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BLED WORKSHOPS
IN PHYSICS
VOL. 23, NO. 1
Proceedings to the 25th [Virtual]
Workshop
What Comes Beyond . . . (p. 162)
Bled, Slovenia, July 4–10, 2022
11 Clifford odd and even objects, offering description
of internal space of fermion and boson fields,
respectively, open new insight into next step beyond
standard model
N. S. Mankoč Borštnik
Department of Physics, University of Ljubljana
SI-1000 Ljubljana, Slovenia
norma.mankoc@fmf.uni-lj.si
Abstract. In a long series of works the author demonstrated, together with collaborators,
that the model named the spin-charge-family theory offers the explanation for all in the stan-
dard model assumed properties of fermion and boson fields, with the families of fermions
and the Higgs’s scalars included. The theory starts with a simple action in ≥ (13 + 1)-
dimensional space-time with massless fermions which interact with massless gravitational
fields only (vielbeins and the two kinds of spin connection fields). The internal spaces of
fermion and boson fields are described by the Clifford odd and even objects, respectively.
The corresponding odd and even ”basis vectors” in a tensor product with the basis in ordi-
nary momentum or coordinate space define the creation and annihilation operators, which
explain the second quantization postulates for fermion and boson fields. The break of the
starting symmetry leads at low energies to the action for families of quarks and leptons and
the corresponding gauge fields, with Higgs’s fields included, offering several predictions
and several explanations of the observed cosmological phenomena. The properties of the
odd dimensional spaces are also discussed.
Povzetek: V dolgem nizu člankov je avtorica, skupaj s sodelavci, pokazala, da ponuja
model, ki ga avtorica poimenuje teorija spinov-nabojev-družin, razlago za vse v standardnem
modelu privzete lastnosti fermionskih in bozonskih polj, vključno z družinami fermionov in
Higgsovimi skalarji. Teorija predpostavi preprosto akcijo v ≥ (13+ 1)-razsežnem prostoru-
času, v kateri fermioni nimajo mase, interagirajo pa samo z brezmasnim gravitacijskim
poljem (tetradani, ki določajo gravitacijsko polje v običajnem prostoru in dvema vrstama
spinskih povezav, ki so umeritvena polja Lorentzovih transformacij v notranjem prostoru
fermionov). Notranji prostor fermionov opiše avtorica z ”bazičnimi vektorji”, ki so lihi
objekti Clifordove algebre, notranji prostor bozonov pa s Cliffordovo sodimi objekti. Us-
trezni lihi in sodi ”bazični vektorji” v tenzorskem produktu z bazo v prostoru gibalnih
količin definirajo kreacijske in anihilacijske operatorje antikomutirajočih fermionskih polj
in komutirajočih bozonskih polj, kar pojasni postulate za drugo kvantizacijo za fermionska
in bozonska polja. Zlomitev začetne simetrije akcije vodi pri nizkih energijah do akcije kot
jo predpostavi standardni model— za družine kvarkov in leptonov in za ustrezna umer-
itvena polja ter za Higgsove skalarje. Teorija ponuja števine napovedi in pojasni vzroke za
kozmološka opaženja. Predstavi tudi lastnosti Cliffordovih objektov v prostorih z lihim
številom dimenzij.
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Title Suppressed Due to Excessive Length 163
Keywords: Second quantization of fermion and boson fields in Clifford space;
beyond the standard model; Kaluza-Klein-like theories in higher dimensional
space, explanation of appearance of families of fermions, scalar fields, fourth
family, dark matter.
11.1 Introduction
The standard model (with massive neutrinos added) has been experimentally con-
firmed without raising any serious doubts so far on its assumptions, which remain
unexplained 1.
The assumptions of 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 ( [1–6].
Among the questions for which the answers are needed are:
i. Where do fermions, quarks and leptons, originate?
ii. Why do family members, quarks and leptons, manifest so different masses if
they all start as massless?
iii. Why are charges of quarks and leptons so different and why have the left
handed family members so different charges from the right handed ones?
iv. Where do antiquarks and antileptons originate?
v. Where do families of quarks and leptons originate and how many families do
exist?
vi. What is the origin of boson fields, of vector fields which are the gauge fields of
fermions?
vii. What is the origin of the Higgs’s scalars and the Yukawa couplings?
viii. How are scalar fields connected with the origin of families and how many
scalar fields determine properties of the so far (and others possibly be) observed
fermions and of weak bosons?
ix. Why have the scalar fields half integer weak and hyper charge? Do possibly
exist also scalar fields with the colour charges in the fundamental representation ?
ix. Could all boson fields, with the scalar fields included, have a common origin?
x. Where does the dark matter originate? Does the dark matter consist of fermions?
xi. Where does the ”ordinary” matter-antimatter asymmetry originate?
xii. Where does the dark energy originate?
xiii. How can we understand the postulates of the second quantized fermion and
boson fields?
xiv. What is the dimension of space? (3+ 1)?, ((d− 1) + 1)?,∞?
xv. Are all the fields indeed second quantized with the gravity included? And
consequently are all the systems second quantized (although we can treat them in
simplified versions, like it is the first quantization and even the classical treatment),
with the black holes included?
xvi. And many others.
1 This introduction is similar to the one appearing in the arxiv:2210.07004. Also most of
sections and subsections are similar. There are, however, some new parts added.
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164 N. S. Mankoč Borštnik
In a long series of works ( [1–3, 5, 23, 25, 27–29, 31, 32] and the references therein),
the author has succeeded, together with collaborators, to find the answer to many
of the above, and also to other open questions of the standard model, as well as to
several open cosmological questions, with the model named the spin-charge-family
theory. The more work is put into the theory the more answers the theory offers.
The theory assumes that the space has more than (3+ 1) dimensions, it must have
d ≥ (13+ 1), so that the subgroups of the SO(13, 1) group, describing the internal
space of fermions by the superposition of odd products of the Clifford objects
γa’s, manifest from the point of view of d = (3+ 1)-dimensional space the spins,
handedness and charges assumed for massless fermions in the standard model.
Correspondingly each irreducible representation of the SO(13, 1) group carrying
the quantum numbers of quarks and leptons and antiquarks and antileptons, rep-
resents one of families of fermions, the quantum numbers of which are determined
by the second kind of the Clifford objects, by γ̃a (by S̃ab (= i
4
{γ̃a, γ̃b}−).
Fermions interact in d = (13+1) with gravity only, with vielbeins (the gauge fields
of momenta) and the two kinds of the spin connection fields, the gauge fields of
the two kinds of the Lorentz transformations in the internal space of fermions, of
Sab(= i
4
{γa, γb}−) and of S̃ab (= i4 {γ̃
a, γ̃b}−).
The theory assumes a simple starting action ( [5] and the references therein) for the
second quantized massless fermion and antifermion fields, and the corresponding
massless boson fields in d = 2(2n+ 1)-dimensional space
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. . (11.1)
Here 2 fα[afβb] = fαafβb − fαbfβa. faα, and the two kinds of the spin connection
fields,ωabα (the gauge fields of Sab) and ω̃abα (the gauge fields of S̃ab), manifest
in d = (3+ 1) as the known vector gauge fields and the scalar gauge fields taking
2 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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Title Suppressed Due to Excessive Length 165
care of masses of quarks and leptons and antiquarks and antileptons and the weak
boson fields [27] 3
While in any even dimensional space the superposition of odd products of γa’s,
forming the Clifford odd ”basis vectors”, offer the description of the internal space
of fermions with the half integer spins, (manifesting in d = (3+ 1) properties of
quarks and leptons and antiquarks and antileptons, with the families included
if d = (13 + 1), the superposition of even products of γa’s, forming the Clifford
even ”basis vectors”, offer the description of the internal space of boson fields
with integer spins, manifesting as gauge fields of the corresponding Clifford odd
”basis vectors”.
From the point of view of d = (3+1) one family of the Clifford odd ”basis vectors”
with 2
d=14
2
−1 members manifest spins, handedness and charges of quarks and
leptons and antiquarks and antileptons appearing in 2
d=14
2
−1 families, while their
Hermitian conjugated partners appear in another group of 2
d
2
−1 members in 2
d
2
−1
families 4.
The Clifford even ”basis vectors” appear in two groups, each with 2
d
2
−1 × 2d2−1
members, with the Hermitian conjugated partners within the same group and
have correspondingly no families. The Clifford even ”basis vectors” manifest from
the point of view of d = (3+ 1) all the properties of the vector gauge fields before
the electroweak break and for the scalar fields causing the electroweak break (as
assumed by the standard model).
Tensor products of the Clifford odd and Clifford even ”basis vectors” (describing
the internal space of fermions and bosons, respectively) with the basis in ordinary
space form the creation operators to which the ”basis vectors” transfer either
anticommutativity or commutativity. The Clifford odd ”basis vectors” transfer
their anticommutativity to creation operators and to their Hermitian conjugated
partners annihilation operators for fermions. The Clifford even ”basis vectors”
transfer their commutativity to creation operators and annihilation operators
for bosons. Correspondingly the anticommutation properties of creation and
annihilation operators of fermions explain the second quantization postulates
of Dirac for fermion fields, while the commutation properties of creation and
annihilation operators for bosons explain the corresponding second quantization
postulates for boson fields 5.
In Sect. 11.2 the Grassmann and the Clifford algebra are explained and creation
and annihilation operators described as a tensor products of the ”basis vectors”
3 Since the multiplication with either γa’s or γ̃a’s changes the Clifford odd ”basis vec-
tors” into the Clifford even objects, and even ”basis vectors” commute, the action for
fermions can not include an odd numbers of γa’s or γ̃a’s, what the simple starting ac-
tion of Eq. (19.1) does not. In the starting action γa’s and γ̃a’s appear as γ0γap̂a or as
γ0γc Sabωabc and as γ0γc S̃abω̃abc.
4 The appearance of the condensate of two right handed neutrinos causes that the number
of the observed families reduces to two at low energies decoupled groups of four groups.
5 The creation and annihilation operators for either fermion or boson fields with the
momenta zero, have no dynamics, and consequently no influence on clusters of fermion
and boson fields.
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166 N. S. Mankoč Borštnik
offering explanation of the internal spaces of fermion (by the Clifford odd algebra)
and boson (by the Clifford even algebra) fields and the basis in ordinary space.
In Subsect. 11.2.1 the ”basis vectors” are introduced and their properties presented.
In Subsect. 11.2.2 the properties of the Clifford odd and even ”basis vectors” are
demonstrated in the toy model in d = (5+ 1). The simplest cases with d = (1+ 1)
and d = (3+ 1) are also added.
In Subsect. 11.2.3 the properties of the creation and annihilation operators for the
second quantized fields are described.
In Sect. 11.3 a short overview of the achievements and predictions so far of the
spin-charge-family theory is presented,
Sect. 11.4 presents what the reader could learn from the main contribution of this
talk.
In Sect. 11.5 the properties of Clifford odd and Clifford even ”basis vectors” in odd
dimensional spaces are presented, demonstrating how much properties of ”basis
vectors” in odd dimensional spaces differ from the properties in even dimensional
spaces.
11.2 Creation and annihilation operators for fermions and
bosons
The second quantization postulates for fermions [16–18] require that the creation
operators and their Hermitian conjugated partners annihilation operators, de-
pending on a finite dimensional basis in internal space, that is on the space of
half integer spins and on charges described by the fundamental representations
of the appropriate groups, and on continuously infinite number of momenta (or
coordinates) ( [5], Subsect. 3.3.1), fulfil anticommutation relations.
The second quantization postulates for bosons [16–18] require that the creation
and annihilation operators, depending on finite dimensional basis in internal
space, that is on the space of integer spins and on charges described by the
adjoint representations of the same groups, and on continuously infinite number
of momenta (or coordinates) ( [5], Subsect. 3.3.1), fulfil commutation relation.
I demonstrate in this talk that using the Clifford algebra to describe the internal
space of fermions and bosons, the creation and annihilation operators which are
tensor products of the internal basis and the momentum/coordinate basis, not only
fulfil the appropriate anticommutation relations (for fermions) or commutation
relations (for bosons) but also have the required properties for either fermion fields
(if the internal space is described with the Clifford odd products of γa’s) or for
boson fields (if the internal space is described with the Clifford even products of
γa’s). The Clifford odd and Clifford even ”basic vectors” correspondingly offer
the explanation for the second quantization postulates for fermions and bosons,
respectively.
There are two Clifford subalgebras which can be used to describe the internal space
of fermions and of bosons, each with 2d members. In each of the two subalgebras
there are 2 × 2d2−1× 2d2−1 Clifford odd and 2 × 2d2−1× 2d2−1 Clifford even ”basic
vectors” which can be used to describe the internal space of fermion fields, the
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Title Suppressed Due to Excessive Length 167
Clifford odd ”basic vectors”, and of boson fields, the Clifford even ”basic vectors”
in any even d. d = (13+ 1) offers the explanation for all the properties of fermion
fields, with families included, and of boson fields which are the gauge fields of
fermion fields.
In any even d, d = 2(2n + 1) or d = 4n, any of the two Clifford subalgebras
offers twice 2
d
2
−1 irreducible representations, each with 2
d
2
−1 members, which
can represent ”basis vectors” and their Hermitian conjugated partners. Each
irreducible representation offers in d = (13 + 1) the description of the quarks
and the antiquarks and the leptons and the antileptons (with the right handed
neutrinos and left handed antineutrinos included in addition to what is) assumed
by the standard model.
