set to be much smaller than the fundamental scale (A solution to the first problem). Also, such pNGB can be sometimes considered as a composite state by the analogy of the pion which can be treated as a pNGB associated with chiral symmetry breaking. In such a scenario, the cut-off scale of the model is lowered to be 0(10) TeV1 and the second problem can become milder. In this talk, based on Ref. [3], we focus on the minimal version of such a scenario with pNGB, so-called Minimal Composite Higgs models (MCHMs)[4] and we study phenomenology in MCHMs. We have found that the double Higgs production process is interesting and powerful to probe MCHMs. 8.2 Model Since there are four real degrees of freedom in a SU(2) doublet, the minimal setup of models where the SM-like Higgs doublet is identified with pNGBs contains four pNGBs. It means that the breaking pattern of global symmetry should include four broken generators. One of the minimal breaking patterns is SO(5) to SO(4). We here consider SO(5)x U(1)x —» SO(4)x U(1)x model. The breaking occurs at the scale f. Associated with this symmetry breaking, four NGBs appears. Since the SU(2)LxU(1)Y subgroup in SO(4)xU(1)X is gauged, the global symmetry is explicitly broken by the gauge coupling. Also, the matter fermions in the SM cannot compose SO(5) multiplet so that the SM Yukawa interactions explicitly break the global symmetry too. Because of these explicit breaking effects, the NGBs become pNGBs, and they get smaller mass compared to the symmetry breaking scale. The gauge interactions in the low energy effective theory are completely determined by the breaking pattern of the global symmetry, while the Yukawa interactions depend on what representations of SO(5) the SM fermions are embedded into. In this review, we consider three cases; all the SM fermions are embedded into 4 dimensional representations (MCHM4), 5 dimensional representations (MCHM5) and 14 dimensional representations (MCHM14) of SO(5). In 1 It means that there can be some intermediate theory which may include a strong dynamics before appearing the final unified theory. 170 T. Shindou each model, the effective Lagrangian for fermion interactions with the SM-like Higgs boson is given by LMCHM4 = Z [nT0 + n r%] vr4) + vq4) [m0 + m1 rii] vt4) r=q,£,u,d,e + ^q4) [Mb + MbriZi] ¥b4) + V(4) [MT + MTriZi] vT4) + h.c. , LMCHMs = z V5 [pn0 + Ztpnii] vr5) + vq5) [m0 + z^z] vt5) r=q,£,u,d,e + Vq5) [Mb + ZtMbZ] Vb5) + vf5 [M0 + ZtMTZ] VT5) + h.c., lMCHM 14 = z vr ,pn0¥r14) r=q,£,u,d,e + (ivr14))pni (vr14)it) + (ivr14)zt)pn2(zvr14)zt) +vq14w0vi14)+(ivq14))M1 (vt14)it)+(ivq14)zt)M2(zvt14)zt) + Vq14)Mbvb14) + (ZVq14))Mb(vb14)Zt) + (ZVq14)Zt)Mb(ZVb14)Zt) +y^'m^4 + (ZV^14))MT(vT14)Zt) + (ZV^14)Zt)MT(ZVT14)Zt) + h.c. , (8.1) where Z is given by Z = sinh/f) (h1,h2,h3,h4,hcot(h/f)) , h = , (8.2) (R) with ha being the pNGBs. Also vr ' denotes the R-dimensional representation into which the SM matter fermion r = q, u, d, £, e is embedded, ri are the gamma matrices in SO(5), and n's and M's are the form factor. In the following, we only focus on the third generation quarks and leptons. In Table 8.1, deviations in the Higgs couplings from the SM predictions are summarized. All the deviations depend on a model parameter £ = v2/f2 where v is the vev of the Higgs boson, and f is the scale where the global symmetry is broken. In the table, we use the scale factors Ka, which are defined by Ka = ga/gaM, where ga denote the coupling constants of the Higgs boson coupling with the weak gauge bosons V = W and Z, matter fermions and the Higgs boson itself as a = hVV, htt, hbb, and hhh. For KhVV, Khtt and Khbb, the abbreviations kv, Kt and Kb, respectively are used. For hhVV couplings, we use the parameter chhVV = ghhVV/ghhw. In the effective theories of the MCHMs, there are new dimension five operators of two Higgs bosons and two fermions such as hhtt. The coupling constant for hhtt is parameterised as ghhtt = cHhttmt/(2v2). The four point interactions such as hhVV and hhtt also play important roles in our analysis. 8 Phenomenological Studies of Models With a Pseudo Nambu... 171 Model KV ChhVV KhHH Kt Kb Kt Chhtt MCHM4 i - 21 1 - 2f, 1 - 2 i - 21 1 - 2 1 - 2 MCHM5 1 - 21 i - 21 1 - 21 1 - 21 -4f, MCHM14 1 9M1+64M2 t 4(3M 1+23M 2) t ' 6M1 + 16M* ^ 3M1+8M2 ^ Table 8.1. Deviations in coupling constants with the Higgs boson in MCHM4, MCHM5 and MCHM14. The formulae in the table are approximated for £, C 1. The table is taken from [3]. £ Î £ Î 140 120 100 80 60 40 20 LHC V7 = 14 TeV PDF MSTW20081O MCHM14 / SM MCHM5 MCHM4 0.00 0.05 0.10 0.15 £ 0.20 0.25 Fig. 8.1. The production cross section of pp —> ggX —> hhX in MCHM4(green), MCHM5(blue) and MCHM14(red) at LHC with the collision energy of 14 TeV. The figure is taken from [3]. 8.3 Numerical results for double Higgs production First, we show the numerical results for the double Higgs boson production at LHC. The double Higgs boson production at LHC is dominated by the gluon fusion process, pp —» ggX —» hhX. In the MCHMs, the cross section is affected by deviations in the top Yukawa coupling constant and the triple Higgs boson coupling constant. In addition to these contributions, the dimension five interaction hhtt enhances the cross section. As a result, the cross section of this process depends on the parameters Kt, Khhh, and chhtt. In Fig. 8.1, the production cross section of pp -> ggX -> hhX in each MCHM at the LHC with Vs = 14 TeV is shown as a function of the compositeness parameter £,. As shown there, the cross section is suppressed in MCHM4, and it is enhanced in MCHM5 and MCHM14. Second, we consider the double Higgs production at an electron-positron collider. This process at the lepton collider is sensitive to the triple Higgs boson coupling hhh and the contact interaction hhVV. Fig. 8.2 shows the a/s dependence of the production cross section of the process e+ e- —» hhvv in MCHM4, MCHM5 and MCHM14 for fixed values of the compositeness parameter £ = 0.1 and 0.2. The cross section of e+ e- —» hhvv is dominated by Z-strahlung which is always suppressed by the scale factors in the MCHMs for a/s < 600 GeV and by W-fusion which is enhanced as a result of unitarity non-cancellation for a/s > 600 GeV. Within the expected accuracy of measurements[5,6], such a specific behaviour 172 T. Shindou 1.000 0.500 Ê 0.100 j| 0.050 è î 0.010 0.005 0.001 Fig. 8.2. The production cross sections for e+ e- —> vvhh in MCHM4 (Left) and in MCHM5 and MCHM14 (Right). The solid curve is for the total cross section. The green(blue) and the brown (magenta) curves are for the case of £, = 0.1 and £, = 0.2, respectively. The dashed and dotted curves show the W-fusion and the Z-strahlung subprocesses, respectively, and the black curves show the SM prediction. The figures are taken from [3]. 0.3 0.25 I °'2 | 0.15 g ^ 0.1 0.05 400 500 600 700 800 900 1000 500 1000 1500 2000 2500 3000 vs[oev] vs[gev] Fig. 8.3. Left: The cross section of e+ e- —> hhZ in the two Higgs doublet model. Right: The cross section of e+ e- —> h.h.vv in the model. Here, the SM-like Higgs boson mass is fixed to be 120 GeV and the masses of extra Higgs bosons are taken to be degenerate as m® = mH = mA = mH±. These figures are taken from Ref. [7]. might be observed by the ^fs scan at the ILC and the CLIC in the cases with a significant size of £,. This fs dependence of the double Higgs boson production cross section in the MCHMs is different from that in other new physics models such as the two Higgs model with a significant deviation of the triple Higgs boson coupling from the SM predction[7] as shown in . Fig. 8.3. In the two Higgs doublet model, large enhancement of the triple Higgs boson coupling enhances the double Higgs boson production cross section via the Z-strahlung, while the cross section by W-fusion contribution is suppressed. This behavior is opposed to the case of MCHMs. i ali 400 600 800 1000 1200 1400 0.001L VS [GeV] 400 600 800 1000 1200 1400 V7 [GeV] 8 Phenomenological Studies of Models With a Pseudo Nambu... 173 8.4 Summary The scenario where the SM-like Higgs boson is identified with pNGBs is attractive new physics model from the view point of gauge hierarchy problem. In this talk, we review a phenomenological study in MCHMs. In particular, we focus on the double Higgs boson production both at LHC and at future lepton collider experiments. We show that MCHMs can be probed by using this process. Especially, the predicted production process at lepton collider e+ e- —» hhvnu shows a specific behavior so that we might be able to distinguish MCHMs from other new physics scenarios by this process unless the parameter £ is too small. Acknowledgements It is partially supported by the Kogakuin University Grant for the project research "Phenomenological study of new physics models with extended Higgs sector". References 1. For a review, see, for example, N. Polonsky, hep-ph/9911329. 2. K. J. Bae, H. Baer, V. Barger and D. Sengupta, Phys. Rev. D 99 (2019) no.11, 115027 doi:10.1103/PhysRevD.99.115027 [arXiv:1902.10748 [hep-ph]]. 3. S. Kanemura, K. Kaneta, N. Machida, S. Odori and T. Shindou, Phys. Rev. D 94 (2016) no.1, 015028 doi:10.1103/PhysRevD.94.015028 [arXiv:1603.05588 [hep-ph]]. 4. D. B. Kaplan and H. Georgi, Phys. Lett. 136B (1984) 183. doi:10.1016/0370-2693(84)91177-8; D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. 136B (1984) 187. doi:10.1016/0370-2693(84)91178-X; H. Georgi, D. B. Kaplan and P. Galison, Phys. Lett. 143B (1984) 152. doi:10.1016/0370-2693(84)90823-2; H. Georgi and D. B. Kaplan, Phys. Lett. 145B (1984) 216. doi:10.1016/0370-2693(84)90341-1; M. J. Dugan, H. Georgi and D. B. Kaplan, Nucl. Phys. B 254 (1985) 299. doi:10.1016/0550-3213(85)90221-4. 5. H. Baer et al., arXiv:1306.6352 [hep-ph]. D. M. Asner et al., arXiv:1310.0763 [hep-ph]. G. Moortgat-Pick et al, Eur. Phys. J. C 75, no. 8,371 (2015); K. Fujii et al., arXiv:1506.05992 [hep-ex]. 6. L. Linssen, A. Miyamoto, M. Stanitzki and H. Weerts, arXiv:1202.5940 [physics.ins-det]. 7. E. Asakawa, D. Harada, S. Kanemura, Y. Okada and K. Tsumura, Phys. Rev. D 82, 115002 (2010). Discussion Section The discussion section is reserved for those open problems presented and discussed during the workshop, that might start new collaboration among participants or at least stimulate participants to start to think about possible solutions of particular open problems in a different way, or to invite new collaborators on the problems, or there was not enough time for discussions and will hopefully be discussed in the next Bled workshop. Since the time between the workshop and the deadline for contributions for the proceedings is very short and includes for most of participants also their holidays, it is not so easy to prepare there presentations or besides their presentations at the workshop also the common contributions to the discussion section. However, the discussions, even if not presented as a contribution to this section, influenced participants' contributions, published in the main section. Contributions in this section might not be yet pedagogically enough written, although they even might be innovative and correspondingly valuable indeed. As it is happening every year also this year quite a lot of started discussions have not succeeded to appear in this proceedings. Organizers hope that they will be developed enough to appear among the next year talks, or will just stimulate the works of the participants. There are seven contributions in this section this year. The author of one contribution presents his own inovative model (which has been started by using the binary code to express the spins and charges of fermions, and correlated later the binary code with the Clifford algebra basis of the spin-charge-family theory), representing the elementary fermions as defects in the periodical tessalations of small charged domains. The relations between the Clifford algebra and the Dirac matrices with the ap-perance of families in (3+1 )-dimensional space, embedded into (5+1 )-dimensional space, so that spin in the fifth and sixth dimensions represents the charge of fermions, are presented. One contribution has started the generalization of the new way of the second quantized fermions in the Clifford space, presented in the talk section, trying to reformulate the cross products of the Hilbert space of indefinite numbers of fermions. The contribution, reviewing the novel string field theory of authors, are pointing out that the possibility for objects to annihilate and create needs to be included. In one contribution it is assumed that neutrinos are composition of Dirac and Majorana neutrinos, fitting correspondingly the parametrization of mass matrices to the experimental data. There is the contribution studyng the possibility that the dark matter particles might decay and annihilate fast enough that the corresponding gamma rays should be observable, but yet they are not because of absorbtion. One contribution considers clusters of primordial black holes, decoupled from the cosmological expansion and therefore heated as compared to the surrounding matter. All discussion contributions are arranged alphabetically with respect to the authors' names. Diskusije Ta razdelek je namenjen odprtim vprašanjem, o katerih smo med delavnico razpravljali in bodo morda privedli do novih sodelovanj med udeleZenci, ali pa so pripravili udeleZence, da razmislijo o moZnih rešitvah odprtih vprašanj na drugačne načine, ali pa bodo k sodelovanju pritegnili katerega od udeleZencev, ali pa ni bilo dovolj casa za diskusijo na doloceno temo in je upati, da bo prišla na vrsto na naslednji blejski delavnici. Ker je cas med delavnico in rokom za oddajo prispevkov zelo kratek, vmes pa so poletne pocitnice, je zelo tezko pripraviti prispevek in se tezje poleg prispevka, v katerem vsak udeleženec predstavi lastno delo, pripraviti še prispevek k temu razdelku. Tako se precejšen del diskusijne bo pojavil v letosnjem zborniku. So pa gotovo vplivale na prispevek marsikaterega udeleženca. Nekateri prispevki se morda niso dovolj pedagosško napisani, so pa vseeno lahko inovativni in zato dragoceni. Organizatorji upamo, da bodo te diskusije do prihodnje delavnice dozorele do oblike, da jih bo mogocše na njej predstaviti. Letos je v tem razdelku sedem prisepvkov. Avtor enega prispevka predstavi svojinovativni model (zacel ga je z uporabo binarne kode za zapis spinov in nabojev fermionov, zapis pa nadgradil s tem, da je povezal binarni zapis s Cliffordovo algebro v teoriji spinov-nabojev-druzin), v katerem osnovne fermione predstavi kot defekte v periodicnem razcepu prostora (teselacijo) na majhne nabite podcelice. Avtorji predstavijo zvezo med Cliffordovo algebro, s katero opisšejo poleg spina in rocnosti tudi družine, in Diracovimi matrikami v (3 + 1 )-razseznem prostoru, ki ga vstavijo v (5 + 1 )-razsezni prostor, tako da spin v peti in sesti dimenziji predstavlja naboj fermiona. Avtorja zelita v njuni novi formulaciji druge kvantizacije, ki pojasni Diracovo drugo kvantizacijo (v predavanjih v tem zborniku pojasnita ta novi predlog druge kvantizacije), posplošiti produkt Hilbertovih prostorov z nedolocenim stevilom fermionov. Avtorja predstavita svojo novo teorijo polj s strunami ter namero, da vkljucita v to teorijo tudi anihilacijo in tvorbo objektov te teorije. V prispevku, ki privzame, da nevtrine sestavljajo Diracovi in Majoranini nevtrini, avtor isce parametrizacijo, ki ustreza eksperimentalnim podatkom. Avtorji prispevka obravnavajo možnost, da delci temne snovi razpadajo in se anihilirajo tako hitro, da bi morali opaziti nastale žarke y, vendar jih zaradi absorbcije ne opazimo. Prispevek obravnava zgruco prvotnih crnih lukenj, ki ni sklopljena s koz-molosko širitvijo vesolja in se zato segreva glede na snov v okolici. Prispevki v tejsekciji so, tako kot prispevki v glavnem delu, urejeni po abecednem redu priimkov avtorjev. Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 179) Bled, Slovenia, July 6-14, 2019 9 Analysis of Programming Tools in Framework of Dark Matter Physics and Concept of New MC-generator * K.M. Belotsky **, A.H. Kamaletdinov *** and E.S. Shlepkina f National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Kashirskoe shosse 31, Moscow, Russia Abstract. We analyse here some programming tools (MC-generators) from viewpoint of their application to the tasks of dark matter (DM) interpretation of cosmic rays puzzles. We shortly describe our tasks, where the main goal is the solution of the problem of suppression of gamma-rays induced by the products of DM decay or annihilation in Galaxy. We show that existing MC-generators do not fully satisfy our task, comparing them, and suggest our own one. Povzetek. Avtorji domnevajo, da delci temne snovi razpadajo in se tudi anihilirajo dovolj pogosto, da bi pri tem nastale Zarke gama morali opaziti. Studirajo procese, ki povzrocajo absorpcijo zarkov gama. Analizirajo obstojeca programska orodja in predlagajo svoje ustreznejse orodje. Keywords: dark matter physics, MC-generators, interaction Lagrangians 9.1 Introduction The necessity of the usage of different MC-programs1 appears in different areas. One of them is connected with dark matter (DM) processes. DM can give signal in cosmic rays (CR) due to their decay or annihilation. Positron anomaly [1,2] or possible excess of electrons and positrons [3] at high energy in CR is one of such subject. DM physics is unknown, what requires a respective flexibility of calculations of the predicted signal in CR e+e-. Realization of this with the help of using some programming tools imposes definite requirements on them about which we will talk. We do not pretend to comprehensive review, we are reviewing it from point of view of our task, what can be useful for many adjacent ones too. * Talk presented by E. Shlepkina ** E-mail: k-belotsky@yandex.ru *** E-mail: kamaletdinov.a.h@yandex.ru J( shlepkinaes@gmail.com 1 MC is decoded as Monte Carlo. Such programs are called as ME (Matrix Element) as well, implying the program tools able to simulate new (high energy) physics process. 180 K.M. Belotsky, A.H. Kamaletdinov and E.S. Shlepkina The physical task itself comes from our previous works [4-14] studying compatibility of DM interpretation of CR e+e- with cosmic gamma-ray data. The main problem is that, when we are trying to explain CR e+e- anomalies we start to contradict to cosmic gamma-ray data even in the framework of, seeming, minimal model case from viewpoint of gamma-ray production. The latter is pure e+e- decay or annihilation mode where gamma appears as (a) FSR (Final State Radiation) and (b) due to interaction of e+e- with interstellar medium. Both contributions seem to be unavoidable. Nonetheless, even in this minimal case we got contradiction with gamma-ray data. There are a few attempts to try to avoid this contradiction (we reviewed them in [5,12,11]), i.e. to suppress gamma coming from DM. It can relate to specifics of space distribution of DM like clumping or existence of dark disk component (supposing that a dominant halo DM component does not produce CR), or specifics in DM interaction. The latter includes both different decay/annihilation modes and Lagrangian of DM particles interaction with ordinary matter. Specifics of DM physics may involve also opportunity of decay of DM particles onto two identical fermions like X++ —» e+e+. In such model it is supposed there exist two types of double charged DM particles, X++ and Y . It is assumed that the last one is in form of electrically neutral bound state states with He, X++ form bound state with Y and decay [15-18]. In case of X++ —» e+e+Y decay, we have factor two of suppression of FSR gamma per one e+ (because they are two in one decay), and also extra suppression is expected due to identity itself of fermions in final state. The last reason takes place explicitly in classical case (dipole radiation of same charged particles is zero) and somehow partially in quantum case - due to so called single photon theorem [19]. All this accounts for necessity to have respective programming tool able to calculate the processes in the aforementioned tasks and, of course, not only. It does not cancel a desirability of analytical calculations. But the latter is often difficult to do and a crosscheck is necessary even when it is possible. It, in its turn, requires opportunity of step by step tracking calculations making with programming tools. We demonstrate here the work of some such tools (Section 9.2). They does not provide identical and, therefore, reliable results at the absolutely same initially set parameters. It related to our tasks. We here come to conclusion of creation of MC (HEP) generator (Section 9.3,9.4) which would allow simple step by step checking of calculation procedure. 9.2 Programming tools analysis As we told, it is impossible to build a model of dark matter in framework of dark halo or dark disk that would completely explain the positron anomaly in cosmic rays. Such attempts will lead to an excess of FSR arising from the decay/annihilation of a dark matter particle into two charged leptons or during the propagation in the interstellar medium. This task requires to create a new physical models that go beyond the Standard Model (BSM). It is necessary to find the most suitable programming tools for such a task that would correspond the following minimum requirements: 9 Analysis of Programming Tools in Framework of Dark Matter Physics... 181 1. the possibility to implement new physical models (BSM), 2. compute a matrix element and squared matrix element in analytical form, 3. the possibility of an explicit description of charge conjugation, 4. high enough precision of calculation. To describe the decay or annihilation of DM particles, taking into account possible FSR, the different programming tools such as MadGraph [20], CompHEP [21], CalcHEP [22] and FormCalc [23] were considered. Implementing BSM models in a generator such as MadGraph requires describing the model using the FeynRules [24] package. FeynRules is a package with Mathematica [25] source code that allows calculating the Feynman rules in momentum space for any physical model of quantum field theory. One of the reasons for using this package is the possibility of describing charge conjugation for fermions, which is necessary in our models. In FeynRules, we started with the following DM models: the simplest model of DM particle X decay on two opposite charged leptons and the model of double charged scalar particles X. In both models particle X is hypothetical long-lived scalar particle with a mass of about 1-3 TeV. Feynman rules for the Lagrangians presented below, which describes the decay of this particle, were tested: L = X^(a + by5)^ + ^y^A^ (9.1) L = X^C(a + by5)^ + X>(a - by5- ^y^A^ (9.2) where a and b are the unknown constant parameters. At the output, sets of model files written in the Universal FeynRules Output (UFO) were obtained that can be used for calculations and modeling of various processes in the MC-generator MadGraph5aMC@NLO. MadGraph is programming tool wich allows calculating cross-sections and squared matrix elements in numerical form. Using the FeynRules model files, several decay modes of the DM particle X, namely, the processes X —» e+e+ and X —» e+e+y, were simulated in this generator. MadGraph allows calculating cross-section, but it does not allow geting the squared matrix element in an analytical form, so this generator does not corresponds to all the previously set requirements. The next two MC-generators that we used in our task are CompHEP and CalcHEP. These tools have attracted our attention since they have the ability to obtain a squared matrix elements. Obtaining the squared matrix elements in analytical form for each of the processes X —» e+ e±, X —» e+e±y, we get the opportunity to monitor the correctness of the results and compare them with those that were obtained manually. To implement our models to CalcHEP, one can use the LanHEP [26] package. LanHEP has been designed as part of the MC-generator CalcHEP. This pack-age,similar to the FeynRules package, is used to generate Feynman rules in a momentum representation based on a given Lagrangian.The output can be written in the form of CalcHEP's model files, which allows to start computing processes in a new physical model. 