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BLED WORKSHOPS
IN PHYSICS
VOL. 22, NO. 1
Proceedings to the 24th [Virtual]
Workshop
What Comes Beyond . . . (p. 226)
Bled, Slovenia, July 5–11, 2021
17 The achievements of the spin-charge-family theory
so far
N.S. Mankoč Borštnik
Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
Abstract. Fifty years ago, the standard model offered an elegant new step towards under-
standing elementary fermion and boson fields, making several assumptions, suggested
by experiments. The assumptions are still waiting for an explanation. There are many
proposals in the literature for the next step. The spin-charge-family theory, proposing a
simple starting action in d ≥ (13 + 1)-dimensional space with fermions interacting with
the gravity only (the vielbeins and the two kinds of the spin connection fields), is offering
the explanation for not only all by the standard model assumed properties of quarks and
leptons and antiquarks and antileptons, with the families included, of the vector gauge
fields, of the Higgs’s scalar and Yukawa couplings, of the appearance of the dark matter,
of the matter-antimatter asymmetry, making several predictions, but explains as well the
second quantization postulates for fermions and bosons by using the odd and the even
Clifford algebra ”basis vectors” to describe the internal space of fermions and bosons,
respectively. Consequently the single fermion and single boson states already anticommute
and commute, respectively. I present in this talk a very short overview of the achievement
of the spin-charge-family theory so far, concluding with presenting not yet solved problems,
for which the collaborators are very welcome.
Povzetek: Pred petdesetimi leti je standardni model, zgrajen na predpostavkah, porojenih iz
rezultatov poskusov, ponudil eleganten nov korak k razumevanju osnovnih fermionskih in
bozonskih polj. V literaturi je veliko predlogov, ki pojasnjujejo predpostavke in ponujajo
nov korak. Teorija spin-charge-family, ki predlaga preprosto začetno akcijo v d ≥ (13 + 1)-
razsežnem prostoru, v kateri si fermioni izmenjujejo samo gravitone (vektorske svežnje in
dve vrsti spinskih povezav), ponuja razlago ne le za vse predpostavke standardnega modela
— za vse lastnosti kvarkov in leptonov ter antikvarkov in antileptonov, ki se pojavljajo v
družinah, za umeritvena vektorska polja, za Higgsove skalarje in Yukawe sklopitve — am-
pak tudi za pojave v vesolju kot so temna snov, nesimetrija med snovjo in antisnovjo, ponudi
vrsto napovedi, ponudi pa tudi pojasnilo za postulate za drugo kvantizacijo za fermione in
bozone. Opis notranjega prostora fermionov in bozonov z liho in sodo Cliffordovo alge-
bro poskrbi, da fermionska stanja antikomutirajo, bozonska pa komutirajo. V predavanju
ponudim kratek pregled dosedanjih dosežkov spin-charge-family teorije, v zaključku pa
predstavim odprta vprašanja. Pri iskanju odgovorov nanje vabim k sodelovanju.
17.1 Introduction
The review article [1] presents a short overview of most of the achievements of the
spin-charge-family theory so far. I shall make use of this article when presenting my
talk.
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17 The achievements of the spin-charge-family theory so far 227
Fifty years ago the standard model offered an elegant new step towards understand-
ing elementary fermion and boson fields by postulating:
a. The existence of massless fermion family members with the spins and charges
in the fundamental representation of the groups, a.i. the quarks as colour triplets
and colouress leptons, a.ii the left handed members as the weak doublets, the
right handed weak chargeless members, a.iii. the left handed quarks differing
from the left handed leptons in the hyper charge, a.iv. all the right handed mem-
bers differing among themselves in hyper charges, a.v. antifermions carrying the
corresponding anticharges of fermions and opposite handedness, a.vi. the fami-
lies of massless fermions, suggested by experiments and required by the gauge
invariance of the boson fields (there is no right handed neutrino postulated, since
it would carry none of the so far observed charges, and correspondingly there is
also no left handed antineutrino allowed in the standard model).
b. The existence of massless vector gauge fields to the observed charges of quarks
and leptons, carrying charges in the adjoint representations of the corresponding
charged groups and manifesting the gauge invariance.
c. The existence of the massive weak doublet scalar higgs, c.i. carrying the weak
charge ±1
2
and the hyper charge ∓1
2
(as it would be in the fundamental represen-
tation of the two groups), c.ii. gaining at some step of the expanding universe
the nonzero vacuum expectation value, c.iii. breaking the weak and the hyper
charge and correspondingly breaking the mass protection, c.iv. taking care of the
properties of fermions and of the weak bosons masses, c.v. as well as the existence
of the Yukawa couplings.
d. The presentation of fermions and bosons as second quantized fields.
e. The gravitational field in d = (3+ 1) as independent gauge field. (The standard
model is defined without gravity in order that it be renomalizable, but yet the
standard model particles are ”allowed” to couple to gravity in the ”minimal”
way.)
The standard model assumptions have been experimentally confirmed without
raising any severe doubts so far, except for some few and possibly statistically
caused anomalies 1, but also by offering no explanations for the assumptions.
The last among the fields postulated by the standard model, the scalar higgs, was
detected in June 2012, the gravitational waves were detected in February 2016.
The standard model has in the literature several explanations, mostly with many
new not explained assumptions. The most popular seem to be the grand unifying
theories [2, 4–18, 59]. At least SO(10)-unifying theories offer the explanation for
the postulates from a.i. to a.iv, partly to b. by assuming that to all the ”fermion”
charges there exist the corresponding vector gauge fields — but does not explain
the assumptions a.v. up to a.vi., c. and d., and does not connect gravity with gauge
vector and scalar fields.
In a long series of works with collaborators ( [19–23, 25, 26, 28–32, 38] and the
references therein), we have found the phenomenological success with the model
named the spin-charge-family theory, with fermions, the internal space of which is
described with the Clifford algebra of all linear superposition of odd products of
1 I think here on the improved standard model, in which neutrinos have non-zero masses,
and the model has no ambition to explain severe cosmological problems.
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228 N.S. Mankoč Borštnik
γa’s in d = (13 + 1), interacting with only gravity ( [38] and references therein).
The spins of fermions from higher dimensions, d > (3 + 1), manifest in d =
(3+ 1) as charges of the standard model, gravity in higher dimensions manifest as
the standard model gauge vector fields as well as the Higgs’s scalar and Yukawa
couplings [26, 31].
Let be added that one irreducible representation of SO(13, 1) contains, if looked
from the point of view of d = (3+1), all the quarks and leptons and antiquarks and
antileptons and just with the properties, required by the standard model, including
the relation between quarks and leptons and handedness and antiquarks and
antileptons of the opposite handedness, as can be reed in Table 5 of App. D,
appearing in the contribution of the same author in this Proceedings [33].
All that in the standard model had to be assumed (extremely effective ”read” from
experiments and also from the theoretical investigations) in the spin-charge-family
theory appear as a possibility from the starting simple action, Eq. (17.15), and
from the assumption that the internal space of fermions are described by the odd
Clifford algebra objects.
One can reed in my second contribution to this Proceedings [33] that the descrip-
tion of the internal space of fermions with the odd Clifford algebra operators γa’s
offers the explanation for the observed quantum numbers of quarks and leptons
and antiquarks and antileptons while unifying spin, handedness, charges and
families. The ”basis vectors” which are superposition of odd products of operators
γa’s, appear in irreducible representations which differ in the quantum numbers
determined by γ̃a’s.
The simple starting action of the spin-charge-family theory offers the explanation
for not only the properties of quarks and leptons and antiquarks and antileptons,
but also for the vector gauge fields, scalar gauge fields, which represent higgs and
explain the Yukawa couplings, and for the scalars, which cause matter/antimatter
asymmetry, the proton decay, while the appearance of the dark matter is explained
by the appearance of two groups of the decoupled families.
It appears, as it is explained in my second contribution to this Proceedings [33],
that the description of the internal space of bosons fields (the gauge fields of the
fermion fields described by the Clifford odd ”basis vectors”) with the Clifford
even ”basis vectors” explains the commutativity and the properties of the second
quantized boson fields, as the description of the internal space of fermion fields
with the Clifford odd ”basis vectors” explains the anticommutativity and the
properties of the second quantized fermion fields.
The description of fermions and bosons with the Clifford odd and Clifford even
”basis vectors”, respectively, makes fermions appearing in families, while bosons
do not. Both kinds of ”basis vectors” contribute finite number, 2
d
2
−1 × 2d2−1,
degrees of freedom to the corresponding creation operators, while the basis of
ordinary space contribute continuously infinite degrees of freedom.
Is the way proposed by the spin-charge-family theory the right way to the next
step beyond the standard model? The theory certainly offers a different view of
the properties of fermion and boson fields and a different view of the second
quantization of both fields than that offered by group theory and the second
quantization by postulates.
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17 The achievements of the spin-charge-family theory so far 229
It has happened so many times in the history of science that the simpler model
has shown up as a more ”powerful” one.
My working hypotheses is that the laws of nature are simple and correspondingly
elegant and that the many body systems around the phase transitions look to
us complicated at least from the point of view of the elementary constituents of
fermion and boson fields.
To this working hypotheses belong also the description of the internal space of
fermions and bosons with the Clifford algebras and the simple starting action for
the (second quantized) massless fermions interacting with the (second quantized) 2
massless bosons, representing gravity only — the vielbeins and the two kinds of
the spin connection fields, the gauge fields of the two kinds of the generators of the
Lorentz transformations Sab(= i
2
(γaγb − γbγa)) and S̃ab(= i
2
(γ̃aγ̃b − γ̃bγ̃a)).
In Sect. 17.2 I shall very shortly overview the Clifford algebra description of
the internal space of fermions, following Ref. [1], and bosons (explained in my
additional contribution to this Proceedings [33]), after the reduction of the two
independent groups of Clifford algebras to only one.
In Sect. 17.3 the definition of the creation and annihilation operators as tensor
products of the ”basis vectors” defined by the Clifford algebra objects and basis in
ordinary space is presented.
In Sect. 17.4 the simple starting action of the spin-charge-family theory is presented
and the achievements of the theory so far discussed.
In Sect. 17.5 the open problems of the spin-charge-family theory are presented, and
the invitation to the reader to participate.
17.2 Clifford algebra and internal space of fermions and bosons
I follow here Ref. [1], Sect. 3 and also my second contribution to this Proceed-
ings [33], Sect. 2.