There are obviously only one kind of fermion fields and correspondingly also of
their gauge fields observed. There is correspondingly no need for two Clifford
subalgebras.
The reduction of the two subalgebras to only one with the postulate in Eq. (19.6),
(Ref. [5], Eq. (38)) solves this problem. At the same time the reduction offers
the quantum numbers for each of the irreducible representations of the Clifford
subalgbebra left, γa’s, when fermions are concerned ( [5] Subsect. 3.2).
Boson fields have no families as it will be demonstrated.
Grassmann and Clifford algebras
The internal space of anticommuting or commuting second quantized fields can
be described by using either the Grassmann or the Clifford algebras [1–3,31]. What
follows is a short overview of Subsect.3.2 of Ref. [5] and of references cited in [5].
In Grassmann d-dimensional space there are d anticommuting (operators) θa, and
d anticommuting operators which are derivatives with respect to θa, ∂
∂θa
,
{θa, θb}+ = 0 , {
∂
∂θa
,
∂
∂θb
}+ = 0 ,
{θa,
∂
∂θb
}+ = δab , (a, b) = (0, 1, 2, 3, 5, · · · , d) . (11.2)
Defining [32]
(θa)† = ηaa
∂
∂θa
, leads to (
∂
∂θa
)† = ηaaθa , (11.3)
with ηab = diag{1,−1,−1, · · · ,−1}.
θa and ∂
∂θa
are, up to the sign, Hermitian conjugated to each other. The identity
is the self adjoint member of the algebra. The choice for the following complex
properties of θa and correspondingly of ∂
∂θa
are made
{θa}∗ = (θ0, θ1,−θ2, θ3,−θ5, θ6, ...,−θd−1, θd) ,
{
∂
∂θa
}∗ = (
∂
∂θ0
,
∂
∂θ1
,−
∂
∂θ2
,
∂
∂θ3
,−
∂
∂θ5
,
∂
∂θ6
, ...,−
∂
∂θd−1
,
∂
∂θd
) . (11.4)
The are 2d superposition of products of θa, the Hermitian conjugated partners of
which are the corresponding superposition of products of ∂
∂θa
.
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168 N. S. Mankoč Borštnik
There exist two kinds of the Clifford algebra elements (operators), γa and γ̃a,
expressible with θa’s and their conjugate momenta pθa = i ∂
∂θa
[2], Eqs. (11.2,
11.3),
γa = (θa +
∂
∂θa
) , γ̃a = i (θa −
∂
∂θa
) ,
θa =
1
2
(γa − iγ̃a) ,
∂
∂θa
=
1
2
(γa + iγ̃a) ,
(11.5)
offering together 2 · 2d operators: 2d are superposition of products of γa and 2d
of γ̃a. It is easy to prove, if taking into account Eqs. (11.3, 11.5), that they form
two anticommuting Clifford subalgebras, {γa, γ̃b}+ = 0, Refs. ( [5] and references
therein)
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a . (11.6)
While the Grassmann algebra offers the description of the ”anticommuting integer
spin second quantized fields” and of the ”commuting integer spin second quan-
tized fields” [5, 35], the Clifford algebras which are superposition of odd products
of either γa’s or γ̃a’s offer the description of the second quantized half integer
spin fermion fields, which from the point of the subgroups of the SO(d − 1, 1)
group manifest spins and charges of fermions and antifermions in the fundamental
representations of the group and subgroups.
The superposition of even products of either γa’s or γ̃a’s offer the description of
the commuting second quantized boson fields with integer spins (as we can see
in [9] and shall see in this contribution) which from the point of the subgroups of
the SO(d− 1, 1) group manifest spins and charges in the adjoint representations
of the group and subgroups.
The following postulate, which determines how does γ̃a’s operate on γa’s, reduces
the two Clifford subalgebras, γa’s and γ̃a’s, to one, to the one described by γa’s [2,
14, 29, 31, 32]
{γ̃aB = (−)B i Bγa} |ψoc > , (11.7)
with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise (−)B =
1 [14], |ψoc > is defined in Eq. (19.8) of Subsect. 11.2.1.
After the postulate of Eq. (19.6) it follows:
a. The Clifford subalgebra described by γ̃a’s looses its meaning for the description
of the internal space of quantum fields.
b. The ”basis vectors” which are superposition of an odd or an even products of
γa’s obey the postulates for the second quantization fields for fermions or bosons,
respectively, Sect.11.2.1.
c. It can be proven that the relations presented in Eq. (19.3) remain valid also after
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Title Suppressed Due to Excessive Length 169
the postulate of Eq. (19.6). The proof is presented in Ref. ( [5], App. I, Statement 3a.
d. Each irreducible representation of the Clifford odd ”basis vectors” described by
γa’s are equipped by the quantum numbers of the Cartan subalgebra members of
S̃ab, chosen in Eq. (19.4), as follows
S03,S12,S56, · · · ,Sd−1 d ,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d ,
Sab = Sab + S̃ab = i (θa ∂
∂θb
− θb
∂
∂θa
) . (11.8)
After the postulate of Eq. (19.6) no vector space of γ̃a’s needs to be taken into
account for the description of the internal space of either fermions or bosons, in
agreement with the observed properties of fermions and bosons. Also the Grass-
mann algebra is reduced to only one of the Clifford subalgebras. The operators γ̃a’s
describe from now on properties of fermion and boson ”basis vectors” determined
by superposition of products of odd or even numbers of γa’s, respectively.
γ̃a’s equip each irreducible representation of the Lorentz group (with the infinites-
imal generators Sab = i
4
{γa, γb}−) when applying on the Clifford odd ”basis
vectors” (which are superposition of odd products of γa
′s) with the family quan-
tum numbers (determined by S̃ab = i
4
{γ̃a, γ̃b}−).
Correspondingly the Clifford odd ”basis vectors” (they are superposition of an
odd products of γa’s) form 2
d
2
−1 families, with the quantum number f, each
family have 2
d
2
−1 members,m. They offer the description of the second quantized
fermion fields.
The Clifford even ”basis vectors” (they are superposition of an even products of
γa’s) have no families as we shall see in what follows, but they do carry both quan-
tum numbers, f andm. They offer the description of the second quantized boson
fields as the gauge fields of the second quantized fermion fields. The generators
of the Lorentz transformations in the internal space of the Clifford even ”basis
vectors” are Sab = Sab + S̃ab.
Properties of the Clifford odd and the Clifford even ”basis vectors” are discussed
in the next subsection.
11.2.1 ”Basis vectors” of fermions and bosons
After the reduction of the two Clifford subalgebras to only one, Eq. (19.6), we
only need to define ”basis vectors” for the case that the internal space of second
quantized fields is described by superposition of odd or even products γa’s 6.
Let us use the technique which makes ”basis vectors” products of nilpotents and
projectors [2, 3, 13, 14] which are eigenvectors of the (chosen) Cartan subalgebra
6 In Ref. [5] the reader can find in Subsects. (3.2.1 and 3.2.2) definitions for the ”basis vectors”
for the Grassmann and the two Clifford subalgebras, which are products of nilpotents and
projectors chosen to be eigenvactors of the corresponding Cartan subalgebra members of
the Lorentz algebras presented in Eq. (19.4).
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170 N. S. Mankoč Borštnik
members, Eq. (19.4), of the Lorentz algebra in the space of γa’s, either in the case
of the Clifford odd or in the case of the Clifford even products of γa’s .
There are d
2
members of the Cartan subalgebra, Eq. (19.4), in even dimensional
spaces.
One finds for any of the d
2
Cartan subalgebra member, Sab or S̃ab, both applying
on a nilpotent
ab
(k) or on projector
ab
[k]
ab
(k):=
1
2
(γa +
ηaa
ik
γb) , (
ab
(k))2 = 0,
ab
[k]:=
1
2
(1+
i
k
γaγb) , (
ab
[k])2 =
ab
[k]
the relations
Sab
ab
(k)=
k
2
ab
(k) , S̃ab
ab
(k)=
k
2
ab
(k) ,
Sab
ab
[k]=
k
2
ab
[k] , S̃ab
ab
[k]= −
k
2
ab
[k] , (11.9)
with k2 = ηaaηbb, demonstrating that the eigenvalues of Sab on nilpotents and
projectors expressed with γa’s differ from the eigenvalues of S̃ab on nilpotents and
projectors expressed with γa’s, so that S̃ab can be used to equip each irreducible
representation of Sab with the ”family” quantum number. 7
We define in even d the ”basis vectors” as algebraic, ∗A, products of nilpotents and
projectors so that each product is eigenvector of all d
2
Cartan subalgebra members.
We recognize in advance that the superposition of an odd products of γa’s, that
is the Clifford odd ”basis vectors”, must include an odd number of nilpotents, at
least one, while the superposition of an even products of γa”s, that is Clifford even
”basis vectors”, must include an even number of nilpotents or only projectors.
To define the Clifford odd ”basis vectors”, we shall see that they have properties
appropriate to describe the internal space of the second quantized fermion fields,
and the Clifford even ”basis vectors”, we shall see that they have properties
appropriate to describe the internal space of the second quantized boson fields,
we need to know the relations for nilpotents and projectors
ab
(k): =
1
2
(γa +
ηaa
ik
γb) ,
ab
[k]:=
1
2
(1+
i
k
γaγb) ,
ab
˜(k): =
1
2
(γ̃a +
ηaa
ik
γ̃b) ,
ab
˜[k]:
1
2
(1+
i
k
γ̃aγ̃b) , (11.10)
7 The reader can find the proof of Eq. (19.7) in Ref. [5], App. (I).
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Title Suppressed Due to Excessive Length 171
which can be derived after taking into account Eq. (19.3)
γa
ab
(k) = ηaa
ab
[−k], γb
ab
(k)= −ik
ab
[−k], γa
ab
[k]=
ab
(−k), γb
ab
[k]= −ikηaa
ab
(−k) ,
γ̃a
ab
(k) = −iηaa
ab
[k], γ̃b
ab
(k)= −k
ab
[k], γ̃a
ab
[k]= i
ab
(k), γ̃b
ab
[k]= −kηaa
ab
(k) ,
ab
(k)
†
= ηaa
ab
(−k) , (
ab
(k))2 = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k] ,
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 ,
ab
˜(k)
†
= ηaa
ab
˜(−k) , (
ab
˜(k))2 = 0 ,
ab
˜(k)
ab
˜(−k)= ηaa
ab
˜[k] ,
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 . (11.11)
Looking at relations in Eq. (19.9) it is obvious that the properties of the ”basis
vectors” which include odd number of nilpotents differ essentially from the ”basis
vectors” which include even number of nilpotents.
One namely recognizes:
i. Since the Hermitian conjugated partner of a nilpotent
ab
(k)
†
is ηaa
ab
(−k) and since
neither Sab nor S̃ab nor both can transform odd products of nilpotents to belong
to one of the 2
d
2
−1 members of one of 2
d
2
−1 irreducible representations (families),
the Hermitian conjugated partners of the Clifford odd ”basis vectors” must belong
to a different group of 2
d
2
−1 members of 2
d
2
−1 families.
Since Sac transforms
ab
(k) ∗A
cd
(k ′) into
ab
[−k] ∗A
cd
[−k ′], while S̃ab transforms
ab
[−k] ∗A
cd
[−k ′] into
ab
(−k) ∗A
cd
(−k ′) it is obvious that the Hermitian conjugated
partners of the Clifford odd ”basis vectors” must belong to the same group of
2
d
2
−1 × 2d2−1 members. Projectors are self adjoint.
ii. Since an odd products of γa’s anticommute with another group of an odd
product of γa, the Clifford odd ”basis vectors” anticommute, manifesting in a
tensor product with the basis in ordinary space together with the corresponding
Hermitian conjugated partners properties of the anticommutation relations postu-
lated by Dirac for the second quantized fermion fields.
The Clifford even ”basis vectors” correspondingly fulfil the commutation relations
for the second quantized boson fields.
iii. The Clifford odd ”basis vectors” have all the eigenvalues of the Cartan subal-
gebra members equal to either ±1
2
or to ± i
2
.
The Clifford even ”basis vectors” have all the eigenvalues of the Cartan subalge-
bra members Sab equal to either ±1 and zero or to ±i and zero.
Let us define odd an even ”basis vectors” as products of nilpotents and projectors
in even dimensional spaces.
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172 N. S. Mankoč Borštnik
a. Clifford odd ”basis vectors”
The Clifford odd ”basis vectors” must be products of an odd number of nilpotents
and the rest, up to d
2
, of projectors, each nilpotent and projector must be the ”eigen-
state” of one of the members of the Cartan subalgebra, Eq. (19.4), correspondingly
are the ”basis vectors” eigenstates of all the members of the Lorentz algebras:
Sab’s determine 2
d
2
−1 members of one family, S̃ab’s transform each member of
one family to the same member of the rest of 2
d
2
−1 families.
Let us name the Clifford odd ”basis vectors” b̂m†f , wherem determines member-
ship of ’basis vectors” in any family and f determines a particular family. The
Hermitian conjugated partner of b̂m†f is named by b̂
m
f = (b̂
m†
f )
†.