182 K.M. Belotsky, A.H. Kamaletdinov and E.S. Shlepkina One of the alternatives to the MC-generators that we considered in framework of this task was FormCalc. FormCalc is the tool wich based on the FORM syntax and implemented as Mathematica package that allows one to calculate Feynman diagrams . Receiving input Feynman diagrams generated by the Fey-nArts (FeynArts [27] tool for generating Feynman diagrams), FormCalc is able to make calculations of the squared matrix element and write it out in Fortran code. The advantage of this program is that one can see some intermediate results, such as squared matrix element. However, FormCalc is a complex modular system of several packages and tools. Figures 9.2 and 9.3 show approximate schemes for working with some MC-generators. The main task at the first stage was the need to determine which programming tools is the most suitable for aforementioned task. An analysis of the above MC-generators was carried out, which consisted in comparing the results obtained from different MC-generators using the same model created using LanHEP. A positive result would be a complete (within the errors) agreement between their results. We considered dependencies of the decay width of the DM particle on its mass (fig. 9.1). These graphs do not show the results obtained from the MadGraph MC-generator, since the decay width obtained using this tool is too large and could not be used in the general analysis. The reason for such deviations has not yet been found. Fig. 9.1. Comparative analysis of MC-generators, using two processes as an example. Left: X e+ e-y , Right: X e+ e+y Figure 9.1 shows the results of the tests. As can be seen, the decay widths for the same model and masses of particle X differ. This deviation motivates us to look for additional verification tools. It is almost impossible to determine the cause of such differences, since in the process of decay modeling it is impossible to obtain any intermediate results, such as, for example, matrix elements, etc. The summary table (table 9.1) of the capabilities of some MC-generators was compiled, as applied, in particular, to BSM processes. Summing up, we can conclude that none of the programming tools we have use are not fully suitable for our task. 9 Analysis of Programming Tools in Framework of Dark Matter Physics... 183 Options CompHEP CalcHEP Madgraph Pythia Implementing of new models + + + - Charge conjugation + + + - Matrix element in analyt- - - - - ical form |M|2 in analytical form + + - - High precision ±2 ± ± + Performance3 ± + ± ± Have an implementation packages4 - + + - Hadronisation - - - + Table 9.1. Comparison of different MC-generators from viewpoint of calculation DM particle processes. Fig. 9.2. Approximate schemes for working with some MC-generators 9.3 Idea of creating of new MC-generator From analysis of existing MC-generators, given above, we come to conclusion that there is so far a necessity of creation of new one adjusted for our (of course, not only) tasks. The proposed new HEP generator allows calculating and displaying all intermediate results of calculations - i.e. analytical form of matrix element, the square of the matrix element in the form of traces of gamma matrices, the square of matrix element in form of kinematic variables and result of integrating the square of the matrix element of the given process over the phase volume. Estimation of intermediate calculation results can be useful for validation of calculation processes and in the phenomenological areas of high energy physics to 2 Hereinafter, the sign ± will mean that this tool does not fit exclusively to our task, but it copes well with other processes and models. 3 Characterizes the speed of calculations 4 New models can be loaded into CalcHEP and MadGraph with the help, for example, FeynRules and LanHEP packages, while in CompHEP one can add new models only by hand. 184 K.M. Belotsky, A.H. Kamaletdinov and E.S. Shlepkina Fig. 9.3. Modular system of FormCalc using understand the contribution of specific Lagrangian terms to the various distributions. In specific of our work on dark matter interaction physics [4-7] we need to estimate why given components of Interaction Lagrangian lead to certain effects. The developing generator is based on FORM symbolic manipulation system [28], which is designed to work with algebraic expressions and constructions. It reads text files containing definitions of mathematical expressions as well as statements which tell it how to manipulate these expressions. It is widely used in the theoretical particle physics community, but it is not restricted to applications in this specific field. FORM "doesn't know" anything about the particle physics processes and calculations of amplitudes and cross sections. Everything that FORM makes - it searches in the string the substrings matching the pattern and replaces them with the developer-specified expressions. Then it leads similar terms and displays the result. User can enter the expression of Lagrangian or the expression of partial term of a perturbation theory series. It is also necessary to explicitly indicate the types of fields used in the Lagrangian and "in" and "out" states. See Figure 9.4. We want to note the monolithic architecture of the developing generator. That is all described above tasks are performed within one single program. The matrix element calculation algorithm is based on the principle of secondary canonical quantization. That is if user enter the expression of lagrangian, program approximate the T-exponent by Teylor series, that give the perturbation theory series. 9 Analysis of Programming Tools in Framework of Dark Matter Physics... 185 User Input i Lagrangian Fig. 9.4. General structure of the developing generator modules is , •. (-iS)2 (-iS)3 -lS = 1 - iS + ^ + + ... (9.3) where S = J d4x L - is the action of model. And take interesting term of this one. After that generator takes the fields of considering model and performs the second quantization5: L = L(*,3^*) * ^ * = ' d3p 1 J^ 72C :(ape-ipx + at,eipx) (9.4) where âp - is the lattice operator such that [âp, âR] = (2n)35(3) (p — q) FORM can perform specified instructions with given expressions taking into account the non-commutativity of variables. Developing generator should include explicitly the permutation rules of the given non-commuting variables in the form of instructions which patterns should be replaced by other expressions. That is the replacing of bosonic rising operators at each iteration schematically looks like: (9.5) âp aq •... -> p q ... • ((2n)35(3) (p - q)- a •... 5 This means that the symbols O are replaced by other text expressions corresponding to operators. e 186 K.M. Belotsky, A.H. Kamaletdinov and E.S. Shlepkina Then program takes the expression of matrix element in form of approximated T-exponent by the Teylor series with second quantization (see Eq.9.4): (out|e-iS|in) = (out|(1 — iS + + + -)|in)- (9.6) Here |in) = fi^, ...a^k|0) and (out| = (0|ap, ...apn are the initial and final states of process which are specified by user and are expressed by specific character sets. Then the program performs the normal ordering of rising operators according to the instructions indicated explicitly in the algorithm and described schematically (9.5) above. One of features of the developing generator is the opportunity for the user to indicate perturbation theory order, as well as choose or enter only the interesting term of perturbation theory for consider only it's contribution. After the matrix element of the process has been calculated - its analytical expression is displayed to the user on the screen (See Figure 9.4 - Matrix element calculation). The part of the program described above has already been developed. The next block of the algorithm in the Figure 9.4 (Squaring of the matrix element) takes an expression for the matrix element, which was calculated in the previous block of the diagram, and builds an expression for hermitian conjugate operator in the form of a specific string of characters. Then the product |M|2 = M • M^ should be reduced to a trace of gamma matrices and displays to the user. After substituting kinematic variables into the obtained expression and taking the trace, integration over the phase volume is performing to obtain the distribution. 9.4 Application of programming tools We compare the results, computed by developing generator with the standard processes of particle physics and the specific processes of our work, previously calculated manually. The results are follows: 1)Two-particle decay of a neutral Dark Matter particle into an electron and a positron user enter the fields X, Y, Y and interaction lagrangian of the model L = XY ( a + by5)Y (9.7) Then he indicates the statistic of fields, that is X - is the scalar field and Y - is the spinor field. This leads to: M = FB(e,k1) • ( a + b • G(5)) • FC(e,k2) • S(X,k3) (9.8) that means: M = u (ki)( a + by5)v(k2) (9.9) 2) Two-parrticle decay of a double charged Dark Matter particle into two positrons. 9 Analysis of Programming Tools in Framework of Dark Matter Physics... 187 Similarly: L — XY (a + by5)Y(c) + H.C. (9.10) with fixed initial and final states as |in >= |X > and fiin >= |e+, e+ > M — —FCT(e,k1) • iG(2) • G(0) • (a + b • G(5)) • FC(e,k2) • S(X,ka)+ (911) +FCT(e,k2) • iG(2) • G(0) • (a + b • G(5)) • FC(e,k1) • S(X,k3) (. ) that means: M — —vT (k1) iy2 Y0 (a + by5) v (k2) + vT (k2) iY2 Y0 (a + by5 )v (k1) (9.12) 9.5 Conclusion Here we considered capabilities of several MC-generators (CompHEP, CalcHEP, MadGraph with applications to some of them such packages as LanHEP, Feyn-Rules and etc. and some modular tools like FormCalc). This was done in framework of our task concerning DM signal search in CR. More concretely, we considered decay of DM particles with different interaction Lagrangians. We see that the considered tools do not quite satisfy our requests. We need some single tool what would allow providing to show "step by step" results of calculations. We suggest it here on the base of code FORM. 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Duhr, "Feynrules-feynman rules made easy," Computer Physics Communications, vol. 180, no. 9, pp. 1614-1641, 2009. 25. S. Wolfram, "The mathematica book," Assembly Automation, 1999. 26. A. Semenov, "Lanhep—a package for automatic generation of feynman rules from the lagrangian," Computer physics communications, vol. 115, no. 2-3, pp. 124-139,1998. 27. T. Hahn, "Generating feynman diagrams and amplitudes with feynarts 3," Computer Physics Communications, vol. 140, no. 3, pp. 418-431, 2001. 28. B. Ruijl, T. Ueda, and J. Vermaseren, "FORM version 4.2," arXiv e-prints, p. arXiv:1707.06453, Jul 2017. Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 190) Bled, Slovenia, July 6-14, 2019 10 Tessellation Approach in Modeling Properties of Physical Vacuum and Fundamental Particles E.G. Dmitrieff * Irkutsk State University, Russia Abstract. The approach of representing fundamental particles by defects in the periodical tessellations built of small electrically-charged domains is discussed in this paper. We give reasons for its use, enumerate the assumptions underlying it, formulate the main tasks that arise with this approach and provide some of solutions for them that we found. Povzetek. Avtor predstavi svoj model za opis lastnosti osnovnih gradnikov snovi - kvarkov, leptonov in njihovih antidelcev ter interakcij med njimi. Elementarne delce predstavi kot defekte v periodicni teselaciji prostora, ki jih dolocajo majhna elektricno nabita obmocja. Pove od kod je crpal vzpodbudo za svoj model, nasteje privzetke, na katerih je model zdrajen ter napovedi, ki jih model ponuja. Keywords: tessellation, bit graph, particle, defect, triple-periodical, satori 10.1 Introduction The Tessellation approach is the denotation for using some analogy between fundamental particles, on one hand, and structure defects in periodical spatial tessellations, on another hand, in calculation of particle properties and speculations about particle physics problems. We do not know exactly, how deep this analogy is, and what causes such a correspondence, but we found this approach useful and productive, and also found it interesting to explore its limits, trying to extend them. We formulated assumptions of the approach while developing several particle models based on bit graphs, aiming to get digital, more calculable by computers, representation of particles instead of usual quantum mechanical one [1]. There are approaches, that have some correspondences to our approach, among them are the Spin-charge-family theory [2], the Cellular automata interpretation [3], [4], and the ether hypothesis [5]. The bit graphs generalize the idea of numbers as bit sequences by allowing not just ordered, i.e. sequential, but also non-ordered and partially-ordered bit combinations. For instance, three bit organized in a closed loop appeared suitable to describe both the color charge of quarks or anti-quarks, and the color absence, characteristic of leptons and anti-leptons [1]. * E-mail: elia@sr.isu.ru 10 Tessellation Approach in Modeling Properties. . . 191 Two three-bit loops was found enough to represent also gluons, weak bosons, and the electrical charge for all the particles. Adding two more bits to the graph, we get a model suitable to describe three fermion families, triplet- and singlet-states of bosons, Higgs scalar and the photon. All these models provide the correct quantum numbers of the corresponding particles as combinations of their bit's values. The only thing one must assume is that the bit's values are not 0 and 1 but +1 or — 6 and they have the physical sense of electric charge. The weak points of our bit-graph models, including the most advanced one, was that they were completely C-symmetrical, and therefore they did not provide the representation of the handedness and the parity asymmetry. To overcome this obstacle, we modified the principal three-bit loop graph, assuming it directed, and, therefore, we get the whole model chiral and CP-symmetrical. The charge conjugation C, meaning exchange of all bit values from 0 to 1 (+1 to — 6) or back, and P, meaning the reverse of all loops' directions, being applied together, turn the model back to the original state. This trick helped, but the bits looked this time rather less like binary digits because they must somehow carry, in addition to the electric charge, some extra information about the direction. According to our eight-bit model, there must be two different versions for all the bosons, one of them more, and another one less symmetrical, which we associated with the triplet and single state of them. The scalar Higgs boson H took, in this model, the place of the singlet Z. Because of CP-symmetry, the same thing also happened to the photon representation, predicting some new scalar chargeless particle taking place of the singlet, or longitudinal, photon. Stacking, like children's blocks, several copies of our Higgs boson model graphs with each other, we found that an electrically and color-neutral filling of space with unlimited size is easily obtained in this way. We associate it with the vacuum condensate. It is chiral because its CP-symmetrical partner is another condensate, which is produced the same way by stacking with each other the copies of longitudinal photon model graphs. We recognized that it can be very effective to consider this condensate as vacuum background, instead of empty free space. It looks like regular periodic directed bit graph, infinite or big enough, consisting of multiple copies of the background bit combination, either Higgs or longitudinal photon. Some of these copies can be easily replaced with other model graphs corresponding to any of known particles, so particles will be just defects in the regular structure, with one or more bits with inverted charge. Since the background is chiral, the left- and right-handed configurations become completely different. As an example, the photon and Z boson, that were CP-partners, went far one from another. Heavy and short-living Z has 6 defect bits in respect to the background while the light-weight and stable photon has only two defected bits. The Higgs boson manifests itself as a scalar neutral particle on the background of longitudinal photon condensate. On its own background it would be non-distinguished from it, and thus experimentally not observable, i.e. non-existent 192 E.G. Dmitrieff - the same way as the longitudinal photon does not exist on the background of itself. That was the first time we think about the space as filled with the regular structure so it can be treated as a start point of our tessellation approach. In contrast to the purely mathematical structure of the bit graph, filling of the space with regions of different charge is a picture that can be called physical. It can be explored to find out what laws can exist in this 'world' and under which of them it will be more similar to ours. 10.2 Assumptions The assumptions we listed below constitute an essential part of the approach. Changing them, we usually get a model that significantly differs from the observables. Generally, they are as follows: • the idea of tessellation, • the statement of electrical charge carried by domains in it, and • grouping of charged domains into triplets and pairs. 10.2.1 The ground state is a domain tessellation The principal assumption of the tessellation approach is to treat the vacuum not as an empty space, either with fluctuations or without them, but, instead, as a dense filling of small regions, or domains. The domains can be either similar or different from each other, and may be either separated or not separated by some kind of walls. These are details that can vary in particular models. This filling, or tessellation, is assumed to be the ground state, so that all fluctuations, defects, geometric distortions should be considered against this background. In principle, the tessellation can be assumed global, crystal-like, or local, similar to some fluid, and even finite, looking like gas of domain clusters. In the last case, though, it is not the tessellation, but something more close to the classical empty space with free distinct particles in it. The liquid tessellation, with just near order of domains, should have some secondary unordered walls separating these ordered regions from each other, that, on our opinion, contradicts to the observations. So we assume the long-ranged, up to the infinity, and, in the first approximation, strictly periodical crystal-like space filling as the basic object for our model. In fact, each defect is the local violation of the periodicity, and the vicinity of a defect also can be slightly distorted. Also, there can be waves of the distortion, but all this is considered as excitations of the ideally periodical ground state. 10.2.2 Domains are electrically charged As the second assumption, we take the statement that the principal difference between domains, and probably the only one, is the difference in their electric charge. We assume it to be either +1/6 or -1/6 in units of proton charge e. 10 Tessellation Approach in Modeling Properties. . . 193 Fulfillment of this requirement is necessary to ensure that all the particles will have their electrical charges proportional to ± 1 only. In this case, the tessellation model gets compatible with the bit graph models we studied before, so we can use the bit values 1 and 0, converted to the charges +1/6 or -1/6, to represent the particle that we want to explore. In fact, it is not mandatory for absolutely all domains to carry these charges: it is only required for those domains that can change their charges individually or along with another domains of the same charge. In case a pair of domain can participate just in mutual charge exchanges, or in case of individual domains that can not change its charge at all, these domains could have any charge as long as they keep compensating each other. However, this is a kind of complication, that we try to avoid. Our 8-bit graph model allows exchange between any pair of bits, so the tessellation, that is compatible to it, must have all the domains charged with either +1/6 or -1/6 only. On our opinion, the scalar electric potential of these charges can play the role of Higgs field in explanation of particle masses, so we do not assume an extra Higgs field for this purpose. The electric field of domains is the only primary field assumed [7]. This hypothetical unification of both fields allows to estimate the domain radius: r - ^ " 36 ■ 137 1246GeV - 8,2 ' ^^ - 1'7 '10-2W (1(U) The whole picture of vacuum as a scalar field, non-zero almost everywhere (excepting walls), looks now close to the vacuum domain model [6] with the difference that domain sizes are not on cosmological but on sub-particle scale. This could, in our opinion, explain the paradox of absence of domain observations while they are predicted as a consequence of symmetry break in the electroweak theory. 10.2.3 Triplets and Pairs It is well known that all the fundamental particles have their electrical charge values in the range from -1 to 1. All the multi-charged particles are considered composite, as bound states. In the tessellation approach we consider this limit as an evidence in favor of assumption that the count of domains that are able to possess simultaneous inversion in the same direction, is exactly three. We suppose them to reside in the tessellation in the close vicinity of each other, most likely being the immediate neighbors. In other words, the tessellation consists of multiple positive- and negative-charged domain triplets, each carrying electric charge of ± 1, and one, two of three domains in a triplet can be defected, i.e. have the charge inverted. This assumption immediately leads to the phenomena of the isotopic symmetry, because it stipulate existence of two variants for each defect configuration, depending on place where it occurs: either instead of positive triplet in the ground 194 E.G. Dmitrieff state tessellation, or instead of negative one. The difference of electrical charge between them is exactly 1, so defects in the positive triplets are down particles; the same defects in negative triples are up. The relocation of the defected triplet from originally negative place to the positive one, is, in fact, its exchange with the positive triplet resided in its place. It causes, besides the transformation of the up particle into down one, the appearance of new positive-charged triple defect in the negative place. It corresponds to the weak boson, so all this exchange should be considered as an example of weak interaction, for instance: u3 —> d-1 + W+. This defect can migrate, exchanging its place with triplets in negative places, or cause the relocation of some defected triplet from the positive place into negative one. In addition to triplets, we assume the possibility of domain pairs. It is the artificial construction, serving as the simplest way to represent several different particles with the same charge. The exchange between domains in a pair affects neither color nor electric charge, but the result combination differs from the original. 10.3 Objectives of the tessellation approach To be applied to problems in particle physics, the tessellation approach requires the concrete suitable tessellation. To calculate energies, including masses, it is necessary to figure out, what is the energy in this case. For the dynamic processes, including interactions, the way of defect migration also should be identified. So the determining of the most optimal structure, obtaining the appropriate Hamiltonian and definition of dynamic may be considered as main objectives for the research. Also it is possible that there are some physical systems, analogous to the tessellations, for instance foams and liquid-liquid mixtures, so the approach could be applied to them, and some observations and experiments with these systems can improve the knowledge of this subject. 