Single fermion states are functions of external coordinates and of internal space of
fermions. If Mab denote infinitesimal generators of the Lorentz algebra in both
spaces, Mab = Lab + Sab, with Lab = xapb − xbpa, pa = i ∂
∂xa
, determining
operators in ordinary space, while Sab are equivalent operators in internal space
of fermions, it follows
{Mab,Mcd}− = i{M
adηbc +Mbcηad −Macηbd − Mbdηac} ,
{Mab, pc}− = −iη
acpb + iηcbpa ,
{Mab, Scd}− = i{S
adηbc + Sad − Sacηbd − Sbdηac} , (17.1)
while the Cartan subalgebra operators of the Lorentz algebra are chosen as
M03,M12,M56, . . . ,Md−1d , (17.2)
2 Since the single fermion states, described by the Clifford odd ”basis vectors”, anticom-
mute due to the anticommuting properties of the Clifford odd ”basis vectors” and the
single boson states, described by the Clifford even ”basis vectors”, correspondingly
commute there are only the second quantized fermion and boson fields.
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230 N.S. Mankoč Borštnik
and will be used to define the basis in both spaces as eigenvectors of the Cartan
subalgebra members. The metric tensor ηab = diag(1,−1,−1, . . . ,−1,−1) for
a = (0, 1, 2, 3, 5, . . . , d) is used.
There are two kinds of anticommuting algebras, the Grassmann algebra θa’s and
pθa’s (= ∂
∂θa
’s), in d-dimensional space with d anticommuting operators θa’s and
with d anticommuting derivatives ∂
∂θa
’s,
{θa, θb}+ = 0 , {
∂
∂θa
,
∂
∂θb
}+ = 0 ,
{θa,
∂
∂θb
}+ = δab , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(θa)† = ηaa
∂
∂θa
, (
∂
∂θa
)† = ηaaθa , (17.3)
where the last line was our choice [32], and the two anticommuting kinds of the
Clifford algebras γa’s and γ̃a’s 3 are expressible with the Grassmann algebra
operators and opposite
γa = (θa +
∂
∂θa
) , γ̃a = i (θa −
∂
∂θa
) ,
θa =
1
2
(γa − iγ̃a) ,
∂
∂θa
=
1
2
(γa + iγ̃a) , (17.4)
offering together 2 · 2d operators: 2d of those which are products of γa and 2d of
those which are products of γ̃a, the same number of operators as of the Grassmann
algebra operators. The two kinds of the Clifford algebras anticommute, fulfilling
the anticommutation relations
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a ,
γaγa = ηaa , γa(γa)† = I , γ̃aγ̃a = ηaa , γ̃a(γ̃a)† = I , (17.5)
where I represents the unit operator. The two kinds of the Clifford algebra objects
are obviously independent.
3 The existence of the two kinds of the Clifford algebras is discussed in [19, 20, 22, 34, 35].
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17 The achievements of the spin-charge-family theory so far 231
The corresponding infinitesimal Lorentz generators are then Sab for the Grass-
mann algebra, and Sab and S̃ab for the two kinds of the Clifford algebras.
Sab =
i
4
(γaγb − γbγa) ,
S̃ab =
i
4
(γ̃aγ̃b − γ̃bγ̃a) ,
Sab = i (θa
∂
∂θb
− θb
∂
∂θa
) ,
{Sab, S̃ab}− = 0 , Sab = Sab + S̃ab ,
{Sab, θe}− = −i (ηae θb − ηbe θa) ,
{Sab, pθe}− = −i (ηae pθb − ηbe pθa) ,
{Sab, γc}− = i(η
bcγa − ηacγb) ,
{S̃ab, γ̃c}− = i(η
bcγ̃a − ηacγ̃b) ,
{Sab, γ̃c}− = 0 , {S̃
ab, γc}− = 0 . (17.6)
The reader can find a more detailed information in Ref. [1] in Sect. 3.
It is useful to choose the ”basis vectors” in each of the two spaces to be products of
eigenstates of the Cartan subalgebra members, Eq. (17.2), of the Lorentz algebras,
(Sab = i
4
(γaγb − γbγa), S̃ab = i
4
(γ̃aγ̃b − γ̃bγ̃a)). The ”eigenstates” of each of
the Cartan subalgebra members, Eqs. (17.4, 17.5), for each of the two kinds of the
Clifford algebras separately can be found as follows,
Sab
1
2
(γa +
ηaa
ik
γb) =
k
2
1
2
(γa +
ηaa
ik
γb) , Sab
1
2
(1+
i
k
γaγb) =
k
2
1
2
(1+
i
k
γaγb) ,
S̃ab
1
2
(γ̃a +
ηaa
ik
γ̃b) =
k
2
1
2
(γ̃a +
ηaa
ik
γ̃b) , S̃ab
1
2
(1+
i
k
γ̃aγ̃b) =
k
2
1
2
(1+
i
k
γ̃aγ̃b) ,(17.7)
k2 = ηaaηbb. The proof of Eq. (17.7) is presented in App. (I) of Ref. [1], Statement
2a. The Clifford ”basis vectors” — nilpotents 1
2
(γa+ η
aa
ik
γb), (1
2
(γa+ η
aa
ik
γb))2 = 0
and projectors 1
2
(1+ i
k
γ̃aγ̃b), (1
2
(1+ i
k
γ̃aγ̃b))2 = 1
2
(1+ i
k
γ̃aγ̃b) — of both algebras
are normalized, up to a phase, as described in the contribution of the same outhor
in this Proceedings [33].
Both, nilpotents and projectors, have half integer spins.
It is useful to introduce the notation for the ”eigenvectors” of the two Cartan
subalgebras as follows, Ref. [34, 35],
ab
(k): =
1
2
(γa +
ηaa
ik
γb) ,
ab
(k)
†
= ηaa
ab
(−k) , (
ab
(k))2 = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k]
ab
[k]: =
1
2
(1+
i
k
γaγb) ,
ab
[k]
†
=
ab
[k] , (
ab
[k])2 =
ab
[k] ,
ab
[k]
ab
[−k]= 0 ,
ab
(k)
ab
[k] = 0 ,
ab
[k]
ab
(k)=
ab
(k) ,
ab
(k)
ab
[−k]=
ab
(k) ,
ab
[k]
ab
(−k)= 0 . (17.8)
The corresponding expressions for nilpotents
ab
˜(k) and projectors
ab
˜[k] follows if we
replace in Eq. (17.8) γa’s by γ̃a’s, the same relation k2 = ηaaηbb is valid for both
algebras.
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232 N.S. Mankoč Borštnik
Let us notice that the ”eigenvectors” of the Cartan subalgebras are equivalent and
the eigenvalues are the same in both algebras: Both algebras have projectors and
nilpotents: ((
ab
[k])2 =
ab
[k] , (
ab
(k))2 = 0), ((
ab
˜[k])2 =
ab
˜[k] , (
ab
˜(k))2 = 0).
In each of the two independent algebras we have two groups of 2
d
2
−1 members
which are eigenvectors of all the Cartan subalgebra members, Eq. (17.2), appearing
in 2
d
2
−1 irreducible representations which have an odd Clifford character — they
are products of an odd number of γa’s (γ̃a’s). These two groups are Hermitian
conjugated to each other. We make a choice of one of the two groups of the Clif-
ford odd ”basis vectors” and name these ”basis vectors” b̂m†f , m describing 2
d
2
−1
members of one irreducible representation, f describing one of 2
d
2
−1 irreducible
representations. The 2
d
2
−1× 2d2−1 members of the second group, Hermitian conju-
gated to b̂m†f , are named as b̂
m
f = (b̂
m†
f )
†.
There are besides two Clifford odd groups in each of the two algebras γa’s and
γ̃a’s, also two Clifford even groups. They are superposition of an even number
of γa’s (γ̃a’s). I named these two 2
d
2
−1× 2d2−1 Clifford even ”basis vectors” Âm†f
and B̂m†f , respectively. Â
m†
f represent gauge vectors of b̂
m†
f , on which they operate.
B̂m†f operate on b̂mf . I discuss their properties in my second contribution of this
Proceedings [33].
The ”basis vectors” of an odd Clifford character, b̂m†f , and their Hermitian conju-
gated partners, b̂mf , fulfil the postulates for second quantized fermions of Dirac,
if we reduce both Clifford algebras to only one [?, 37, 38], while keeping all the
relations, presented in Eq. (17.5), valid. Let us make a choice of γa’s and postulate
the application of γ̃a’s on B which is a superposition of any products of γa’s as
follows
{γ̃aB = (−)B i Bγa} |ψoc > , (17.9)
with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise (−)B =
1 [35], |ψoc > is defined in Eq. (17.10). (Sects. (2.1, 2.2 in [33]) and Sects. (3.2.2, 3.2.3
in [1])).
The vacuum state |ψoc > is defined as follows
|ψoc >=
2
d
2
−1∑
f=1
b̂mf ∗A b̂
m†
f | 1 > , (17.10)
for one of the membersm, anyone, of the odd irreducible representation f, with
| 1 >, which is the vacuum without any structure, the identity, ∗A means the
algebraic product. It follows that b̂mf ∗A |ψoc >= 0 and b̂
m†
f ∗A |ψoc >= |ψ
m
f >.
After the postulate of Eq. (17.9) ”basis vectors” b̂m†f which are superposition of an
odd products of γa’s (represented by an odd number of nilpotents, the rest are
projectors) obey all the fermions second quantization postulates of Dirac. There are
S̃ab which dress the irreducible representations with the family quantum numbers
of the Cartan subalgebra members (S̃03, S̃12, S̃56, . . . , S̃d−1d), Eq. (17.2).
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17 The achievements of the spin-charge-family theory so far 233
{b̂mf , b̂
m ′†
f ′ }∗A+ |ψoc > = δ
mm ′ δff ′ |ψoc > ,
{b̂mf , b̂
m ′
f ′ }∗A+ |ψoc > = 0 · |ψoc > ,
{b̂m†f , b̂
m ′†
f ′ }∗A+ |ψoc > = 0 · |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
b̂mf ∗A|ψoc > = 0 · |ψoc > , (17.11)
with (m,m ′) denoting the ”family” members and (f, f ′) denoting ”families”, ∗A
represents the algebraic multiplication of b̂†mf and b̂
m
f with the vacuum state
|ψoc > of Eq. (17.10) and among themselves, taking into account Eq. (17.5).