Let us start in d = 2(2n + 1) with the ”basis vector” b̂1†1 which is the product
of only nilpotents, all the rest members belonging to the f = 1 family follow
by the application of S01, S03, . . . , S0d, S15, . . . , S1d, S5d . . . , Sd−2d. The algebraic
product mark ∗A is skipped.
d = 2(2n + 1) ,
b̂1†1 =
03
(+i)
12
(+)
56
(+) · · ·
d−1 d
(+) ,
b̂2†1 =
03
[−i]
12
[−]
56
(+) · · ·
d−1 d
(+) ,
· · ·
b̂2
d
2
−1†
1 =
03
[−i]
12
[−]
56
(+) . . .
d−3 d−2
[−]
d−1 d
[−] ,
· · · . (11.12)
The Hermitian conjugated partners of the Clifford odd ”basis vector” b̂m†1 , pre-
sented in Eq. (11.12), are
d = 2(2n + 1) ,
b̂11 =
03
(−i)
12
(−) · · ·
d−1 d
(−) ,
b̂21 =
03
[−i]
12
[−]
56
(−) · · ·
d−1 d
(−) ,
· · ·
b̂2
d
2
−1†
1 =
03
[−i]
12
[−]
56
(−)
78
[−] . . .
d−3 d−2
[−]
d−1 d
[−] ,
· · · . (11.13)
In d = 4n the choice of the starting ”basis vector”with maximal number of nilpo-
tents must have one projector
d = 4n ,
b̂1†1 =
03
(+i)
12
(+) · · ·
d−1 d
[+] ,
b̂2†1 =
03
[−i]
12
[−]
56
(+) · · ·
d−1 d
[+] ,
· · ·
b̂2
d
2
−1†
1 =
03
[−i]
12
[−]
56
(+) . . .
d−3 d−2
[−]
d−1 d
[+] ,
. . . . (11.14)
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The Hermitian conjugated partners of the Clifford odd ”basis vector” b̂m†1 , pre-
sented in Eq. (11.14), follow if all nilpotents
ab
(k) are transformed into ηaa
ab
(−k).
For either d = 2(2n + 1) or for d = 4n all the 2
d
2
−1 families follow by applying
S̃ab’s on all the members of the starting family. (Or one can find the starting b̂1f for
all families f and then generate all the members b̂mf from b̂
1
f by the application of
S̃ab on the starting member.)
It is not difficult to see that all the ”basis vectors” within any family as well as
the ”basis vectors” among families are orthogonal, that is their algebraic product
is zero, and the same is true for the Hermitian conjugated partners, what can be
proved by the algebraic multiplication using Eq.(19.9).
b̂m†f ∗A b̂
mԠ
f‘ = 0 , b̂
m
f ∗A b̂m‘f‘ = 0 , ∀m,m ′, f, f‘ . (11.15)
If we require that each family of ”basis vectors”, determined by nilpotents and
projectors described by γa’s, carries the family quantum number determined by
S̃ab and define the vacuum state on which ”basis vectors” apply as
|ψoc >=
2
d
2
−1∑
f=1
b̂mf ∗A b̂
m†
f | 1 > , (11.16)
it follows that the Clifford odd ”basis vectors” obey the relations
b̂mf ∗A |ψoc > = 0. |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
{b̂mf , b̂
m ′
f‘ }∗A+|ψoc > = 0. |ψoc > ,
{b̂m†f , b̂
m ′†
f‘ }∗A+|ψoc > = 0. |ψoc > ,
{b̂mf , b̂
m ′†
f }∗A+|ψoc > = δ
mm ′ δff‘|ψoc > , (11.17)
while the normalization< ψoc|b̂
m ′†
f ′ ∗A b̂
m†
f ∗A |ψoc >= δmm
′
δff ′ is used and the
anticommutation relation mean {b̂m†f , b̂
m ′†
f‘ }∗A+ = b̂
m†
f ∗A b̂
m ′†
f‘ + b̂
m ′†
f‘ ∗A b̂
m†
f .
If we write the creation and annihilation operators as the tensor, ∗T , products
of ”basis vectors” and the basis in ordinary space, the creation and annihilation
operators fulfil the Dirac’s anticommutation postulates since the ”basis vectors”
transfer their anticommutativity to creation and annihilation operators. It turns
out that not only the Clifford odd ”basis vectors” offer the description of the
internal space of fermions, they offer the explanation for the second quantization
postulates for fermions as well.
Table 11.1, presented in Subsect. 11.2.2, illustrates the properties of the Clifford
odd ”basis vectors” on the case of d = (5+ 1).
b. Clifford even ”basis vectors”
The Clifford even ”basis vectors” must be products of an even number of nilpotents
and the rest, up to d
2
, of projectors, each nilpotent and projector in a product must
be the ”eigenstate” of one of the members of the Cartan subalgebra, Eq. (19.4),
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174 N. S. Mankoč Borštnik
correspondingly are the ”basis vectors” eigenstates of all the members of the
Lorentz algebra: Sab’s and S̃ab’s generate from the starting ”basis vector” all
the 2
d
2
−1× 2d2−1 members of one group which includes as well the Hermitian
conjugated partners of any member. 2
d
2
−1 members of the group are products of
projectors only. They are self adjoint.
There are two groups of Clifford even ”basis vectors”with 2
d
2
−1×2d2−1 members
each. The members of one group are not connected with the members of another
group by either by Sab’s or S̃ab’s or both.
Let us name the Clifford even ”basis vectors” iÂm†f , where i = (I, II) denotes that
there are two groups of Clifford even ”basis vectors”, while m and f determine
membership of ’basis vectors” in any of the two groups, I or II. Let me repeat that
the Hermitian conjugated partner of any ”basis vector” appears either in the case
of IÂm†f or in the case of IIÂ
m†
f within the same group.
Let us write down the Clifford even ”basis vectors” as a product of an even number
of nilpotents and the rest of projectors, so that the Clifford even ”basis vectors”
are eigenvectors of all the Cartan subalgebra members, and let us name them as
follows
d = 2(2n+ 1)
IÂ1†1 =
03
(+i)
12
(+) · · ·
d−1d
[+] , IIÂ1†1 =
03
(−i)
12
(+) · · ·
d−1d
[+] ,
IÂ2†1 =
03
[−i]
12
[−]
56
(+) · · ·
d−1d
[+] , IIÂ2†1 =
03
[+i]
12
[−]
56
(+) · · ·
d−1d
[+] ,
IÂ3†1 =
03
(+i)
12
(+)
56
(+) · · ·
d−3d−2
[−]
d−1d
(−) , IIÂ3†1 =
03
(−i)
12
(+)
56
(+) · · ·
d−3d−2
[−]
d−1d
(−) ,
. . . . . .
d = 4n
IÂ1†1 =
03
(+i)
12
(+) · · ·
d−1d
(+) , IIÂ1†1 =
03
(−i)
12
(+) · · ·
d−1d
(+) ,
IÂ2†1 =
03
[−i]
12
[−i]
56
(+) · · ·
d−1d
(+) , IIÂ2†1 =
03
[+i]
12
[−i]
56
(+) · · ·
d−1d
(+) ,
IÂ3†1 =
03
(+i)
12
(+)
56
(+) · · ·
d−3d−2
[−]
d−1d
[−] , IIÂ3†1 =
03
(−i)
12
(+)
56
(+) · · ·
d−3d−2
[−]
d−1d
[−]
. . . . . . (11.18)
There are 2
d
2
−1 × 2d2−1 Clifford even ”basis vectors” of the kind IÂm†f and there
are 2
d
2
−1 ×2d2−1 Clifford even ”basis vectors” of the kind IIÂm†f .
Table 11.1, presented in Subsect. 11.2.2, illustrates properties of the Clifford odd
and Clifford even ”basis vectors” on the case of d = (5 + 1). Looking at this
particular case it is easy to evaluate properties of either even or odd ”basis vectors”.
I shall present here the general results which follow after careful inspection of
properties of both kinds of ”basis vectors”.
The Clifford even ”basis vectors” belonging to two different groups are orthogonal
due to the fact that they differ in the sign of one nilpotent or one projectors, or the
algebraic products of members of one group give zero according to Eq. (19.9).
IÂm†f ∗A IIÂ
m†
f = 0 =
IIÂm†f ∗A IÂ
m†
f . (11.19)
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Title Suppressed Due to Excessive Length 175
The members of each of this two groups have the property
I,IIÂm†f ∗A I,IIÂ
m ′†
f‘ → { I,IIÂm†f‘ , only one for ∀f‘ ,or zero . (11.20)
Two ”basis vectors” IÂm†f and IÂ
m ′†
f ′ , the algebraic product, ∗A, of which gives
nonzero contribution, ”scatter” into the third one IÂm†f‘ . The same is true also for
the ”basis vectors” IIÂm†f .
Let us write the commutation relations for Clifford even ”basis vectors” taking
into account Eq. (11.20).
i. In the case that IÂm†f ∗A IÂ
m ′†
f‘ → IÂm†f‘ and IÂm ′†f‘ ∗A IÂm†f = 0 it follows
{IÂm†f , IÂ
m ′†
f‘ }∗A − →
{
IÂm†f‘ , (if IÂ
m†
f ∗A IÂ
m ′†
f‘ → IÂm†f‘
and IÂm
′†
f‘ ∗A IÂ
m†
f = 0) ,
(11.21)
ii. In the case that IÂm†f ∗A IÂ
m ′†
f‘ → IÂm†f‘ and IÂm ′†f‘ ∗A IÂm†f → IÂm ′†f it
follows
{IÂm†f , IÂ
m ′†
f‘ }∗A − →
{
IÂm†f‘ − IÂ
m ′†
f , (if
IÂm†f ∗A IÂ
m ′†
f‘ → IÂm†f‘
and IÂm
′†
f‘ ∗A IÂ
m†
f → IÂm ′†f ) ,(11.22)
iii. In all other cases we have
{IÂm†f , IÂ
m ′†
f‘ }∗A − = 0 . (11.23)
{IÂm†f , IÂ
m ′†
f‘ }∗A − means
IÂm†f ∗A IÂ
m ′†
f‘ −
IÂm
′†
f‘ ∗A IÂ
m†
f .
It remains to evaluate the algebraic application, ∗A, of the Clifford even ”basis
vectors” IÂm†f on the Clifford odd ”basis vectors” b̂
m ′†
f‘ . One finds
IÂm†f‘ ∗A b̂
m ′†
f → { b̂m†f ,or zero . (11.24)
For each IÂm†f there are among 2
d
2
−1 × 2d2−1 members of the Clifford odd ”basis
vectors” (describing the internal space of fermion fields) 2
d
2
−1 members, b̂m
′†
f‘ ,
fulfilling the relation of Eq. (11.24). All the rest (2
d
2
−1 × (2d2−1 − 1), give zero
contributions.
Eq. (11.24) clearly demonstrates that IÂm†f transforms the Clifford odd ”basis
vector” in general into another Clifford odd ”basis vector”, transfering to the
Clifford odd ”basis vector” an integer spin.
We can obviously conclude that the Clifford even ”basis vectors” offer the descrip-
tion of the gauge fields to the corresponding fermion fields.
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176 N. S. Mankoč Borštnik
While the Clifford odd ”basis vectors” offer the description of the internal space
of the second quantized anticommuting fermion fields, appearing in families, the
Clifford even ”basis vectors” offer the description of the internal space of the
second quantized commuting boson fields, having no families and manifesting as
the gauge fields of the corresponding fermion fields.
11.2.2 Example demonstrating properties of Clifford odd and even ”basis
vectors” for d = (1+ 1), d = (3+ 1), d = (5+ 1)
‘
Subsect. 11.2.2 demonstrates properties of the Clifford odd and even ”basis vectors”
in special cases when d = (1+ 1), d = (3+ 1), and d = (5+ 1).
Let us start with the simplest case:
d=(1+1)
There are 4 (2d=2) ”eigenvectors” of the Cartan subalgebra members S01 and S01
of the Lorentz algebra Sab and Sab , Eq. (19.4), representing one Clifford odd
”basis vector” b̂1†1 =
01
(+i) (m=1), appearing in one family (f=1) and correspondingly
one Hermitian conjugated partner b̂11 =
01
(−i) 8 and two Clifford even ”basis vector”
IA1†1 =
01
[+i] and IIA1†1 =
01
[−i], each of them is self adjoint.
Correspondingly we have two Clifford odd
b̂1†1 =
01
(+i) , b̂11 =
01
(−i)
and two Clifford even
IA1†1 =
01
[+i] , IIA1†1 =
01
[−i]
”basis vectors”.
The first two Clifford odd ”basis vectors” are Hermitian conjugated to each other.
I make a choice that b̂1†1 is the ”basis vector”, the second Clifford odd object is
its Hermitian conjugated partner. Defining the handedness as Γ (1+1) = γ0γ1 it
follows, using Eq. (19.5), that Γ (1+1) b̂1†1 = b̂
1†
1 , which means that b̂
1†
1 is the right
handed ”basis vector”.
We could make a choice of left handed ”basis vector” if choosing b̂1†1 =
01
(−i), but
the choice of handedness would remain only one.
Each of the two Clifford even ”basis vectors” is self adjoint ((I,IIA1†1 )† = I,IIA
1†
1 ).
8 It is our choice which one,
01
(+i) or
01
(−i), we chose as the ”basis vector” b̂1†1 and which
one is its Hermitian conjugated partner. The choice of the ”basis vector” determines the
vacuum state |ψoc >, Eq. (19.8). For b̂1†1 =
01
(+i), the vacuum state is |ψoc >=
01
[−i] (due to
the requirement that b̂1†1 |ψoc > is nonzero) which is the Clifford even object.