10.3.1 Finding the optimal structure There are a lot of mathematically possible different spatial fillings that, in principle, can be used in the tessellation approach. Each of them provides, as its defect combinations, the spectrum of possible fundamental particles. Some of them are better than others, i.e. their defect combinations looks more similar to the particles found in the real world. So there should be one or several tessellations that provide the best correspondence to experimental data. So, Determining of the optimal structure is the first and main task of the tessellation approach. We examined five structures, in the following order: • 1-dimensional probe tessellation of 8-bit 'V' bit graphs • simple cubic grid (NaCl type), • body-centered cubic grid (CsCl), • Weaire-Phelan [8] structure, or A15 phase [9] (|3-W, NbaSn) [10], and 10 Tessellation Approach in Modeling Properties. . . 195 • 4-dimensional 'Satori' structure [11], built as alternation of two modified A15 grids. All the structures are compatible with, but not limited by, our 8-bit model. In all these cases we considered electrically-neutral grids containing equal quantities of positive- and negative-charged domains in their nodes. In the simple cubic grid, to ensure both the neutrality, and also the CP-symmetry, we used as node's charge its parity, calculated as product of its row's, column's and layer's parities. Since all subsequent grids can be produced from the (hyper-)cubic grid performing shifts of its rows, columns and/or layers, the parity is still defined for their nodes so we distribute the charge in the same way. To obtain the domain structure from the grid, we use the Voronoi diagram [12] built for the nodes. In case of simple cubic grid, the Voronoi diagram is also simple cubic, dual to the original. In case of body-centered grid, the Voronoi diagram is the Kelvin structure, the tessellation of equal tetrakaidecahedra, each of them is truncated octahedron. In both cases, the structure is not chiral, so both even and odd domains have identical shape and spatial orientation. The key difference of the Weaire-Phelan structure in respect to simple and body-centered grids is that in it the domains of different parity have different orientation, being mirror reflections of each other. Moreover, there are two different kinds of domains: for each three tetrakaidekahedra of three different orientations, there is one dodecahedron. Each translation unit consist of two equilateral triangles built from tetrakaidecahedra and two dodecahedra of opposite parities. So it is obviously compatible with the 8-bit graph model, while the first two are not. The last tessellation that is 4-dimensional, now it is constructed but not well-studied yet. We needed the four-dimensional structure in order to have any model of three-dimensional defect motion (see below). Like A15, from which it is derived, it has minimal wall pro cell ratio, but, in contrast to it, is built of the domains having the same shape. 10.3.2 Constructing the Hamiltonian The electrical charge of particles, factorized into ones' complement bit representation, define most of the quantum numbers as bit combinations: weak charge, hyper-charge, baryon- and lepton-numbers, and matter type (matter-or-anti-matter bit). The unary triplet-bit-loop represents the color charge. So, it is easy to determine bit combinations and corresponding defects for the properties that influence on the electric charge. It is more difficult to guess the possible combinations, that would represent the equal-charged particles of both handedness-es, different spin, members of three (or more) families, or possessing boson and fermion kind of statistic. For instance, they are up, charm, and top quarks, or W- boson, tau, muon, and electron. However, it can be done, following the symmetry of the tessellation structure. But the problem of particle masses, which are very different, very special, and do not manifest any dependence on the particle's charge, on our opinion, can be 196 E.G. Dmitrieff solved in the tessellation approach just by applying some additional assumption about mass origin. Since there is nothing in the model but spatially distributed electrical charge, the mass of particle, which appears as some difference in the distribution structure in respect to background one, should depend on this difference, that can be expressed analytically in geometric terms. We start with choosing of the suitable definition for mass. The best one, on our opinion, is to treat as the particle's mass, the part of energy, associated with it regardless of its state of motion and of its interactions with other particles. It is preferable to the inertial mass definition, because it does not depend on motion, and to the gravitational one, since it does not require more than one particle. It means that if we prepare the model containing one non-moving defect, corresponding to a particle, in the infinite periodic tessellation, and calculate the difference in energy between pure and defected models, we should get the particle's mass. The tensor field of tessellation distortion, that might emerge around the defect, as we suppose, should be associated with the gravitational field of the particle. In this approach, the field of gravity is not created by mass nor by energy, but it is an essential part of the energy, and particularly, of mass. Following it, we should consider the total mass as split in two different parts: one of them is connected to the changes of not only sizes but also of the topology of domain walls, that is occurred in the place of defect; while another part is connected to the minor residual changes in shape of domains around it, that retain the tessellation topology, but can spread on rather bigger distances. Both parts are supposed to be able to exchange their energy and minimize it. So, obtaining the appropriate Hamiltonian is the second task of the approach, essential for its application to mass and energy prediction. The energy function could depend on domains' and walls' volume, area, curvature, thickness, charge density and so on. To check it, we calculate the Hamiltonian for the sample pure background tessellation (that should be as large as possible, ideally infinite). After that we figure out how the appearance of the particular defect rearranges the tessellation components in-place and in the vicinity, and calculate the Hamiltonian again, this time for the defected tessellation. The difference we treat as defect energy, which should be equivalent to the particle mass in absence of interactions and movements (in the reference frame where the domain centers are motionless). In addition to the mass calculation, the Hamiltonian can play another significant role. Both the initial assumption about existing of the domain tessellation, and choosing concrete structure for it, need some physical grounding for them, aside of their usefulness in explaining or predicting the particle and vacuum properties. We suppose that the energy depends on the structure shape so that it has the locally or globally minimum corresponding to the tessellation in the ground state. Taking the Hamiltonians gradient as analogue of tension force, we can allow the model to relax under it, and do not care anymore about maintaining of correct form of domains. In the most preferable case, we can omit the step of choosing the shape of tessellation, allowing the Hamiltonian minimization to self-assemble the tessellation. 10 Tessellation Approach in Modeling Properties. . . 197 This task looks rather real because, for instance, the tessellation A15 is an example of extremal case: it has minimal known wall area to given domain volume ratio among all 3-dimensional equal-volumed tessellations. Nevertheless, the use of just such a Hamiltonian is not necessary: for simplicity, tessellation can be given imperatively, by the coordinates of points, or analytically, for example by a trigonometric or exponential function. We have considered some simple rules of calculating energy, as follows: • The simplest hypothesis is to estimate the energy as being proportional to the count of bits or domains that are inverted with respect to the ground state. Its advantage is that it can be applied to infinite or even to the finite bit graphs regardless of their structure. The results are mostly qualitative, and can only be considered valid for a few cases. For instance, the smallest but non-zero masses must correspond to the photon and neutrino because they are represented with just two inverted bits. The most heavy particle should be Higgs boson, built from eight defects. Z corresponds to six defects while triplet-W does with five ones. So the mass ratios should be ^ = 5 = 0,625, ^ = § = 0,75, while experimental values are 0.643 and 0.728 H • Considering two kinds of bits, that reside in triplets and in pairs, as different, and treating solo changes of domain in pairs as having no influence on the mass, we could improve these results. This caused us to move from bit models to tessellations, where we can take in account the geometric properties. • In the polyhedral approximation of A15 structure, constructed from domains of two parities (and, of course, two corresponding charges), there are three kinds of faces of different area, and they can separate domains of either equal or opposite charge. We supposed that the energy is proportional to area of the domain walls and it is different for two types of wall: for double-layered walls between opposite-charged domains, containing zero-charged film in their core, and for walls between domains of the same charges: these walls supposed to have another structure, without zero surface inside. The particle, as combinations of several defects, define the configuration of walls, that can be calculated manually, even without computer simulation, just by counting faces of particular type. For A15 model, this energy calculation leads to existence of massless, low-massive, and highly massive particles. The massless particles correspond to inversions in dodecahedra, that have six equal pentagonal faces of each type, and after recharging they have six equal faces of each type, again. Since the changes can be in both directions, and the difference between arithmetic mean of two face's area and the third face is very small, the particles containing combinations, compensating each other, are lite-weight. Others are massive. We could not reproduce all the known masses in this simple scheme, but slightly varying the tessellation geometry, we found some defect combinations, that simultaneously give correct quantum numbers and also correct masses, for the photon, neutrinos, electron, weak bosons and Higgs. 198 E.G. Dmitrieff Gluon threads in mesons, supposed as 1-dimensional condensates of diagonal (rr, gg, bb) gluons, also appear massless excepting their ends. The solo gluons, not stacked in threads, have in this schema sufficient masses on GeV scale, so the conception of threads is preferable. Quarks do not look like individual particles, but as indispensable ends of diagonal gluon thread or, for closed non-diagonal one, as sites where it changes its direction. Some mass values, for example 105.65MeV for the muon, could not be represented this way unless we allow not just even but also odd count of changed faces, even though they always appear in pairs. This can mean that the second family should be considered in dynamics only, as oscillation or combination of two forms, having both even but different changed faces count, producing odd arithmetic mean. So by now we have not suggested the Hamiltonian that we could call ultimate nor close to it. The task seems to be complex because it should allow to take in account the particle's motion, including relativistic case. 10.3.3 Dynamics, time and motion To be able to represent dynamic effects we needed at least the tessellation that can get changed. However, we did not see that such an ability is present in any of the three-dimensional tessellations that we considered. Both the ground state, and the defects, manifest their tendency to be stable, motionless, especially under the Hamiltonian minimization. Nothing forces the defects to jump into another locations and also nothing causes them to keep jumping conserving their momentum or velocity. Cellular automaton as 4d tessellation One thing we could do is to consider consequential 'snapshots' of the same tessellation, where the defects took different places, 'moving' in the same sense as 'move' the motionless frames on a film. By assuming some external, additional rules of the jumps we could get the working model that would be a kind of cellular automaton. Geometrically, the cellular automaton build on the basis of three-dimensional Kelvin or A15 structure is the infinite four-dimensional tessellation with the dedicated direction, that is the direction of computation, orthogonal to the other three. Each 3-dimensional domain turns in it into the 4-dimensional cylinder or prism. From the viewpoint of the tessellation approach, there is no reason to believe that the shape of prism or cylinder is the best shape for the domain in the tessellation, suitable for the modeling. Instead, we should get one step back and suggest some 4-dimensional tessellation that would be 'good' or may be 'the best' according to its abilities to reproduce the phenomena we want in our model. Cross-sections with 'moving' domains On another hand, observing the 2-dimen-sional cross-sections1 of the 3-dimensional A15 model, we found out that its 1 In the trigonometrical approximation, with p = ^(sinz(1 — cosx)(1 + cosy) + sinx(1 — cos y)(1 + cos z) + sin y(1 — cos z)(1 + cos x)) 10 Tessellation Approach in Modeling Properties. . . 199 sequential cross-sections, that can be taken continually, look like a cartoon film, showing perpetually moving two-dimensional domains, even in the pure non-defected tessellations. The character of movement could be described as kind of oscillation or rotation, but since the similar-charged domains are indistinguishable, when they meet, they can exchange, so the movement also can be treated as directed relocation of domains on any distance and in any direction with the limited velocity. Any defect, occurred in this tessellation, in order to conserve its charge, must participate in its neighborhood's movement. Otherwise, it would overlay with other domain of the same charge, causing the double-charged domain, or mutually cancel the domain of opposite charge, forming the domain with reduced or zero charge. Both cases violate the principal assumption of the domain behavior, postulating their constant charge. So the charge conservation can be treated as the cellular automaton law, determining domain migration into the appropriate place on the each step. Hypothetical speculations about modeling movements and time Each time when the defected domain meets two neighbors of the opposite charge, it must choose, which place to take. Manipulating with this choice, we can control the movement: if it happens predominantly in one direction, than the defect moves there; otherwise it moves randomly or oscillating, keeping close to the point of origin. The small distortions of the domain's walls shape, caused by last choice made, can play role of the short-term memory, keeping some information about it, and make influence on the next upcoming choice. This possibility turns the process to be analog of Markov chain and allows keeping the movement direction, for instance, with the mechanism similar to the Bresenham's line algorithm. We also supposed that the number of situations of making some choice of direction, can play role of the own time for the particle, that influences on the probability of the particle's decay. Propagating with high velocities, close to the limit, defected domains have less freedom in choosing direction, that can be treated by the low-velocity observer as the time dilation of the quickly propagating particle. Unfortunately, the effect of 'moving' domains could not be used directly to represent the movement in the 3-dimensional model, because it reduces the dimension count by one, so in each temporal moment, i.e. cross-section, the model space is flat. Combining the idea of cellular automaton, as a 4-dimensional tessellation, with the observations of movement-like behavior of domains in flat cross-sections, we supposed that there exists a 4-dimensional tessellation allowing cross-sections, which in turn are 3-dimensional tessellations, able to represent known set of particles, and the movement observed from within 3d sections is a certain process in 4d one, equivalent to sampling successive sections in some direction with strict conservation of charge for each domain in the section. So the third task of the tessellation approach can be formulated as to find the appropriate 4-dimensional tessellation. It must offer the same possibilities 200 E.G. Dmitrieff as 3-dimensional ones, but, additionally, provide the way to represent momentum and, ideally, the law that causes domains to conserve it. 4d tessellation 'Satori' Since the most successive 3-dimensional model was the optimal space tessellation, we looked for the references to optimal tessellation in the 4-dimensional space, but did not find any. So we analyzed the way how the optimal tessellations in 2 and 3 dimensions are build, and found out that they are relaxed Voronoi diagrams of square or cubic point grids, with some nodes shifted on the unit half-size along the rows, columns or through layers. We noted that the optimal 3d node grid is produced from two isomeric optimal 2d grids (in one of them each second row is shifted while in another one the points are shifted in each second column). Being placed in alternating adjacent layers, they offer possibility to perform additional shift of 4 points along the straight lines orthogonal to the layers, so the ratio of shifted points raises from 0 in 1 dimension through -2 in 2d up to 4 in 3d, and the calculated value of the optimality criterion2 was reduced, which meant compaction. This procedure also produces two 3d-isomers, depending on selection of even-odd or odd-even order of 2d isomers used. Following this way, we repeated the same operation once more, placing two alternating isomeric 3d grids in adjacent spaces. Doing so, we got all the remaining non-shifted 4 points disposing on straight lines perpendicular to the spatial layers, so we could perform the ultimate shift along these lines. Calculating the Voronoi diagram (using the qhull package [13]), we found out that it consists of all the regions having the same size and the same shape. They are 78-verticed polytopes, with 26 3d faces, two of which are distorted dodecahedra while the remaining 24 are nonahedra. They have 4 orthogonal orientations, that can be defined by the vector connecting centers of their dodecahedral 3d-faces. Polytopes of the same orientation stack together sharing dodecahedral 3d-faces along each of four orthogonal axis. Even and odd polytopes are alternating along the stack, being the mirror reflections of each other. Calculating the optimality criterion, we found it3 « 4.9% less than in 3d, so since all the points are yet shifted, it is impossible to get more compact tessellation with the same way. It means that, probably, this 4d tessellation that we called 'Satori' is the most compact one in all the Euclidean spaces. ' The optimality criterion we calculate as c — —a ,d 1 where d is the space dimension r 3 d- d ND d— 1 count, Dd-1 is the hyper-area of walls in the sample of N domains, and Dd is the hypervolume of the sample. It has the value of 1 for simple hyper-cubic grids in all dimensions. The optimal flat honeycomb has c — \J2373 ~ 0.93060 while non-relaxed A15 has era 0.882825. 3 c — I 1 + 7 J§) ra 0.83943 10 Tessellation Approach in Modeling Properties. . . 201 Checking the cross-sections4, we made sure that they keep the 'moving' behavior of domains, now in three dimensions. The section is to be made orthogonal to one of the axis . In contrast, when the section is performed orthogonal to the diagonal of the Cartesian reference frame, the 'movement' looses its stochastic character, keeping all domains in 4-beat oscillating near points close to their centers. The new structure is made of equal domains so it is supposed to be stable under the relaxation with the tension applied with suitable Hamiltonian. 4d Cylinder tessellation With all its advantages, the Satori structure has at least two drawbacks that make us look for improvements. First, there is no more D-type domains that had equal count of neighbors of both parities, which allowed us to easily build models for massless particles using them. Now each domain shares two dodecahedral 3d-faces with two its neighbors, so even in mutual charge exchange between two neighbors the opposite 3d-faces would change their kind, that we usually treat as a sign of some mass connected with such a defect. Second, the tessellation looks having the lack of causality from the viewpoint of observer inside 3d cross-section. Propagating in some direction, the process of cross-sectioning can meet regions, containing other defects, that for the 3d observer would be miracle artifacts, appearing from nowhere and violating the conservation laws. We see that the possible solution for both problems listed above is the restriction in one of four dimensions with only one translating unit, turning the tessellation into the 4-dimensional cylinder, infinite in three dimensions but periodical in the fourth one. In this case two domains of opposite parity lying along the periodical axis would share both dodecahedral 3d-faces, so they both will remain intact in the mutual charge exchanges. The process of cross-sectioning is limited now with only four domain layers, so it cannot meet anything that does not exists in these layers. That ensures the same reality for both 4d and 3d observers. The sectioning process degenerates to the directed oscillation or rotation between four 3d-spaces, schematically shown below: in which states of domains in each space depend only on states of domains in two previous spaces, and also influence only on the state of domains in two subsequent spaces (rules for even and odd spaces are different due to their different structures). 4 In the trigonometric approximation of the 'Satori' structure that we constructed having extremal points in the domain centers: p = ^ (sin x(cos y—cos z+cos t—cos y cos z cos t) + siny (cos z—cosx + cos t—cos zcos x cos t) + sin z(cosx—cosy + cos t—cosx cosy cos t) — sint(cosx + cosy + cosz + cosxcosy cosz)) 202 E.G. Dmitrieff 10.4 Discussion It is not obvious whether the tessellation approach is compatible with the known 'no-go' theorems. For instance, it should not be considered as deterministic because it is based on bit graphs, which are multivalent, producing multiple eigenvalues as result of the serialization, which corresponds to the quantum measurement. Also, it offers some combination of spatial and internal degrees of freedom so it is interesting to check against the Coleman-Mandula theorem. 10.5 Conclusion The tessellation approach that we define and discuss in this paper allow us to formulate and solve problems of the particle modeling. Some of them have also the general mathematical meaning, for instance the problem of multi-dimensional filling optimality and measurement of information that tessellation holds. References 1. E.G. Dmitrieff: Experience in modeling properties of fundamental particles using binary codes, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 19th Workshop 'What Comes Beyond the Standard Models', Bled, 11. - 19. July 2016. 2. N.S. Mankoc Borstnik: Fermions and Bosons in the Expanding Universe by the Spin-charge-family theory, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 20th Workshop 'What Comes Beyond the Standard Models', Bled, July 9 - 17 2017. 3. G. 't Hooft, "The Cellular Automaton Interpretation of Quantum Mechanics. A View on the Quantum Nature of our Universe, Compulsory or Impossible?" arXiv:1405.1548 [quant-ph]. 4. Stephen Wolfram: A new kind of science, Wolfram Media, 2002 ISBN: 1579550088 5. D. Weaire et al., The Kelvin Problem, Taylor & Francis, 1996. 6. Ya. B. Zeldovich, I. Yu. Kobzarev, and L. B. Okun': Cosmological consequences of a spontaneous breakdown of a discrete symmetry: Zh. Eksp. Teor. Fiz. 67, 3-11 (July 1974) [ Sov. Phys. JETP 40,1 (1974)]. 