# Clifford odd and even objects offer description of internal space of
fermions and bosons, respectively, opening new insight into the second
quantization of fields

N.S. Mankoč Borštnik,1
1Department of Physics, University of Ljubljana
SI-1000 Ljubljana, Slovenia
<EMAIL_ADDRESS>
###### Abstract

In a long series of works the author has demonstrated that the model named the
spin-charge-family theory offers the explanation for all in the standard model
assumed properties of the fermion and boson fields, as well as for many of
their so far observed properties if the space-time is $\geq(13+1)$ while
fermions interact with gravity only. In this talk, I briefly report on the so
far achievements of the theory. The main contribution demonstrates the offer
of the Clifford odd and even objects for the description of the internal
spaces of fermion (Clifford odd) and boson (Clifford even) fields, which is
opening up a new understanding of the second quantization postulates for the
fermion and boson fields: The ”basis vectors” determined by the Clifford odd
objects demonstrate all the properties of the internal space of fermions and
transfer their anticommutativity to their creation and annihilation operators,
while the ”basis vectors” determined by the Clifford even objects demonstrate
all the properties of the internal space of boson fields and transfer their
commutativity to their creation and annihilation operators. The toy model with
$d=(5+1)$ illustrates the statements.

##  1 Introduction

The standard model (improved with neutrinos having non zero masses) has been
experimentally confirmed without raising any serious doubts so far on its
assumptions, which remain unexplained, waiting for the explanation.

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, 2, 3, 4, 5, 6] and many others.

Among the questions which must be answered are:
i. Where do fermions, quarks and leptons, with their families included
originate and why do their internal spaces differ so much from the internal
spaces of boson fields?
ii. 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?
iii. Where do antiquarks and antileptons originate?
iv. Where do families of quarks and leptons originate and how many families do
exist?
v. Why do family members – quarks and leptons — manifest so different masses
if they all start as massless?
vi. What is the origin of boson fields, of vector fields which are the gauge
fields of fermions, and of the Higgs’s scalars and the Yukawa couplings.
vii. 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 masses of weak bosons?
viii. Do possibly exist also scalar fields with the colour charges in the
fundamental representation (like the Higgs’s scalars are doublets with respect
to the weak charge) and where, if they are, do they manifest?
ix. Can it be that all boson fields, with the scalar fields, included, have a
common origin?
x. Where does the dark matter originate?
xi. Where does the ”ordinary” matter-antimatter asymmetry originate?
xii. Where does the dark energy originate and why is it so small?
xiii. How can we understand the second quantized fermion and boson fields?
xiv. What is the dimension of space? $(3+1)?$, $((d-1)+1)?$, $\infty?$
xv. Are all the fields indeed second quantized with the gravity included? And
consequently all the systems are 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.

In a long series of works ([7, 8, 9, 21, 23, 25, 26, 27, 29, 30] and the
references therein), the author has succeeded, together with collaborators, to
find the answer to the above, and also other open questions of the standard
model and also to several open cosmological questions, with the model named
the spin-charge-family theory.

The theory assumes that the space has more than $(3+1)$ dimensions, it must
have $d\geq(13+1)$, 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 $\gamma^{a}$’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, represents one of families of
fermions, the quantum numbers of which are determined by the second kind of
the Clifford objects, by $\tilde{\gamma}^{a}$ (by $\tilde{S}^{ab}$
$(=\frac{i}{4}\,\\{\tilde{\gamma}^{a},\,\tilde{\gamma}^{b}\\}_{-}$, indeed).

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 $S^{ab}=\frac{i}{4}\,\\{\gamma^{a},\,\gamma^{b}\\}_{-}$ and of
$\tilde{S}^{ab}$
$(=\frac{i}{4}\,\\{\tilde{\gamma}^{a},\,\tilde{\gamma}^{b}\\}_{-}$.

The theory assumes a simple starting action ([10] 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

$\displaystyle{\cal A}\,$ $\displaystyle=$
$\displaystyle\int\;d^{d}x\;E\;\frac{1}{2}\,(\bar{\psi}\,\gamma^{a}p_{0a}\psi)+h.c.+$
$\displaystyle\int\;d^{d}x\;E\;(\alpha\,R+\tilde{\alpha}\,\tilde{R})\,,$
$\displaystyle p_{0a}$ $\displaystyle=$ $\displaystyle
f^{\alpha}{}_{a}p_{0\alpha}+\frac{1}{2E}\,\\{p_{\alpha},Ef^{\alpha}{}_{a}\\}_{-}\,,$
$\displaystyle p_{0\alpha}$ $\displaystyle=$ $\displaystyle
p_{\alpha}-\frac{1}{2}S^{ab}\omega_{ab\alpha}-\frac{1}{2}\tilde{S}^{ab}\tilde{\omega}_{ab\alpha}\,,$
$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\\{f^{\alpha[a}f^{\beta
b]}\;(\omega_{ab\alpha,\beta}-\omega_{ca\alpha}\,\omega^{c}{}_{b\beta})\\}+h.c.\,,$
$\displaystyle\tilde{R}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\\{f^{\alpha[a}f^{\beta
b]}\;(\tilde{\omega}_{ab\alpha,\beta}-\tilde{\omega}_{ca\alpha}\,\tilde{\omega}^{c}{}_{b\beta})\\}+h.c.\,.$
(1)