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Let us notice, taking into account Eqs. (19.5, 19.9), that
{b̂11(≡
01
(−i)) ∗A b̂1†1 (≡
01
(+i))}|ψoc >=
IIA1†1 (≡
01
[−i])|ψoc >= |ψoc > ,
{b̂1†1 (≡
01
(+i)) ∗A b̂11(≡
01
(−i))}|ψoc >= 0 ,
IA1†1 (≡
01
[+i]) ∗A b̂11(≡
01
(+i))|ψoc >= b̂
1
1(≡
01
(+i))|ψoc > ,
IA1†1 (≡
01
[+i]) b̂11(≡
01
(−i))|ψoc >= 0 .
We find that
IA1†1 ∗A IIA
1†
1 = 0 =
IIA1†1 ∗A IA
1†
1 .
From the case d = (3+ 1) we can learn a little more:
d=(3+1)
There are 16 (2d=4) ”eigenvectors” of the Cartan subalgebra members (S03, S12)
and (S03,S12) of the Lorentz algebras Sab and Sab , Eq. (19.4), in d = (3+ 1).
There are two families (2
4
2
−1, f=(1,2)) with two (2
4
2
−1, m=(1,2)) members each of
the Clifford odd ”basis vectors” b̂m†f , with 2
4
2
−1 × 2 42−1 Hermitian conjugated
partners b̂mf in a separate group (not reachable by S
ab).
There are 2
4
2
−1 × 2 42−1 members of the group of IAm†f , which are Hermitian
conjugated to each other or are self adjoint, all reachable by Sab from any starting
”basis vector IA1†1 .
And there is another group of 2
4
2
−1×2 42−1 members of IIAm†f , again either Hermi-
tian conjugated to each other or are self adjoint. All are reachable from the starting
vector IIA1†1 by the application of Sab.
Again we can make a choice of either right or left handed Clifford odd ”basis
vectors”, but not of both handedness. Making a choice of the right handed ”basis
vectors”
f = 1 f = 2
S̃03 = i
2
, S̃12 = −1
2
, S̃03 = − i
2
, S̃12 = 1
2
, S03, S12
b̂1†1 =
03
(+i)
12
[+] b̂1†2 =
03
[+i]
12
(+) i
2
1
2
b̂2†1 =
03
[−i]
12
(−) b̂2†2 =
03
(−i)
12
[−] − i
2
−1
2
,
we find for the Hermitian conjugated partners of the above ”basis vectors”
S03 = − i
2
, S12 = 1
2
, S03 = i
2
, S12 = −1
2
, S̃03, S̃12
b̂11 =
03
(−i)
12
[+] b̂12 =
03
[+i]
12
(−) − i
2
−1
2
b̂21 =
03
[−i]
12
(+) b̂22 =
03
(+i)
12
[−] i
2
1
2
.
Let us notice that if we look at the subspace SO(1, 1) with the Clifford odd ”basis
vectors” with the Cartan subalgebra member S03 of the space SO(3, 1), and neglect
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178 N. S. Mankoč Borštnik
the values of S12, we do have b̂1†1 =
03
(+i) and b̂2†2 =
03
(−i), which have opposite
handedness Γ (1,1) in d = (1+1), but they have different ”charges” S12 in d = (3+1).
In the whole internal space all the Clifford odd ”basis vectors” have only one
handedness.
We further find that |ψoc >= 1√2 (
03
[−i]
12
[+] +
03
[+i]
12
[+]). All the Clifford odd ”basis
vectors” are orthogonal: b̂m†f ∗A b̂
m ′†
f ′ = 0.
For the Clifford even ”basis vectors” we find two groups of either self adjoint
members or with the Hermitian conjugated partners within the same group. The
two groups are not reachable by S03. We have for IAm†f ,m = (1, 2), f = (1, 2)
S03 S12 S03 S12
IA1†1 =
03
[+i]
12
[+] 0 0 , IA1†2 =
03
(+i)
12
(+) i 1
IA2†1 =
03
(−i)
12
(−) −i −1 , IA2†2 =
03
[−i]
12
[−] 0 0 ,
and for IIAm†f ,m = (1, 2), f = (1, 2)
S03 S12 S03 S12
IIA1†1 =
03
[+i]
12
[−] 0 0 , IIA1†2 =
03
(+i)
12
(−) i 1
IIA2†1 =
03
(−i)
12
(+) −i 1 , IIA2†2 =
03
[−i]
12
[+] 0 0 .
The Clifford even ”basis vectors” have no families. IAm†f ∗A IA
m ′†
f‘ = 0, for any
(m, m’, f, f ‘).
d = (5+ 1)
In Table 11.1 the 64 (= 2d=6) ”eigenvectors” of the Cartan subalgebra members
of the Lorentz algebra Sab and Sab, Eq. (19.4), are presented. The Clifford odd
”basis vectors”, they appear in 4 (= 2
d=6
2
−1) families, each family has 4members,
are products of an odd number of nilpotents, that is either of three nilpotents or
of one nilpotent. They appear in Table 11.1 in the group named odd I b̂m†f . Their
Hermitian conjugated partners appear in the second group named odd II b̂mf .
Within each of these two groups, the members are orthogonal, Eq. (11.15), which
means that the algebraic product of b̂m†f ∗A b̂
m ′†
f‘ = 0 for all (m,m
′, f, f‘). This can
be checked by using relations in Eq. (19.9). Equivalently, the algebraic products of
their Hermitian conjugated partners are also orthogonal among themselves. The
”basis vectors” and their Hermitian conjugated partners are normalized as follows
< ψoc|b̂
m
f ∗A b̂m
′†
f‘ |ψoc >= δ
mm ′δff‘ , (11.25)
since the vacuum state |ψoc >= 1√
2
d=6
2
−1
(
03
[−i]
12
[−]
56
[−] +
03
[−i]
12
[+]
56
[+] +
03
[+i]
12
[−]
56
[+]
+
03
[+i]
12
[+]
56
[−]) is normalized to one: < ψoc|ψoc >= 1.
The longer overview of the properties of the Clifford odd ”basis vectors” and their
Hermitian conjugated partners for the case d = (5+ 1) can be found in Ref. [5].
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Title Suppressed Due to Excessive Length 179
The Clifford even ”basis vectors” are products of an even number of nilpo-
tents, of either two or none in this case. They are presented in Table 11.1 in
two groups, each with 16 (= 2
d=6
2
−1 × 2d=62 −1) members, as even IAm†f and
even IIAm†f . One can easily check, using Eq. (19.9), that the algebraic product
IAm†f ∗A IIA
m ′†
f‘ = 0, ∀ (m,m ′, f.f‘), Eq. (11.19). The longer overview of the Clif-
ford even ”basis vectors” and their Hermitian conjugated partners for the case
d = (5+ 1)- can be found in Ref. [9].
While the Clifford odd ”basis vectors” are (chosen to be) right handed, Γ (5+1) = 1,
have their Hermitian conjugated partners opposite handedness 9
While the Clifford odd ”basis vectors” have half integer eigenvalues of the Cartan
subalgebra members, Eq.(19.4), that is of S03, S12, S56 in this particular case of
d = (5 + 1), the Clifford even ”basis vectors” have integer spins, obtained by
S03 = S03 + S̃03, S12 = S12 + S̃12, S56 = S56 + S̃56.
Let us check what does the algebraic application, ∗A, of IÂm†f=3,m = (1, 2, 3, 4),
presented in Table 11.1 in the third column of even I, do on the Clifford odd ”basis
vectors” b̂m=1†f=1 , presented as the first odd I ”basis vector” in Table 11.1. This can
easily be evaluated by taking into account Eq. (19.5) for anym.
IÂm†3 ∗A b̂
1†
1 (≡
03
(+i)
12
[+]
56
[+]) :
IÂ1†3 (≡
03
[+i]
12
[+]
56
[+]) ∗A b̂1†1 (≡
03
(+i)
12
[+]
56
[+])→ b̂1†1 , selfadjoint
IÂ2†3 (≡
03
(−i)
12
(−)
56
[+]) ∗A b̂1†1 → b̂2†1 (≡ 03[−i] 12(−) 56[+]) ,
IÂ3†3 (≡
03
(−i)
12
[+]
56
(−)) ∗A b̂1†1 → b̂3†1 (≡ 03[−i] 12[+] 56(−)) ,
IÂ4†3 (≡
03
[+i]
12
(−)
56
(−)) ∗A b̂1†1 → b̂4†1 (≡ 03(+i) 12(−) 56(−)) . (11.26)
The sign→means that the relation is valid up to the constant. IÂ1†3 is self adjoint,
the Hermitian conjugated partner of IÂ2†3 is IÂ
1†
4 , of
IÂ3†3 is IÂ
1†
2 and of
IÂ4†3 is
IÂ1†1 .
We can conclude that the algebraic, ∗A, application of IÂm†3 (≡
03
(−i)
12
[+]
56
(−)) on b̂1†1
leads to the same or another family member of the same family f = 1, namely to
b̂m†1 ,m = (1, 2, 3, 4).
Calculating the eigenvalues of the Cartan subalgebra members, Eq. (19.4), before
and after the algebraic multiplication, ∗A, one sees that IÂm†3 carry the integer
eigenvalues of the Cartan subalgebra members, namely of Sab = Sab + S̃ab, since
they transfer when applying on the Clifford odd ”basis vector” to it the integer
eigenvalues of the Cartan subalgebra members, changing the Clifford odd ”basis
vector” into another Clifford odd ”basis vector”.
We therefore find out that the algebraic application of IÂm†3 , m = 1, 2, 3, 4, on
b̂1†1 transforms b̂
1†
1 into b̂
m†
1 , m = (1, 2, 3, 4). Similarly we find that the algebraic
application of IÂm4 , m = (1, 2, 3, 4) on b̂2†1 transforms b̂
2†
1 into b̂
m†
1 ,m = (1, 2, 3, 4).
9 The handedness Γ (d), one of the invariants of the group SO(d), with the infinitesi-
mal generators of the Lorentz group Sab, is defined as Γ (d) = αεa1a2...ad−1ad S
a1a2 ·
Sa3a4 · · · Sad−1ad , with α chosen so that Γ (d) = ±1.
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180 N. S. Mankoč Borštnik
Table 11.1: 2d = 64 ”eigenvectors” of the Cartan subalgebra of the Clifford odd
and even algebras — the superposition of odd and even products of γa’s — in
d = (5+ 1)-dimensional space are presented, divided into four groups. The first
group, odd I, is chosen to represent ”basis vectors”, named b̂m†f , appearing in
2
d
2
−1 = 4 ”families” (f = 1, 2, 3, 4), each ”family” with 2
d
2
−1 = 4 ”family” mem-
bers (m = 1, 2, 3, 4). The second group, odd II, contains Hermitian conjugated
partners of the first group for each family separately, b̂mf = (b̂
m†
f )
†. Either odd I or
odd II are products of an odd number of nilpotents, the rest are projectors. The
”family” quantum numbers of b̂m†f , that is the eigenvalues of (S̃
03, S̃12, S̃56), are
for the first odd I group written above each ”family”, the quantum numbers of the
members (S03, S12, S56) are written in the last three columns. For the Hermitian
conjugated partners of odd I, presented in the group odd II, the quantum numbers
(S03, S12, S56) are presented above each group of the Hermitian conjugated part-
ners, the last three columns tell eigenvalues of (S̃03, S̃12, S̃56). The two groups with
the even number of γa’s, even I and even II, each has their Hermitian conjugated
partners within its own group, have the quantum numbers f, that is the eigen-
values of (S̃03, S̃12, S̃56), written above column of four members, the quantum
numbers of the members, (S03, S12, S56), are written in the last three columns.