7. E.G. Dmitrieff: The Hypothesis of Unity of the Higgs Field With the Coulomb Field, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 19th Workshop 'What Comes Beyond the Standard Models', Bled, 11. - 19. July 2016. 8. D.Weaire, R.Phelan: A counter-example to Kelvin's conjecture on minimal surfaces, Phil. Mag. Lett., (1994) 69:107-110, doi:10.1080/09500839408241577 9. Hartmann, Hellmuth; Ebert, Fritz; Bretschneider, Otto (1931). "Elektrolysen in Phosphatschmelzen. I. Die elektrolytische Gewinnung von a- und |-Wolfram". Zeitschrift fur anorganische und allgemeine Chemie. 198: 116. doi:10.1002/zaac.19311980111. 10. E.G.Dmitrieff: On triple-periodic electrical charge distribution as a model of physical vacuum and fundamental particles, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 21th Workshop 'What Comes Beyond the Standard Models', Bled, 23. - 29. June 2018 11. Elia Dmitrieff: On 4-dimensional equi-hypervolumed tessellation with possible the smallest interface hyper-area.: Phil. Mag. Lett. 10 Tessellation Approach in Modeling Properties. . . 203 12. Liebling, Thomas; Pournin, Lionel (2012). "Voronoi diagrams and Delaunay triangulations: ubiquitous Siamese twins". Optimization Stories. Documenta Mathematica. Extra Volume ISMP. pp. 419-431. 13. Qhull package. http://www.qhull.org/ 14. Coleman, Sidney; Mandula, Jeffrey (1967). "All Possible Symmetries of the S Matrix". Physical Review. 159 (5): 1251. Bibcode:1967PhRv..159.1251C. doi:10.1103/PhysRev.159.1251. Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 204) Bled, Slovenia, July 6-14, 2019 11 Mass Matrix Parametrization for Pseudo-Dirac Neutrinos A. Gorin1'2 * 1 National Research Nuclear University MEPHI (Moscow Engineering Physics Institute), 115409 Moscow, Russia, 2INR RAS (Institute for Nuclear Research of the Russian Academy of Sciences), 108840, Troitsk, Russia Abstract. An overview of pseudo-Dirac neutrino framework is given starting from general spinor phenomenology. The framework is then tested by simulation of oscillations for T2K experiment parameters. Two possible derivations [7] and [8] of oscillation parameters are indicated to have the same result. Povzetek. Avtor poda pregled modela psevdo Diracovega nevtrina v okviru splosne fenomenologije spinorjev. Model preizkusi s simulacijo oscilacij za parametre poskusa TK2. Pokaze, da izpeljavi [7] in [8] privedeta do enakega rezultata. Keywords: neutrino oscillations, sterile neutrinos, pseudo-Dirac neutrinos, neutrino oscillation experiments 11.1 Introduction Massive neutrinos directly indicate presence of physics beyond the Standard model (BSM). Precise measurements of neutrino oscillations provide the possibility to probe various BSM theories. Since the absolute values of neutrino masses are currently beyond direct measurements various experiments are focused on the standard neutrino model (vSM) oscillation parameters - square mass differences Am2 and 6-phase. Some experiments however reported the existence of anomalies in experimental data. These anomalies can find explanation in theories with additional neutrino interactions, most notably the sterile neutrinos. Recently a number of short-baseline reactor experiments declared an observation of sterile neutrinos with the significance of 3o\ However the observations are not entirely compatible to each other. The matter is under investigation in the ongoing STEREO, PROSPECT, SoLid and Neutrino-4 experiments. Experimental evidences suggesting sterile neutrino with mass ~ 1 eV can be explained in the simplest way in 3+1 neutrino model. * E-mail: gorin@inr.ru 11 Mass Matrix Parametrization for Pseudo-Dirac Neutrinos 205 Standard unitary 3+1 data fit suffers from strong tension between MINOS and MINOS+ bound on disappearance [2] and LSND&MiniBooNE —» VJ appearance [3,3,4]. There are two ways to approach this problem. First possibility is to consider 3+1 non-unitary mixing scenario [5]. It can be used to explain short-baseline disappearance experiments however the anomalies observed in LSND and MiniBooNE experiments [6] remain unexplained. Second possibility is addressing to more than one sterile neutrino. 3+2 scenario can be studied in general framework of 3 active and 3 sterile neutrino. Here we are probing the pseudo-Dirac scenario with 3 active and 3 sterile neutrinos. In Section 11.2 we will describe how pseudo-Dirac neutrinos naturally arise when the neutrino is a composition of Dirac and Majorana spinors. In Section 11.3 we will show that pseudo-Dirac neutrinos can be effectively described by three parameters. Then the mass matrix can be effectively diago-nalized which we show using two different approaches. Then we will plot the oscillation probability for pseudo-Dirac scenario against pure Dirac neutrinos for the setup of T2K experiment. In Section 11.4 we will discuss what can be further done to address the problem of streile neutrinos and neutrino mass generation. 11.2 General spinor formalism Lagrangian mass term for two spinors x and n has the form Lmass = 2 (Xn) M Q (11.1) where mass is given by M = ^M and M, A, B are 2x2 matrices. For the most general free field case we can write down "Weyl-Majorana-Dirac equation" io-^a^L -nD,Rmo,R^R -nLmL(i°"2)^L = 0 (11 2) is- nD,L^D,L^L - nRmR(iCT2)^R = 0 with non-negative mass terms m and phase terms n = e1^ from unitary group + i^A , = /^5 + iW can be transformed into the form [1]: U(1). Mrnrng TU = and = + , ^ = + J^) this equation n [1]: + MM2® = 0 (11.3) where ® = ..^8)T Now let us illustrate only the simple case mD,L = mD,R = mD. For this case general spinor mass matrix is positive semi-definite Hermitian matrix of the form M2 = /Mr 0 0 A \ 0 Mr -A 0 0 -B Ml 0 V B 0 0 MLJ (11.4) 206 A. Gorin where Mr = (Vi + mR -+V2 ML = (Vi + mL ^ ) V2 V! +mRy y V2 V! + mLy B _ / Hi M-2 j, a = (k 0 ) and mDmL + mDmR = k > 0 and moreover VH-2 -m / \0 -kj D faDml +mditiR = m mD = v1 +iv2. This matrix has four doubly degenerate eigenvalues. Considering real and positive mR and mD and complex mL we are down to just two eigenvalues. Now consider x and n in 11.1 to be the left- and right-handed neutrino fields vl and vr. "We can work with two Majorana neutrinos if we stipulate vr = vlc. Then M = ( mL mo ) There are three commonly known special cases for the ^mo mR J values of the elements of this matrix: • First case is mL = mR. In this scenario we have a pair of eigenvalues mD ± mL and mixing angle between vl and vr is given by tan29 = m2,mmL = n. No active-sterile oscillations are realized in this case. • Second case is mL = mR = 0. In this scenario we have a pure Dirac neutrino. • Last case is mL,mR ^ mD. This scenario is referred to as pseudo-Dirac case. In general, neutrino can have Majorana and Dirac parts £d + m = £D + £L +£R (115) Lmass Lmass 1 Lmass 1 Lmass l11'^ and Dirac neutrino can be represented as two Majorana neutrinos. Left-handed neutrinos are concerned active while right-handed are sterile i.e. they are singlets under SU(2)L x U(1)Y. For the Pseudo-Dirac neutrino the symmetry of mass matrix is not the symmetry of the weak interaction. It is easy to obtain Pseudo-Dirac neutrino decomposition ^±l = ?. )= —(nil ± in2lm —(nil ± e^L) y/2 vni ± imy v2 Vl ^±R = - (-ia2(n0 ± in2)) = —(ncl ± ^nclm -(ncl ± eivNcL) 2 2 2 (11.6) for a pair of almost degenerate mass Majorana neutrino with opposite CP sign and lepton number not being conserved in higher order weak interaction. Because of the small value of mass matrix distortions the mixing angle between two Majorana neutrinos is ~ n. 11.3 Modeling 11.3.1 Mass matrix diagonalization For chirality preserving processes it is suffice to diagonalize M^M. We will now consider two possibilities - M2 and M diagonalization and show that in the leading order they provide the same result for pseudo-Dirac neutrinos. 11 Mass Matrix Parametrization for Pseudo-Dirac Neutrinos 207 In general, 6x6 mass matrix diagonalization gives 15 mixing angles, multiple violating CP phases and 6 eigenvalues. Under Pseudo-Dirac assumption this can be approximated by ordinary 3x3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [7]. , t ™ „ ^ n,t vm^mL + mRmo mDmD MtM mDmD mLmD + mDmR^ (117) \mDmL + mRmD mDmD J consider bi-unitary transformation URmDUL = diag(mi ,m2,m3) = m then V = (U0 UR) and Vt(MtM)V = ( T m2 T ULmLULm + mURmRUR) (11.8) \mUL mLUL + URmRURm m2 J If we completely ignore off-diagonal parts then it is just Dirac scenario with doubly-degenerate eigenvalues. Otherwise in the first order approximation each pair takes the form ( mi en-T2a ^eimi m2 Now we obtain 6 mass eigenstates v1S = ^ (v1L + e1Vi v1R) v1A = —72 (v1L — e1Vi v1R) such that e1Vi = y^ for decomposition 11.6 and mass eigenvalues given by m2s,A = m2 ± eimi. Another method for diagonalization M itself is completely removing left-(anded Majorana spinor part of the Dirac one - mass matrix takes the form M = 0 m,D 1 In [8] it is shown that the appropriate diagonalizing transformation is mD Msy given in form ( ) ( ) v = ± ("u 1)(—i. 0 <-) where U diagonalizes mD and i = U(e/2 + e), £T = —£ and Ms = 2emD — emD + mDe. This produces M = V .mV where ( ) = / mD (1 + e) 0 m V 0 —mD(1 — e) Now m2 in the leading order have the eigenvalues m2 ± e1m1 which are the same as in the previous case. 11.3.2 Probing the pseudo-Dirac scenario With these eigenvalues we can write down the oscillation probability in terms of ordinary PMNS matrix. Assume that mass eigenvalues splitting for pseudo-Dirac neutrino is given by m2S A = m2 ± e1m1. Using the results from [7] it is easy to model v^ —» ve oscillation probability which is 208 A. Gorin P(Va > Vß) = 4 3 2 2 m2s m2a X Ußj (e1 mVt + e1 mE j=i ;)U aj (11.10) To illustrate potentially observable differences between Dirac and pseudo-Dirac scenario we will simulate oscillations for T2K experiment parameters: • L = 295 km and E < 2 GeV. • 5 = -n and sin2e12 = 0.307 sin2e23 = 0.5 sin2e13 = 0.218. • Am22 = 7.53 • 10-5eV2 Am23 = 2.44 • 10-3eV2. normal mass hierarchy. This allows us to probe the impact of small Majorana additives. Please also note that energy spectrum now depends on the absolute mass of neutrino because of the splitting. First we will model the situation where et = 0.1, Fig. 11.1. Please note that neutrino beam in T2K experiment has energy distribution with maximum at 0.6 GeV and almost all neutrinos have energy in the interval 0.5 ^ 1 GeV. So we cannot make any assumptions considering pseudo-Dirac neutrinos using only T2K data. 2 L.GeV Fig. 11.1. Pseudo-Dirac neutrino v^ —> ve oscillation probability compared to pure Dirac scenario for T2K experiment parameters and naive assumptions for pseudo-Dirac mass eigenvalues. Let us illustrate the difference in energy spectrum for more realistic et parameters. In Fig. 11.2 we have taken mi = 0.01 eV, ei = 2.6 • 10-3,e2 = 4.0 • 10-3 and e3 = 5.0 • 10-3 proportional to mass squares differences. 11 Mass Matrix Parametrization for Pseudo-Dirac Neutrinos 209 E. GeV Fig. 11.2. Pseudo-Dirac neutrino v^ —> ve oscillation probability compared to pure Dirac scenario for T2K experiment parameters for more realistic mass splittings. 11.4 Discussion and Conclusion Now we are in the situation where combined experimental data from atmospheric, reactor and accelerator neutrino experiments is in good agreement with 3 active neutrino model for the first three oscillation peaks. Upcoming experiments can provide more experimental data thus clarifying the situation. Long-baseline experiments can provide precise values of vSM oscillation parameters and provide enough data to determine the neutrino mass hierarchy. Short-baseline experiments can either improve their statistics and cancel out all anomalies or successfully approve that the vSM needs expansion. Using precise |3 -decay and K-capture measurements it would be arguably possible to measure neutrino masses directly or at least put a constraints on them. pp and 0vpp observations as well as atmospheric, solar, galactic and extra-galactic neutrino experiments are important for probing different neutrino mass generation mechanisms. It is also important to consider theoretical models for processes in early Universe - the constraints from these models are generally less strict than from direct observations but still helpful either for a cross-checking or for limiting the potential of exotic mass generation and mixing models. Here we presented the derivation of pseudo-Dirac neutrino from general spinor formalism. For the parameters of T2K experiment the probability of v^ —» ve oscillation was modeled. The current setup of the experiment however is not sensitive to differences in Dirac and pseudo-Dirac oscillations. 210 A. Gorin It was shown that in the leading order approximation PD neutrino can be effectively described by three e parameters of mass splitting - it is valid for M2 and M diagonalization. There are questions arising naturally in the context of neutrino mass generation mechanism. First question is whether it is suffice to consider pseudo-Dirac neutrino to fit observations or general framework is needed? This question will be addressed by the future observations. Second question is about the compatibility of particular mass generation mechanism with pseudo-Dirac scenario in particular and it's rigidity to possible observational data as a whole. Which mechanisms are the best candidates, Yukawa coupling or multiple scalar fields (like in Zee model) or maybe even geometric models of mass generation? Acknowledgements I am grateful to M.Yu. Khlopov for an invitation to XXII Bled Workshop and to the organizing committee of the Workshop for an opportunity to make a talk via internet. References 1. A. Aste: Weyl, Majorana and Dirac fields from a unified perspective, Symmetry, 8(9), 87 (2016). 2. P. Adamson et al: Search for sterile neutrinos in MINOS and MINOS+ using a two-detector fit, Phys. Rev. Lett. 122, 091803 (2019). 3. S. Gariazzo, C. Giunti, M. Laveder, Y.F. Li: Model-Independent ve Short-Baseline Oscillations from Reactor Spectral Ratios, Phys. Lett. B 782,13 (2018). 4. M. Dentler et al: Updated global analysis of neutrino oscillations in the presence of eV-scale sterile neutrinos, J. High Energ. Phys. 2018: 10 (2018). 5. C. Giunti: Short-baseline neutrino oscillations with 3+1 non-unitary mixing, Phys. Lett. B 795, 236 (2019). 6. MiniBooNE Collaboration, A. A. Aguilar-Arevalo et al: Significant Excess of ElectronLike Events in the MiniBooNE Short-Baseline Neutrino Experiment, Phys. Rev. Lett. 121, 221801 (2018). 7. M. Kobayashi, C.S. Lim: Pseudo-Dirac Scenario for Neutrino Oscillations, Phys. Rev. D 64, 013003 (2001). 8. A. de Gouvea, W.C. Huang, J. Jenkins: Pseudo-Dirac Neutrinos in the New Standard Model, Phys. Rev. D 80, 073007 (2009). Bled Workshops in Physics Vol. 20, No. 2 JLV Proceedings to the 22nd Workshop What Comes Beyond ... (p. 211) Bled, Slovenia, July 6-14, 2019 12 Relations Between Clifford Algebra and Dirac Matrices * D. Lukman3, M. Komendyak2, N.S. Mankoc Borštnik1 1 University of Ljubljana, Slovenia 2 Department of Physics, University of Warwick, CV4 7AL, UK 3CAMTP —Center for Applied Mathematics and Theoretical Physics, University of Maribor, Slovenia Abstract. In the spin-charge-family theory [2-7] there are Vn e N, 2d Clifford operators, forming the vector space. Space can have for given n e N dimension d = 2(2n + 1) or 4n. Half of them are Clifford odd operators with the properties of fermion creation and annihilation operators for 2d-1 family members of 2 d-1 families, fulfilling for each momentum pk the anticommutation relations for the second quantized fermions [8]. Families in Clifford space are reachable by Sab = 1 yayb, a = b and family members by Sab = 2yayb, a = b. In this paper the basis in d = (3 +1) Clifford space is discussed, chosen in a way that the matrix representation of ya and of generators of the Lorentz transformations in internal space, Sab, coincide for each family quantum number, determined by Sab, with Dirac matrices. The appearance of charges in Clifford space is discussed by embedding d = (3 + 1) space into d = (5 + 1 )-dimensional space. Povzetek. V teoriji spina-naboja-druzin [2-7] je v d dimenzionalnem prostoru 2d Cliffor-dovih operatorjev, ki določajo vektorski prostor. Teorija izbere d > (13 + 1). Ce uredimo vektorski prostor tako, da so vektorji lastni vektorji Cartanove podalgebre Lorentzove grupe, izpolnjujejo lihi Cliffordovi vektorji 2d-1 druzin s po 2d-1 clani vse Diracove pogoje za fermione v drugi kvantizaciji. Druzinske clane dolocajo generatorji Lorentzove grupe Sab (= 2yayb, a = b), družine pa Sab = 2yayb, a = b. V tem prispevku predstavijo avtorji bazo v d = (3 + 1) razseznem Cliffordovem prostoru ter matricno upodobitev za operatorje ya, Sab, Sab, ya ter Sab. d = (3 + 1) razsezzni Cliffordov prostor vgradijo v prostor d = (5 + 1) ter komentirajo pojav naboja fermionov v d = (3 + 1) . 12.1 Introduction In the Grassmann graded algebra of anticommuting coordinates 0a there are in d-dimensional space 2d vectors, which define, together with the corresponding derivatives , two kinds of the Clifford algebra objects: ya and ya [2,6-8], both with the anticommutation properties of the Dirac ya matrices, while the * Talk presented by N.S. Mankoc Borstnik 212 D. Lukman, M. Komendyak and N.S. Mankoc Borstnik anticommutators among Ya and Yb are equal to zero. {Ya ,Yb}+ = 2nab = {Y a,Yb}+ , (Ya,^b}+ = 0, (Ya)t = naa Ya, (Y a )f = naa Ya, Sab = 4(YaYb - YbYa), Sab = 4(YaYb - YbYa), {Sab,Sab}+ = 0, (a, b) = (0,1,2,3,5, ■■■ ,d). (12.1) The two Clifford algebras, Ya's and Ya's, are obviously completely independent and form two independent spaces, each with 2d vectors [9]. Sacrificing the space of Ya's by defining YaB(Ya) = (-)B iBYa , (12.2) with (-)B = -1, if B is an odd product of Ya's, otherwise (-)B = 1 [7], we end up with vector space of 2d degrees of freedom, defined by Ya's only. A general vector can correspondingly be written as d B = ao + ^ aa,a2 ...ak Yai Ya2 •••Yak l^o > , at < at+i ,k = 1,...,d (12.3) k=1 where |^o > is the vacuum state. We arrange these vectors as products of nilpotents and projectors ab 1 naa ab (k): = 2 (Ya + n^), ((k))2 = 0. ab 1 i ab ab [k]:= 2 (1 + kYaYb), ([k])2 =M , (12.4) where k2 = naanbb. Their Hermitian conjugated values follow from Eq. (12.1). ab t ab abt ab (k) = naa (-k), [k] =[k] . (12.5) Vectors in Clifford space are chosen to be eigenstates of the Cartan subalgebra, Eq. (12.6), of the generators of the Lorentz transformations Sab in the internal space of Ya's. S03 s12 s56 ■ ■ ■ Sd-1 d S03,S 12,S56, ■■■ ,Sd-1 d, (12.6) ab ab ab ab with the eigenvalues Sab (k)= 2k (k), Sab [k] = 1 k [k]. All the relations of Eq. (12.1) remain unchanged after the assumption of Eq. (12.3), while each irreducible representation of the Lorentz algebra Sab receives the additional quantum number f, defined by S ab. ab k ab ab k ab Sab (k)= ^ (k), Sab (k) = ^ (k), ab k ab ab k ab Sab [k] = - [k], Sab [k]= -- [k] . (12.7) 12 Relations Between Clifford Algebra and Dirac Matrices 213 Eq. (12.7) demonstrates that the eigenvalues of Sab on nilpotents and projectors generated by Ya's differ from the eigenvalues of Sab. States, which are products of projectors and nilpotents, have well defined handedness of both kinds, r(d) and F(d). r(d) : = (i)d/2 H (VnaaYa), if d = 2n, a F(d) :=(i)d/2 H (Vn^Ya), if d = 2n. (12.8) a The spin-charge-family theory [2-7] of N.S. Mankoc Borštnik uses products of nilpotents, 1 (ya + n""Yb), and projectors, 2(1 + kYaYb), to define 2d vectors in this space of the Clifford graded algebra [3-5]. In this theory Sab determine in d = (3 + 1 ) space, which is a part of d = (13 + 1 )-dimensional space, spins and charges of quarks and leptons, while S ab determine families of quarks and leptons. It is interesting to notice ([9,8] and references therein): Vectors, which are superposition of odd products of nilpotents and projectors, anticommute fulfilling the anticommutation relations postulated by Dirac [1]for second quantized fermions, explaining correspondingly Dirac's postulate [9,8]. In Sect. 12.2 the properties of products of nilpotents and projectors are discussed, arranged in eigenvectors of the Cartan subalgebra, defining the internal vector space of fermions in d-dimensional space when d = (3 + 1 )-dimensional space is embedded into d = (5 +1 )-dimensional space, so that the spin in d = (5,6) determines the charge of fermions in d = (3 + 1 ). In Sect. 12.2.3 the matrix representation of vectors are presented. 12.2 Properties of vectors in Clifford space In Refs. [9,8] the fact that the Clifford vectors, spanned by products of an odd number of Ya's, fulfill the anticommutation relations postulated by Dirac for the second quantized fermions, explains these Dirac's anticommutation relations. Let us see on the case that d = (5 + 1) how this happens. Let us denote vectors in d = (5 + 1), presented in Table 12.1 as products of three nilpotents or projectors or both, by b^t, m = (ch, s), the member quantum number m includes the charge, ch and the spin s. The corresponding Hermitian conjugated partner is denoted by (b^) = b^. The first member m = (1,1) of the first family a, which is the product 03 12 56 of three nilpotents, is correspondingly denoted by bal M = (+i) (+) | (+). All ( 2 , 2 J the rest vectors of the family f = a follow by the application of Sab. The families f = (b, c, d) follow from f = a by the application of Sab. The Hermitian conjugated partners follow by the application of Eq. (12.1). Table 12.1, taken from Table IV of Ref. [8], represents four families of Clifford odd vectors and their Hermitian conjugated partners. All the families have the same quantum numbers m of the corresponding members, (S03, S12, S56), each family carries its own family quantum number f. 214 D. Lukman, M. Komendyak and N.S. Mankoc Borstnik f (amily) m (ch, s) 6ft b m em S03 S12 S56 r3 + 1 S 03 S 12 S 56 03 12 56 56 12 03 a 1 (1 1) ( 2 , 2 ) ( + ( + ) ( + ) ( —) (—) |( ) (—) i) i 2 1 2 1 2 1 i 2 1 2 1 2 0 12 56 56 12 03 a 2 (1__L ) ( 2 , 2 ) [ —i [ —] ( + ) (—) —) | [ ] i] i 2 1 2 1 2 1 i 2 1 2 1 2 03 12 56 56 12 03 a 3 ( _ 1 1) ( 2 , 2 ) [ —i ( + ) [ ] [ —] |( ) ( —) i] i 2 1 2 1 2 i 2 1 2 1 2 03 12 56 56 12 03 a 4 I 1 1 1 ( _2, _1) ( + [ —] [ ] —] [ —] i) 22 1 — 2 1 — 2 2 1 22 1 22 03 12 56 6 12 03 b 1 ( 1 1 ) [ + [+] ( + ) (—) —) [+] +i] i 2 1 2 1 2 1 i — 2 1 — 2 1 2 03 12 56 56 12 03 b 2 (1__1) ( 2 , 2 ) (—i) ( —) ( + ) ( —) (—) |( ) (+) +i) —2 1 — 2 1 2 1 —22 1 —22 1 22 03 12 56 56 12 03 b 3 ( — 1 1) ( 2 , 2 ) (—i) [+] [ ] —] [+] +i) i 2 1 2 1 2 i 2 1 2 1 2 03 12 56 56 12 03 b 4 (_1__1) ( 2 , 2 ) [+ ( —) [ ] [ —] — (+) +i] i 2 1 2 1 2 i 2 1 2 1 2 03 12 56 56 12 03 c 1 (1 1) ( 2 , 2 ) [+ ( + ) | [+] [+] |( ) ( —) +i] i 2 1 2 1 2 1 i 2 1 2 1 2 03 12 56 56 12 03 c 2 (1__1) ( 2 , 2 ) (—i) [ —] | [+] +] [ —] +i) i 2 1 2 1 2 1 i 2 1 2 1 2 03 12 56 56 12 03 c 3 ( 1 1 ) (—i) ( + ) |( ) ( —) (+) |( ) (—) +i) —2 1 2 1 — 2 — 2 1 22 1 — 22 03 12 56 56 12 03 c 4 (_1__1) ( 2 , 2 ) [+ ][ ] |( ) (—) +) |[ ] +i] i 2 1 2 1 2 i 2 1 2 1 2 03 12 56 6 12 03 d 1 ( 1 1 ) ( 2 , 2 ) ( + [+] | [+] +] [+] i) 2 2 2 1 22 —2 — 22 03 12 56 56 12 03 d 2 (1__1) ( 2 , 2 ) [ —i ( —) | [+] [+ |( ) (+ [ —] i 2 1 2 1 2 1 i 2 1 2 1 2 0 12 56 56 12 03 d 3 ( — 1 1) ( 2 , 2 ) [ —i [+] | ( ) (—) +) [+] i] i 2 1 2 1 2 i 2 1 2 1 2 03 12 56 56 12 03 d 4 (_1__1) ( 2 , 2 ) ( + ( —) | ( ) ( —) (+) —) (+) i) i 2 1 2 1 2 i 2 1 2 1 2 Table 12.1. The basic creation operators, which are sums of odd products of y s — bm, — (ch, s), ch represents the spin in d — (5, 6), manifesting in d — (3 + 1) as the charge, and s represents the spin in d=(1,2), according to the choice of the Cartan subalgebra, Eq. (12.6) — and their annihilation partners — bm — are presented for the d — (5 + 1)-dimensional case. The basic creation operators are the products of nilpotents and projectors, which are the "eigenstates" of the Cartan subalgebra generators, (S03, S12, S56) and (S03, S12, S56), presented in Eq. (12.6). The Clifford odd parts of creation operators, belonging to d — (3 +1) space, are marked. Half of vectors, the eigenvectors of the Cartan subalgebra, Eq. (12.6), which are products of nilpotents and projectors, are odd products of Ya's and half of them are even products of Ya's. On Table 12.1 only Clifford odd vectors are presented. Let us make a choice of the vacuum state [6-9]. (In the case of a general even d the normalization factor is , d , since the vacuum states, generated v 2T—1 by projectors only, follows from the starting products of f projectors, let say 03 12 56 d-1 d [—i] [-] | [-] [-] ), by changing all possible pairs of [-]...