Ref. ( [33], Sects. 2.4 and 3) presents the starting study of properties of the second
quantized boson fields, the internal space of which is represented by the ”basis
vectors” Âm†f which appear as the gauge fields of the second quantized fermion
fields the internal space of which is described by the ”basis vectors” b̂m†f .
We pay attention on even dimensional spaces, d = 2(2n + 1) or d = 4n, n ≥ 0,
only.
17.3 Creation and annihilation operators
Here Sect. 3.3 of Ref. [1] is roughly followed.
Describing fermion fields as the creation b̂s†f (p⃗) and annihilation b̂
s
f(p⃗) operators
operating on the vacuum state we make tensor products, ∗T , of 2
d
2
−1× 2d2−1
Clifford odd ”basis vectors” b̂m†f and of continuously infinite basis in ordinary
space determined by b̂†
p⃗
{b̂s†f (p⃗) =
∑
m
cmsf(p⃗) b̂
†
p⃗ ∗T b̂
m†
f } |ψoc > ∗T |0p⃗ > , (17.12)
where p⃗ determines the momentum in ordinary space with p0 = |⃗p| and s deter-
mines all the rest of quantum numbers. The state |ψoc > ∗T |0p⃗ > is considered as
the vacuum for a starting single particle state from which one obtains the other
single particle state by the operators, b̂p⃗, which pushes the momentum by an
amount p⃗, in a tensor product with b̂m†f . We have
|⃗p > = b̂†
p⃗
| 0p > , < p⃗ | =< 0p | b̂p⃗ ,
< p⃗ | p⃗ ′ > = δ(p⃗− p⃗ ′) =< 0p |b̂p⃗ b̂
†
p⃗ ′ | 0p > ,
leading to
b̂p⃗ ′ b̂
†
p⃗
= δ(p⃗ ′ − p⃗) , (17.13)
since we normalize < 0p | 0p >= 1 to identity.
The ”basis vectors” b̂m†f which are products of an odd number of nilpotent, the
rest to d
2
are then projectors, anticommute, transferring the anticommutativity
to the creation operators b̂s†f (p⃗) and correspondingly also to their Hermitian
conjugated partners annihilation operators b̂sf(p⃗), Eq. (17.12). The creation and
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234 N.S. Mankoč Borštnik
annihilation operators then fulfil the anticommutation relations of the second
quantized fermions explaining the postulates of Dirac
{b̂s
′
f‘ (p⃗
′) , b̂s†f (p⃗)}+ |ψoc > |0p⃗ > = δ
ss ′δff ′ δ(p⃗
′ − p⃗) |ψoc > |0p⃗ > ,
{b̂s
′
f‘ (p⃗
′) , b̂sf(p⃗)}+ |ψoc > |0p⃗ > = 0 |ψoc > |0p⃗ > ,
{b̂s
′†
f ′ (p⃗
′) , b̂s†f (p⃗)}+ |ψoc > |0p⃗ > = 0 |ψoc > |0p⃗ > ,
b̂s†f (p⃗) |ψoc > |0p⃗ > = |ψ
s
f(p⃗) >
b̂sf(p⃗) |ψoc > |0p⃗ > = 0 |ψoc > |0p⃗ >
|p0| = |⃗p| . (17.14)
Statement The description of the internal space of fermions with the superposition of odd
products of γa’s, that is with the clifford odd ”basis vectors”, not only explains the Dirac’s
postulates of the second quantized fermions but also explains the appearance of families of
fermions.
Ref. [33] is offering the explanation for the second quantized commuting boson
fields (described by the ”basis vectors” of an even number of nilpotents, the rest
are projectors), they are the gauge fields of the anticommuting fermion fields
(described by the ”basis vectors” of an odd number of nilpotents).
17.4 Achievements so far of spin-charge-family theory
Here Sects. (6, 7.2.2 and 7.3.1) of Ref. [1], which review shortly the achievements
so far of the spin-charge-family theory, are followed.
The main new achievement of this theory in the last few years is the recognition
that the description of the internal space of fermion fields with the Clifford algebra
objects in d > (3+ 1) not only offers the explanation for all the assumptions of the
standard model for fermion and boson fields, with the appearance of families for
fermion fields and the properties of the corresponding vector and scalar gauge
fields included, but also get to know, that the anticommuting property of the inter-
nal space of fermions takes care of the second quantization properties of fermions,
so that the second quantized postulates are not needed. The second quantized
properties of fermions origin in their internal space and are transferred to creation
and annihilation operators. This year contribution to Proceedings Ref. [33] offers
the recognition that also commuting properties of the second quantized boson
fields origin in the internal space of bosons.
Describing the internal space of bosons by the Clifford even ”basis vectors”,
written in terms of the Clifford even number of γa’s, these Clifford even ”basis
vectors”, Âm†f , applying on fermion states transform the ”basis vectors” b̂
m†
f either
into another ”basis vectors” b̂m
′†
f with the same family quantum number f, or if
written in terms of the Clifford even number of γ̃a’s, ^̃A
m†
f , transform b̂
m ′†
f to b̂
m†
f‘ ,
keeping the family member quantum number m unchanged and changing the
family quantum number to f‘. 4 This topic, started in Ref. [33], needs further study.
4 The first operation happens if the internal space of bosons is described by ”basis vectors”
which are even products of nilpotents of the kind Âm†f =
03
(−i)
12
(−)
56
[+] · · ·
d−1 d
[+] ), in this
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17 The achievements of the spin-charge-family theory so far 235
The spin-charge-family theory proposes a simple action for interacting second
quantized massless fermions and the corresponding gauge fields in d = (13+ 1)-
dimensional space as
A =
∫
ddx E
1
2
(ψ̄ γap0aψ) + h.c.+∫
ddx E (αR+ α̃ R̃) ,
p0a = f
α
ap0α +
1
2E
{pα, Ef
α
a}− ,
p0α = pα −
1
2
Sabωabα −
1
2
S̃abω̃abα ,
R =
1
2
{fα[afβb] (ωabα,β −ωcaαω
c
bβ)}+ h.c. ,
R̃ =
1
2
{fα[afβb] (ω̃abα,β − ω̃caα ω̃
c
bβ)}+ h.c. . (17.15)
Here 5 fα[afβb] = fαafβb − fαbfβa.
This simple action in d = (13+ 1)-dimensional space,
i. in which massless fermions interact with the massless gravitation fields only
(with the vielbeins and the two kinds of the spin connection fields, the gauge fields
of Sab and S̃ab, respectively),
ii. together with the assumption that the internal space of the second quantized
fermions are described by the Clifford odd ”basis vectors” (what explains after
the break of symmetries at low energies the appearance of quarks and leptons
and antiquarks and antileptons of the standard model and the existence of families,
predicting the number of families [46]),
iii.and the internal space of the second quantized boson fields are described by
the Clifford even ”basis vectors”, offers the explanations for
iv. not only all the assumptions of the standard model — for properties of quarks
and leptons and antiquarks and antileptons (explaining the relations among spins,
handedness and charges of fermions and antifermions [23, 44]) and for the appear-
ance of families of quarks and leptons [34, 35, 42],
v. for the second quantized postulates of Dirac [36, 37],
vi. for the appearance of the vector gauge fields to the corresponding fermion
fields [26],
particular case two nilpotents form ”basis vectors”, the second operation happens if
all the nilpotents
ab
(k) and projectors
cd
[k] are replaced by the corresponding
ab
˜(k) and
ab
˜[k],
respectively.
5 fαa are inverted vielbeins to eaα with the properties eaαfαb = δab, eaαfβa = δβα, E =
det(eaα). Latin indices a, b, ..,m, n, .., s, t, .. denote a tangent space (a flat index), while
Greek indices α, β, .., µ, ν, ..σ, τ, .. denote an Einstein index (a curved index). Letters from
the beginning of both the alphabets indicate a general index (a, b, c, .. and α, β, γ, .. ),
from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m,n, .. and
µ, ν, ..), indexes from the bottom of the alphabets indicate the compactified dimensions
(s, t, .. and σ, τ, ..). We assume the signature ηab = diag{1,−1,−1, · · · ,−1}.
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236 N.S. Mankoč Borštnik
vii. for the appearance of gauge scalars explaining the interactions among fermions
belonging to different families [26, 28, 29, 31, 39–41, 46], and correspondingly of the
appearance of the higgs scalar and Yukawa couplings,
viii. predicting the number of families — the fourth one to the observed three [46],
ix. predicting the second group of four families the stable of which explains the
appearance of the dark matter [23, 45],
x. predicting additional gauge fields,
xi. predicting additional scalar fields, which explain the existence of matter-
antimatter asymmetry [25],
and several others.
The manifoldM(13+1) breaks at high scale ∝ 1016 GeV or higher first toM(7+1) ×
M(6) due to the appearance of the scalar condensate (so far just assumed, not yet
proven that it appears spontaneously) of the two right handed neutrinos with the
family quantum numbers of the group of four families, which does not include
the observed three families bringing masses (of the scale of break ∝ 1016 GeV or
higher) to all the gauge fields, which interact with the condensate [25].
Since the left handed spinors — fermions — couple differently (with respect to
M(7+1)) to scalar fields than the right handed ones, the break can leave massless
and mass protected 2((7+1)/2−1)(= 8) families [49]. The rest of families get heavy
masses 6.
The manifoldM(7+1) × SU(3)×U(1) breaks further by the scalar fields, presented
in Sect. 17.4.2, toM(3+1)× SU(3)×U(1) at the electroweak break. This happens
since the scalar fields with the space index (7, 8), Subsubsect. 17.4.2, they are a part
of a simple starting action Eq.(17.15), gain the constant values (the nonzero vacuum
expectation values independent of the coordinates in d = (3 + 1)). These scalar
fields carry with respect to the space index the weak charge ±1
2
and the hyper
charge ∓1
2
[23, 25], Sect. 17.4.2, just as required by the standard model, manifesting
with respect to S̃ab and Sab additional quantum numbers.
Let us point out that all the fermion fields (with the families of fermions and the
neutrinos forming the condensate included), the vector and the scalar gauge fields,
offering explanation for by the standard model postulated ones, origin in the simple
starting action.
The starting action, Eq. (17.15), has only a few parameters. It is assumed that the
coupling of fermions toωabc’s can differ from the coupling of fermions to ω̃abc’s,
The reduction of the Clifford space, Eq. 17.9, causes this difference. The additional
breaks of symmetries influence the coupling constants in addition.