′′basis vectors ′′ m f = 1 f = 2 f = 3 f = 4
(S̃03, S̃12, S̃56) → ( i
2
,− 1
2
,− 1
2
) (− i
2
,− 1
2
, 1
2
) (− i
2
, 1
2
,− 1
2
) ( i
2
, 1
2
, 1
2
) S03 S12 S56
odd I b̂
m†
f
1
03
(+i)
12
[+]
56
[+]
03
[+i]
12
[+]
56
(+)
03
[+i]
12
(+)
56
[+]
03
(+i)
12
(+)
56
(+) i
2
1
2
1
2
2 [−i](−)[+] (−i)(−)(+) (−i)[−][+] [−i][−](+) − i
2
− 1
2
1
2
3 [−i][+](−) (−i)[+][−] (−i)(+)(−) [−i](+)[−] − i
2
1
2
− 1
2
4 (+i)(−)(−) [+i](−)[−] [+i][−](−) (+i)[−][−] i
2
− 1
2
− 1
2
(S03, S12, S56) → (− i
2
, 1
2
, 1
2
) ( i
2
, 1
2
,− 1
2
) ( i
2
,− 1
2
, 1
2
) (− i
2
,− 1
2
,− 1
2
) S̃03 S̃12 S̃56
03 12 56 03 12 56 03 12 56 03 12 56
odd II b̂m
f
1 (−i)[+][+] [+i][+](−) [+i](−)[+] (−i)(−)(−) − i
2
− 1
2
− 1
2
2 [−i](+)[+] (+i)(+)(−) (+i)[−][+] [−i][−](−) i
2
1
2
− 1
2
3 [−i][+](+) (+i)[+][−] (+i)(−)(+) [−i](−)[−] i
2
− 1
2
1
2
5 −1 −1
4 (−i)(+)(+) [+i](+)[−] [+i][−](+) (−i)[−][−] − i
2
1
2
1
2
(S̃03, S̃12, S̃56) → (− i
2
, 1
2
, 1
2
) ( i
2
,− 1
2
, 1
2
) (− i
2
,− 1
2
,− 1
2
) ( i
2
, 1
2
,− 1
2
) S03 S12 S56
03 12 56 03 12 56 03 12 56 03 12 56
even I IAm
f
1 [+i](+)(+) (+i)[+](+) [+i][+][+] (+i)(+)[+] i
2
1
2
1
2
2 (−i)[−](+) [−i](−)(+) (−i)(−)[+] [−i][−][+] − i
2
− 1
2
1
2
3 (−i)(+)[−] [−i][+][−] (−i)[+](−) [−i](+)(−) − i
2
1
2
− 1
2
4 [+i][−][−] (+i)(−)[−] [+i](−)(−) (+i)[−](−) i
2
− 1
2
− 1
2
(S̃03, S̃12, S̃56) → ( i
2
, 1
2
, 1
2
) (− i
2
,− 1
2
, 1
2
) ( i
2
,− 1
2
,− 1
2
) (− i
2
, 1
2
,− 1
2
) S03 S12 S56
03 12 56 03 12 56 03 12 56 03 12 56
even II IIAm
f
1 [−i](+)(+) (−i)[+](+) [−i][+][+] (−i)(+)[+] − i
2
1
2
1
2
2 (+i)[−](+) [+i](−)(+) (+i)(−)[+] [+i][−][+] i
2
− 1
2
1
2
3 (+i)(+)[−] [+i][+][−] (+i)[+](−) [+i](+)(−) i
2
1
2
− 1
2
4 [−i][−][−] (−i)(−)[−] [−i](−)(−) (−i)[−](−) − i
2
− 1
2
− 1
2
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Title Suppressed Due to Excessive Length 181
The algebraic application of IÂm2 , m = (1, 2, 3, 4) on b̂3†1 transforms b̂
3†
1 into
b̂m†1 ,m = (1, 2, 3, 4). And the algebraic application of
IÂm1 , m = (1, 2, 3, 4) on b̂4†1
transforms b̂4†1 into b̂
m†
1 ,m = (1, 2, 3, 4).
The statement of Eq. (11.24) is therefore demonstrated on the case of d = (5+ 1).
It remains to stress and illustrate in the case of d = (5+ 1) some general properties
of the Clifford even ”basis vector” IÂm†f when they apply on each other. Let us
denote the self adjoint member in each group of ”basis vectors” of particular f as
IÂm0†f . We easily see that
{IÂm†f , IÂ
m ′†
f }− = 0 , if (m,m
′) 6= m0 orm = m0 = m ′ ,∀ f ,
IÂm†f ∗A IÂ
m0†
f → IÂm†f , ∀m, ∀ f . (11.27)
In Table 11.1 we see that in each column of either even IÂm†f or of evenIIÂ
m†
f
there is one self adjoint I,IIÂm0†f . We also see that two ”basis vectors” IÂ
m†
f and
IÂm
′†
f of the same f and of (m,m
′) 6= m0 are orthogonal. We only have to take
into account Eq. (19.9), which tells that
ab
(k)
ab
[k]= 0 ,
ab
[k]
ab
(k)=
ab
(k) ,
ab
(k)
ab
[−k]=
ab
(k) ,
ab
[k]
ab
(−k)= 0.
These relations tell us that IÂ1†4 ∗AIÂ
2†
3 =
IÂ1†3 , what illustrates Eq. (11.23), while
IÂ2†3 ∗AIÂ
1†
4 =
IÂ2†4 illustrating Eq. (11.22), while IÂ
1†
3 ∗AIÂ
2†
4 = 0 illustrates
Eq. (11.21).
Table 11.2 presents the Clifford even ”basis vectors” IÂm†f for d = (5+ 1) with the
properties: i.They are products of an even number of nilpotents,
ab
(k), with the
rest up to d
2
of projectors,
ab
[k]. ii. Nilpotents and projectors are eigenvectors of
the Cartan subalgebra members Sab = Sab + S̃ab, Eq. (19.4), carrying the integer
eigenvalues of the Cartan subalgebra members.
iii. They have their Hetmitian conjugated partners within the same group of IÂm†f
with 2
d
2
−1 × 2d2−1 members.
iv. They have properties of the boson gauge fields. When applying on the Clifford
odd ”basis vectors” (offering the description of the fermion fields) they transform
the Clifford odd ”basis vectors” into another Clifford odd ”basis vectors”, trans-
ferring to the Clifford odd ”basis vectors” the integer spins with respect to the
SO(d− 1, 1) group, while with respect to subgroups of the SO(d− 1, 1) group they
transfer appropriate superposition of the eigenvalues (manifesting the properties
of the adjoint representations of the corresponding groups).
To demonstrate that the Clifford even ”basis vectors” have properties of the gauge
fields of the corresponding Clifford odd ”basis vectors” we study properties of the
SU(3) ×U(1) subgroups of the Clifford odd and Clifford even ”basis vectors”.
We present in Eqs. (11.28, 11.29) the superposition of members of Cartan subalgebra,
Eq. (19.4), for Sab for the Clifford odd ”basis vectors”, for the subgroups SO(3, 1)×U(1)
(N3± , τ) and for the subgroups SU(3) ×U(1): (τ ′, τ3, τ8). The same relations can be used
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182 N. S. Mankoč Borštnik
also for the corresponding operators determining the ”family” quantum numbers (Ñ3± , τ̃)
of the Clifford odd ”basis vectors’, if Sab’s are replaced by S̃ab’s. For the Clifford even
objects Sab(= Sab + S̃ab) must replace Sab.
N3±(= N
3
(L,R)) :=
1
2
(S12 ± iS03) , τ = S56 , (11.28)
τ3 :=
1
2
(−S1 2 − iS0 3) , τ8 =
1
2
√
3
(−iS0 3 + S1 2 − 2S5 6) ,
τ ′ = −
1
3
(−iS0 3 + S1 2 + S5 6) . (11.29)
Let us, for example, algebraically apply IÂ23 (≡
03
(−i)
12
(−)
56
[+]), denoted by  on
Table 11.2, carrying (τ3 = 0, τ8 = − 1√
3
, τ ′ = 2
3
), represented also on Fig. 11.2 by
, on the Clifford odd ”basis vector” b̂1†1 (≡
03
(+i)
12
(+)
56
(+)), presented on Table 11.1,
with (τ3 = 0, τ8 = 0, τ ′ = −1
2
), as we can calculate using Eq. (11.29) and which
is represented on Fig. 11.1 by a square as a singlet. IÂ23 transforms b̂1†1 (by trans-
ferring to b̂1†1 (τ
3 = 0, τ8 = − 1√
3
, τ ′ = 2
3
)) to b̂1†2 with (τ
3 = 0, τ8 = − 1√
3
, τ ′ = 1
6
),
belonging on Fig. 11.1 to the triplet, denoted by ©. The corresponding gauge
fields, presented on Fig. 11.2, if belonging to the sextet, would transform the triplet
of quarks among themselves.
τ3
τ8
τ'
(1/2,1/2√3,1/6)
(0,0,-1/2)
(-1/2,1/2√3,1/6)
(0,-1/√3,1/6)
Fig. 11.1: Representations of the subgroups SU(3) and U(1) of the group SO(5, 1),
the properties of which appear in Table 11.1, are presented. (τ3, τ8 and τ ′) can be
calculated if using Eqs.(11.28, 11.29). On the abscissa axis, on the ordinate axis
and on the third axis the eigenvalues of the superposition of the three Cartan
subalgebra members, τ3, τ8, τ ′ are presented. One notices one triplet, denoted by
© with the values τ ′ = 1
6
, (τ3 = −1
2
, τ8 = 1
2
√
3
, τ ′ = 1
6
), (τ3 = 1
2
, τ8 = 1
2
√
3
, τ ′ =
1
6
), (τ3 = 0, τ8 = − 1√
3
, τ ′ = 1
6
), respectively, and one singlet denoted by the square.
(τ3 = 0, τ8 = 0, τ ′ = −1
2
). The triplet and the singlet appear in four families.
In the case of the group SO(6) (SO(5, 1)indeed), manifesting as SU(3)×U(1) and
representing the SU(3) colour group and U(1) the ”fermion” quantum number,
embedded into SO(13, 1) the triplet would represent quarks and the singlet lep-
tons. The corresponding gauge of the fields, presented on Fig. 11.2, if belonging to
the sextet, would transform the triplet of quarks among themselves, changing the
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Title Suppressed Due to Excessive Length 183
colour and leaving the ”fermion” quantum number equal to 1
6
.
τ(1,0,0)(-1,0,0)
(1/2,√3/2,0)(-1/2,√3/2,0)
(-1/2,-√3/2,0) (1/2,-√3/2,0)
(0,1/√3,-2/3)
(-1/2,-1/(2√3),-2/3)
(1/2,-1/(2√3),-2/3)
(1/2,1/(2√3),2/3)(-1/2,1/(2√3),2/3)
(0,-1/√3,2/3)
τ
τ
3
8
'
Fig. 11.2: The Clifford even ”basis vectors” IÂmf , in the case that d = (5 + 1),
are presented with respect to the eigenvalues of the commuting operators of
the subgroups SU(3) and U(1) of the group SO(5, 1): τ3 = 1
2
(−S12 − iS03),
τ8 = 1
2
√
3
(S12− iS03− 2S56), τ ′ = −1
3
(S12− iS03+S56). Their properties appear
in Table 11.2. The abscissa axis carries the eigenvalues of τ3, the ordinate axis
of τ8 and the third axis the eigenvalues of τ ′, One notices four singlets with
(τ3 = 0, τ8 = 0, τ ′ = 0), denoted by©, representing four self adjoint Clifford even
”basis vectors” IÂmf , one sextet of three pairs with τ ′ = 0, Hermitian conjugated
to each other, denoted by 4 (with (τ ′ = 0, τ3 = −1
2
, τ8 = − 3
2
√
3
) and (τ ′ =
0, τ3 = 1
2
, τ8 = 3
2
√
3
) ), respectively, by ‡ (with (τ ′ = 0, τ3 = −1, τ8 = 0) and
(τ ′ = 0, τ3 = 1, τ8 = 0), respectively, and by ⊗ (with (τ ′ = 0, τ3 = 1
2
, τ8 = − 3
2
√
3
)
and (τ ′ = 0, τ3 = −1
2
, τ8 = 3
2
√
3
) ), respectively, and one triplet, denoted by ??
with (τ ′ = 2
3
, τ3 = 1
2
, τ8 = 1
2
√
3
), by • with (τ ′ = 2
3
, τ3 = −1
2
, τ8 = 1
2
√
3
), and
by  with (τ ′ = 2
3
, τ3 = 0, τ8 = − 1√
3
), as well as one antitriplet, Hermitian
conjugated to the triplet, denoted by ?? with (τ ′ = −2
3
, τ3 = −1
2
, τ8 = − 1
2
√
3
), by •
with (τ ′ = −2
3
, τ3 = 1
2
, τ8 = − 1
2
√
3
), and by with (τ ′ = −2
3
, τ3 = 0, τ8 = 1√
3
).
We can see that IÂm†3 with (m = 2, 3, 4), if applied on the SU(3) singlet b̂
1†
1 with
(τ ′ = −1
2
, τ3 = 0, τ8 = 0), transforms it to b̂m=2,3,4)†1 , respectively, which are
members of the SU(3) triplet. All these Clifford even ”basis vectors” have τ ′ equal
to 2
3
, changing correspondingly τ ′ = −1
2
into τ ′ = 1
6
and bringing the needed
values of τ3 and τ8.
In Table 11.2 we find (6+ 4) Clifford even ”basis vectors” IÂm†f with τ‘ = 0. Six of
them are Hermitian conjugated to each other — the Hermitian conjugated partners
are denoted by the same geometric figure on the third column. Four of them are
self adjoint and correspondingly with (τ ′ = 0, τ3 = 0, τ8 = 0), denoted in the third
column of Table 11.2 by©. The rest 6 Clifford even ”basis vectors” belong to one
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184 N. S. Mankoč Borštnik
triplet with τ ′ = 2
3
and (τ3, τ8) equal to [(0,− 1√
3
), (−1
2
, 1
2
√
3
), (1
2
, 1
2
√
3
)] and one
antitriplet with τ ′ = −2
3
and ((τ3, τ8) equal to [(−1
2
,− 1
2
√
3
), (1
2
,− 1
2
√
3
), (0, 1√
3
)].
Each triplet has Hermitian conjugated partner in antitriplet and opposite. In
Table 11.2 the Hermitian conjugated partners of the triplet and antitriplet are
denoted by the same signum: (IÂ1†1 , IÂ
4†
3 ) by ??, (
IÂ1†2 , IÂ
3†
3 ) by •, and (IÂ
2†
3 ,
IÂ1†4 ) by .
The octet and the two triplets are presented in Fig. 11.2.