[-], with [—i] included, to [+]...[+], leading therefore to 2d-1 summands. 1 03 12 56 03 12 56 03 12 56 03 12 56 | ^ > — ()2 ([-i] [-] | [-] + [+i] [+] | [-] + [+i] [-] | [+] + [-i] [+] | [+]) | 1 > . 2 (12.9) 12 Relations Between Clifford Algebra and Dirac Matrices 215 It then follows that emi^o > = if >, em i^o > = 0 i^o >, iem f}+ = sff' smm ,i^o >, iem ,emt}+=o i^o >, {bmf}+=oi ^o >, V m and V f. (12.10) Eq. (12.10) represents all the requirements for the second quantized fermions. 12.2.1 Action The action for a free massless fermion is needed and the corresponding equations of motion to take into account the ordinary space as well. The Lorentz invariant action for a free massless fermion in Clifford space is well known A = ddx 1 (^y0 YaPa^)+ h.c., (12.11) Pa = i dda' leading to the equations of motion YaPai = 0, (12.12) which fulfill also the Klein-Gordon equation YaPaYbPbi ^ > = papai^ >= 0, (12.13) for each of the basic vectors i >= bm i >. (y0 appears in the action to take care of the Lorentz invariance of the action.) Solutions of equtions of motion, Eq. (12.12), for a free massless fermions with momentum pa = (p0,p1 ,p2,p3,0,0) and a particular charge ±2, are superposition of vectors with spin 1 and — 1, multiplied by the plane wave e-ipa*a. Coefficients in superposition depend on the momentum pa. 12.2.2 Creation and annihilation operators in d = (3 + 1) space embedded in d = (5 + 1) The creation and annihilation operators of Table 12.1 are all of an odd Clifford character (they are superposition of odd products of Ya's). The rest of 24 creation operators of an even Clifford character can be found in Refs. [9,8]. ab ab Taking into account Eq. (12.1) one recognizes that Ya transform (k) into [—k], ab ab ab ab never to [k], while Ya transform (k) into [k], never to [—k] ab ab ab ab ab ab ab ab Ya (k)= naa [—k], Yb (k)= -ik [—k], Ya [k]=(—k), Yb [k]= -iknaa (-k), ab ab ab ab ab ab ab ab Y~a (k) = —inaa [k], Y~b (k)= —k [k], Y~a [k]= i (k), Y~b [k]= —knaa (k)(12.14) 216 D. Lukman, M. Komendyak and N.S. Mankoc Borstnik With the knowledge presented in Eq. (12.14) it is not difficult to reproduce Table 12.2, representing vectors that belong to d = (3 + 1) space. Vectors carry no charge and have either an odd or an even Clifford character. Multiplying these vectors by the appropriate charge (that is by either the nilpotent— if the d = (3 +1) part has an even Clifford character — or the projector — if the d = (3 + 1) part has an odd Clifford character — both must be the eigenfunction of S56) we end up with the Clifford odd vectors from Table 12.1. The properties of vectors of Table 12.2 are analyzed in details in order that the correspondence with the Dirac y matrices in d = (3 + 1) space would be easy to recognize. Superposition of vectors with the spin ±2 (either Clifford even or odd) solve the equations of motion, Eq. (12.12), for free massless fermions. As seen in Table 12.2 Ya as well as Ya change the handedness of states. Sab, which do not belong to Cartan subalgebra, generate all the states of one representation of particular handedness, Eq. (12.8), and particular family quantum number. Sab, which do not belong to Cartan subalgebra, transform a family member of one family into the same family member of another family, Ya change the family quantum number as well as the handedness F(3+1Eq. (12.8). Dirac matrices Ya and Sab do not distinguish among the families: Corresponding family members of any family have the same properties with respect to Sab and Ya, manifesting for d = (3 + 1) space four times twice 2 x 2 by diagonal matrices, which are, up to a phase, identical. The operators Ya and Sab are correspondingly four times 4 x 4 matrices. One finds that half of vectors of Table 12.2 are Hermitian conjugated to a 03 12 each other. In the Clifford odd part of Table 12.2 one finds that b^L^ 4) ([—i] (+) 03 12 d c, 03 12 03 12 , (+i) [—]) have as the Hermitian conjugated partners 0^=2 (—](—), (—i) [—]), re- 03 12 03 12 spectively. And bmt=(3 4) ((—i) [+], [+i] (—)) have as the Hermitian conjugated 03 12 03 12 partners ^=1 ((+i) [+], [+i] (+)), respectively. The vacuum state for the d = (3 + 1) case is correspondingly: 03 12 03 12 03 12 03 12 (^ )2 ([—i][—] + [+i][+] + [+i][—] + [—i][+]). 2 b 03 12 Embedding bm=3 (=[—i] (+)) into odd part of Table 12.1 the creation operator 03 12 12 extends into [—i] (+) [—], manifesting in d = (3 + 1) the charge 1 2 12.2.3 Ya matrices in d = (3 + 1) There are 24 = 16 basic states in d = (3 + 1), presented in Table 12.2. They all can be found as well as a part of states in Table 12.1 with either nilpotent or projector, expressing the charge, added. We make a choice of products of nilpotents and projectors, which are eigenstates of the Cartan subalgebra operators, Eq. (12.6), as presented in Eqs. (12.7). The family members of a family are reachable by either Sab or by Ya, and represent twice two vectors of definite handedness r(d) in d = (3 + 1). Different families are reachable by either S ab or by Ya. Each state carries correspondingly 4>in YO^m Yl Y2^m Y3^m YO^m Yl Y2^m Y3^m S03 s12 s03 s12 p3 + 1 p3+1 (+i)(+) M>4 -ia|>? -ill)? -ia|>? t ? 1 -> t ? 1 1 1 hi] H M>4 -l|>4 h|>2 h|>2 t ? i -> t ? i -> 1 1 [-«(+) m h|>3 h|>3 t ? i -> t ? i ^ -1 1 M>4 (+i)H M>2 M>2 -w -h|>4 M>4 -h|>4 t i ? t 2 i 7 -1 1 [+i] [+] M>4 M>4 ia|>? t 1 ? t 2 1 7 1 1 M>2 (-i)H M>4 -h|>3 -l|>4 h|>2 -h|>2 -l|>2 -h|>2 t ? i -> t ? I ^ 1 1 (-i)[+] -l|>2 -h|>2 h|>3 -m -4-3 -h|>3 t ? i -> t ? I -> -1 1 [+«(-) M>2 M>2 t 2 i 7 t 2 I 7 -1 1 [+«(+) -1|>4 -h|>4 iatf t ? 1 -> t ? 1 ^ 1 -1 M>f (-i)H M>4 h|>3 -1|>4 M>2 t I ? t 2 I 7 1 -1 (-i)(+) M>f h|>2 M>3 t ? I -> t ? I ^ -1 -1 M>4 [+i] H M>f -ia|>? M>f -l|>4 -h|>4 t I ? t 2 I 7 -1 -1 (+i)[+] M -h|>4 M h|>ib -ill)? —a*" t ? 1 -> t ? 1 ^ 1 -1 ^ MK-) M -4>3d h|>f t 2 I 7 t 2 I 7 1 -1 4>3d hi] h] M -M t ? 1 -> t ? 1 -> -1 -1 M>4 (+i)(-) M>2 -ill)? M>2 M>4 -h|>4 -l|>4 M>4 t 2 1 2 t 2 1 2 -1 -1 Table 12.2. In this table 2d = 16 vectors, describing internal space of fermions in d = (3 + 1), are presented. Each vector carries the family member quantum number f = (a, b, c, d) — determined by S03 and S12, Eqs. (12.7) — and the family quantum number m — determined by S03 and S12, Eq. (12.6). Vectors if)^ are obtained by applying bJl on the vacuum state. Vectors — that is the family members of any family — split into even (they are sums of products of even number of ya's) and odd (they are sums of products of odd number of ya's). If these vectors are embedded into the vectors of d = (5 + 1) (by gaining the appropriate nilpotent or projector), they gain charges. The Clifford odd parts of vectors are marked, entering into Table 12.1. 218 D. Lukman, M. Komendyak and N.S. Mankoc Borstnik quantum numbers of the two kinds of the Cartan subalgebra. In Table 12.2 also r(3+1) (= -4iS03S12) and r(3+1) (= -4iS03S12) are presented. When once the basic states are chosen and Table 12.2 is made it is not difficult to find the matrix representations for the operators (ya, Sab, Ya, Sab, r(3+1), r(3+1'). They are obviously 16 x 16 matrices with a 4 x 4 diagonal or off diagonal or partly diagonal and partly off diagonal substructure. Let us define, to simplify the notation, the unit 4 x 4 submatrix and the submatrix with all the matrix elements equal to zero as follows 1 = 1 0 0 1 0 = 00 00 (12.15) We also use (2 x 2) Pauli matrices 01 10 0 -i i 0 10 0 -1 (12.16) It is easy to find the matrix representations for y0, y1 , y2 and y3 from Table 12.2 Y Y / 0 a0 ' a0 0 0 0 0 0 a1 „1 A 0 0 a0 a0 0 0 0 0 Y2 = 0 0 0 — a 0 0 0 0 /0 a3 -a3 0 Y V 0 —a a1 0 0 0 0 0 a2 — a2 0 0 0 0 a3 a3 0 0 0 0 0 a0 0 0 0 0a (12.17) a 0 0 0 0 a2 a2 0 0 0 a3 3 0 0 0 0 0 0 0a a1 0 0 0 0 0 —a a2 0 0 0 0 0a —a3 0 (12.18) a 0 (12.19) (12.20) manifesting the 4 x 4 substructure along the diagonal of 16 x 16 matrices. The representations of the Ya do not appear in the Dirac case. They manifest the off diagonal structure as follows / Y 0 3 0 la 0 —la 0 0 0 la3 0 0 0 0 0 la3 0 0 —la3 0 \ (12.21) 0 2 3 ( ( ( 0 0 o o 0 a 0 0 o 0 a 0 3 0 0 3 0 12 Relations Between Clifford Algebra and Dirac Matrices 219 / Y Y2 = 0 0 -iff3 0 0 iff3 0 ( 0 0 -ff3 0 0 ff3 0 0 0 iff3 0 0 -iff3 0 0 0 ff3 0 0 -ff3 3 Y iff3 0 0 0 iff iff3 03 0 0 iff3 V -iff3 0 0 iff3 0 0 0 ff3 0 0 -ff3 0 0 0 0 0 0 iff3 0 0 -iff3 0 iff3 0 0 -iff3 0 0 o \ ff3 0 0 ff3 0 0 0\ 0 iff3 0 0 -iff3 (12.22) (12.23) 0 (12.24) / Matrices Sab have again along the diagonal the 4 x 4 substructure, as expected, manifesting the repetition of the Dirac 4 x 4 matrices, up to a phase, since the Dirac Sab do not distinguish among families. / S01 = 0 -2 ° 0 i ff1 \ (12.25) (-2 ff S02 = 0 i ff2 S03 = 0 -2 ° -2 ff3 0 0 0 -2 ff2 0 0 0 0 ff3 0 ff2 (12.26) (12.27) /1 ff3 0 S12 = i ff3 0 0 0 ff3 0 i ff3/ (12.28) 0 D 0 0 0 — ^ o 2 0 0 0 ^ o 2 0 0 2 ff 0 o 0 2 i1 0 D 2 2 0 0 0 i2 0 ^ o 2 0 2 0 ff 2 i2 0 ff 2 2 0 (J 2 0 3 D 2 0 0 0 3 0 ff 2 0 i 0 ff 2 3 0 -=r O 2 0 3 0 D 2 0 0 1 ff3 0 2 1 ff3 0 2 1 ff3 0 2 1 ff3 0 2 0 0 220 D. Lukman, M. Komendyak and N.S. Mankoc Borstnik /1 a2 0 n n S13 = 1 a2 /1 a1 0 S23 = 1 a1 0 1 a 0 0 0 ra2 0 0 - 2 a 0 0 0 a1 0 0 - 2 a 0 (12.29) r3+1 =-4is03s12 = 01 -01 0 0 0 1 -01 0 0 0 0 -0 000 0 0 0 La1 0 1 a1/ 0 \ 0 (12.30) 0 10 0 -1 (12.31) The operators Sab have again off diagonal 4 x 4 substructure, except S03 and S12, which are diagonal. S01 = S02 = S03 = S12 = S13 = S23 = ( 0 0 0 —2 A 0 0 — i 1 2 1 0 0 —i 1 2 1 0 0 V i 1 2 1 0 0 0 ) 0 0 0 11\ 0 0 11 2 1 0 0 1 2 10 0 I-2 10 0 0 n i 0 0 0 0 —i 1 2 1 0 0 0 0 i 1 2 0 0 0 0 2 V (11 0 0 0 0 11 2 1 0 0 0 0- 11 2 1 0 0 0 0 11/ 0 0 0 21\ 0 0 i 1 2 1 0 0 — i 1 2 1 0 0 U i 0 0 0 0 0 0 1 0 0 11 2 1 0 0 11 2 1 0 0 I-2 10 0 0 (12.32) (12.33) (12.34) (12.35) (12.36) (12.37) 1 a2 0 2 0 1 a2 0 2 0 2 1 a2 0 0 1 a2/ 0 1 _1 0 — ^ o 2 1 a1 0 2 12 Relations Between Clifford Algebra and Dirac Matrices 221 F 3+1 = —4iS 03S12 = /1 0 0 0 0 —1 0 0 0 0 —1 0 \0 00 1 (12.38) 12.3 Conclusions We present in this contribution the matrix representations of operators applying on the basis, defined by the creation and annihilation operators in d-dimensional Clifford space — d = 2(2n + 1), or 4n, n is a positive integer. We make a choice of d = (3 + 1) and d = (5 + 1). Creation and annihilation operators, which define the vector space, are in our case products of nilpotents and projectors (applying on the vacuum state, Eq. (12.9)), which are eigenvectors of the Cartan subalgebra, Eq. (12.6), of the Lorentz algebra of S ab, as well as of the corresponding Cartan subalgebra, Eq. (12.6), of the Lorentz algebra of Sab. Creation and annihilation operators are Hermitian conjugated to each other. We make a choice of the creation operators by choosing the vacuum state, Eq. (12.9), to be the sum of the Clifford odd (they are superposition of an odd number of Ya's) annihilation operators multiplying their Hermitian conjugated partners from the left hand side. Sab generate 2d -1 family members of a particular family of an odd Clifford character, S ab generate the corresponding 2d-1 families. The Hermitian conjugation determines their 2 d -1 x 2 d -1 partners (which are reachable also by YaY a). The Clifford even representations follow from the odd 2d-1 vectors by the application of Ya's or Ya's. There are correspondingly 2d vectors in d-dimensional space (d = 2(2n + 1 ),4n). The Clifford even operators keep the Clifford character unchanged. Ya's and Ya's change the Clifford character of vectors — from odd to even or opposite. Embedding SO(3 + 1) into SO(d), d > (3 + 1), d even, spins in d > (5 + 1) manifest in d = (3 + 1) as charges. One can check that the creation operators of an odd Clifford character and their Hermitian conjugated partners, applied on the vacuum state, Eq.(12.9), fulfill the anticomutation relations for the second quantized fermions, Eq. (12.10), postulated by Dirac, what explains the Dirac's second quantization postulates. One can also observe the appearance of families, used in the spin-charge-family theory for the explanation of families of quarks and leptons [3-5]. In this contribution the matrix representations for operators (ya's, Sab's, Ya's, Sab's) are presented for the basis in which creation operators are eigenstates of the Cartan subalgebras of both kinds, Eq. (12.7). It is discussed how do Clifford odd and even products of nilloptents and projectors in (3 + 1) become a part of creation and annihilation operators of an odd Clifford character in d = (5 + 1), manifesting the spin in a = (5,6) as the charge in d = (5 + 1). There are 24 = 16 basic vectors in d = (3 + 1) and correspondingly all the matrices have dimension 16 x 16, which are for the operators, determined by Ya's, by diagonal and for the operators, determined by Ya's, off diagonal. We keep the 222 D. Lukman, M. Komendyak and N.S. Mankoc Borstnik Clifford odd and the Clifford even vectors as the basic vectors. We treat in the Clifford odd part the creation and annihilation operators as they would all define the vector space, to point out, that if space of d = (3 + 1) is embedded into d > 6 , all the parts, even and odd contribute to the enlarged vector space as factors. References 1. P.A.M. Dirac, "The Quantum Theory of the Electron", Proc. Roy. Soc. Al17 (1928) 610. 2. N.S. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. Math. Phys. 34, 3731-3745 (1993). 3. N.S. Mankoc Borstnik, "Can spin-charge-family theory explain baryon number non conservation?", Phys. Rev. D 91 (2015) 6, 065004 ID: 0703013. doi:10.1103; [arxiv:1409.7791, arXiv:1502.06786v1]. 4. N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic Dynamics of articles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1607.01618v2]. 5. N.S. Mankoc Borstnik, "The explanation for the origin of the higgs scalar and for the Yukawa couplings by the spin-charge-family theory", J. of Mod. Phys. 6 (2015) 2244-2274. 6. N.S. Mankoc Borstnik, H.B.F. Nielsen, "How to generate spinor representations in any dimension in terms of projection operators", J. of Math. Phys. 43 (2002) 5782, [hep-th/0111257]. 7. N.S. Mankoc Borstnik, H.B.F. Nielsen, "How to generate families of spinors", J. of Math. Phys. 44 4817 (2003) [hep-th/0303224]. 8. "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v1v2]. 9. N.S. Mankoc Borstnik, H.B.F. Nielsen, "Understanding the second quantization of fermions in Clifford and in Grassmann space" New way of second quantization offermions — Part I and Part II, Proceedings to the 22nd Workshop "What comes beyond the standard models", 6 - 14 of July, 2019, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana, December 2019. 10. D. Lukman and N.S. Mankoc Borstnik, "Representations in Grassmann space and fermion degrees of freedom", [arXiv:1805.06318 ]. Bled Workshops in Physics Vol. 20, No. 2 JLV Proceedings to the 22nd Workshop What Comes Beyond ... (p. 223) Bled, Slovenia, July 6-14, 2019 13 Second Quantization as Cross Product N.S. Mankoc Borštnik1 and H.B.F. Nielsen2 * 1 Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 2Niels Bohr Institute, University of Copenhagen, Blegdamsvej 15-21, Copenhagen 0, Denmark Abstract. In the contributions [4,5] of this proceedings the new way of the second quantization of fermions is proposed, inspired by the fact that the Clifford and Grassmann algebra by themselves offer basis in internal space, presented as creation operators on the corresponding vacuum state, which together with their Hermitian conjugated annihilation partners fulfill all the requirements for the second quantized fermions, provided that the part of the basis in the ordinary space is orthogonal. In the Hilbert space of indefinite number of fermions it is assumed that each fermion has to distinguish from all the others either in ordinary or in internal space or in both spaces. The purpose of this contribution is to generalize this last requirement for either fermions or bosons. Povzetek. V prispevkih [4,5] tega zbornika predstavita avtorja nov nacin druge kvanti-zacije fermionov. Cliffordov in Grassmanov prostor ponudita namrec bazo v notranjem prostoru fermionov, ki jo določajo kreacijski operatorji na vakuumskem stanju, ti pa skupaj s Hermitsko andjungiranimi operatorji (annihilacijskimi operatorji) izpolnjujejo vse Diracove zahteve za fermione v drugi kvantizaciji pod pogojem, da je baza v prostoru gibalnih kolicin ortogonalna. V Hilbertovem prostoru nedolocenega stevila fermionov mora vsakemu fermionu ustrezati drugacen notranji prostor ali drugacna gibalna kolicina, V drugi kvantizaciji je Hilbertov prostor direkten produkt neskoncne mnozice Hilbertovih prostorov za izbrano vrednost gibalne kolicine. Namen tega prispevka je posplošiti ta drugi del zahteve tako za fermione kot za bozone. Keywords: second quantization, bosons, fermions, cross product 13.1 Introduction We present in this contribution the possibility to make a new step in the new way of the second quantization of fermions, presented in the contributions [4,5] of this proceedings, for indefinite number of fermions and bosons. It is the purpose of the present discussion to seek to use such a formulation of second quantized theories to generalize them to possibly quite new types of second quantization like theories. This is inspired from the type of theory put forward by one of us as being unification theory of spin, charges and families [1-5] * H.B. Nielsen presented the talk. 224 N.S. Mankoc Borštnik and H.B.F. Nielsen Second quantization as Cross product It is rather trivial and welknown that a second quantized (free) theory of bosons has a second quantized Hilbert space, that can be written as a Cartesian cross product over an (infinite) set of (smaller) Hilbert spaces, each of which is attached for example to the momentum, and tells how many particles have just this momentum. Simplest case: A scalar without internal degrees of freedom If we think of a charged scalar - like - it may be natural to even include in our "momentum" also the sign of the energy and use that as the 'factorisation parameter" p. We like to do it as abstract and general as possible, so we now use the letter p and you can think of it as "(factorization) parameter" or as momentum as you like. In the case we take the "factorization parameter" p to be: p = (p, sign(E)). (13.1) The general form as factorized space: The Hilbert space for the second quantized boson system can always be written like H = 0 Hp. (13.2) p In the example, where p = (p, sign(E)), the Hilbert space Hp is actually that of harmonic oscillator for which the number operator counts the number of particles with just the p-specification p. (Here we stepped too fast over the Dirac sea for bosons problem, but that is not so crucial just now; just think of antiparticles instead, when formally sign(E) < 0.) Dream of generalization(s) In the formulation as the Cartesian product H = 0 Hp (13.3) p one could dream about making a new - and perhaps interesting theory - by replacing the Hilbert spaces that are factors in the Cartesian product such as Hp by some Hilbert spaces with a different structure, e.g. different dimensionality. E.g. Could we decide that all these harmonic oscillators could only be excited up to their 7th level, after that it would not be possible to put more in with a given p ? 13 Second Quantization as Cross Product 225 We could of course postulate such a "theory" but it would be rather strange physically. A postulate of only up to 7 particles per p would violate locality In a big universe particles with the same momentum are so far from each other that one cannot from locality feel if there are more or less than 7 particles in the same momentum eigenstate. If we use x instead of p then locality would be automatic. If one thinks of a discretized (d-1)-space, i.e. really a (perhaps a bit irregular) lattice, and take the state of the universe to be described by the a state in the Hilbert space H, then factorization of the type H = 0 Hx (13.4) x i.e. where we as "factorization parameter" use the spatial position x - the lattice point, if discretized - this Cartesian product would be automatically suited for locality, one should just only provide it with local interaction, but could for the structure and operators acting on the single factors Hx be very free since everything would be o.k.. Usual second quantization for the Norma's spin-charge-family theory Once one has decided on the inner degrees of freedom, the statistics - fermion or boson - and of dimension of space time and thus of the dimension of the momentum vectors, one would than think that there is only one way to second quantize. This way will then turn out in the boson case to indeed be of the form that the full second quantized Hilbert space H takes the product form, and thus be written in the product way. However, if one starts by a product form and has not gotten it via the standard procedure, then we would feel a priori unsafe if this would be a physically meaningful way or not. It probably depends strongly on the details. A couple of trivialities on component numbers i. A Dirac (rather Weyl) massless spinor in an even number d of space time dimensions has 22-1 components. ii. In Norma's spin-charge-family theory ([3] and the references therein) there is not only the usual Dirac spin index with 2 2 -1 components, but a quite analogous family index again with the 22-1 components. So in this model the number of components could be marked by two Dirac indices, or instead using another but equivalent formalism with projection and nilpotent "operators". But in any case of these two formalisms the number of components for a full fermion particle is the square of the number for an ordinary Dirac construction. The number of components is therefore 2d-2. One can learn in Ref. [4,5] in this proceedings that: 226 N.S. Mankoc Borštnik and H.B.F. Nielsen a. Only operators of an odd character can offer the second quantization fermions. b. The operators of an odd character split into two parts, Hermitian conjugated to each other. iii. If we ignore momentum and look at one single momentum only, then the number of different states one could produce by having for this single momentum various possible numbers with the 2d-2 different components filled or unfilled would be 22 . Let us add that the rest of possibilities belong to either the Her-mitian conjugated partners or have the evenness Character and do not fulfil the anticommutation relations for fermions (and probably even not for bosons. In any case the number is much much more than the number of components. Standard second quantization procedure in factor language Before telling this standard procedure of quantizing fermions by the factorization into the Cartesian product of "subHilbert" spaces, we have to admit that one cannot do that without some essential modification, which we though postpone to discuss below in the section called "The problem of fermions". However, we are for the moment interested in reaching to the point, where we can see the problems when one attempts to make a new way of second quantizing by postulating some algebraic structure for the operators acting on the "subHilbert" spaces Hp going into the Cartesian product. For this problem presumably the statistics being fermion or boson statistics may however not matter so much, so our postponing is not so crucial for that. iv. Let us first look for a fixed momentum p and calculate which states are needed to describe the possibilities for filling with the allowed number of particles (up to one for fermions, and up to infinity for bosons) all the internal states. v. Then we construct the Hilbert space Hp,of which is just the number of different ways of filling particles into the different combinations of internal states. vi. Then finally you can take the Cartesian product and get the genuine Hilbert space for the full second quantized theory. , d_2 Standard way dim(Hp) = 22 for Norma's theory. Since there are (2d-2)2 component combinations, namely say 2d-2 genuine Dirac components, and 2d-2 family index values, there for assumed fermion-statistic 22 possibilities for filling or not filling these 2d-2 difference internal states. Thus the Hilbert-space for only one momentum should have the dimension d_2 dim (Hp) = 22 . (13.5) (Notice that this space Hp thus has a much bigger dimension than the space of single particle internal states, which has only dimension = 2d-2.) 