The breaks of symmetries is under consideration for quite a long time and has not
yet been finished.
6 A toy model [49, 52, 53] was studied in d = (5 + 1)-dimensional space with the action
presented in Eq. (17.15), The break from d = (5 + 1) to d = (3 + 1)× an almost S2 was
studied for a particular choice of vielbeins and for a class of spin connection fields. While
the manifold M(5+1) breaks into M(3+1) times an almost S2 the 2((3+1)/2−1) families
remain massless and mass protected. Equivalent assumption, although not yet proved
how does it really work, is made also for the d = (13 + 1) case. This study is in progress
quite some time.
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All the observed properties of fermions, of vector gauge fields and scalar gauge
fields follow from the simple starting action, while the breaks of symmetries
influence the properties of fermion and boson fields as well.
17.4.1 Properties of interacting massless fermions as manifesting in
d = (3 + 1) before electroweak break
One irreducible representation of SO(13, 1) includes all the left handed and right
handed quarks and leptons and antiquarks and antileptons as one can see in Table 5
of Ref. [33] in this Proceedings or in Table 7 of Ref. [1]. In both tables fermion
”basis vectors” are represented by odd numbers of nilpotents and their properties
analysed from the point of view of the standard model subgroups SO(3, 1)×SU(2)×
SU(2) × SU(3) × U(1) of the group SO(13, 1). Quarks and leptons as well as
antiquarks and antileptons appear with handedness as required by the standard
model.
One easily notices that quarks and leptons have the same content of the subgroup
SO(7, 1), distinguishing only in SU(3)×U(1) content of SO(6): all the quarks, left
and right handed, have the ”fermion” τ4 equal to 1
6
and appear in three colours,
all the leptons, left and right handed, have τ4 equal to −1
2
and are colourless.
Also antiquarks and antileptons have the same content of the subgroup SO(7, 1)
(which is different from the one of quarks and leptons), and differ in SU(3)×U(1)
content of SO(6), all the antiquarks, left and right handed, have τ4 equal to −1
6
and appear in three anticolours, all the antileptons have τ4 equal to 1
2
and are
anticolourless.
Let us notice also that since there are two SU(2) weak charges the right handed
neutrinos and the left handed antineutrinos have non zero the second SU(2)II
weak charge and interact with the SU(2)II weak field. Both have the standard model
hyper charge Y = τ4 + τ23 equal to zero. Let me point out that this particular
property are offered also by the SO(10) unifying model [59], but with the manifold
M(3+ 1) decoupled from charges. (Comments can be found in Ref. [1], Sect. 7).
The expressions for the generators of the Lorentz transformations of subgroups
SO(3, 1)× SU(2)× SU(2)× SU(3)×U(1) of the group SO(13, 1) can be found in
App. 17.6 (also in Eqs. (39-41) of Ref. [33] or in Eqs. (85-89) of Ref. [1]).
The condensate, presented in Table 17.2 (Table 6 of Ref. [1]), makes one of the two
weak SU(2) fields massive and causes the break of symmetries fromM(13+1) to
M(7+1) × SU(3) × U(1) [49, 52, 53], leaving only two decoupled groups of four
families massless, 2
7+1
2
−1 = 8. The reader can find these two groups of families in
Table 17.1 (from Table 5 of Ref. [1]).
Table 17.1 presents ”basis vectors” (b̂m†f , Eq. (17.11)) for eight families of the
right handed u-quark of the colour (1
2
, 1
2
√
3
) and the right handed colourless ν-
lepton. The SO(7,1) content of the SO(13, 1) group are in both cases identical, they
distinguish only in the SU(3) and U(1) subgroups of SO(6). All the members
of any of these eight families of Table 17.1 follows from either the u-quark or
the ν-lepton by the application of Sab. Each family carries the family quantum
numbers, determined by the Cartan subalgebra of S̃ab in Eq. (17.2) and presented
in Table 17.1.
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238 N.S. Mankoč Borštnik
The two groups of families are after the break of symmetries decoupled since
{ÑiL, Ñ
j
R}− = 0 ,∀(i, j), {τ̃1 i, τ̃2 j}− = 0 ,∀(i, j), {ÑiL,R, τ̃1,2 j}− = 0 ,∀(i, j), while
{Sab, S̃cd }− = 0, since {γa, γ̃a}− = 0, Eq. (17.5).
Table 17.1: Eight families of the ”basis vectors” b̂m†f , of û
c1†
R — the right handed
u-quark with spin 1
2
and the colour charge (τ33 = 1/2, τ38 = 1/(2
√
3)), appearing
in the first line of Table 7 in Ref. [1], or Table 5 in Ref. [33] — and of the colourless
right handed neutrino ν̂†R of spin
1
2
, appearing in the 25th line of Table 7 in Ref. [1],
or Table 5 in Ref. [33] — are presented in the left and in the right part of this
table, respectively. Table is taken from [31]. Families belong to two groups of four
families, one (I) is a doublet with respect to ( ⃗̃NL and ⃗̃τ1) and a singlet with respect
to ( ⃗̃NR and ⃗̃τ2), App. 17.6 (Eqs. (85-88) of Ref. [1]), the other group (II) is a singlet
with respect to ( ⃗̃NL and ⃗̃τ1) and a doublet with respect to ( ⃗̃NR and ⃗̃τ2). All the
families follow from the starting one by the application of the operators (ѱR,L,
τ̃(2,1)±). The generators (N±R,L, τ
(2,1)±) transform û†1R to all the members of one
family of the same colour charge. The same generators transform equivalently the
right handed neutrino ν̂†1R to all the colourless members of the same family.
τ̃13 τ̃23 Ñ3
L
Ñ3
R
τ̃4
I û
c1†
R1
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R1
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) − 1
2
0 − 1
2
0 − 1
2
I û
c1†
R2
03
[+i]
12
(+) |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R2
03
[+i]
12
(+) |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) − 1
2
0 1
2
0 − 1
2
I û
c1†
R3
03
(+i)
12
[+] |
56
(+)
78
[+] ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R3
03
(+i)
12
[+] |
56
(+)
78
[+] ||
9 10
(+)
11 12
(+)
13 14
(+) 1
2
0 − 1
2
0 − 1
2
I û
c1†
R4
03
[+i]
12
(+) |
56
(+)
78
[+] ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R4
03
[+i]
12
(+) |
56
(+)
78
[+] ||
9 10
(+)
11 12
(+)
13 14
(+) 1
2
0 1
2
0 − 1
2
II û
c1†
R5
03
[+i]
12
[+] |
56
[+]
78
[+] ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R5
03
[+i]
12
[+] |
56
[+]
78
[+] ||
9 10
(+)
11 12
(+)
13 14
(+) 0 − 1
2
0 − 1
2
− 1
2
II û
c1†
R6
03
(+i)
12
(+) |
56
[+]
78
[+] ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R6
03
(+i)
12
(+) |
56
[+]
78
[+] ||
9 10
(+)
11 12
(+)
13 14
(+) 0 − 1
2
0 1
2
− 1
2
II û
c1†
R7
03
[+i]
12
[+] |
56
(+)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R7
03
[+i]
12
[+] |
56
(+)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) 0 1
2
0 − 1
2
− 1
2
II û
c1†
R8
03
(+i)
12
(+) |
56
(+)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] ν̂
†
R8
03
(+i)
12
(+) |
56
(+)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) 0 1
2
0 1
2
− 1
3
It is the assumption that the eight families from Table 17.1 remain massless after
the break of symmetry from SO(13, 1) to SO(7, 1)× SO(6), made after we proved
for the toy model [49, 52] that the break of symmetry can leave some families of
fermions massless, while the rest become massive. But we have not yet proven the
masslessness of the 2
7+1
2
−1 families after the break from SO(13, 1) to SO(7, 1)×
SO(6).
The break from the starting symmetry SO(13, 1) to SO(7, 1) × SU(3) × U(1) is
supposed to be caused by the appearance of the condensate of two right handed
neutrinos with the family quantum numbers of the upper four families, that is of
the four families, which do not contain the three so far observed families, at the
energy of ≥ 1016 GeV. This condensate is presented in Table 17.2.
To see how do gravitational fields — vielbeins and the two spin connection fields,
the gauge fields of Sab and S̃ab, respectively — contribute to dynamics of fermion
fields and after the electroweak break also to the masses of twice four families
and the vector gauge field let us rewrite the fermion part of the action, Eq. (17.15),
in the way that the fermion action manifests in d = (3 + 1), that is in the low
energy regime before the electroweak break, by the standard model postulated
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17 The achievements of the spin-charge-family theory so far 239
Table 17.2: The condensate of the two right handed neutrinos νR, with the quantum
numbers of the VIIIth family, Table 17.1, coupled to spin zero and belonging to
a triplet with respect to the generators τ2i, is presented, together with its two
partners. The condensate carries τ⃗1 = 0, τ23 = 1, τ4 = −1 and Q = 0 = Y.