Table 11.2: The Clifford even ”basis vectors” IÂm†f , each of them is the product
of projectors and an even number of nilpotents, and each is the eigenvector of
all the Cartan subalgebra members, S03, S12, S56, Eq. (19.4), are presented for
d = (5+ 1)-dimensional case. Indexesm and f determine 2
d
2
−1 × 2d2−1 different
members IÂm†f . In the third column the ”basis vectors” IÂ
m†
f which are Hermitian
conjugated partners to each other (and can therefore annihilate each other) are
pointed out with the same symbol. For example, with ?? are equipped the first
member with m = 1 and f = 1 and the last member of f = 3 with m = 4. The
sign© denotes the Clifford even ”basis vectors” which are self adjoint (IÂm†f )†
= IÂm
′†
f‘ . It is obvious that
† has no meaning, since IÂm†f are self adjoint or are
Hermitian conjugated partner to another IÂm
′†
f‘ . This table represents also the
eigenvalues of the three commuting operators N 3L,R and S56 of the subgroups
SU(2) × SU(2) × U(1) of the group SO(5, 1) and the eigenvalues of the three
commuting operators τ3, τ8 and τ ′ of the subgroups SU(3)×U(1).
f m ∗ IÂm†
f
S03 S12 S56 N3
L
N3
R
τ3 τ8 τ ′
I 1 ??
03
[+i]
12
(+)
56
(+) 0 1 1. 1
2
1
2
− 1
2
− 1
2
√
3
− 2
3
2 4
03
(−i)
12
[−]
56
(+) −i 0 1 1
2
− 1
2
− 1
2
− 3
2
√
3
0
3 ‡
03
(−i)
12
(+)
56
[−] −i 1 0 1 0 −1 0 0
4 ©
03
[+i]
12
[−]
56
[−] 0 0 0 0 0 0 0 0
II 1 •
03
(+i)
12
[+]
56
(+) i 0 1 − 1
2
1
2
1
2
− 1
2
√
3
− 2
3
2 ⊗
03
[−i]
12
(−)
56
(+) 0 −1 1 − 1
2
− 1
2
1
2
− 3
2
√
3
0
3 ©
03
[−i]
12
[+]
56
[−] 0 0 0 0 0 0 0 0
4 ‡
03
(+i)
12
(−)
56
[−] i −1 0 −1 0 1 0 0
III 1 ©
03
[+i]
12
[+]
56
[+] 0 0 0 0 0 0 0 0
2 
03
(−i)
12
(−)
56
[+] −i −1 0 0 −1 0 − 1√
3
2
3
3 •
03
(−i)
12
[+]
56
(−) −i 0 −1 1
2
− 1
2
− 1
2
1
2
√
3
2
3
4 ??
03
[+i]
12
(−)
56
(−) 0 −1 −1 − 1
2
− 1
2
1
2
1
2
√
3
2
3
IV 1 
03
(+i)
12
(+)
56
[+] i 1 0 0 1 0 1√
3
− 2
3
2 ©
03
[−i]
12
[−]
56
[+] 0 0 0 0 0 0 0 0
3 ⊗
03
[−i]
12
(+)
56
(−) 0 1 −1 1
2
1
2
− 1
2
3
2
√
3
0
4 4
03
(+i)
12
[−]
56
(−) i 0 −1 − 1
2
1
2
1
2
3
2
√
3
0
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Title Suppressed Due to Excessive Length 185
Fig. 11.2 represents the 2
d
2
−1 × 2d2−1 members IÂmf of the Clifford even ”basis
vectors” for the case that d = (5+ 1). The properties of IÂmf are presented also in
Table 11.2. There are in this case again 16members. Manifesting the structure of
subgroups SU(3)×U(1) of the group SO(5, 1) they are represented as eigenvectors
of the superposition of the Cartan subalgebra members (S03,S12,S56), that is with
τ3 = 1
2
(−S12 − iS03), τ8 = 1
2
√
3
(S12 − iS03 − 2S56), and τ ′ = −1
3
(S12 − iS03 +
S56). There are four self adjoint Clifford even ”basis vectors” with (τ3 = 0, τ8 =
0, τ ′ = 0), one sextet of three pairs Hermitian conjugated to each other, one triplet
and one antitriplet with the members of the triplet Hermitian conjugated to the
corresponding members of the antitriplet and opposite. These 16 members of the
Clifford even ”basis vectors” IÂmf are the boson ”partners” of the Clifford odd
”basis vectors” b̂m†f , presented in Fig. 11.1 for one of four families, anyone. The
reader can check that the algebraic application of IÂmf , belonging to the triplet,
transforms the Clifford odd singlet, denoted on Fig. 11.1 by a square, to one of the
members of the triplet, denoted on Fig. 11.1 by the circle©.
Looking at the boson fields IÂm†f from the point of view of subgroups SU(3)×U(1)
of the group SO(5+ 1) we will recognize in the part of fields forming the octet the
colour gauge fields of quarks and leptons and antiquarks and antileptons.
11.2.3 Second quantized fermion and boson fields the internal spaces of
which are described by the Clifford basis vectors.
We learned in the previous subsection that in even dimensional spaces (d =
2(2n+ 1) or d = 4n) the Clifford odd and the Clifford even ”basis vectors”, which
are the superposition of the Clifford odd and the Clifford even products of γa’s,
respectively, offer the description of the internal spaces of fermion and boson
fields.
The Clifford odd algebra offers 2
d
2
−1 ”basis vectors” b̂m†f , appearing in 2
d
2
−1
families (with the family quantum numbers determined by S̃ab = i
2
{γ̃a, γ̃b}−),
which together with their 2
d
2
−1× 2d2−1 Hermitian conjugated partners b̂mf fulfil
the postulates for the second quantized fermion fields, Eq. (11.17) in this paper,
Eq.(26) in Ref. [5], explaining the second quantization postulates of Dirac.
The Clifford even algebra offers 2
d
2
−1× 2d2−1 ”basis vectors” of IÂm†f (and the
same number of IIÂm†f ) with the properties of the second quantized boson fields
manifesting as the gauge fields of fermion fields described by the Clifford odd
”basis vectors” b̂m†f .
The Clifford odd and the Clifford even ”basis vectors” are chosen to be products
of nilpotents,
ab
(k) (with the odd number of nilpotents if describing fermions and
the even number of nilpotents if describing bosons), and projectors,
ab
[k]. Nilpotents
and projectors are (chosen to be) eigenvectors of the Cartan subalgebra members
of the Lorentz algebra in the internal space of Sab for the Clifford odd ”basis
vectors” and of Sab(= Sab + S̃ab) for the Clifford even ”basis vectors”.
To define the creation operators, either for fermions or for bosons besides the
”basis vectors” defining the internal space of fermions and bosons also the basis in
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186 N. S. Mankoč Borštnik
ordinary space in momentum or coordinate representation is needed. Here Ref. [5],
Subsect. 3.3 and App. J is overviewed.
Let us introduce the momentum part of the single particle states. The longer
version is presented in Ref. [5] in Subsect. 3.3 and in App. J.
|~p > = b̂†~p | 0p > , < ~p | =< 0p | b̂~p ,
< ~p |~p ′ > = δ(~p− ~p ′) =< 0p |b̂~p b̂
†
~p ′ | 0p > ,
leading to
b̂ ~p ′ b̂
†
~p = δ(
~p ′ − ~p) , (11.30)
with the normalization < 0p | 0p >= 1. While the quantized operators ~̂p and
~̂x commute {p̂i , p̂j}− = 0 and {x̂k , x̂l}− = 0, it follows for {p̂i , x̂j}− = iηij. One
correspondingly finds
< ~p |~x > = < 0~p | b̂~p b̂
†
~x|0~x >= (< 0~x | b̂~x b̂
†
~p |0~p >)
†
{b̂†~p , b̂
†
~p ′ }− = 0 , {b̂~p, b̂~p ′ }− = 0 , {b̂~p, b̂
†
~p ′ }− = 0 ,
{b̂†~x, b̂
†
~x ′ }− = 0 , {b̂~x, b̂~x ′ }− = 0 , {b̂~x, b̂
†
~x ′ }− = 0 ,
{b̂~p, b̂
†
~x}− = e
i~p·~x 1√
(2π)d−1
, , {b̂~x, b̂
†
~p}− = e
−i~p·~x 1√
(2π)d−1
, (11.31)
.
The internal space of either fermion or boson fields has the finite number of ”basis
vectors”, 2
d
2
−1 × 2d2−1, the momentum basis is continuously infinite.
The creation operators for either fermions or bosons must be a tensor product, ∗T ,
of both contributions, the ”basis vectors” describing the internal space of fermions
or bosons and the basis in ordinary, momentum or coordinate, space.
The creation operators for a free massless fermion of the energy p0 = |~p|, belonging
to a family f and to a superposition of family membersm applying on the vacuum
state |ψoc > ∗T |0~p > can be written as ( [5], Subsect.3.3.2, and the references
therein)
b̂s†f (~p) =
∑
m
csmf(~p) b̂
†
~p ∗T b̂
m†
f , (11.32)
where the vacuum state for fermions |ψoc > ∗T |0~p > includes both spaces, the
internal part, Eq.(19.8), and the momentum part, Eq. (11.30) (in a tensor product
for a starting single particle state with zero momentum, from which one obtains
the other single fermion states of the same ”basis vector” by the operator b̂†~p which
pushes the momentum by an amount ~p 10).
10 The creation operators and their Hermitian conjugated partners annihilation operators in
the coordinate representation can be read in [5] and the references therein: b̂s†f (~x, x
0) =∑
m b̂
m†
f
∫+∞
−∞ dd−1p(√2π)d−1 cmsf (~p) b̂†~p e−i(p0x0−ε~p·~x) ( [5], subsect. 3.3.2., Eqs. (55,57,64)
and the references therein).
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Title Suppressed Due to Excessive Length 187
The creation operators fulfil the anticommutation relations for the second quan-
tized fermion fields
{b̂s
′
f‘ ( ~p ′) , b̂
s†
f (~p)}+ |ψoc > |0~p > = δ
ss ′δff ′ δ(~p
′ − ~p) |ψoc > |0~p > ,
{b̂s
′
f‘ ( ~p ′) , b̂
s
f(~p)}+ |ψoc > |0~p > = 0 . |ψoc > |0~p > ,
{b̂s
′†
f ′ (
~p ′) , b̂s†f (~p)}+ |ψoc > |0~p > = 0 . |ψoc > |0~p > ,
b̂s†f (~p) |ψoc > |0~p > = |ψ
s
f(~p) >
b̂sf(~p) |ψoc > |0~p > = 0 . |ψoc > |0~p >
|p0| = |~p| . (11.33)
The creation operators b̂s†f (~p)) and their Hermitian conjugated partners annihila-
tion operators b̂sf(~p), creating and annihilating the single fermion states, respec-
tively, fulfil when applying on the vacuum state, |ψoc > ∗T |0~p >, the anticommu-
tation relations for the second quantized fermions, postulated by Dirac (Ref. [5],
Subsect. 3.3.1, Sect. 5). 11
To write the creation operators for boson fields we must take into account that
boson gauge fields have the space index α, describing the α component of the
boson field in the ordinary space 12. We therefore add the space index α as follows
IÂm†fα (~p) = b̂
†
~p ∗T Cmfα IÂ
m†
f . (11.34)
We treat free massless bosons of momentum ~p and energy p0 = |~p| and of partic-
ular ”basis vectors” IÂm†f ’s which are eigenvectors of all the Cartan subalgebra
members 13, Cmfα carry the space index α of the boson field. Creation operators
operate on the vacuum state |ψocev > ∗T |0~p >with the internal space part just a
constant, |ψocev >= | 1 >, and for a starting single boson state with a zero momen-
tum from which one obtains the other single boson states with the same ”basis
vector” by the operators b̂†~p which push the momentum by an amount ~p, making
also Cmfα depending on ~p.
For the creation operators for boson fields in a coordinate representation we find
using Eqs. (11.30, 11.31)
IÂm†fα (~x, x0) =
∫+∞
−∞
dd−1p
(
√
2π)d−1
IÂm†fα (~p) e−i(p
0x0−ε~p·~x)|p0=|~p| . (11.35)
11 The anticommutation relations of Eq. (11.33) are valid also if we replace the vacuum state,
|ψoc > |0~p >, by the Hilbert space of Clifford fermions generated by the tensor product
multiplication, ∗TH , of any number of the Clifford odd fermion states of all possible
internal quantum numbers and all possible momenta (that is of any number of b̂s †f (~p) of
any (s, f,~p)), Ref. ( [5], Sect. 5.).
12 In the spin-charge-family theory the Higgs’s scalars origin in the boson gauge fields with
the vector index (7, 8), Ref. ( [5], Sect. 7.4.1, and the references therein).
13 In general the energy eigenstates of bosons are in superposition of IÂm†f . One exam-
ple, which uses the superposition of the Cartan subalgebra eigenstates manifesting the
SU(3)×U(1) subgroups of the group SO(6), is presented in Fig. 11.2.
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188 N. S. Mankoč Borštnik
To understand what new does the Clifford algebra description of the internal space
of fermion and boson fields, Eqs. (11.34, 11.35, 11.32), bring to our understanding
of the second quantized fermion and boson fields and what new can we learn
from this offer, we need to relate
∑
ab c
abωabα and
∑
mf
IÂm†f Cmfα , recognizing
that IÂm†f Cmfα are eigenstates of the Cartan subalgebra members, whileωabα are
not.