13 Second Quantization as Cross Product 227 We ignored at first equations of motion. We have to modify the above simplified proposal by: vii. Notice that using the momentum energy relation E2 - p2 = 0 (13.6) we have for each (d-1)-momentum p two values for the energy E of the particle, so that we should let, as already mentioned, as a possibility p =(p,E), (13.7) meaning a doubling of the space of momenta to be used. viii. Let us take into account that the (free) equation of motion (the Dirac equation, the Weyl equation indeed) for a choice of energy E = iv^p2 only allow a subspace of the internal space of states for the (single) particle, (pM = 0. (13.8) Standard second quantization as product over (p, sign(E)). Letting an index emr denote that we have restricted the single particle sates to the states obeying the equations of motion (emr = "equation of motion restricted") we write the true standard second quantized Hilbert space Hemr = H(p,s1gn(E)),emr, (13.9) (p,sign(E)) where now H(p,s1gn(E)),emr is constructed from space of single particle internal states obeying the Dirac equation and having E = sign(E) v^p2, which because of the restriction by the equation of motion has only half the dimensionality of 2d/2-1 in the simple Dirac case or half of 2d-2 in the case with families. So dim(H(p,sign(E)),emr) = 22"-1 /2 = 2^ . (13.10) 13.2 The problem of fermions Yet a problem for Cartesian product form for fermions. For just constructing the Hilbert space we could claim that this Cartesian product procedure is o.k. even for fermions, but for the creation and annihilation operators or the field operators for fermions there is a problem more: If we take a true Cartesian product and let it be understood that the creation and annihilation operators for a state with (p, sign(E)) = p alone shall act on the Cartesian product factor Hp, then we cannot make suchfermion creation or annihilation operators for different p anticommute! Operators acting alone on different Cartesian product factors will namely always commute. 228 N.S. Mankoc Borštnik and H.B.F. Nielsen Suggested trick to solve the anticommutation problem: Use operators (—1 )Fp, where Fp is the fermion number for the fermions in the Cartesian factor Hp. That is to say to construct the "true creation or annihilation operators" -b"(i; p) or b(i; p) - for the p Cartesian factor we modify the truly "local ones", c" (i; p) and c(i; p) defined so as to only act on the Cartesian factor Hp, not touching the other factors, by multiplying it with a lot of factors of the form (—1 )Fp'. Associate in fact to each essentially momentum p a subset of this kind of essential momenta B(p) and define b"(i; p) = b(i; p) = n (—dfp' p 'eB(p) n (—dfp' p 'eB(p) c"(i; p) c(i;p) (13.11) (13.12) 13.3 Dream of Algebra Although we for fermions must introduce the modification from c"(i; p) to b"(i; p) in order to achieve the anticommutativity of the annihilation operators b(i; p), when we build up the Hilbert space construction from a Cartesian product, we might dream of using this Cartesian product idea to make a generalization of the algebra for the operators acting on one of these Hilbert spaces Hp (we could call them factor-Hilbert spaces) from which the Cartesian product is made up to a more general algebra, say F. That is to say we imagine an algebra F consisting of operators acting on the Hilbert space Hp. We can easily think of e.g. a couple operators/elements f, g G F, which e.g. anticommute {f, g}+ = 0. Of course we shall then have such algebra elements for every factor-Hilbert-space Hp, and correspondingly we should of course distinguish analogous algebra elements related to different factor-Hilbert-spaces or equivalently different p as we decided to enumerate these factor-Hilbert-spaces. That is to say we should write f(p) for the operator of a given structure in F when it acts on Hp. But now if we do not even make the modification of inserting the (—1 )Fp' -factors when in the ordering we had to have p and p' were in a certain relative order - say p' < p - then of course any f(p) and any g(p) at one p will commute with any f(p') and any g(p') at another "momentum" p' = p, independent of how f and g for the same p may happen to commute or anticommute. In other words we cannot prevent the commutation due to independent factor-Hilbert-spaces for the operators, what ever we take the local algebra to be, i.e. it does not modify this commutation to let the operators say anticommute locally, it does not help even if say {f(p), g(p)}+ = 0 to prevent [f(p), g(p')] = 0 for p = p'. 13 Second Quantization as Cross Product 229 13.3.1 Even with (—1 )Fp-factors Even if we improve our purely Cartesian product construction with the (—1 )Fp -factors as above, it will not bring us to get the commutation or anticommutation to progress from the "local" to the inter p commutator or anticommutator so easily. If we indeed include the type of factor (from (13.11,13.12)) being the product over the factors (—1 )Fp' for all p' which are say "smaller" in the ordering than the p considered, then we will achieve that we get anticommutation all operators g(p) say at p with all the ones at another place p' provided both operators carry a fermion number in the sense that they shift the value of the fermion number Fp for their factor Hilbert space by their action. So if e.g. two operators are fermionic in this quantum number F sense and even if they commuted when at the same site, they will anticommute when they are at different sites. If oppositely they anticommute locally they will again anticommute when at different sites(= different p's). The conclusion from the remarks just above should be: Using the starting point of the Cartesian product and only modifying by the extra factor of the type from equations (13.11,13.12) the commutation versus anticommutation of operators associated with different p-values depend alone on: a. the fermion number of the operators, b. from whether one introduce transformation (13.11,13.12) above at all or not. But it does not depend on on how the algebra elements considered may commute or not in the "local algebra" i.e. for the same p-value. 13.3.2 More generally: The above proposed method for making fermion-fields on the basis of a Cartesian product by means of an ordering of all the p-values is really not very attractive. In fact such an ordering does not match well with the topological structure of a momentum space or a position space except for the spatial dimension being dspaUai = 1. In higher dimensions you rather have to use the axiom of choice to even see that there exists such an ordering. We also need such a construction if we would like to make fermionization, and then this only by axiom of choice found ordering would not seem attractive at all either. So attempting to generalize this method of constructing fermion fields from a Cartesian product is highly called for. Now if there is in the theory some sort of gauge freedom one might not require quite as strict the properties of the extra factors introduced to convert the a priori commuting fields appearing from operators acting on different factor-Hilbert-spaces from (13.11.13.12). If one allows more freedom in the construction then one might optimistically hope to construct such factors to convert the boson-commuting operators into fermion ones to have some continuity and thus compatibility with the topology of a higher dimensional space(than just dimension =1). We here at first write down the type of transformation to be made to construct fermions from commuting fields in a general way. Then one may investigate how 230 N.S. Mankoc Borštnik and H.B.F. Nielsen much one needs to require about the multiplying factors U(p, p') converting the bosons to fermions so to speak. Unfortunately we have not come far in developing these conditions, but just the thought of looking at it more generally might turn out useful: bt(i; p) = b(i; p) = n u(p,p')1 p 'eB(p) n u(p,p ) p 'eB(p) ct(i; p) c(i; p) Not even crudely local unless the modification by U(x, x ) inessential. So there should preferably be a "gauge" transformation which could be the effect of the modification U(x, x') or "jump over correction"-replacement. Natural that the U(x,x') depends on the direction from x to x', and thus is a function of a point on thee sphere Sd-2. Also the 'gauge"like modifications must lie in a group G. So need map Sd-2 —» G. 13.3.3 Anyons To exercise constructing other statistics than bosons from the Cartesian product one would of course like to exercise with two spatial dimensions because this is the first case after the one spatial dimension case in which there are essentially no problem and fermionization is already well done. But now just 2 spatial dimensions is the interesting case in which also Leinaas Myhrheim or anyon statistics is possible[6]. With the suspicion of the gauge symmetries being important in allowing a more developed choice of the conversion factors U(p, p') a first exercise might be to even construct a system of anyons or first just a pair by electromagnetic ingredients. Acknowledgement The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory, the author H.B.N. thanks the Niels Bohr Institute for being allowed to staying as emeritus, both authors thank DMFA and Matjaz Breskvar of Beyond Semiconductor for donations, in particular for sponsoring the annual workshops entitled "What comes beyond the standard models" at Bled. 13 Second Quantization as Cross Product 231 Fig. 13.1. Anyons as electric magnetic made. References 1. N. Mankoc Borstnik, "Spin connection as a superpartner of a vielbein", Phys. Lett. B 292 (1992) 25-29. 2. N. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. of Math. Phys. 34 (1993) 3731-3745. 3. N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Rela-tivistic Dynamics of Particles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981, arXiv:1607.01618v2]. 4. N.S. Mankoc Borstnik and H.B. Nielsen, "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v1v2]. 5. N.S. Mankoc Borstnik, H.B.F. Nielsen, "Understanding the second quantization of fermions in Clifford and in Grassmann space" New way of second quantization offermions — Part I and Part II, Proceedings to the 22nd Workshop "What comes beyond the standard models", 6 - 14 of July, 2019, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana, December 2019. 6. Leinaas, Jon Magne; Myrheim, Jan (11 January 1977). "On the theory of identical particles" (PDF). Il Nuovo Cimento B. 37 (1): 1-23. Bibcode:1977NCimB..37....1L. doi:10.1007/BF02727953. Wilczek, Frank (4 October 1982). "Quantum Mechanics of Fractional-Spin Particles" (PDF). Physical Review Letters. 49 (14): 957-959. Bibcode:1982PhRvL..49..957W. doi:10.1103/PhysRevLett.49.957. Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 232) Bled, Slovenia, July 6-14, 2019 14 Novel String Field Theory and Remaining Problems * H.B.F. Nielsena and M. Ninomiyab aNiels Bohr Institute, Copenhagen University, 17 Blegdamsvej, Copenhagen 0, DK-2100 Denmark b Department of Physical Sciences, College of Science and Engineering, Ritsumeikan University, Shiga, 525 -8577 Abstract. We review our Novel String Field theory, and then it is pointed out very important remaining problem. Povzetek. Avtorja na kratko predstavita njuno Novo teorijo polj s strunami in obravnavata kljucen problem te teorije. Keywords: string field theory 14.1 Introduction It has been shown that Super string Theories loop corrections constructed in the theories do not lead to ultraviolet divergences, contrary to the conventional field theories. Nevertheless they have so good physical properties, e.g. revealing Regge trajectories. Thus they are considered serious candidates for the theory of Nature. From the point of view of the Novel String Field theory advocated by H.B. Nielsen and M. Ninomiya[1], they can considered the strings composite from an infinite number of what are called "objects" - to some extend similar to C. Thorn's string bits[2]. But they deviate by the fact that the objects correspond to a description of the right and left variables on the string t - ct and t + ct respectively, while C. Thorn rather discretizes the ct variable. We figure out that in the field theory typical diagrams of 2 particles 2 particles scattering processes in perturbation as Fig 14.1. If we took the particles to be closed strings the usual string theory formulation would lead to the corresponding string pictures in Fig 14.2. But it happens that if we consider the particles in Fig. 14.1 open strings, then in our formalism with cyclically ordered chains of objects, the second line could actually also represent the topological structure of the developments in our formalism (for open strings then). In the string theory the diagrams of the first few diagram of 2 closed strings —» 2 closed strings processes are given by Fig2 above. * Talk presented by H.B.N. at 22nd Bled workshop July 6. - 14., 2019, Bled, Slovenia 14 Novel String Field Theory and Remaining Problems 233 sC y \ X.or... A B W ib) (c) (cij Fig. 14.1. Feynman diagrams for scatterings; Incoming particles are denoted as A and B, while intermediate ones are denoted as X, Y, Z • • • respectively. Fig. 14.2. The topological structure of the developments in our formalism (for open strings). String theory actually avoids the ultraviolet divergences even in the loop corrections such as (d), corresponding to the loop corrections (c) and (d). In quantum field theory one has such divergences. However, in the string theories the loop corrections, e.g. (d') falls off exponentially with a squared of the external momenta expression. 14.2 Analogy with parton model. One can consider the string as composed objects of infinitely many constituents such as partons [3]. Thus they have Bjorken's [4] variable x = 0. In deep inelastic scattering one often uses the concept that a hadron (e.g. proton) is composed of partons as a bound state; see e.g. Fig 14.3. When Bjorken x is non-zero one can obtain for sufficiently high collision energy large transverse momenta — jets — for scattering of constituent partons with x = 0. Such scattering could again cause ultraviolet divergences, so to realize our analogy of getting rid of ultraviolet divergences for the bound, we should usu bound states with all Bjorken variables x = 0. 14.3 Some characteristic features of the novel string field theory Our novel SFT [1] is a kind of string-bit theory similar to that of C.Thorn [2]; but we use the right moving and left moving fields XR and XL respectively. And that each of them are functions of the variables t — ct and t + ct, contrary to those of Thorn who uses the genuine string parameter ct. 234 H.B.F. Nielsen and M. Ninomiya In deep ineLastic scattering one often uses the concept of a hadron / proton is composed of partons as a bound state: ' Total p of bound state, ( x--Bjorken x) Partons move for Large total momentum p with a fraction x*p. Fig. 14.3. The constituents i = 1, 2,.. carry the longitudinal momentum x_i*p where the longitudinal momentum of the total bound state is p. Thus our constituents objects are associated rather with wave packets running along the string back and forth. It turns out that our constituents equal to objects do not change at all. Thus scattering is exchange of objects rather than interpreted as collisions of the objects. Other aspects of our SFT [1] is the following: At first straight and resting string, you may produce a wave-packet in just one direction until it reflects at the end, and run back (see Fig 14.4 ). Shaking an at first straight and resting string you may prod uce a wave-packet moving in just ONE direction, untiLL it reflects at the end. Fig. 14.4. Producing a wave packet. The whole way around in a period would correspond to a run both forth and back and thus have the topology of S1: see Fig 14.5. 14.4 One of the great points in our novel SFT: Objects do not change We now stress that one of the great points of our novel SFT is that it corresponds to the fact that the wave packets run along the string without any change, we 14 Novel String Field Theory and Remaining Problems 235 A Little wave- packet of p hon ons wouLd run a Long the string, fjrst one way and then be reflected at an end and run back. The wole way arround in a period wouLd correspond to a run both forth and back and have the topology of Fig. 14.5. The topology of S1. arrange that our objects - which describe these wave structures as moving along -do not change in time at all. Thus our description of several string theory (= a string field theory has no development in the object formulation.The string theory is so to speak solved in terms of objects. This is the great hallmark of our novel SFT: Nothing moves. All scattering (etc.) is fake. To form the cyclically ordered chains of objects corresponding to moving forth and backward along the open string we need a cyclic ordering of a series of objects. We could describe that by a successor function f that is mapping one object to the next one in the chain: f(object1) = object1 + 2(modN) (14.1) (we only consider, due to a technical detail, with an even number in the cyclic chain series) 14.5 Conclusion and future outlook We have constructed a String Field theory called "Novel String Field theory" by using objects. The strings are in our theory considered as bound states of several objects. In our theory we can derive the Veneziano amplitude with recourse to exchange objects between incoming strings. However we did not introduce the possibility for objects to annihilate and create will be our subject to be investigated in our String Field theory. 236 H.B.F. Nielsen and M. Ninomiya \j Fig. 14.6. Cyclicaly ordered chains(—> indicates f map). Acknowledgement One of us (H.B.N.) acknowledges the Niels Bohr Institute for allowing him to work as emeritus. M. Ninomiya acknowledges the Niels Bohr Institute and Niels Bohr International academy for giving him very good hospitality during his stay. M.N. also acknowledges Yuri at college Sugawara Lab. of Science and Engineering, Department of physics sciences, Ritsumeikan University, Biwa Lake Campus for allowing him as a visiting Researcher. References 1. Novel SFT with also negative energy constituents/objects gives Veneziano Amplitude, H.B.Nielsen , Masao Ninomiya, arXiv: 1705.01739, JHEP Feb. 2018, 2018:97; An idea of newSFT -Liberating right and left movers -, Proc. of the 14th workshop, "What comes beyond the Standard Models" July 11 - 21, Bled, Slovenia (2011) arXiv:1309.2430; A novel SF solving string theory by liberating left and right movers, JHEP 05 (2014) 026 [arXiv 1211.1454]. 2. C. B. Giles and C. B. Thorn, A lattice approach to string theory, Phys.Rev. D 16 (1977) 366. C. B. Thorn, On the derivation of dual models from field theory, Phys. Lett. B 70 (1977) 85. O.Bergman and C. B. Thorn, String bit models for superstring. Phys.Rev. D 52 (1995) 5980; C. B. Thorn, Space from String bits, JHEP 11 (2014) 110[arXiv: 1407.8142] 3. P. Feynman, The behavior of hadron collisions, at extreme energies, in proceedings of High energy collisions: Third International Conference at Stony Brook, N.Y.. Gordon & Breach pp 237 -249. 4. J. Bjorken and E. Paschos, Inelastic electron -proton and y-proton scattering and structure of Nucleon, Phys. Rev. 185(5), 1975- 1982 (1969) Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 237) Bled, Slovenia, July 6-14, 2019 15 Local Temperature Distribution in the Vicinity of Gravitationally Bound Objects in the Expanding Universe * P.M. Petriakova ** and S.G. Rubin *** National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Kashirskoe shosse 31, Moscow, Russia Abstract. We consider a cluster of Primordial Black Holes which is decoupled from the cosmological expansion (Hubble flow) and this region is heated as compared to the surrounding matter. The increased temperature inside the region can be explained by several mechanisms of Primordial Black Holes formation. We study the temperature dynamics of the heated region of Primordial Black Holes cluster. Povzetek. Avtorja obravnavata gruco prvotnih črnih lukenj, ki ni sklopljena s kozmolosko širitvijo (Hubblovim tokom) v obmocju segretem glede na snov, ki obmocje obdaja. Povisano temperaturo lahko pojasnita z vec mehanizmi nastanka prvotnih crnih lukenj. Obravnavata gibanje temperature segretega obmocšja grucše prvotnih cšrnih lukenj. Introduction The idea of the Primordial Black Holes (PBH) formation was predicted five decades ago [1]. Although they have not yet been identified in observations but some astrophysical effects can be attributed to PBH: supermassive black holes in early quasars. Therefore till now, PBH give information about processes in the Early Universe only in the form of restrictions on the primordial perturbations [2] and on physical conditions at different epochs. It is important now to describe and develop in detail models of PBH formation and their possible effects in cosmology and astrophysics. There are several models of PBH formation. PBH can be formed during the collapses of adiabatic (curvature) density perturbations in relativistic fluid [3]. They could be formed as well at the early dust-like stages [4] and rather effectively on stages of dominance of dissipative superheavy metastable particles owing to a rapid evolution of star-like objects that such particles form [5]. There is also an exciting model of PBH formation from the baryon charge fluctuations [6]. Another set of models uses the mechanism of domain walls formation and evolution with the subsequent collapse [7]. Quantum fluctuations of a scalar field near a potential * Talk presented by P. Petriakova ** E-mail: petriakovapolina@gmail.com *** E-mail: sergeirubin@list.ru 238 P.M. Petriakova and S. G.Rubin maximum or saddle point during inflation lead to the formation of closed domain walls [12]. After the inflation is finished, the walls could collapse into black holes in the final state. There is a substantial amount of the inflationary models containing a potential of appropriate shape. The most known examples are the natural inflation [8] and the hybrid inflation [9] (and its supergravity realization [10]). The landscape string theory provides us with a wide class of the potentials with saddle points, see review [11] and references within. Heating of the surrounding matter is the inherent property of the domain wall mechanism of PBH cluster formation. While collapsing the domain wall partially transfers its kinetic energy to the ambient matter. It would allow to distinguish different models by observations. 15.1 The first Chapman-Enskog approximation According to the discussion above, PBH are gathering into the clusters with heated media inside them. It is assumed that after decoupling from the cosmological expansion the temperature of gas inside the cluster and its density is higher than that around the cluster. These factors can ignite a new chain of nuclear reactions changing chemical composition of the matter in given region. We are going to study the rate of temperature spreading into surrounding space and the temperature distribution within the cluster. The temperature dynamics is described by the appropriate equations in the framework of the Chapman-Enskog procedure. The Chapman-Enskog method [13] makes it possible to obtain a solution to the transport equation and it can be applied to the relativistic transport equation in general case. The applicability condition of this method: macroscopic wavelengths should be significantly greater than the mean free path. This excludes the propagation velocity that is faster than the thermal velocity of particles [14]. Using this method, we can find linear laws for flows, thermodynamic forces and expressions for transfer coefficients based on the solution of linearized transfer equation. After that we apply this linear laws to continuity, energy and motion equations. This leads to the relativistic Navier-Stokes equations which form a closed system for hydrodynamic variables. In the first approximation various irreversible flows are linearly related to non-uniformities present in the system. In this case the relativistic generalization of the Fourier-law for the heat flux and the linear expression for the viscous pressure tensor has the form (c = h = kB = 1) T hn nqv = +nvA^vvffuff (15.2) Iq = A(VT - hnv^p) (15.1) A - the heat conductivity, n - the shear viscosity, nv - the volume viscosity, V^ = A^v9v , A^v = g^v - u^uY and this operator acts as a projector: A^vuv = 0. They are designed to select two hydrodynamic four-velocity expressions proposed by Eckart and Landau-Lifshitz. We will use the definition of Eckart [15] which relates the hydrodynamic four-velocity directly to the particle four-flow N^ N^ u^ = , . (15.3) VNv Nv 15 Local Temperature Distribution in the Vicinity of Gravitationally... 