The triplet carries τ̃4 = −1, τ̃23 = 1 and Ñ3R = 1, Ñ
3
L = 0, Ỹ = 0, Q̃ = 0. The
family quantum numbers of quarks and leptons are presented in Table 17.1. The
definition of the operators τ⃗1, ⃗̃τ1, τ⃗2, ⃗̃τ2, τ4, τ̃4, N3R, Ñ
3
R, N
3
L, Ñ
3
L, Q, Y, Q̃, Ỹ can be
found in App. 17.6 (and in Ref. [1], Eqs. (85-88) or in Eqs. (39-41) of Ref. [33]).
state S03 S12 τ13 τ23 τ4 Y Q τ̃13 τ̃23 τ̃4 Ỹ Q̃ Ñ3L Ñ
3
R
(|νVIII1R >1 |ν
VIII
2R >2) 0 0 0 1 −1 0 0 0 1 −1 0 0 0 1
(|νVIII1R >1 |e
VIII
2R >2) 0 0 0 0 −1 −1 −1 0 1 −1 0 0 0 1
(|eVIII1R >1 |e
VIII
2R >2) 0 0 0 −1 −1 −2 −2 0 1 −1 0 0 0 1
properties, while manifesting the properties which make the spin-charge-family
theory a candidate to go beyond the standard model:
i. The spins, handedness, charges and family quantum numbers of fermions are
determined by the Cartan subalgebra of Sab and S̃ab, and the internal space of
fermions is described by the Clifford ”basis vectors” b̂m†f .
ii. Couplings of fermions to the vector gauge fields, which are the superposi-
tion of gauge fields ωstm, Sect. 17.4.2, with the space index m = (0, 1, 2, 3) and
with charges determined by the Cartan subalgebra of Sab and S̃ab (Sabωcde =
i(ωadeη
bc −ωbdeη
ac) and equivalently for the other two indexes of ωcde gauge
fields, manifesting the symmetry of space (d − 4)), and couplings of fermions
to the scalar gauge fields [19, 20, 23, 29, 31, 38, 41, 42, 45, 46] with the space index
s ≥ 5 and the charges determined by the Cartan subalgebra of Sab and S̃ab (as
explained in the case of the vector gauge fields), and which are superposition of
eitherωsts or ω̃abts, Sect. 17.4.2
Lf = ψ̄γm(pm −
∑
A,i
gAiτAiAAim )ψ+
{
∑
s=7,8
ψ̄γsp0s ψ}+
{
∑
t=5,6,9,...,14
ψ̄γtp0t ψ} , (17.16)
where p0s = ps − 12S
s ′s"ωs ′s"s −
1
2
S̃abω̃abs, p0t = pt − 12S
t ′t"ωt ′t"t −
1
2
S̃abω̃abt,
with m ∈ (0, 1, 2, 3), s ∈ (7, 8), (s ′, s") ∈ (5, 6, 7, 8), (a, b) (appearing in S̃ab)
run within either (0, 1, 2, 3) or (5, 6, 7, 8), t runs ∈ (5, . . . , 14), (t ′, t") run either
∈ (5, 6, 7, 8) or ∈ (9, 10, . . . , 14). The spinor function ψ represents all family mem-
bers of all the 2
7+1
2
−1 = 8 families.
The first line of Eq. (17.16) determines in d = (3+1) the kinematics and dynamics
of fermion fields, coupled to the vector gauge fields [23, 26, 31]. The vector gauge
fields are the superposition of the spin connection fields ωstm, m = (0, 1, 2, 3),
(s, t) = (5, 6, · · · , 13, 14), and are the gauge fields of Sst, Sect. 17.4.2.
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The operators τAi (τAi =
∑
a,b c
Ai
ab S
ab, Sab are the generators of the Lorentz
transformations in the Clifford space of γa’s) are presented in Eqs. (17.27, 17.28) of
App. 17.6. They represent the colour charge, τ⃗3, the weak charge, τ⃗1, and the hyper
charge, Y = τ4 + τ23, τ4 is the ”fermion” charge, originating in SO(6) ⊂ SO(13, 1),
τ23 belongs together with τ⃗1 of SU(2)weak to SO(4) (⊂ SO(13+ 1)).
One fermion irreducible representation of the Lorentz group contains, as seen in Table 7
of Ref. [1] or in Table 5 of Ref. [33], quarks and leptons and antiquarks and antileptons,
belonging to the first family in Table 17.1.
Let us repeat again that the SO(7, 1) subgroup content of the SO(13, 1) group is the
same for the quarks and leptons and the same for the antiquarks and antileptons.
Quarks distinguish from leptons, and antiquarks from antileptons, only in the
SO(6) ⊂ SO(13, 1) part, that is in the colour (τ33, τ38) part and in the ”fermion”
quantum number τ4. The quarks distinguish from antiquarks, and leptons from
antileptons, in the handedness, in the SU(2)I (weak), SU(2)II, in the colour part
and in the τ4 part, explaining the relation between handedness and charges of
fermions and antifermions, postulated in the standard model 7.
All the vector gauge fields, which interact with the condensate, presented in
Table 17.2, become massive, Sect. 17.4.2. The vector gauge fields not interacting with
the condensate — the weak, colour, hyper charge and electromagnetic vector gauge fields
— remain massless, in agreement with by the standard model assumed gauge fields
before the electroweak break 8.
After the electroweak break, caused by the scalar fields, the only conserved charges
are the colour and the electromagnetic charge Q = τ13 + Y (Y = τ4 + τ23). All the
rest interact with the scalar fields of the constant value.
The second line of Eq. (17.16) is the mass term, responsible in d = (3+ 1) for the
masses of fermions and of the weak gauge field (originating in spin connection
fieldsωstm). The interaction of fermions with the scalar fields with the space index
s = (7, 8) (to these scalar fields particular superposition of the spin connection
fields ωabs and all the superposition of ω̃abs with the space index s = (7, 8)
and (a, b) = (0, 1, 2, 3) or (a, b) = (5, 6, 7, 8) contribute), which gain the constant
values in d = (3+ 1), makes fermions and antifermions massive.
The scalar fields, presented in the second line of Eq. (17.16), are in the standard model
interpreted as the higgs and the Yukawa couplings, Sect. 17.4.2, predicting in the
spin-charge-family theory that there must exist several scalar fields 9.
These scalar gauge fields split into two groups of scalar fields. One group of two
triplets and three singlets manifests the symmetry S̃U(2)
(S̃O(3,1),L)
×S̃U(2)
(S̃O(4),L)
7 Ref. [30] points out that the connection between handedness and charges for fermions and
antifermions, both appearing in the same irreducible representation, explains the triangle
anomalies in the standard model with no need to connect ”by hand” the handedness and
charges of fermions and antifermions.
8 The superposition of the scalar gauge fields ω̃st7 and ω̃st8, which at the electroweak
break gain constant values in d = (3 + 1), bring masses to all the vector gauge fields,
which couple to these scalar fields.
9 The requirement of the standard model that there exist the Yukawa couplings, speaks by
itself that there must exist several scalar fields explaining the Yukawa couplings.
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17 The achievements of the spin-charge-family theory so far 241
×U(1). The other group of another two triplets and the same three singlets mani-
fests the symmetry S̃U(2)
(S̃O(3,1),R)
×S̃U(2)
(S̃O(4),R)
×U(1).
The three U(1) singlet scalar gauge fields are superposition of ωs ′t ′s, s = (7, 8),
(s ′, t ′) = (5, 6, · · · , 14), with the sums of Ss ′t ′ arranged into superposition of τ13,
τ23 and τ4. The three triplets interact with both groups of quarks and leptons and
antiquarks and antileptons [39–41, 45–48].
Families of fermions from Table 17.1, interacting with these scalar fields, split
as well into two groups of four families, each of these two groups are coupled
to one of the two groups of scalar triplets while all eight families couple to the
same three singlets. The scalar gauge fields, manifesting S̃U(2)L,R × S̃U(2)L,R, are
the superposition of the gauge fields ω̃abs, s = (7, 8), (a, b) = either (0, 1, 2, 3) or
(5, 6, 7, 8), manifesting as twice two triplets.
17.4.2 Vector and scalar gauge fields before electroweak break
The second line of Eq. (17.15) represents the action for the gauge fields Agf
Agf =
∫
ddx E (αR+ α̃ R̃) ,
R =
1
2
{fα[afβb] (ωabα,β −ωcaαω
c
bβ)}+ h.c. ,
R̃ =
1
2
{fα[afβb] (ω̃abα,β − ω̃caα ω̃
c
bβ)}+ h.c. . (17.17)
It is proven in Ref. [26] that the vector and the scalar gauge fields manifest in
d = (3+ 1), after the break of the starting symmetry, as the superposition of spin
connection fields, when the space (d−4) manifest the assumed symmetry. fβa and
eaα are vielbeins and inverted vielbeins respectively, eaαfβa = δ
β
α, eaαfαb = δab,
E = det(eaα).
Varying the action of Eq. (17.17) with respect to the spin connection fields the
expression for the spin connection fieldsωabe follows
ωab
e =
1
2E
{eeα ∂β(Ef
α
[af
β
b]) − eaα ∂β(Ef
α
[bf
βe]) − ebα∂β(Ef
α[efβa])}
+
1
4
{Ψ̄(γe Sab − γ[aSb]
e)Ψ}
−
1
d− 2
{δea[
1
E
edα∂β(Ef
α
[df
β
b]) + Ψ̄γdS
d
b Ψ]
− δeb[
1
E
edα∂β(Ef
α
[df
β
a]) + Ψ̄γdS
d
a Ψ]} . (17.18)
Replacing Sab in Eq. (17.18) with S̃ab, the expression for the spin connection fields
ω̃ab
e follows.
If there are no spinors (fermions) present, Ψ = 0, then either ωabe or ω̃abe are
uniquely expressed with the vielbeins.
Spin connection fields ωabe represent vector gauge fields to the corresponding
fermion fields if index e is m = (0, 1, 2, 3). If e ≥ 5 the spin connection fields
manifest in d = (3+ 1) as scalar gauge fields.
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It is proven in Ref. [26] 10 that in spaces with the desired symmetry the vielbein
can be expressed with the gauge fields,
fσm =
∑
A
τ⃗Aσ A⃗Am ,
τAiσ =
∑
st
cAist (esτ f
σ
t − etτ f
σ
s)x
τ ,
AAim =
∑
st
cAistω
st
m ,
τAi =
∑
st
cAist S
st ,
{τAi, τBj}− = iδ
ABfAijkτAk . (17.19)
The vector gauge fields AAim of τAi represent in the spin-charge-family theory all the
observed gauge fields, as well as the additional non observed vector gauge fields,
which interacting with the condensate gain heavy masses.
The scalar (gauge) fields, carrying the space index s = (5, 6, . . . , d), offer in the
spin-charge-family for s = (7, 8) the explanation for the origin of the Higgs’s scalar
and the Yukawa couplings of the standard model, while scalars with the space index
s = (9, 10, . . . , 14) offer the explanation for the proton decay, as well as for the
matter/antimatter asymmetry in the universe.
In the scalar gauge fields besidesωsts ′ also ω̃abs contribute.
The explicit expressions for cAiab, and correspondingly for τAi, and AAia , are
written in Sects. 4.2.1. and 4.2.2 of Ref. [1].
2.a Vector gauge fields.
All the vector gauge fields are in the spin-charge-family theory expressible with the
spin connection fieldsωstm as
AAim =
∑
s,t
cAist ω
st
m , (17.20)
with
∑
A,i τ
AiAAim =
∑∗
a,b S
abωabm,∗ means that summation runs over (a, b)
respecting the symmetry SO(7, 1)× SU(3)×U(1), with SO(7, 1) breaking further
to SO(3, 1)× SU(2)I × SU(2)II.
The vector gauge fields are namely analysed from the point of view of the possibly
observed fields in d = (3+ 1) space: besides gravity, the colour SU(3), the weak
SU(2)I, the second SU(2)II and theU(1)τ4 - the vector gauge field of the ”fermion”
quantum number τ4.