The gravity fields, the vielbeins and the two kinds of the spin connection fields,
faα, ωabα, ω̃abα, respectively, are in the spin-charge-family theory (unifying spins,
charges and families of fermions and offering not only the explanation for all the
assumptions of the standard model but also for the increasing number of phenomena
observed so far) the only boson fields in d = (13+1), observed in d = (3+1) besides
as gravity also as all the other boson fields with the Higgs’s scalars included [27].
We therefore need to relate
{
1
2
∑
ab
Sabωabα}
∑
m
βmf b̂m†f (~p) relate to {
∑
m ′f ′
IÂm
′†
f ′ Cm
′f ′
α }
∑
m
βmf b̂m†f (~p) ,
∀f and∀βmf ,
Scd
∑
ab
(cabmfωabα) relate to Scd (IÂm†f Cmfα ) ,
∀ (m, f),
∀ Cartan subalgebra memberScd .(11.36)
Let be repeated that IÂm†f are chosen to be the eigenvectors of the Cartan subal-
gebra members, Eq. (19.4). Correspondingly we can relate a particular IÂm†f Cmfα
with such a superposition ofωabα’s which is the eigenvector with the same values
of the Cartan subalgebra members as there is a particular IÂm†f Cmfα . We can do
this in two ways:
i. Using the first relation in Eq. (11.36). On the left hand side of this relation Sab’s
apply on b̂m†f part of b̂
m†
f (~p). On the right hand side
IÂm†f apply as well on the
same ”basis vector” b̂m†f .
ii. Using the second relation, in which Scd apply on the left hand side onωabα’s
Scd
∑
ab
cabmfωabα =
∑
ab
cabmf i (ωcbαη
ad −ωdbαη
ac +ωacαη
bd −ωadαη
bc),(11.37)
on eachωabα separately; cabmf are constants to be determined from the second
relation, where on the right hand side of this relation Scd(= Scd + S̃cd) apply on
the ”basis vector” IÂm†f of the corresponding gauge field.
Let us conclude this section by pointing out that either the Clifford odd ”basis
vectors” b̂m†f or the Clifford even ”basis vectors”
iÂm†f , i = (I, II) have in any
even d 2
d
2
−1 ×2d2−1 members, whileωabα as well as ω̃abα have each for each α
d
2
(d − 1) members. It is needed to find out what new does this difference bring
into the - unifying theories of the Kaluza-Klein theories are.
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Title Suppressed Due to Excessive Length 189
11.3 Short overview and achievements of spin-charge-family
theory
The spin-chare-family theory [1, 2, 23, 25, 27–32] is a kind of the Kaluza-Klein theo-
ries [27, 38–45] since it is built on the assumption that the dimension of space-time
is ≥ (13+ 1) 14, and that the only interaction among fermions is the gravitational
one (vielbeins, the gauge fields of momenta, and two kinds of the spin connection
fields, the gauge fields of Sab and of S̃ab 15).
This theory assumes as well that the internal space of fermion and boson fields are
described by the Clifford odd and Clifford even algebra, respectively [6, 7] 16.
The theory is offering the explanation for all the assumptions of the standard model,
unifying not only charges, but also spins, charges and families, [36, 37, 46, 48, 51]
and consequently offering the explanation for the appearance of families of quarks
and leptons and antiquarks and antileptons, of vector gauge fields [27], of Higgs’s
scalar field and the Yukawa couplings [28, 30, 32, 36], for the differences in masses
among quarks and leptons [46, 51], for the matter-antimatter asymmetry in the
universe [51], for the dark matter [49], making several predictions.
The spin-charge-family theory shares with the Kaluza-Klein like theories their weak
points, like: a. Not yet solved the quantization problem of the gravitational
field 17. b. The spontaneous symmetry breaking which would at low energies
manifest the observed almost massless fermions [30, 32, 34, 39]. The spontaneously
break of the starting symmetry of SO(13+ 1) with the condensate 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 ( [19], Table III), ( [5], Table 6)
bringing masses of the scale ∝ 1016 GeV or higher to all the vector and scalar
gauge fields, which interact with the condensate [25] is promising to show the
right way [32–34].
The scalar fields (scalar fields are the spin connection fields with the space index
α higher than (0, 1, 2, 3)) with the space index (7, 8) offer, after gaining constant
non zero vacuum values, the explanation for the Higgs’s scalar and the Yukawa
couplings. They namely determine the mass matrices of quarks and leptons and
antiquarks and antileptons. In Refs. [24,27] it is pointed out that the spin connection
14 d = (13 + 1) is the smallest dimension for which the subgroups of the group SO(13, 1)
offer the description of spins and charges of fermions assumed by the standard model and
correspondingly also of boson gauge fields.
15 If there are no fermions present both spin connection fields are expressible with vielbeins (
[5], Eq. (103)).
16 Fermions and bosons internal spaces are assumed to be superposition of odd products of
γa’s (fermion fields) or of even products of γa’s (boson fields) what offers the explanation
for the second quantized postulates of Dirac [16]. The ”basis vectors” of the internal
spaces namely determine anticommutativity or commutativity of the corresponding
creation and annihilation operators.
17 The description of the internal space of fermions and bosons as superposition of odd
(for fermion fields) or even (for boson fields) products of the Clifford objects γa’s seems
very promising in looking for a new way to second quantization of all fields, with gravity
included, as discussed in this talk.
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190 N. S. Mankoč Borštnik
gauge fields do manifest in d = (3+1) as the ordinary gravity and all the observed
vector and scalar gauge fields.
The spin-charge-family theory assumes a simple starting action for second quantized
massless fermion and the corresponding gauge boson fields in d = (13 + 1)-
dimensional space, presented in Eq. (19.1).
The fermion part of the action, Eq. (19.1), can be rewritten in the way that it
manifests in d = (3+ 1) in the low energy regime before the electroweak break by
the standard model postulated properties of: i. Quarks and leptons and antiquarks
and antileptons with the spins, handedness, charges and family quantum numbers.
Their internal space is described by the Clifford odd ”basis vectors” which are
eigenvectors of the Cartan subalgebra of Sab and S̃ab, Eqs. (19.4, 11.29, 11.28).
ii. Couplings of fermions to the vector gauge fields, which are the superposition
of gauge fieldsωstα, Sect. 6.2 in Ref. [5], with the space index α = (0, 1, 2, 3) and
with the charges determined by the Cartan subalgebra of Sab and S̃ab manifesting
the symmetry of space (d − 4), and to the scalar gauge fields [1, 2, 23, 24, 26, 29,
31, 36, 37, 48–50] with the space index α ≥ 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 eitherωstα or ω̃abα,
Lf = ψ̄γm(pm −
∑
A,i
gAiτAiAAim )ψ+
{
∑
s=7,8
ψ̄γsp0s ψ}+
{
∑
t=5,6,9,...,14
ψ̄γtp0t ψ} , (11.38)
where p0s = ps − 12S
s ′s"ωs ′s"s −
1
2
S̃abω̃abs, p0t = pt − 12S
t ′t"ωt ′t"t −
1
2
S̃abω̃abt,
with p0s = eαs p0α,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
members of all the 2
7+1
2
−1 = 8 families.
The first line of Eq. (11.38) determines in d = (3+1) the kinematics and dynamics
of fermion fields coupled to the vector gauge fields [23, 27, 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, Subsect. (6.2.1) of Ref. [5].
The reader can find in Sect. 6 of Ref. [5] a quite detailed overview of the properties
which the massless fermion and boson fields appearing in the simple starting
action, Eq. (19.1), (the later only as gravitational fields) manifest in d = (3+ 1) as
all the observed fermions — quarks and leptons and antiquarks and antileptons
in each family — appearing in twice four families, with the lower four families
including the observed three families of quarks and leptons and antiquarks and
antileptons. The higher four families offer the explanation for the dark matter [49].
Table 5 and Eq. (110) of Ref. [5] explain that the scalar fields with the space index
α = (7, 8) carry the weak charge τ13 = ±1
2
and the hyper charge Y = ∓1
2
, just as
assumed by the standard model.
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Title Suppressed Due to Excessive Length 191
Masses of families of quarks and leptons are determined by the superposition of
the scalar fields, Eq. (108-120) of Ref. [5], appearing in two groups, each of them
manifesting the symmetry SU(2)×SU(2) ×U(1) 18.
The scalar gauge fields with the space index (7, 8) determine correspondingly the
symmetry of mass matrices of quarks and leptons ( [5], Eq. (111)) which appear
in two groups as the scalar fields do [49, 51]. In Table 5 in Ref. [5]) the symmetry
SU(2)× SU(2)×U(1) for each of the two groups is presented and explained.
Although spontaneous symmetry braking of the starting symmetry has not (yet
consistently enough) been studied and the coupling constants of the scalar fields
among themselves and with quarks and leptons are not yet known, the known
symmetry of mass matrices, presented in Eq. (111) of Ref. [5], enables to determine
parameters of mass matrices from the measured data of the 3 × 3 sub mixing
matrices and the masses of the measured three families of quarks and leptons.
Although the known 3× 3 submatrix of the unitary 4× 4 matrix enables to deter-
mine 4 × 4 matrix, the measured 3 × 3 mixing sub matrix is even for quarks far
accurately enough measured, so that we only can predict the matrix elements of
the 4× 4mixing matrix for quarks if assuming that masses (times c2) of the fourth
family quarks are heavy enough, that is above one TeV [46, 49]. The new measure-
ments of the matrix elements among the observed 3 families agree better with the
predictions obtained by the sspin-charge-family theory than the old measurements.
The reader can find predictions in Refs. ( [50, 51]) and the overview in Ref. ( [5],
Subsect. 7.3.1).
The upper group of four families offers the explanation for the dark matter, to which
the quarks and leptons from the (almost) stable of the upper four families mostly
contribute. The reader can find the report on this proposal for the dark matter origin
in Ref. [49] and a short overview in Subsect. 7.3.1 of [5], where the appearance,
development and properties of the dark matter are discussed. The upper four
families predict nucleons of very heavy quarks with the nuclear force among
nucleons which is correspondingly very different from the known one [49, 52].
Besides the scalar fields with the space index α = (7, 8), which manifest in
d = (3 + 1) as scalar gauge fields with the weak and hyper charge ±1
2
and
∓1
2
, respectively, and which gaining at low energies constant values make fam-
ilies of quarks and leptons and the weak gauge field massive, there are in the
starting action, Eqs. (19.1), additional scalar gauge fields with the space index
α = (9, 10, 11, 12, 13, 14). They are with respect to the space index α either triplets
or antitriplets causing transitions from antileptons into quarks and from antiquarks
into quarks and back.
18 The assumption that the symmetry SO(13, 1) first breaks into SU(3)× U(1) × SO(7, 1)
makes that quarks and leptons distinguish only in the part SU(3) × U(1), while the
SO(7, 1) part is identical separately for quarks and leptons and separately for antiquarks
and antileptons. Table 7 of Ref. [5], presenting one family, which includes quarks and
leptons and antiquarks and antileptons, manifests these properties. The ωabα, with
the space index (7, 8) carry with respect to the flat index ab only quantum numbers
Q, Y, τ4, (Q (= τ13+Y),τ13 (= 1
2
(S56−S78), Y (= τ4+ τ23) and τ4 = − 1
3
(S9 10+S11 12+
S13 14), the flat index (ab) of ω̃abα, with the space index (7, 8), includes all (0, 1, . . . , 8)
correspondingly forming the symmetry SU(2)×SU(2) ×U(1).
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192 N. S. Mankoč Borštnik
Their properties are presented in Ref. [25] and briefly in Table 9 and Fig. 1 of
Ref. [5].
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 [34].
In d = (3 + 1)-dimensional space — at low energies — the gauge gravitational
fields manifest as the observed vector gauge fields [27], which can be quantized in
the usual way.
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. (19.1) — reproduce in d = (3 + 1)
the observed properties of quarks and leptons [23, 25, 27–32], the observed vector
gauge fields, the scalar field and the Yukawa couplings, 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 on the spin-charge-family theory seems promising.
11.4 Conclusions
In the spin-charge-family theory [1, 2, 5, 23, 25, 27–32] the Clifford odd algebra is
used to describe the internal space of fermion fields. The Clifford odd ”basis
vectors” — the superposition of odd products of γa’s — in a tensor product
with the basis in ordinary space form the creation and annihilation operators, in
which the anticommutativity of the ”basis vectors” is transferred to the creation
and annihilation operators for fermions, offering the explanation for the second
quantization postulates for fermion fields.
The Clifford odd ”basis vectors” have all the properties of fermions: Half integer
spins with respect to the Cartan subalgebra members of the Lorentz algebra in the
internal space of fermions in even dimensional spaces (d = 2(2n+ 1) or d = 4n),
as discussed in Subsects. (11.2.1, 11.2.3).
With respect to the subgroups of the SO(d−1, 1) group the Clifford odd ”basis vec-
tors” appear in the fundamental representations, as illustrated in Subsects. 11.2.2.
In this article it is demonstrated that the Clifford even algebra is offering the
description of the internal space of boson fields. The Clifford even ”basis vectors”
— the superposition of even products of γa’s — in a tensor product with the basis
in ordinary space form the creation and annihilation operators which manifest the
commuting properties of the second quantized boson fields, offering explanation
for the second quantization postulates for boson fields [9]. The Clifford even ”basis
vectors” have all the properties of bosons: Integer spins with respect to the Cartan
subalgebra members of the Lorentz algebra in the internal space of bosons, as
discussed in Subsects. (11.2.1, 11.2.3).