239 The relativistic equation of motion and equation of energy are given by [16] hnDu^ = V^p-A£VffnVff + (hn)-1 Vvp- (15.4) -(AVDI^ + I£VvUv + I^Vv u^) nDe = —pV^u^ + - + 2I^Du^. (15.5) After linearization, the energy equation is reduced to DT_ 1 T cv V„u^ - - fv2T - — V2p u p V hn (15.6) where we have taken into account the linear laws (15.1) and (15.2), V2 stands for V2 = V^V^ and D = u^. If the hydrodynamic four-velocity is constant and p = nT (we will see it in the next section) the energy equations reduce to the relativistic heat-conduction equation: ncvDT = -a( V2T - hTn V2p). (15.7) 15.2 The thermodynamic values The equilibrium distribution function with no external fields takes the form of the Juttner distribution function 1 — puu. (2 n f(p) = ^Ta exp( T " )• (15.8) It allows to calculate the particle four-flow in equilibrium 1 N^ = (2n)3 d3P U (V — P^MR OA -pp-pu exp^—t— J • (15.9) The Juttner distribution function outlines one direction in space-time. As a result, it must be proportional to four-velocity, where the proportionally factor of this relation is the particle density n 2 n puuuexp^—t—) • (15.10) The integral is a scalar and it can be calculated at selected u^ = (1,0,0,0). This result can be expressed in the modified Bessel function of the second kind 4nm2T f m^ (2n) n = -^r M T expi T ) . (15.11) 3 240 P.M. Petriakova and S. G.Rubin We can obtain the equilibrium pressure following the same reasoning to calculate the energy-momentum tensor in equilibrium: 11 p = — f=— 3 4nm2T2 / m\ / u\ -- M m exp( = nT. rd-^p^pv A^f(p) = (15.12) (2n) 3 V TT rVT Hence, if we identify T with the temperature of the system the standard scheme of thermodynamics could be clearly seen. Using recurrence relation for the modified Bessel function of the second kind and taking into account particle density the expression has the form K3 (m/T) e = mK^mTT)— T. (15.13) Considering the result of (15.12) for pressure we can find the enthalpy per particle - K-jm/T) TXT K3 (m/T) h =e+pn = mK-(m7TT—T+T = mK^rnTT). (15.14) The heat capacity per particle at constant volume by definition cv = 3e/3T. (15.15) We can get asymptotic behaviour of these values for large arguments of the modified Bessel function of the second kind (which corresponds to the case of low temperatures) and for small arguments (which corresponds to the case of massless particles). For small values of temperature we have the asymptotic ratio for large arguments (w = m/T): 1 pK Kn (w) ~ — \ -— J ew V2w 4n2 — 1 (4n2 — 1)(4n2 — 9) 1 + 8w + 2!(8w)2 +... (15.16) It allows to obtain the enthalpy per particle: 5 15 T2 h = m +-T + ^ —+ ... (15.17) 2 8 m and to derive the caloric equation of state of relativistic perfect gas (15.13) and the heat capacity per particle at constant volume (15.15): 3 15 T2 e = m +-T + ^ —+ ... (15.18) 2 8 m 9e 3 15 T cv ... (15.19) oT 2 4 m Massless particles are essential in relativistic kinetic theory. For this purpose we should expand our formulas in this special case. The results can be obtained 15 Local Temperature Distribution in the Vicinity of Gravitationally... 241 by taking the limit in m —» 0 with the asymptotic relation for the modified Bessel function of the second kind: lim wnKn(w) = 2n-1(n- 1)! (15.20) e = 3T , h = 4T , cv = 3 . (15.21) In this case, with the caloric equation of state of relativistic gas and p = nT we can obtain well-known expression for pressure for massless particles: p = en/3. We can find Fourier differential equation of the heat conduction in the non-relativistic case. Following expression (15.17) in the case of low temperatures (T C m) and considering the ratio p = nT we obtain ncvDT = -A (V2T - hnv2p) ~ -Ay2T. (15.22) In this case, the heat-conduction equation allows an infinite propagation velocity. Although this feature is already present in the non-relativistic theory in the relativistic theory it becomes a paradox: the thermal disturbances can not propagate faster than the speed of light. This paradox is easily resolved in the framework of the Chapman-Enskog procedure. In fact the restriction inherent in the Chapman-Enskog method (the macroscopic wave lengths has to be much greater than the mean free path) prevents the existence of propagation velocities faster than the thermal velocity of particles. 15.3 Thermal equilibrium We should check the applicability of our results by estimating to what extent the electron-proton-photon plasma is close to kinetic equilibrium before and during recombination. All our previous calculations were made under the assumption that the distribution functions have equilibrium form and all components have the same temperature equal to the photon temperature. To make sure that the temperature of electron-proton component coincides with the photon temperature we have to study the following effect. The effective temperature of photons would decrease in time slower than that of electrons and protons. Thus we have to check that energy transfer from photons to electrons and protons is sufficiently fast. Electrons get energy from photons via Compton scattering process that occurs with Thomson cross section. The time between two subsequent collisions of a given electron with photons is 1 t =--(15.23) nyoj here oT - the Compton cross section and nY - the number density of photons. For energy transfer the time te in which an electron obtains kinetic energy of the same order of magnitude as temperature due to the Compton scattering should be found. We note that the typical energy transfer in a collision of a slow electron with a low energy photon is actually suppressed for estimation of this time. The 242 P.M. Petriakova and S. G.Rubin estimation for number of scattering events needed to heat up a moving electron is given by [17] / T \ 2 me N ^- —. (15.24) We can obtain the time of electron heating [17] me te(t) - nt(t)---—. (15.25) ny(TJctjT At the moment of recombination TE(TTec) — 6 yrs. It is much smaller than the Hubble time and energy transfer from photons to electrons is efficient. Thus electrons and protons have the photon temperature. What about the heating of protons? Doing the same procedure (with mp substituted for me in (25)) we obtain that process of direct interaction of proton with photons is irrelevant. Since the Thomson cross section is proportional to m-2 the time for protons is larger by a factor (mp/me )3 and this time is larger than the Hubble time. Energy transfer to protons occurs due to elastic scattering of electrons off protons. The energy transfer time [17] is memp / 3T \ 3/2 te (T) - ^-m e2 , " . , — (15.26) 16nne(T)a2 ln(6TrD/a)\mey here rD = J --and during recombination TE(TTec) - 104 s and this time is y4nnea very small compared to the Hubble time at recombination. The estimation done for protons is valid for electrons as well with me substituted for mp and numeric factor. This means that electrons and protons have equilibrium distribution functions with temperature equal to photon temperature. 15.4 Transport coefficients The divergence of the collision integrals is the main difficulty encountered when applying the transport equation to plasma. The many particle correlations which provide the Debye shielding are not included in the transport equation due to the long range nature of electromagnetic interaction. In the Standard Model of the Universe Compton scattering between photons and electrons was the dominant mechanism for energy and momentum transfer in the radiation-dominated era (RD-stage). It seems worthy to present a quantitative description of the non-equilibrium processes that can be expected in a hot photon gas coupled to plasma by Compton scattering. In case of low temperatures we have the following expression for heat conductivity [16] 4xY 1 A = - (15.27) 5Xe 0"T 15 Local Temperature Distribution in the Vicinity of Gravitationally... 243 here x - the fraction of particles, ctt - the Compton cross section and the ratio of electron and photon number densities through baryon-to-photon ratio with electric neutrality of the Universe nB o ne = — = 0.6 10-9 . (15.28) 15.5 Dependence of the equation on the rate of expansion of space We should set the form of operators included in the equation (15.7). If the matter of the surrounding space is stationary as a whole then the four-velocity takes the form U|x = (1,0,0,0) hence D = u^ = 3t. We need to make the following substitution: V1 = A|V3V —» A|VDV in order to take into account the expansion of space. For a scalar field covariant differentiation is simply partial differentiation: = 9 ^ for a covariant vector we have: D|Av = 3|AV - r^A a and for a contravariant vector: D|AV — 3|AV + H^A0) The Christoffel symbols of the second kind: r^V — gaCT(gVCT + 3Vg|CT — 3a9|v)/2. Thus our operator V2 is explicitly dependent on the metric V2 — V^V| — V^9|vVv — A|VDvg|vAvk3k . (15.29) The rate of temperature spreading into the surrounding space will be calculated with respect to the Friedmann-Lemaitre-Robertson-Walker metric. The metric tensor in this case has the form giv — diag(1, —a2(t), —a2(t)r2, —a2(t)r2 sin2 9). (15.30) The scale factor a(t) can be found from Friedmann equations (cA 8n dp p aJ — 8nGP, da— —3£cc (15.31) here £ — 4/3 (1) for RD-stage (MD-stage). Finally we get the following dependence for scale factor a(t) = /8nGpo 1 + TV 3°(t - to) 2/3f, (15.32) obtained under the conditions a(t0) = 1, p(t0) = 0.53 • 10 5 GeV/cm3, t0 ~ 14 • 109 yrs- the age of Universe. 15.6 Final statement of the problem and result of calculation We consider spherical symmetry for simplicity. The heat-conduction equation (15.7) with expression (15.12) for pressure and in case of stationary matter takes the form ncv 3 / \ T (r, t) -, ^T 3l(T (r,t^ = -y2T (r,t) + t^ V2nT (r,t) (15.33) 244 P.M. Petriakova and S. G.Rubin with boundary conditions ( 3 = 0, 3 ^T (r,t) 3r or r=o _ (15.34) -r ( w ' out T (r,t)|T=TO = — here the dependence for scale factor a(t) is taken from (15.32). The initial condition is T (r, 0) = Tin exp(-r2/r0) + Tout (15.35) here Tin and Tout - temperatures of matter inside cluster and the surrounding space respectively, r0- temperature distribution parameter. In general the obtained expressions can also be used in calculations at the RD-stage (stage of radiation dominance) and the MD-stage (stage of the matter domination). For this purpose the expression for the scale factor (15.32) should be taken at different £ and with modified heat conductivity. Presumably the cluster of primordial black holes virializes at the end of the RD-stage. It makes sense to estimate its cooling before the end of this stage. We need to choose specific values of the following parameters: • temperature distribution parameter r0 = 1 pc; • temperature inside the area Tin = 100 keV; • temperature of the surrounding space Tout = 1 keV; • dependence a(t) in boundary condition is selected for RD-stage; • for enthalpy and heat capacity we should select forms in case of low temperatures (15.17) and (15.19) accordingly. Using numerical simulation in MAPLE by the BackwardEuler method with the interval of spatial points on a discrete grid 1/60 we have Fig.15.1. As can be seen from the figure, the gravitationally bound region almost completely retains temperature which was obtained during the formation at the RD-stage. The next step is to determine what happens with this heated region at the MD-stage. 15.7 Estimation for MD-stage We will be interested in the internal temperature of the gravitationally bound region during the MD-stage. At the end of the RD-stage we have a region with higher temperature. It is possible to ignite a new chain of nuclear reactions changing chemical composition of the matter in given region. The temperature inside the cluster can be calculated in Minkowski space and we can find the dependence of the thermal conductivity on temperature in the non-relativistic case. The thermal diffusivity by definition is given by 1 À Texe 3.16 ,, X =-= 3.16-^ -Te5/2 (15.36) neCv mecv 2V2n/meqeZne(T) ln(6TrD/a) e 1 Here the values are expressed in the CGS system and the temperature in eV 15 Local Temperature Distribution in the Vicinity of Gravitationally... 245 The thermal diffusivity in pc year is X(T ) 2.3 ■ 10-14 A^eieV) (15.37) The calculated value allows to retain the increased temperature inside the cluster until the recombination starts. The heat is conserved within a region starting from the moment of its formation. Thus, there are significant prerequisites for anomalies in the chemical composition of this region which makes sense to consider in future. Conclusion We investigated the temperature dynamics of the heated region around the primordial black holes cluster. For this purpose the relativistic heat-conduction equation (without convective terms) was considered taking into account the expansion of space in the framework of the Chapman-Enskog relativistic procedure. The numerical solution was found with the corresponding initial and boundary conditions. According to our calculations, the gravitationally bound region almost completely retains temperature which was obtained during the formation. At the MD-stage the increased temperature inside the cluster is conserved until then recombination will start. Thus, there are significant prerequisites for anomalies in the chemical composition of this region. In prospect, we are going to study possible anomalies in the chemical content of the region with comparison to the observed data. Acknowledgement The authors are grateful to K. Belotsky and A. Kirillov for helpful discussions. The work of S.G.R. is supported by the grant RFBR (N 19-02-00930) and is performed according to the Russian Government Program of Competitive Growth of Kazan Fig. 15.1. Numerical calculation results of local temperature distribution at the stage of radiation dominance. 246 P.M. Petriakova and S. G.Rubin Federal University. The work was also supported by the Ministry of Education and Science of the Russian Federation, MEPhI Academic Excellence Project (contract N 02.a03.21.0005, 27.08.2013). References 1. Y. B. Zel'dovich and I. D. Novikov, The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model, Sov. Astronomy 10 (1967) 602. 2. A. S. Josan, A. M. Green and K. A. Malik: Generalized constraints on the curvature perturbation from primordial black holes, Phys. Rev. D 79 (2009) 103520. 3. B. J. Carr, The primordial black hole mass spectrum, Astrophys. J. 201 (1975) 1-19. 4. M. Y. Khlopov and A. G. Polnarev, Primordial black holes as a cosmological test of grand unification, Phys. Lett. B 97 (1980) 383-387. 5. O. K. Kalashnikov and M. Y. Khlopov, On the possibility of a test of the cosmology of asymptotically free SU(5) theory, Phys. Lett. B 127 (1983) 407-412. 6. A. Dolgov and J. Silk, Baryon isocurvature fluctuations at small scales and baryonic dark matter, Phys. Rev. D 47 (1993) 4244-4255. 7. V. A. Berezin, V. A. Kuzmin and 1.1. Tkachev, Thin-wall vacuum domain evolution, Phys. Lett. B 120 (1983) 91-96. 8. K. Freese, J. A. Frieman and A. V. Olinto, Natural inflation with pseudo Nambu-Goldstone bosons, Phys. Rev. Lett. 65 (1990) 3233-3236. 9. J. E. Kim, H. P. Nilles and M. Peloso, Completing natural inflation, J. Cosmol. Astropart. Phys. 1 (2005) 005, [hep-ph/0409138]. 10. A. Linde and A. Riotto, Hybrid inflation in supergravity, Phys. Rev. D 56 (1997) R1841-R1844, [hep-ph/9703209]. 11. D. Lust, Seeing through the string landscape - a string hunter's companion in particle physics and cosmology, J. High Energ. Phys. 3 (2009) 149, [0904.4601]. 12. K. M. Belotsky, V. I. Dokuchaev, Y. N. Eroshenko, E. A. Esipova, M. Yu. Khlopov, L. A. Khromykh, A. A. Kirillov, V. V. Nikulin, S. G. Rubin, I. V. Svadkovsky: Clusters of primordial black holes, Eur. Phys. J., C79 no.3, 246 (2019) [1807.06590]. 13. S. Chapman and T. G. Cowling, Cambridge University Press (1970). 14. H. D. Weymann, Amer. J. Phys., 488 (1967). 15. C. Eckart, Phys. Rept., 919 (1929). 16. S. R. Groot, W. A. Leewen, Ch. G. Weert: Relativistic kinetic theory. Principles and Applications, "North-Holland", 1980. 17. D. S. Gorbunov, V. A. Rubakov: Introduction to the theory of the Early Universe: Hot Big Bang Theory, "New Jersey: World Scientific", 2017. Virtual Institute of Astroparticle Physics Presentation Bled Workshops in Physics Vol. 20, No. 2 A Proceedings to the 22nd Workshop What Comes Beyond ... (p. 249) Bled, Slovenia, July 6-14, 2019 16 The Platform of Virtual Institute of Astroparticle Physics for Studies of BSM Physics and Cosmology M.Yu. Khlopov 1'2'3'4 1 Centre for Cosmoparticle Physics "Cosmion" 2 National Research Nuclear University "Moscow Engineering Physics Institute", 115409 Moscow, Russia 3 APC laboratory 10, rue Alice Domon et Leonie Duquet 75205 Paris Cedex 13, France 4 Institute of Physics, Southern Federal University Stachki 194, Rostov on Don 344090, Russia Abstract. Being a unique multi-functional complex of science and education online, Virtual Institute of Astroparticle Physics (VIA) operates on website http://viavca.in2p3.fr/site.html. It supports presentation online for the most interesting theoretical and experimental results, participation online in conferences and meetings, various forms of collaborative scientific work as well as programs of education at distance, combining online videoconferences with extensive library of records of previous meetings and Discussions on Forum. Since 2014 VIA online lectures combined with individual work on Forum acquired the form of Open Online Courses. Aimed to individual work with students the Course is not Massive, but the account for the number of visits to VIA site converts VIA in a specific tool for MOOC activity. VIA sessions are now a traditional part of Bled Workshops' programme. At XXII Bled Workshop they involved not only remote presentations but also online streaming of most of the talks and discussions, supporting world-wide propagation of the main ideas, presented at this meeting. Special VIA sessions were dedicated at the XXII Bled Workshop to scientific debuts of students. Povzetek. Virtual Institute of Astroparticle Physics (VIA, http://viavca.in2p3.fr/site.html), ponuja direktne predstavitve najbolj zanimivih in aktualnih teoretičnih spoznanj ter eksperimentalnih rezultatov, odprtih diskusij na konferencah, delavnicah, videokonferencah in drugih srecanjih, ponuja tudi izobraŽevanje preko spleta. Na svoji spletni strani hrani zapis vseh predavanj, diskusij in drugih dogodkov. Po letu 2014 so postala predavanja z odrtimi diskusijami na daljavo tudi izobrazevalni tecaji, ki vkljucujejo tudi individualno mentorstvo za posamezne studente (MOOC). Seje VIA so postale tradicionalen del programa vsakoletnih Blejskih delavnic z nalovom "Kako preseci oba standardna modela, elektrosibkega in barvnega ter kozmoloskega". Ponujajo ne le ziv (sproten) prenos predavanj in diskusij, ampak omogocajo raziskovalcem iz vseh laboratorijev po svetu, ki jih teme delavnice zanimajo, komentarje, vprasanja. 22. delavnica je ponudila podoktorskim studentom na posebni seji prve predstavitve raziskovalnih spoznanj. Keywords: astroparticle physics, physics beyond the Standard model, e-learning, e-science, MOOC 250 M.Yu. Khlopov 16.1 Introduction Studies in astroparticle physics link astrophysics, cosmology, particle and nuclear physics and involve hundreds of scientific groups linked by regional networks (like ASPERA/ApPEC [1,2]) and national centers. The exciting progress in these studies will have impact on the knowledge on the structure of microworld and Universe in their fundamental relationship and on the basic, still unknown, physical laws of Nature (see e.g. [3,4] for review). The progress of precision cosmology and experimental probes of the new physics at the LHC and in nonaccelerator experiments, as well as the extension of various indirect studies of physics beyond the Standard model involve with necessity their nontrivial links. Virtual Institute of Astroparticle Physics (VIA) [5] was organized with the aim to play the role of an unifying and coordinating platform for such studies. Starting from the January of 2008 the activity of the Institute takes place on its website [6] in a form of regular weekly videoconferences with VIA lectures, covering all the theoretical and experimental activities in astroparticle physics and related topics. The library of records of these lectures, talks and their presentations was accomplished by multi-lingual Forum. Since 2008 there were 207 VIA online lectures, VIA has supported distant presentations of 132 speakers at 27 Conferences and provided transmission of talks at 74 APC Colloquiums. In 2008 VIA complex was effectively used for the first time for participation at distance in XI Bled Workshop [7]. Since then VIA videoconferences became a natural part of Bled Workshops' programs, opening the virtual room of discussions to the world-wide audience. Its progress was presented in [8-17]. Here the current state-of-art of VIA complex, integrated since 2009 in the structure of APC Laboratory, is presented in order to clarify the way in which discussion of open questions beyond the standard models of both partcile physics and cosmology were presented at the XXII Bled Workshop with the of VIA facility to the world-wide audience. Active involvement of young scientists in VIA sessions and discussions and VIA streaming of virtually all the talks were specific new features of VIA activity at XXII Bled Workshop. 16.2 VIA structure and activity 16.2.1 VIA activity The structure of the VIA complex is illustrated by the Fig. 16.1. The home page, presented on this figure, contains the information on the coming and records of the latest VIA events. The upper line of menu includes links to directories (from left to right): with general information on VIA (About VIA); entrance to VIA virtual rooms (Rooms); the library of records and presentations (Previous), which contains records of VIA Lectures (Previous Lectures), records of online transmissions of Conferences (Previous Conferences), APC Colloquiums (Previous APC Colloquiums), APC Seminars (Previous —> APC Seminars) and Events (Previous Events); Calender of the past and future VIA events (All events) and VIA Forum (Forum). In the upper right angle there are links to Google search engine 16 The Platform of Virtual Institute of Astroparticle Physics... 