10 We presented in Ref. [26] the proof, that the vielbeins fσm (Einstein index σ ≥ 5, m =
0, 1, 2, 3) lead in d = (3 + 1) to the vector gauge fields, which are the superposition of
the spin connection fields ωstm: fσm =
∑
A A⃗
A
m τ⃗
Aσ
τ x
τ, with AAim =
∑
s,t c
Ai
stω
st
m,
when the metric in (d − 4), gστ, is invariant under the coordinate transformations xσ
′
=
xσ +
∑
A,i,s,t ε
Ai (xm) cAist E
σst(xτ) and
∑
s,t c
Ai
st E
σst = τAiσ, while τAiσ solves the
Killing equation:Dσ τAiτ +DττAiσ = 0 (Dσ τAiτ = ∂σ τAiτ − Γτ
′
τστ
Ai
τ ′ ). And similarly also for
the scalar gauge fields.
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17 The achievements of the spin-charge-family theory so far 243
Due to the interaction with the condensate the second SU(2)II (one superposition
of the third component of SU(2)II and of the U(1)τ4 vector gauge fields and the
rest two components of the SU(2)II vector gauge field) become massive, while the
colour SU(3), the weak SU(2)I, the second superposition of the third component
of SU(2)II and the U(1)τ4 , forming the hyper charge vector gauge field, remain
massless. That is: All the vector gauge fields, as well as the scalar gauge fields of
Sab and of S̃ab, which interact with the condensate, become massive.
The effective action for all the massless vector gauge fields, the gauge fields which
do not interact with the condensate and remain therefore massless, before the
electroweak break, equal to
∫
d4x {−1
4
FAimn F
Aimn }, with the structure constants
fAijk concerning the colour SU(3), weak SU(2) and hyper charge U(1) groups [26].
All these relations are valid as long as spinors and vector gauge fields are weak
fields in comparison with the fields which force (d− 4) space to be (almost) curled,
Ref. [50]. When all these fields, with the scalar gauge fields included, start to be
comparable with the fields (spinors or scalars), which determine the symmetry of
(d− 4) space, the symmetry of the whole space changes.
The electroweak break, caused by the constant (non zero vacuum expectation)
values of the scalar gauge fields, carrying the space index s = (7, 8), makes the
weak and the hyper charge gauge fields massive. The only vector gauge fields
which remain massless are, besides the gravity, the electromagnetic and the colour
vector gauge fields — the observed three massless gauge fields.
2.b. Scalar gauge fields in d = (3+ 1).
The starting action of the spin-charge-family theory offers scalar fields of two kinds:
a. Scalar fields, taking care of the masses of quarks and leptons have the space index
s = (7, 8) and carry with respect to this space index the weak charge τ13 = ±1
2
and the hyper charge Y = ∓1
2
, Table 17.3, Eq. (17.23). With respect to the index Ai,
determined by the relation τAi =
∑
ab c
Ai
abS
ab and τ̃Ai =
∑
ab c
Ai
abS̃
ab, that
is with respect to Sab and S̃ab, they carry charges and family charges in adjoint
representations.
b. There are in the starting action of the spin-charge-family theory, Eq. (17.15), scalar
fields, which transform antileptons and antiquarks into quarks and leptons and
back. They carry space index s = (9, 10, . . . , 14), They are with respect to the space
index colour triplets and antitriplets, while they carry charges τAi and τ̃Ai in
adjoint representations.
Following Refs. [1, 31, 38] I shall review both kinds of scalar fields. 11
2.b.i Scalar gauge fields determining scalar higgs and Yukawa couplings
Making a choice of the scalar index equal to s = (7, 8) (the choice of (s = 5, 6)
would also work) and allowing all superposition of ω̃ãb̃s, while with respect to
11 Let us demonstrate how do the infinitesimal generators Sab apply on the spin connections
fields ωbde (= fαe ωbdα) and ω̃b̃d̃e (= f
α
e ω̃b̃d̃α), on either the space index e or any
of the indices (b, d, b̃, d̃) SabAd...e...g = i (ηaeAd...b...g − ηbeAd...a...g) (Section IV. and
Appendix B in Ref. [31]).
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ωabs only the superposition representing the scalar gauge fields A
Q
s , AYs and A4s ,
s = (7, 8) (or any three superposition of these three scalar fields) may contribute.
Let us use the common notation AAis for all the scalar gauge fields with s = (7, 8),
independently of whether they originate inωabs — in this case Ai = (Q,Y, τ4) —
or in ω̃ãb̃s. All these gauge fields contribute to the masses of quarks and leptons
and antiquarks and antileptons after gaining constant values (nonzero vacuum
expectation values).
AAis represents (A
Q
s , A
Y
s , A
4
s ,
⃗̃A1̃s ,
⃗̃A
ÑL̃
s ,
⃗̃A2̃s ,
⃗̃A
ÑR̃
s ) ,
τAi represents (Q, Y, τ4, ⃗̃τ1, ⃗̃NL, ⃗̃τ2, ⃗̃NR) . (17.21)
Here τAi represent all the operators which apply on fermions. These scalars with
the space index s = (7, 8), they are scalar gauge fields of the generators τAi and
τ̃Ai, are expressible in terms of the spin connection fields, App. 17.6 (Ref. [31],
Eqs. (10, 22, A8, A9)).
All the scalar fields with the space index (7, 8) carry with respect to this space
index the weak and the hyper charge (∓1
2
, ±1
2
), respectively, all having therefore
properties as required for the higgs in the standard model.
To make the scalar fields the eigenstates of τ13 = 1
2
(S56 − S78) and to check their
properties with respect to Y (= τ4+τ23 = (1
2
(S56+S78)− 1
3
(S9 10+S11 12+S13 14))
andQ (= τ13 + Y) we need to apply the operators τ13, Y andQ on the scalar fields
with the space index s = (7, 8), taking into account the relation SabAd...e...g =
i (ηaeAd...b...g − ηbeAd...a...g).
Let us rewrite the second line of Eq. (17.16), paying no attention to the momentum
ps , s ∈ (5, . . . , 8), when having in mind the lowest energy solutions manifesting
at low energies.∑
s=(7,8),A,i
ψ̄ γs (−τAiAAis )ψ =
−
∑
A,i
ψ̄ {
78
(+) τAi (AAi7 − iA
Ai
8 )+
78
(−) (τAi (AAi7 + iA
Ai
8 ) }ψ ,
78
(±)= 1
2
(γ7 ± i γ8 ) , AAi78
(±)
:= (AAi7 ∓ iAAi8 ) , (17.22)
with the summation over A and i performed, with AAis representing the scalar
fields (AQs , AYs , A4s) determined by ωs ′,s ′′,s , as well as (Ã4̃s ,
⃗̃A1̃s ,
⃗̃A2̃s ,
⃗̃AÑRs and
⃗̃AÑLs ), determined by ω̃a,b,s , s = (7, 8).
The application of the operators τ13, Y and Q on the scalar fields (AAi7 ∓ iAAi8 )
with respect to the space index s = (7, 8), gives
τ13 (AAi7 ∓ iAAi8 ) = ±
1
2
(AAi7 ∓ iAAi8 ) ,
Y (AAi7 ∓ iAAi8 ) = ∓
1
2
(AAi7 ∓ iAAi8 ) ,
Q (AAi7 ∓ iAAi8 ) = 0 . (17.23)
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Since τ4, Y, τ13 and τ1+, τ1− give zero if applied on (AQs ,AYs andA4s) (with respect
to the quantum numbers (Q, Y, τ4)), and since Y,Q, τ4 and τ13 commute with
the family quantum numbers, one sees that the scalar fields AAis ( =(A
Q
s , AYs ,
AY
′
s , Ã4̃s , Ã
Q̃
s , ⃗̃A1̃s ,
⃗̃A2̃s ,
⃗̃AÑRs ,
⃗̃AÑLs )), s = (7, 8), rewritten as AAi78
(±)
= (AAi7 ∓ iAAi8 ) ,
are eigenstates of τ13 and Y, having the quantum numbers of the standard model
Higgs’s scalar.
These superposition of AAi78
(±)
are presented in Table 17.3 as two doublets with
respect to the weak charge τ13, with the eigenvalue of τ23 (the second SU(2)II
charge) equal to either −1
2
or +1
2
, respectively.
Table 17.3: The two scalar weak doublets, one with τ23 = −1
2
and the other with
τ23 = +1
2
, both with the ”fermion” quantum number τ4 = 0, are presented. In this
table all the scalar fields carry besides the quantum numbers determined by the
space index also the quantum numbers A and i from Eq. (17.21). The table is taken
from Ref. [31].
name superposition τ13 τ23 spin τ4 Q
AAi78
(−)
AAi7 + iA
Ai
8 +
1
2
− 1
2
0 0 0
AAi56
(−)
AAi5 + iA
Ai
6 −
1
2
− 1
2
0 0 -1
AAi78
(+)
AAi7 − iA
Ai
8 −
1
2
+ 1
2
0 0 0
AAi56
(+)
AAi5 − iA
Ai
6 +
1
2
+ 1
2
0 0 +1
It is not difficult to show that the scalar fields AAi78
(±)
are triplets as the gauge fields
of the family quantum numbers ( ⃗̃NR, ⃗̃NL, ⃗̃τ2, ⃗̃τ1 or singlets as the gauge fields of
Q = τ13 + Y, Q ′ = − tan2 ϑ1Y +τ13 and Y ′ = − tan2 ϑ2τ4 + τ23.
Table 17.1 represents two groups of four families. It is not difficult to see that ѱL
and τ̃1± transform the first four families among themselves, leaving the second
group of four families untouched, while ѱR and τ̃
2± do not influence the first
four families and transform the second four families among themselves. All the
scalar fields with s = (7, 8) ”dress” the right handed quarks and leptons with the
hyper charge and the weak charge so that they manifest charges of the left handed
partners.
The mass matrices 4× 4, representing the application of the scalar gauge fields on
fermions of each of the two groups, have the symmetry SU(2)× SU(2)×U(1) of
the form as presented in Eq. (17.24) 12. The influence of scalar fields on the masses
of quarks and leptons depends on the coupling constants and the masses of the
12 The symmetry SU(2) × SU(2) × U(1) of the mass matrices, Eq. (17.24), is expected to
remain in all loop corrections [47].