With respect to the subgroups of the SO(d− 1, 1) group the Clifford even ”basis
vectors” manifest the adjoint representations, as illustrated in Subsect. 11.2.2.
There are two kinds of anticommuting algebras [2]: The Grassmann algebra,
offering in d-dimensional space 2 . 2d operators (2d θa’s and 2d ∂
∂θa
’s, Hermitian
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Title Suppressed Due to Excessive Length 193
conjugated to each other, Eq. (11.3)), and the two Clifford subalgebras, each with
2d operators named γa’s and γ̃a’s, respectively, [2, 13, 14], Eqs. (11.2-19.3).
The operators in each of the two Clifford subalgebras appear in two groups of
2
d
2
−1× 2d2−1 of the Clifford odd operators (the odd products of either γa’s in one
subalgebra or of γ̃a’s in the other subalgebra), which are Hermitian conjugated to
each other: In each Clifford odd group of any of the two subalgebras there appear
2
d
2
−1 irreducible representation each with the 2
d
2
−1 members and the group of
their Hermitian conjugated partners.
There are as well the Clifford even operators (the even products of either γa’s
in one subalgebra or of γ̃a’s in another subalgebra) which again appear in two
groups of 2
d
2
−1× 2d2−1 members each. In the case of the Clifford even objects the
members of each group of 2
d
2
−1× 2d2−1 members have the Hermitian conjugated
partners within the same group, Subsect. 11.2.1, Table 11.1.
The Grassmann algebra operators are expressible with the operators of the two
Clifford subalgebras and opposite, Eq. (11.5). The two Clifford subalgebras are
independent of each other, Eq. (19.3), forming two independent spaces.
Either the Grassmann algebra [15, 20] or the two Clifford subalgebras can be used
to describe the internal space of anticommuting objects, if the superposition of odd
products of operators (θa’s or γa’s, or γ̃a’s) are used to describe the internal space
of these objects. The commuting objects must be superposition of even products
of operators (θa’s or γa’s or γ̃a’s).
No integer spin anticommuting objects have been observed so far, and to describe
the internal space of the so far observed fermions only one of the two Clifford odd
subalgebras are needed.
The problem can be solved by reducing the two Clifford sub algebras to only one,
the one (chosen to be) determined by γab’s. The decision that γ̃a’s apply on γa as
follows: {γ̃aB = (−)B i Bγa} |ψoc >, Eq. (19.6), (with (−)B = −1, if B is a function
of an odd products of γa’s, otherwise (−)B = 1) enables that 2
d
2
−1 irreducible
representations of Sab = i
2
{γa , γb}− (each with the 2
d
2
−1 members) obtain the
family quantum numbers determined by S̃ab = i
2
{γ̃a , γ̃b}−.
The decision to use in the spin-charge-family theory in d = 2(2n + 1), n ≥ 3
(d ≥ (13+1) indeed), the superposition of the odd products of the Clifford algebra
elements γa’s to describe the internal space of fermions which interact with the
gravity only (with the vielbeins, the gauge fields of momenta, and the two kinds of
the spin connection fields, the gauge fields of Sab and S̃ab, respectively), Eq. (19.1),
offers not only the explanation for all the assumed properties of fermions and
bosons in the standard model, with the appearance of the families of quarks and
leptons and antiquarks and antileptons ( [5] and the references therein) and of the
corresponding vector gauge fields and the Higgs’s scalars included [27], but also
for the appearance of the dark matter [49] in the universe, for the explanation of the
matter/antimatter asymmetry in the universe [25], and for several other observed
phenomena, making several predictions [37, 47, 48, 50].
Recognition that the use of the superposition of the even products of the Clifford
algebra elements γa’s to describe the internal space of boson fields, what appear
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194 N. S. Mankoč Borštnik
to manifest all the properties of the observed boson fields, as demonstrated in this
articles, makes clear that the Clifford algebra offers not only the explanation for
the postulates of the second quantized anticommuting fermion fields but also for
the postulates of the second quantized boson fields.
The relations in Eq. (11.36)
{
1
2
∑
ab
Sabωabα}
∑
m
βmf b̂m†f (~p) relate to {
∑
m ′f ′
IÂm
′†
f ′ Cm
′f ′
α }
∑
m
βmf b̂m†f (~p) ,
∀f and∀βmf ,
Scd
∑
ab
(cabmfωabα) relate to Scd (IÂm†f Cmfα ) ,
∀ (m, f),
∀ Cartan subalgebra memberScd ,
offers the possibility to replace the covariant derivative p0α
p0α = pα −
1
2
Sabωabα −
1
2
S̃abω̃abα
in Eq. (19.1) with
p0α = pα −
∑
mf
IÂm†f ICmfα −
∑
mf
I ^̃A
m†
f
IC̃mfα ,
where the relation among I ^̃A
m†
f
IC̃mfα and II
^̃A
m†
f
IIC̃mfα with respect to ωabα and
ω̃abα, not discussed directly in this article, needs additional study and explana-
tion.
Although the properties of the Clifford odd and even ”basis vectors” and corre-
spondingly of the creation and annihilation operators for fermion and boson fields
are, hopefully, clearly demonstrated in this article, yet the proposed way of the
second quantization of fields, the fermion and the boson ones, needs further study
to find out what new can the description of the internal space of fermions and
bosons bring in understanding of the second quantized fields.
Let be added that in even dimensional spaces the Clifford odd ”basis vectors”
carry only one handedness, either right or left, depending on the definition of
handedness and the choice of the ”basis vectors”. Their Hermitian conjugated
partners carry opposite handedness. The ”basis vectors” in the subspace of the
whole space do have both handedness. In odd dimensional spaces (d = (2n+ 1))
the operator of handedness is a superposition of an odd products of γa’s. The
eigenstates of the operator of handedness must be therefore the superposition of
the Clifford odd and the Clifford even ”basis vectors”. These eigenstates can have
either right or left handed. The properties of ”basis vectors” in odd dimensional
spaces are demonstrated in the App. 11.5 of this contribution for d = 1 and
d = (2+ 1) spaces.
It looks like that this study, showing up that the Clifford algebra can be used to
describe the internal spaces of fermion and boson fields in an equivalent way,
offering correspondingly the explanation for the second quantization postulates
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Title Suppressed Due to Excessive Length 195
for fermion and boson fields, is opening the new insight into the quantum field
theory, since studies of the interaction of fermion fields with boson fields and of
boson fields with boson fields so far looks very promising.
The study of properties of the second quantized boson fields, the internal space of
which is described by the Clifford even algebra, has just started and needs further
consideration. Studying properties of ”basis vectors” in odd dimensional spaces
might help to understand anomalies of quantum fields.
11.5 Examples demonstrating properties of Clifford odd and
even ”basis vectors” in odd dimensional spaces for d = (1),
d = (2+ 1)
The spin-charge-family theory, using even dimensional spaces, d = (13+ 1) indeed,
offers the explanation for all the assumptions of the standard model, explaining as
well the postulates for the second quantization of fermion and boson fields. The
internal space of fermions is in this theory described by ”basis vectors” which
are superposition of odd products of γa’s while the internal space of bosons is
described by ”basis vectors” which are superposition of even products of γa’s.
Subsect. 11.2.2 demonstrates properties of the Clifford odd and even ”basis vectors”
in special cases when d = (1+ 1), d = (3+ 1), and d = (5+ 1).
Let us discuss here odd dimensional spaces, which have very different properties:
i. While in even dimensional spaces the Clifford odd ”basis vectors” have 2
d
2
−1
members m in 2
d
2
−1 families f, b̂m†f , and their Hermitian conjugated partners
appear in a separate group of 2
d
2
−1 members in 2
d
2
−1 families, there are in odd
dimensional spaces some of the 2
d
2
−1 × 2d2−1 = 2d−2 Clifford odd ”basis vectors”
self adjoint and have correspondingly some of the Hermitian conjugated partners
in another group with 2d−2 members.
ii. In even dimensional spaces the Clifford even ”basis vectors” iÂm†f , i = (1, 2),
appear in two orthogonal groups, each with 2
d
2
−1× 2d2−1 members and each with
the Hermitian conjugated partners within the same group, 2
d
2
−1 of them are self
adjoint. In odd dimensional spaces the Clifford even ”basis vectors” appear in two
groups, each with 2
d
2
−1 × 2d2−1 = 2d−2 members, which are either self adjoint or
have their Hermitian conjugated partners in another group. Not all the members
of one group are orthogonal to the members of another group, only the self adjoint
ones are orthogonal.
iii. While b̂m†f have in even dimensional spaces one handedness only (either right
or left, depending on the definition of handedness), in odd dimensional spaces the
operator of handedness is a Clifford odd object, still commuting with Sab, which
is the product of odd number of γa’s and correspondingly transforms the Clifford
odd ”basis vectors” into Clifford even ”basis vectors” and opposite. Correspond-
ingly are the eigenvectors of handedness the superposition of the Clifford odd and
the Clifford even ”basis vectors”. Correspondingly there are in odd dimensional
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196 N. S. Mankoč Borštnik
spaces right handed and left handed eigenvectors of the operator of handedness.
Let us illustrate the above mentioned properties of the ”basis vectors” in odd
dimensional spaces, starting with the simplest case:
d=(1)
There is one Clifford odd ”basis vector”
b̂1†1 = γ
0
and one Clifford even ”basis vectors”
iÂ1†1 = 1 .
The operator of handedness Γ (0+1) = γ0 transforms b̂1†1 into identity
iÂ1†1 and
iÂ1†1 into b̂
1†
1 .
The two eigenvectors of the operator of handedness are
1√
2
(γ0 + 1) ,
1√
2
(γ0 − 1) ,
with the handedness (+1,−1), that is of right and left handedness. respectively.
d=(2+1)
There are twice 2d=3−2 = 2 Clifford odd ”basis vectors”. We chose as the Cartan
subalgebra member S01 of Sab: b̂1†1 =
01
[−i] γ2, b̂2†1 =
01
(+i), b̂1†2 =
01
(−i), b̂2†2 =
01
[+i] γ2,
with the properties
f = 1 f = 2
S̃01 = i
2
S̃01 = − i
2
, S01
b̂1†1 =
01
[−i] γ2 b̂1†2 =
01
(−i) − i
2
b̂2†1 =
01
(+i) b̂2†2 =
03
[+i] γ2 i
2
,
b̂1†1 and b̂
2†
2 are self adjoint (up to a sign), b̂
2†
1 =
01
(+i) and b̂1†2 =
01
(−i) are Hermitian
conjugated to each other.
In odd dimensional spaces the ”basis vectors” are not separated from their Hermi-
tian conjugated partners and are correspondingly not well defined.
The operator of handedness is (chosen up to a sign to be) Γ (2+1) = iγ1γ2γ2.
There are twice 2(d=3)−2 = 2Clifford even ”basis vectors”. We choose as the Cartan
subalgebra member S01: IÂ1†1 =
01
[+i], IÂ2†1 =
01
(−i) γ2, IIÂ1†2 =
01
[−i], IIÂ2†2 =
01
(+i) γ2,
with the properties
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Title Suppressed Due to Excessive Length 197
S01 S01
IÂ1†1 =
01
[+i] 0 IIÂ1†2 =
01
[−i] 0
IÂ2†1 =
01
(−i) γ2 −i IIÂ2†2 =
03
(+i) γ2 i,
IÂ1†1 =
01
[+i] and IIÂ1†2 =
01
[−i] are self adjoint, IÂ2†1 =
01
(−i) γ2 and IIÂ2†2 =
03
(+i) γ2
are Hermitian conjugated to each other.
In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are
not orthogonal.
Let us find the eigenvectors of the operator of handedness Γ (2+1) = iγ0γ1γ2. Since
it is the Clifford odd object its eigenvectors are superposition of Clifford odd and
Clifford even ”basis vectors”.
It follows
Γ (2+1){
01
[−i] ±i
01
[−i] γ2} = ∓{
01
[−i] ±i
01
[−i] γ2} ,
Γ (2+1){
01
(+i) ±i
01
(+i) γ2} = ∓{
01
(+i) ±i
01
(+i) γ2} ,
Γ (2+1){
01
[+i] ±i
01
[+i] γ2} = ±{
01
[+i] ±i
01
[+i] γ2} ,
Γ (2+1){
01
(−i) γ2 ± i
01
(−i)} = ±{
01
(−i) γ2 ± i
01
(−i)} ,
We can conclude that neither Clifford odd nor Clifford even ”basis vectors” have
in odd dimensional spaces the properties which they demonstrate in even dimen-
sional spaces.
i. In odd dimensional spaces the ”basis vectors” are not separated from their
Hermitian conjugated partners and are correspondingly not well defined, that
is we can not define creation and annihilation operators as a tensor products of
”basis vectors” and basis in momentum space.
In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are
not orthogonal, only self adjoint ”basis vectors” are orthogonal, the rest of ”basis
vectors” have their Hermitian conjugated partners in another group.
ii. The Clifford odd operator of handedness allows left and right handed superpo-
sition of Clifford odd and Clifford even ”basis vectors”.
11.5.1 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”.
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