251 Next regular Lecture's 11 November8, 2019 16h - 17h Paris Lecture by Arthur-George Suvorov Title of lecture: "Neutron star superspace: quantifying the difference between stellar structures in general relativity and beyond" Language of lecture: GW170814-: A three-detector observation of gravitational waves from a, binary black hole coal See the Article Previous Lecture 11 Lecture by Igor Nikitin "White holes" See All presentations results lá See the Article GW170104-: Observation of a 50-solar-mass binary black hole coalescence at redshift0.2 See the Article XENON1T Experiment See the Article Beyond the standard model Astropartide physics Cosmology Gravitational waves experiments Astrophysics cosmovia.org in the News Applications Facebook Partners of VIA Contact î» UNIVERSITÉ ? DE GENÈVE Fig. 16.1. The home page of VIA site 252 M.Yu. Khlopov (Search in site) and to contact information (Contacts). The announcement of the next VIA lecture and VIA online transmission of APC Colloquium occupy the main part of the homepage with the record of the most recent VIA events below. In the announced time of the event (VIA lecture or transmitted APC Colloquium) it is sufficient to click on "to participate" on the announcement and to Enter as Guest (printing your name) in the corresponding Virtual room. The Calender shows the program of future VIA lectures and events. The right column on the VIA homepage lists the announcements of the regularly up-dated hot news of Astroparticle physics and related areas. In 2010 special COSMOVIA tours were undertaken in Switzerland (Geneva), Belgium (Brussels, Liege) and Italy (Turin, Pisa, Bari, Lecce) in order to test stability of VIA online transmissions from different parts of Europe. Positive results of these tests have proved the stability of VIA system and stimulated this practice at XIII Bled Workshop. The records of the videoconferences at the XIII Bled Workshop are available on VIA site [18]. Since 2011 VIA facility was used for the tasks of the Paris Center of Cos-mological Physics (PCCP), chaired by G. Smoot, for the public programme "The two infinities" conveyed by J.L.Robert and for effective support a participation at distance at meetings of the Double Chooz collaboration. In the latter case, the experimentalists, being at shift, took part in the collaboration meeting in such a virtual way. The simplicity of VIA facility for ordinary users was demonstrated at XIV Bled Workshop in 2011. Videoconferences at this Workshop had no special technical support except for WiFi Internet connection and ordinary laptops with their internal webcams and microphones. This test has proved the ability to use VIA facility at any place with at least decent Internet connection. Of course the quality of records is not as good in this case as with the use of special equipment, but still it is sufficient to support fruitful scientific discussion as can be illustrated by the record of VIA presentation "New physics and its experimental probes" given by John Ellis from his office in CERN (see the records in [19]). In 2012 VIA facility, regularly used for programs of VIA lectures and transmission of APC Colloquiums, has extended its applications to support M.Khlopov's talk at distance at Astrophysics seminar in Moscow, videoconference in PCCP, participation at distance in APC-Hamburg-Oxford network meeting as well as to provide online transmissions from the lectures at Science Festival 2012 in University Paris7. VIA communication has effectively resolved the problem of referee's attendance at the defence of PhD thesis by Mariana Vargas in APC. The referees made their reports and participated in discussion in the regime of VIA videoconference. In 2012 VIA facility was first used for online transmissions from the Science Festival in the University Paris 7. This tradition was continued in 2013, when the transmissions of meetings at Journees nationales du Developpement Logiciel (JDEV2013) at Ecole Politechnique (Paris) were organized [21]. In 2013 VIA lecture by Prof. Martin Pohl was one of the first places at which the first hand information on the first results of AMS02 experiment was presented [20]. 16 The Platform of Virtual Institute of Astroparticle Physics... 253 In 2014 the 100th anniversary of one of the foundators of Cosmoparticle physics, Ya. B. Zeldovich, was celebrated. With the use of VIA M.Khlopov could contribute the programme of the "Subatomic particles, Nucleons, Atoms, Universe: Processes and Structure International conference in honor of Ya. B. Zeldovich 100th Anniversary" (Minsk, Belarus) by his talk "Cosmoparticle physics: the Universe as a laboratory of elementary particles" [22] and the programme of "Conference YaB-100, dedicated to 100 Anniversary of Yakov Borisovich Zeldovich" (Moscow, Russia) by his talk "Cosmology and particle physics" [23]. In 2015 VIA facility supported the talk at distance at All Moscow Astrophysi-cal seminar "Cosmoparticle physics of dark matter and structures in the Universe" by Maxim Yu. Khlopov and the work of the Section "Dark matter" of the International Conference on Particle Physics and Astrophysics (Moscow, 5-10 October 2015). Though the conference room was situated in Milan Hotel in Moscow all the presentations at this Section were given at distance (by Rita Bernabei from Rome, Italy; by Juan Jose Gomez-Cadenas, Paterna, University of Valencia, Spain and by Dmitri Semikoz, Martin Bucher and Maxim Khlopov from Paris) and its work was chaired by M.Khlopov from Paris [28]. In the end of 2015 M. Khlopov gave his distant talk "Dark atoms of dark matter" at the Conference "Progress of Russian Astronomy in 2015", held in Sternberg Astronomical Institute of Moscow State University. In 2016 distant online talks at St. Petersburg Workshop "Dark Ages and White Nights (Spectroscopy of the CMB)" by Khatri Rishi (TIFR, India) "The information hidden in the CMB spectral distortions in Planck data and beyond", E. Kholupenko (Ioffe Institute, Russia) "On recombination dynamics of hydrogen and helium", Jens Chluba (Jodrell Bank Centre for Astrophysics, UK) "Primordial recombination lines of hydrogen and helium", M. Yu. Khlopov (APC and MEPHI, France and Russia)"Nonstandard cosmological scenarios" and P. de Bernardis (La Sapiensa University, Italy) "Balloon techniques for CMB spectrum research" were given with the use of VIA system [29]. At the defense of PhD thesis by F. Gregis VIA facility made possible for his referee in California not only to attend at distance at the presentation of the thesis but also to take part in its successive jury evaluation. Since 2018 VIA facility is used for collaborative work on studies of various forms of dark matter in the framework of the project of Russian Science Foundation based on Southern Federal University (Rostov on Don). In September 2018 VIA supported online transmission of 17 presentations at the Commemoration day for Patrick Fleury, held in APC [30]. The discussion of questions that were put forward in the interactive VIA events is continued and extended on VIA Forum. Presently activated in En-glish,French and Russian with trivial extension to other languages, the Forum represents a first step on the way to multi-lingual character of VIA complex and its activity. Discussions in English on Forum are arranged along the following directions: beyond the standard model, astroparticle physics, cosmology, gravitational wave experiments, astrophysics, neutrinos. After each VIA lecture its pdf presentation together with link to its record and information on the discussion during it are put in the corresponding post, which offers a platform to continue discussion in replies to this post. 254 M.Yu. Khlopov 16.2.2 VIA e-learning, OOC and MOOC One of the interesting forms of VIA activity is the educational work at distance. For the last eleven years M.Khlopov's course "Introduction to cosmoparticle physics" is given in the form of VIA videoconferences and the records of these lectures and their ppt presentations are put in the corresponding directory of the Forum [24]. Having attended the VIA course of lectures in order to be admitted to exam students should put on Forum a post with their small thesis. In this thesis students are proposed to chose some BSM model and to study the cosmological scenario based on this chosen model. The list of possible topics for such thesis is proposed to students, but they are also invited to chose themselves any topic of their own on possible links between cosmology and particle physics. Professor's comments and proposed corrections are put in a Post reply so that students should continuously present on Forum improved versions of work until it is accepted as admission for student to pass exam. The record of videoconference with the oral exam is also put in the corresponding directory of Forum. Such procedure provides completely transparent way of evaluation of students' knowledge at distance. In 2018 the test has started for possible application of VIA facility to remote supervision of student's scientific practice. The formulation of task and discussion of progress on work are recorded and put in the corresponding directory on Forum together with the versions of student's report on the work progress. Since 2014 the second semester of the course on Cosmoparticle physics is given in English and converted in an Open Online Course. It was aimed to develop VIA system as a possible accomplishment for Massive Online Open Courses (MOOC) activity [25]. In 2016 not only students from Moscow, but also from France and Sri Lanka attended this course. In 2017 students from Moscow were accompanied by participants from France, Italy, Sri Lanka and India [26]. The students pretending to evaluation of their knowledge must write their small thesis, present it and, being admitted to exam, pass it in English. The restricted number of online connections to videoconferences with VIA lectures is compensated by the wide-world access to their records on VIA Forum and in the context of MOOC VIA Forum and videoconferencing system can be used for individual online work with advanced participants. Indeed Google Analytics shows that since 2008 VIA site was visited by more than 242 thousand visitors from 153 countries, covering all the continents by its geography (Fig. 16.2). According to this statistics more than half of these visitors continued to enter VIA site after the first visit. Still the form of individual educational work makes VIA facility most appropriate for PhD courses and it is planned to be involved in the International PhD program on Fundamental Physics, which can be started on the basis of Russian-French collaborative agreement. In 2017 the test for the ability of VIA to support fully distant education and evaluation of students (as well as for work on PhD thesis and its distant defense) was undertaken. Steve Branchu from France, who attended the Open Online Course and presented on Forum his small thesis has passed exam at distance. The whole procedure, starting from a stochastic choice of number of examination ticket, answers to ticket questions, discussion by professors in the absence of student and announcement of result of exam to him was recorded and put on VIA Forum [27]. 16 The Platform of Virtual Institute of Astroparticle Physics... 255 r*.V - 4 _> "-I, Fig. 16.2. Geography of VIA site visits according to Google Analytics In 2019 in addition to individual supervisory work with students the regular scientific and creative VIA seminar is in operation aimed to discuss the progress and strategy of students scientific workin the field of cosmoparticle physics. 16.2.3 Organisation of VIA events and meetings First tests of VIA system, described in [5,7-9], involved various systems of videoconferencing. They included skype, VRVS, EVO, WEBEX, marratech and adobe Connect. In the result of these tests the adobe Connect system was chosen and properly acquired. Its advantages are: relatively easy use for participants, a possibility to make presentation in a video contact between presenter and audience, a possibility to make high quality records, to use a whiteboard tools for discussions, the option to open desktop and to work online with texts in any format. Initially the amount of connections to the virtual room at VIA lectures and discussions usually didn't exceed 20. However, the sensational character of the exciting news on superluminal propagation of neutrinos acquired the number of participants, exceeding this allowed upper limit at the talk "OPERA versus Maxwell and Einstein" given by John Ellis from CERN. The complete record of this talk and is available on VIA website [31]. For the first time the problem of necessity in extension of this limit was put forward and it was resolved by creation of a virtual "infinity room", which can host any reasonable amount of participants. Starting from 2013 this room became the only main virtual VIA room, but for specific events, like Collaboration meetings or transmissions from science festivals, special virtual rooms can be created. This solution strongly reduces the price of the licence for the use of the adobeConnect videoconferencing, retaining a possibility for creation of new rooms with the only limit to one administrating Host for all of them. The ppt or pdf file of presentation is uploaded in the system in advance and then demonstrated in the central window. Video images of presenter and participants appear in the right window, while in the lower left window the list of all the attendees is given. To protect the quality of sound and record, the participants are required to switch out their microphones during presentation and 256 M.Yu. Khlopov to use the upper left Chat window for immediate comments and urgent questions. The Chat window can be also used by participants, having no microphone, for questions and comments during Discussion. The interactive form of VIA lectures provides oral discussion, comments and questions during the lecture. Participant should use in this case a "raise hand" option, so that presenter gets signal to switch out his microphone and let the participant to speak. In the end of presentation the central window can be used for a whiteboard utility as well as the whole structure of windows can be changed, e.g. by making full screen the window with the images of participants of discussion. Regular activity of VIA as a part of APC includes online transmissions of all the APC Colloquiums and of some topical APC Seminars, which may be of interest for a wide audience. Online transmissions are arranged in the manner, most convenient for presenters, prepared to give their talk in the conference room in a normal way, projecting slides from their laptop on the screen. Having uploaded in advance these slides in the VIA system, VIA operator, sitting in the conference room, changes them following presenter, directing simultaneously webcam on the presenter and the audience. If the advanced uploading is not possible, VIA streaing is used - external webcam and microphone are directed to presenter and screen and support online streaming. 16.3 VIA Sessions at XXII Bled Workshop VIA sessions of XXII Bled Workshop continued the tradition coming back to the first experience at XI Bled Workshop [7] and developed at XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX and XXI Bled Workshops [8-17]. They became a regular part of the Bled Workshop's program. In the course of XXII Bled Workshop, the list of open questions was stipulated, which was proposed for wide discussion with the use of VIA facility. The list of these questions was put on VIA Forum (see [32]) and all the participants of VIA sessions were invited to address them during VIA discussions. During the XXII Bled Workshop the announcement of VIA sessions was put on VIA home page, giving an open access to the videoconferences at VIA sessions. Though the experience of previous Workshops principally confirmed a possibility to provide effective interactive online VIA videoconferences even in the absence of any special equipment and qualified personnel at place, VIA Sessions were directed at I Workshop by M.Khlopov at place. Only laptop with microphone and webcam together with WiFi Internet connection was proved to support not only attendance, but also VIA presentations and discussions. Starting from the Openening of the Workshop VIA streaming of most of the talks was arranged for distant participents. This new form of VIA transmission that avoids the necessity upload presentations in advance made possible to convert VIA sessions with a very limited set of talks to online streaming of practically all the conference accompanied by its record in the VIA library [33]. In the framework of the program of XXII Bled Workshop, E. Kiritsis, gave his talk "Emergent gravity (from hidden sector)" (Fig. 16.4), from Paris (see records in [33]). 16 The Platform of Virtual Institute of Astroparticle Physics... 257 Fig. 16.3. VIA streaming of Opening of XXII Bled Workshop by Norma Mankoc- Borstnik Fig. 16.4. VIA talk "Emergent gravity (from hidden sector)" by E. Kiritsis from Paris at XXII Bled Workshop 258 M.Yu. Khlopov The talks "Conspiracy of BSM Physics and BSM Cosmology" by Maxim Yu. Khlopov (Fig. 16.5) "Experimental consequences of spin-charge family theory" by Norma Mankoc-Borstnik (Fig. 16.6), as well as virtually all other talks were transmitted from Bled in the regime of streaming, inviting distant participants to join the discussion and extending the creative atmosphere of these discussions to the world-wide audience. Fig. 16.5. VIA talk by Maxim Yu. Khlopov "Conspiracy of BSM Physics and BSM Cosmology" at XXII Bled Workshop Two special VIA sessions provided remote presentation of students' scientific debuts in BSM physics and cosmology as it was the talk by Valery Nikulin (Fig. 16.7) who could not attend the Workshop, but could manage to present his interesting results with the use of VIA facility. The records of all these lectures and discussions can be found in VIA library [33]. 16.4 Conclusions The Scientific-Educational complex of Virtual Institute of Astroparticle physics provides regular communication between different groups and scientists, working in different scientific fields and parts of the world, the first-hand information on the newest scientific results, as well as support for various educational programs at distance. This activity would easily allow finding mutual interest and organizing task forces for different scientific topics of astroparticle physics and related topics. It can help in the elaboration of strategy of experimental particle, nuclear, astro-physical and cosmological studies as well as in proper analysis of experimental data. It can provide young talented people from all over the world to get the highest level education, come in direct interactive contact with the world known 16 The Platform of Virtual Institute of Astroparticle Physics... 259 Fig. 16.6. VIA talk "Dark matter, Matter-antimatter and spin-charge-family theory" by Norma Mankoc-Borstnik at XXII Bled Workshop Fig. 16.7. VIA talk "Inflationary limits on the size of compact extra space" by Valery Nikulin at XXII Bled Workshop 260 M.Yu. Khlopov scientists and to find their place in the fundamental research. These educational aspects of VIA activity is now being evolved in a specific tool for International PhD programme for Fundamental physics. Involvement of young scientists in creative discussions was an important step of VIA activity at XXII Bled Workshop. VIA applications can go far beyond the particular tasks of astroparticle physics and give rise to an interactive system of mass media communications. VIA sessions became a natural part of a program of Bled Workshops, maintaining the platform of discussions of physics beyond the Standard Model for distant participants from all the world. This discussion can continue in posts and post replies on VIA Forum. The experience of VIA applications at Bled Workshops plays important role in the development of VIA facility as an effective tool of e-science and e-learning. Acknowledgements The initial step of creation of VIA was supported by ASPERA. I am grateful to P.Binetruy, J.Ellis and S.Katsanevas for permanent stimulating support, to J.C. Hamilton for support in VIA integration in the structure of APC laboratory, to K.Belotsky, A.Kirillov, M.Laletin and K.Shibaev for assistance in educational VIA program, to A.Mayorov, A.Romaniouk and E.Soldatov for fruitful collaboration, to M.Pohl, C. Kouvaris, J.-R.Cudell, C. Giunti, G. Cella, G. Fogli and F. DePaolis for cooperation in the tests of VIA online transmissions in Switzerland, Belgium and Italy and to D.Rouable for help in technical realization and support of VIA complex. The work was supported by grant of Russian Science Foundation (project N-18-12-00213). I express my gratitude to the Organizers of Bled Workshop N.S. Mankoc Borstnik, D. Lukman and H.Nielsen for cooperation in the organization of VIA Sessions at XXII Bled Workshop. References 1. http://www.aspera-eu.org/ 2. http://www.appec.org/ 3. M.Yu. Khlopov: Cosmoparticle physics, World Scientific, New York -London-Hong Kong - Singapore, 1999. 4. M.Yu. Khlopov: Fundamentals of Cosmic Particle Physics, CISP-Springer, Cambridge, 2012. 5. M. Y. Khlopov, Project of Virtual Institute of Astroparticle Physics, arXiv:0801.0376 [astro-ph]. 6. http://viavca.in2p3.fr/site.html 7. M. Y. Khlopov, Scientific-educational complex - virtual institute of astroparticle physics, Bled Workshops in Physics 9 (2008) 81-86. 8. M. Y. Khlopov, Virtual Institute of Astroparticle Physics at Bled Workshop, Bled Workshops in Physics 10 (2009) 177-181. 9. M. Y. Khlopov, VIA Presentation, Bled Workshops in Physics 11 (2010) 225-232. 10. M. Y. Khlopov, VIA Discussions at XIV Bled Workshop, Bled Workshops in Physics 12 (2011) 233-239. 16 The Platform of Virtual Institute of Astroparticle Physics... 261 11. M. Y. .Khlopov, Virtual Institute of astroparticle physics: Science and education online, Bled Workshops in Physics 13 (2012) 183-189. 12. M. Y. .Khlopov, Virtual Institute of Astroparticle physics in online discussion of physics beyond the Standard model, Bled Workshops in Physics 14 (2013) 223-231. 13. M. Y. .Khlopov, Virtual Institute of Astroparticle physics and "What comes beyond the Standard model?" in Bled, Bled Workshops in Physics 15 (2014) 285-293. 14. M. Y. .Khlopov, Virtual Institute of Astroparticle physics and discussions at XVIII Bled Workshop, Bled Workshops in Physics 16 (2015) 177-188. 15. M. Y. .Khlopov, Virtual Institute of Astroparticle Physics — Scientific-Educational Platform for Physics Beyond the Standard Model Bled Workshops in Physics 17 (2016) 221-231. 16. M. Y. .Khlopov: Scientific-Educational Platform of Virtual Institute of Astroparticle Physics and Studies of Physics Beyond the Standard Model. Bled Workshops in Physics 18 (2017) 273-283. 17. M. Y. .Khlopov: The platform of Virtual Institute of Astroparticle physics for studies of BSM physics and cosmology. Bled Workshops in Physics 19 (2018) 383-394. 18. In http://viavca.in2p3.fr/ Previous - Conferences - XIII Bled Workshop 19. In http://viavca.in2p3.fr/ Previous - Conferences - XIV Bled Workshop 20. In http://viavca.in2p3.fr/ Previous - Lectures - Martin Pohl 21. In http://viavca.in2p3.fr/ Previous - Events - JDEV 2013 22. In http://viavca.in2p3.fr/ Previous - Conferences - Subatomic particles, Nucleons, Atoms, Universe: Processes and Structure International conference in honor of Ya. B. Zeldovich 100th Anniversary 23. In http://viavca.in2p3.fr/ Previous - Conferences - Conference YaB-100, dedicated to 100 Anniversary of Yakov Borisovich Zeldovich 24. In http://viavca.in2p3.fr/ Forum - Discussion in Russian - Courses on Cosmoparticle physics 25. In http://viavca.in2p3.fr/ Forum - Education - From VIA to MOOC 26. Inhttp://viavca.in2p3.fr/ Forum - Education - Lectures of Open Online VIA Course 2017 27. In http://viavca.in2p3.fr/ Forum - Education - Small thesis and exam of Steve Branchu 28. http://viavca.in2p3.fr/ Previous - Conferences - The International Conference on Particle Physics and Astrophysics 29. http://viavca.in2p3.fr/ Previous - Conferences - Dark Ages and White Nights (Spec-troscopy of the CMB) 30. http://viavca.in2p3.fr/ Previous - Events - Commemoration day for Patrick Fleury. 31. In http://viavca.in2p3.fr/ Previous - Lectures - John Ellis 32. In http://viavca.in2p3.fr/ Forum - CONFERENCES BEYOND THE STANDARD MODEL - XXII Bled Workshop "What comes beyond the Standard model?" 33. In http://viavca.in2p3.fr/ Previous - Conferences - XXII Bled Workshop "What comes beyond the Standard model?" Blejske Delavnice Iz Fizike, Letnik 20, št. 2, ISSN 1580-4992 Bled Workshops in Physics, Vol. 20, No. 2 Zbornik 22. delavnice 'What Comes Beyond the Standard Models', Bled, 6. -14. julij 2019 Proceedings to the 22nd workshop 'What comes Beyond the standard Models', Bled, July 6.-14., 2019 Uredili Norma Susana Mankoc Borštnik, Holger Bech Nielsen in Dragan Lukman Izid publikacije je finančno podprla Javna agencija za raziskovalno dejavnost RS iz sredstev drzzavnega proracuna iz naslova razpisa za sofinanciranje domacih znanstvenih periodicnih publikacij Brezplacni izvod za udelezzence Tehnicni urednik Matjaz Zaverenik Zalozilo: DMFA - zaloznistvo, Jadranska 19,1000 Ljubljana, Slovenija Natisnila tiskarna Itagraf v nakladi 100 izvodov Publikacija DMFA stevilka 2105