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scalar fields, determining parameters of the mass matrix
Mα =


−a1 − a e d b
e∗ −a2 − a b d
d∗ b∗ a2 − a e
b∗ d∗ e∗ a1 − a


α
, (17.24)
with α representing family members — quarks and leptons [39–41, 46, 48]. In
Subsect. 17.4.3 the predictions of the spin-charge-family theory following from the
symmetry of mass matrices of Eq. (17.24) are discussed.
The spin-charge-family theory treats quarks and leptons in equivalent way. The
differences among family members occur due to the scalar fields (Q · AQ78
(±)
, Y ·
AQ78
(±)
, τ4 ·A478
(±)
) [46, 48].
Twice four families of Table 17.1, with the two groups of two triplets applying
each on one of the two groups of four families and one group of three singlets
applying on all eight families, i. offer the explanation for the appearance of the
Higgs’s scalar and Yukawa couplings of the observed three families, predicting
the fourth family to the observed three families and several scalar fields, ii. predict
that the stable of the additional four families with much higher masses that the
lower four families contributes to the dark matter.
2.b.ii Scalar gauge fields causing transitions from antileptons and antiquarks
into quarks and leptons [25]
Besides the scalar fields with the space index s = (7, 8) which manifest in d =
(3 + 1) as scalar gauge fields with the weak and hyper charge ±1
2
and ∓1
2
, re-
spectively, and which gaining at low energies constant values cause masses of
families of quarks and leptons and of the weak gauge field, there are in the start-
ing action, Eqs. (17.15, 17.16), additional scalar gauge fields with the space index
t = (9, 10, 11, 12, 13, 14). They are with respect to the space index t either triplets or
antitriplets causing transitions from antileptons into quarks and from antiquarks
into quarks and back. These scalar fields are in Eq. (17.16) presented in the third
line.
These scalar fields are offering the explanation for the matter/antimater asymme-
try in the universe, and might be responsible for proton decay and lepton number
nonconservation. The reader is kindly ask to read the article [25], for a short review
one can see the Refs. [1, 23].
17.4.3 Predictions of spin-charge-family theory
Let me say that the fact that the simple starting action, Eq. (17.15) — in which
fermions interact with gravity only (the vielbeins and the two kinds of the spin
connection fields), while the internal spaces of fermions and bosons are describ-
able by the ”basis vectors” which are superposition of odd or even products of
Clifford algebra operators γa’s, respectively — offers the explanation for all the
assumptions of the standard model and for the second quantized postulates for
fermions and bosons, while unifying all the so far known forces, with gravity
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included, predicting new vector gauge fields, new scalar gauge fields and new
families of fermions, gives a hope that the spin-charge-family theory is offering the
right next step beyond the standard model.
i. The existence of the lower group of four families predicts the fourth family
to the observed three, which should be seen in next experiments. The masses of
quarks of these four families are determined by several scalar fields, all with the
properties of the scalar higgs, some of them of which might also be observed.
The symmetry [46, 47], Eq. (17.24), and the values of mass matrices of the lower
four families are determined with two triplet scalar fields, ⃗̃A1̃
78
(±)
and ⃗̃AÑL̃
78
(±)
, and
three singlet scalar fields, AQ
78
(±)
, AY
78
(±)
, A4
78
(±)
, Eq. (17.21), explaining the Higgs’s
scalar and Yukawa couplings of the standard model, Refs. [23, 27, 31, 46, 48] and
references therein.
Any accurate 3 × 3 submatrix of the 4 × 4 unitary matrix determines the 4 × 4
matrix uniquely. Since neither the quark and (in particular) nor the lepton 3× 3
mixing matrix are measured accurately enough to be able to determine three
complex phases of the 4× 4mixing matrix, we assume (what also simplifies the
numerical procedure) [39–41, 45, 46] that the mass matrices are symmetric and
real and correspondingly the mixing matrices are orthogonal. We fitted the 6
free parameters of each family member mass matrix, Eq. (17.24), to twice three
measured masses (6) of each pair of either quarks or leptons and to the 6 (from
the experimental data extracted) parameters of the corresponding 4× 4 mixing
matrix.
I present here the old results for quarks only, taken from Refs. [46]. The accuracy
of the experimental data for leptons are not yet large enough that would allow any
meaningful prediction 13. It turns out that the experimental [54] inaccuracies are
for the mixing matrices too large to tell trustworthy mass intervals for the quarks
masses of the fourth family members 14. Taking into account the calculations
of Ref. [54] fitting the experimental data (and the meson decays evaluations in
the literature as well as our own evaluations) the authors of the paper [46] very
roughly estimate that the fourth family quarks masses might be pretty above 1
TeV.
Since the matrix elements of the 3 × 3 submatrix of the 4 × 4 mixing matrix de-
pend weakly on the fourth family masses, the calculated mixing matrix offers the
prediction to what values will more accurate measurements move the present ex-
13 The numerical procedure, explained in the paper [46], to fit free parameters of the mass
matrices to the experimental data within the experimental inaccuracy of the mixing
matrix elements of the so far observed quarks (the inaccuracy of masses do not influence
the results very much) is tough.
14 We have not yet succeeded to repeat the calculations presented in Refs. [46] with the
newest data from Ref. [55]. Let us say that the accuracy of the mixing matrix even for
quarks remains in Ref. [55] far from needed to predict the masses of the fourth two quarks.
For the chosen masses of the four family quarks the mixing matrix elements are expected
to slightly change in the direction proposed by Eq. (17.25).
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248 N.S. Mankoč Borštnik
perimental data and also the fourth family mixing matrix elements in dependence
of the fourth family masses, Eq. (17.25):
Vud will stay the same or will very slightly decrease; Vub and Vcs, will still lower;
Vtd will lower, and Vtb will lower; Vus will slightly increase; Vcd will (after
decreasing) slightly rise; Vcb will still increase and Vts will (after decreasing)
increase.
In Eq. (17.25) the matrix elements of the 4× 4 mixing matrix for quarks are pre-
sented, obtained when the 4× 4mass matrices respect the symmetry of Eq. (17.24)
while the parameters of the mass matrices are fitted to the (exp) experimental
data [54], Ref. [46]. The two choices of the fourth family quark masses are used
in the calculations: mu4 = md4 = 700 GeV (scf1) and mu4 = md4 = 1 200 GeV
(scf2). In parentheses, ( ) and [ ], the changes of the matrix elements are presented,
which are due to the changes of the top mass within the experimental inaccuracies:
with the mt = (172 + 3 × 0.76) GeV and mt = (172 − 3 × 0.76), respectively (if
there are one, two or more numbers in parentheses the last one or more numbers
are different, if there is no parentheses no numbers are different) [arxiv:1412.5866].
|V(ud)| =


exp 0.97425± 0.00022 0.2253± 0.0008 0.00413± 0.00049
scf1 0.97423(4) 0.22539(7) 0.00299 0.00776(1)
scf2 0.97423[5] 0.22538[42] 0.00299 0.00793[466]
exp 0.225± 0.008 0.986± 0.016 0.0411± 0.0013
scf1 0.22534(3) 0.97335 0.04245(6) 0.00349(60)
scf2 0.22531[5] 0.97336[5] 0.04248 0.00002[216]
exp 0.0084± 0.0006 0.0400± 0.0027 1.021± 0.032
scf1 0.00667(6) 0.04203(4) 0.99909 0.00038
scf2 0.00667 0.04206[5] 0.99909 0.00024[21]
scf1 0.00677(60) 0.00517(26) 0.00020 0.99996
scf2 0.00773 0.00178 0.00022 0.99997[9]


.
(17.25)
Let me conclude that according to Ref. [46] the masses of the fourth family lie
much above the known three. The larger are masses of the fourth family the
larger are Vu1d4 in comparison with Vu1d3 and the more is valid that Vu2d4 <
Vu1d4 , Vu3d4 < Vu1d4 . The flavour changing neutral currents are correspondingly
weaker.
Let be noticed that the prediction of Ref. [56], Vu1d4 > Vu1d3 , Vu2d4 < Vu1d4 ,
Vu3d4 < Vu1d4 , agrees with the prediction of Refs. [46].
In Ref. [48] the authors discuss the question why the existence of the fourth family
is not (at least yet) in contradiction with the experimental data.
ii. The theory predicts the existence of several scalar fields. To the lower four fami-
lies two triplets and three singlets contribute, to the upper four families the same
three singlets and different two triplets contribute, Eq. (17.21), Sects. 17.4.2, 17.4.2.
Some superposition of the three singlets and two triplets contributing to masses
and to mixing matrices of quarks and leptons of the lower four families will be
observed, representing so far the observed scalar higgs and Yukawa couplings.
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17 The achievements of the spin-charge-family theory so far 249
iii. The theory predicts the existence of besides the additional scalar fields also the
additional vector gauge fields of very high mass, Sects. 17.4.2, 17.4.2.
iv. The theory predicts the existence of the upper four families of quarks and
leptons and antiquarks and antileptons, Table 17.1, with the same family members
charges, Table 7 of Ref [1], as are the charges of the lower four families, interacting
correspondingly with the same vector gauge fields. At low energies the upper four
families are decoupled from the lower four families.
The masses of the upper four families are determined by the two triplets (⃗̃A2̃
78
(±)
, ⃗̃A
ÑR̃
78
(±)
)
and three singlets (AQ
78
(±)
, AQ
′
78
(±)
, AY
′
78
(±)
), the same singlets contribute also to masses
of the lower four families, Sect. 17.4.2.
The stable of the upper four families offers the explanation for the appearance of
the dark matter in our universe.
Since the masses of the upper four families are much higher than the masses of
the lower four families, the ”nuclear” force among the baryons and mesons of
these quarks and antiquarks differ a lot from the nuclear force of the baryons and
fermions of the lower four families.
A rough estimation of properties of baryons of the stable fifth family members, of
their behaviour during the evolution of the universe and when scattering on the
ordinary matter, as well as a study of possible limitations on the family properties
due to the cosmological and direct experimental evidences are done in Ref. [45].
In Ref. [57] the weak and ”nuclear” scattering of such very heavy baryons by
ordinary nucleons is studied, showing that the cross section for such scattering
is very small and therefore consistent with the observation of experiments so far,
provided that the quark mass of this baryon is about 100 TeV or above.
In Ref. [45] a simple hydrogen-like model is used to evaluate properties of baryons
of these heavy quarks, with one gluon exchange determining the force among the
constituents of the fifth family baryons 15.
The authors of Ref. [45] study the freeze out procedure of the fifth family quarks
and antiquarks and the formation of baryons and antibaryons up to the tempera-
ture kbT = 1 GeV, when the colour phase transition starts which depletes almost
all the fifth family quarks and antiquarks, while the colourless fifth family neu-
trons with very small scattering cross section decouples long before (at kbT = 100
GeV).
The cosmological evolution suggests for the mass limits the range 10 TeV< mq5 <
a few · 102 TeV and for the scattering cross sections 10−8 fm2 < σc5 < 10−6 fm2.
The measured density of the dark matter does not put much limitation on the
properties of heavy enough clusters 16.
15 The weak force and the electromagnetic force start to be at small distances due to heavy
masses of quarks of the same order of magnitude as the colour force.
16 In the case that the weak interaction determines the cross section of the neutron n5, the
interval for the fifth family quarks would be 10 TeV < mq5 c
2 < 105 TeV.
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250 N.S. Mankoč Borštnik
The DAMA/LIBRA experiments [60] limit, provided that they measure the heavy
fifth family clusters, the quark mass in the interval: 200TeV < mq5 < 10
5 TeV,
Ref. [45].
Baryons of the fifth family are heavy, forming small enough clusters with small
enough scattering amplitude among themselves and with the ordinary matter to
be the candidate for the dark matter.
Masses of the stable fifth family of quarks and leptons are much above the fourth
family members.
Although the upper four families carry the weak (of two kinds) and the colour
charge, these group of four families are completely decoupled from the lower four
families up to the < 1016 GeV when the breaks of symmetries are expected to
recover.
17.5 Conclusions
The spin-charge-family theory [1, 20, 21, 23, 36, 37, 42, 44] assumes in d = (13 + 1)-
dimensional space a simple action, Eq. (17.15), for the massless fermions and for
the massless vielbeins and the two kinds of spin connection fields, with which
fermions interact. The description of the internal space of fermions with ”basis
vectors” which are superposition of an odd products of the Clifford algebra objects
and of bosons with ”basis vectors” which are superposition of an even products of
the Clifford algebra objects offers the explanation for spins, charges and families
of fermions and their vector and scalar gauge fields, as required by the standard
model, while explaining as well the second quantization postulates for fermions
and bosons.
Some of the predictions of the spin-charge-family theory can experiments soon
confirm and correspondingly confirm (or reject) the theory. Because the theory
offers meaningful answers to many open questions in physics of elementary
fermion and boson fields and in cosmology and because the theory offers more
and more answers the more effort and work is put into it, it might very well be
that the theory does offer the right next step beyond the standard model.
The description of fermions and bosons, both second quantized, with the Clifford
odd and the Clifford even ”basis vectors”, respectively, clarifies how strongly are
all the properties of elementary fields determined by the internal space of fields,
and that the internal space of fermions not only unifies spin, handedness, all the
charges and families of fermions but manifests as well the strong connections with
the corresponding boson vector and scalar gauge fields.
The theory obviously needs more collaborators as it is necessary to find answers
to questions, like:
i. What is the dimension of space time? In any dimension d = 2(2n + 1) there
namely exist fermions of only one handedness, as discussed in Ref. [33], while in
any subspace of this space there are fermions of both handedness. i.a. How can
we look for anomalies of Kaluza-Klein theories in higher dimensions? i.b. As well
as for the renormalizability?
ii. The spontaneous breaks of symmetries, from the starting one to the final ones,
must carefully be done. ii.a. The breaks from any d = 2(2n + 1) in steps to the
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17 The achievements of the spin-charge-family theory so far 251
observable d = (3+ 1) must be done, following the number of massless families
of fermions and the appearance of the vector and scalar gauge fields in each step.
So far we studied only the breaks of symmetry for the toy models [49, 51, 52],
starting with d = (5+ 1). ii.b. To learn more the electroweak break with the scalar
fields defined in d = 2(2n + 1), n = 3, with the space index (7, 8) [49–52] needs
additional treating.
iii. The second quantization of fermion and boson fields with the description of
the internal space of fermions and bosons by the Clifford odd and even ”basis
vectors”, respectively, is opening a new insight in to quantum field theory. Ref. [33]
presents only the first step to the second quantization of bosons by the Clifford
even ”basis vectors”. A further study is needed.
iv. One irreducible representation of the Lorentz group in the internal space of
fermions, Table 7 in Ref. [1] and Table 5 in Ref. [33], includes all the quarks and
leptons and antiquarks and antileptons observed so far (with not yet observed
the right handed neutrinos and the left handed antineutrinos included). No Dirac
sea is needed. iv.a. Additional studies of masses of fermions and antifermions in
addition to those of Refs. [46, 58] are needed.
v. So far only three families of quarks and leptons have been observed. The spin-
charge-family theory predicts the fourth family to the observed three, very weakly
coupled to the observed three with masses a few TeV or higher. Although the
accurately known 3 × 3 submatrix of the 4 × 4 unitary matrix determines the
4× 4 matrix uniquely, even the quarks mixing matrix is known far non accurately
enough to enable prediction of masses of the fourth family, Ref. [46]. v.a. A further
study of the properties of the 4 × 4 mixing matrix as following from the mass
matrices of quarks and leptons with the known symmetries (what reduces the
number of free parameters to be fitted to the experimental data) is needed and the
way of improving the experimental accuracy needs to be suggested. v.b.The proof
that the symmetry of mass matrices S̃U(2)× S̃U(2)×U(1) keeps in all orders of
loop corrections, presented in Ref. [58], must be checked.
vi. There are scalar fields which are colour triplets and antitriplets, predicted
by the spin-charge-family theory [25], which transform antileptons into quarks
and antiquarks into quarks and back, causing in the expanding universe matter-
antimatter asymmetry. The study is needed to see their influence on the lepton
number non conservation.
vii. A study of the coupling constants of fermions to the corresponding gauge
vector and scalar fields in comparison with those of SO(10) and SO(13 + 1) is
needed.
viii. The masses of the upper four families after the electroweak break and the
influence of the neutrino condensate on their masses must be studied. viii.a. The
behaviour of the stable fifth family members, their ”freezing” out and formation
of neutral objects, interacting with the weak force, is needed and their contribution
to the dark matter. viii.b. As well as the contribution of the heavy neutrinos to the
xdark matter.
ix. If the spin-charge-family theory is the right next step beyond the standard model, it
is worthwhile to find out what it has in common with all the theories and models
which seems to be promising.
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252 N.S. Mankoč Borštnik
x. And many more.
17.6 Infinitesimal generators of subgroups of SO(13, 1)group
The relations are taken from Ref. [1].
The reader can calculate all the quantum numbers of Table 5 in Ref. [33] and of Table 17.1,
if taking into account the generators of the two SU(2) (⊂ SO(3, 1) ⊂ SO(7, 1) ⊂ SO(13, 1))
groups, describing spins and handedness of fermions, their two kinds of the weak charges,
the colour charges, the ”fermion” charge, as well as the family quantum numbers.
One needs
N⃗±(= N⃗(L,R)) :=
1
2
(S23 ± iS01, S31 ± iS02, S12 ± iS03) , ⃗̃N±(= ⃗̃N(L,R)) :=
1
2
(S̃23 ± iS̃01 ,
(17.26)
the generators of the two SU(2) (SU(2)⊂ SO(4)⊂ SO(7, 1) ⊂ SO(13, 1)) groups, describing
the weak charge, τ⃗1, and the second kind of the weak charge, τ⃗2, of fermions and the
corresponding family quantum numbers
τ⃗1 : =
1
2
(S58 − S67, S57 + S68, S56 − S78) , τ⃗2 :=
1
2
(S58 + S67, S57 − S68, S56 + S78) ,
⃗̃τ1 : =
1
2
(S̃58 − S̃67, S̃57 + S̃68, S̃56 − S̃78) , ⃗̃τ2 :=
1
2
(S̃58 + S̃67, S̃57 − S̃68, S̃56 + S̃78) ,
(17.27)
and the generators of SU(3) and U(1) subgroups of SO(6) ⊂ SO(13, 1), describing the
colour charge and the ”fermion” charge of fermions as well as the corresponding family
quantum number τ̃4
τ⃗3 :=
1
2
{S9 12 − S10 11 , S9 11 + S10 12, S9 10 − S11 12 , S9 14 − S10 13,
S9 13 + S10 14 , S11 14 − S12 13 , S11 13 + S12 14,
1√
3
(S9 10 + S11 12 − 2S13 14)} ,
τ4 := −
1
3
(S9 10 + S11 12 + S13 14) ,
τ̃4 := −
1
3
(S̃9 10 + S̃11 12 + S̃13 14) .
(17.28)
The (chosen) Cartan subalgebra operators, determining the commuting operators in the
above equations, is presented in Eq. (17.2).
The hypercharge Y and the electromagnetic charge Q and the corresponding family quan-
tum numbers then follows as
Y := τ4 + τ23 , Q := τ13 + Y , Y ′ := −τ4 tan2 ϑ2 + τ23 , Q ′ := −Y tan2 ϑ1 + τ13 , ,
Ỹ := τ̃4 + τ̃23 , Q̃ := Ỹ + τ̃13 , Ỹ ′ := −τ̃4 tan2 ϑ2 + τ̃23 , Q̃ ′ = −Ỹ tan2 ϑ1 + τ̃13 . .
(17.29)
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17 The achievements of the spin-charge-family theory so far 253
Below are some of the above expressions written in terms of nilpotents and projectors
N±+ = N
1
+ ± iN2+ = −
03
(∓i)
12
(±) , N±− = N1− ± iN2− =
03
(±i)
12
(±) ,
ѱ+ = −
03
˜(∓i)
12
˜(±) , ѱ− =
03
˜(±i)
12
˜(±) ,
τ1± = (∓)
56
(±)
78
(∓) , τ2∓ = (∓)
56
(∓)
78
(∓) ,
τ̃1± = (∓)
56
˜(±)
78
˜(∓) , τ̃2∓ = (∓)
56
˜(∓)
78
˜(∓) . (17.30)
Acknowledgment
The author 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 by offering the room and computer facili-
ties and Matjaž Breskvar of Beyond Semiconductor for donations, in particular
for the annual workshops entitled ”What comes beyond the standard models”.
Thanks also to all the participants of the Annual workshops ”What comes beyond
the standard models” for helpful discussions.
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