Here 111$f^{\alpha}{}_{a}$ are inverted vielbeins to $e^{a}{}_{\alpha}$ with
the properties
$e^{a}{}_{\alpha}f^{\alpha}{\\!}_{b}=\delta^{a}{\\!}_{b},\;e^{a}{\\!}_{\alpha}f^{\beta}{\\!}_{a}=\delta^{\beta}_{\alpha}$,
$E=\det(e^{a}{\\!}_{\alpha})$. Latin indices $a,b,..,m,n,..,s,t,..$ denote a
tangent space (a flat index), while Greek indices
$\alpha,\beta,..,\mu,\nu,..\sigma,\tau,..$ denote an Einstein index (a curved
index). Letters from the beginning of both the alphabets indicate a general
index ($a,b,c,..$ and $\alpha,\beta,\gamma,..$ ), from the middle of both the
alphabets the observed dimensions $0,1,2,3$ ($m,n,..$ and $\mu,\nu,..$),
indexes from the bottom of the alphabets indicate the compactified dimensions
($s,t,..$ and $\sigma,\tau,..$). We assume the signature
$\eta^{ab}=diag\\{1,-1,-1,\cdots,-1\\}$. $f^{\alpha[a}f^{\beta b]}=f^{\alpha
a}f^{\beta b}-f^{\alpha b}f^{\beta a}$. $f^{a}_{\alpha}$, and the two kinds of
the spin connection fields, $\omega_{ab\alpha}$ (the gauge fields of $S^{ab}$)
and $\tilde{\omega}_{ab\alpha}$ (the gauge fields of $\tilde{S}^{ab}$),
manifest in $d=(3+1)$ as the known vector gauge fields and the scalar gauge
fields taking care of masses of quarks and leptons and antiquarks and
antileptons and the weak boson fields [25] 222 Since the multiplication with
either $\gamma^{a}$’s or $\tilde{\gamma}^{a}$’s changes the Clifford odd
”basis vectors” into the Clifford even objects, and even ”basis vectors”
commute, the action for fermions can not include an odd numbers of
$\gamma^{a}$’s or $\tilde{\gamma}^{a}$’s, what the simple starting action of
Eq. (1) does not. In the starting action $\gamma^{a}$’s and
$\tilde{\gamma}^{a}$’s appear as $\gamma^{0}\gamma^{a}\hat{p}_{a}$ or as
$\gamma^{0}\gamma^{c}\,S^{ab}\omega_{abc}$ and as
$\gamma^{0}\gamma^{c}\,\tilde{S}^{ab}\tilde{\omega}_{abc}$.

While in any even dimensional space the superposition of odd products of
$\gamma^{a}$’s, forming the Clifford odd ”basis vectors”, offer the
description of the internal space of fermions with the half integer spins, the
superposition of even products of $\gamma^{a}$’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 $2^{\frac{d}{2}-1}$ members manifest spins, handedness and
charges of quarks and leptons and antiquarks and antileptons appearing in
$2^{\frac{d}{2}-1}$ families, while their Hermitian conjugated partners appear
in another group of $2^{\frac{d}{2}-1}$ members in $2^{\frac{d}{2}-1}$
families.

The Clifford even ”basis vectors” appear in two groups, each with
$2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ 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” 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. The Clifford even
”basis vectors” transfer their commutativity to creation operators and
annihilation operators. 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.

The spin-charge-family theory is offering answers to most of the open
questions presented above. The more work is put into the theory the more
answers the theory offers.

In Sect. 2 the Grassmann and the Clifford algebra is explained and creation
and annihilation operators described as a tensor products of the ”basis
vectors” 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. 2.1 the ”basis vectors” are introduced and their properties
presented.

In Subsect. 2.2 the properties of the Clifford odd and even ”basis vectors”
are demonstrated in the toy model in $d=(5+1)$.

In Subsect. 2.3 the properties of the creation and annihilation operators for
the second quantized fields are described.

In Sect. 3 a short overview of the achievements and predictions so far of the
spin-charge-family theory is presented,

Sect. 4 presents what the reader could learn from the main contribution of
this talk.

##  2 Creation and annihilation operators for fermions and bosons

The second quantization postulates for fermions [14, 15, 16] require that the
creation operators and their Hermitian conjugated partners annihilation
operators, depending 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) ([10], Subsect. 3.3.1), fulfil anticommutation
relations.

The second quantization postulates for bosons [14, 15, 16] 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) ([10], 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 $\gamma^{a}$’s) or of boson fields (if the
internal space is described with the Clifford even products of
$\gamma^{a}$’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 $2^{d}$ members. In each of the two
subalgebras there are 2 $\times$ $2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$
Clifford odd and 2 $\times$ $2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$
Clifford even ”basic vectors” which can be used to describe the internal space
of fermion fields, the 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 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 $2^{\frac{d}{2}-1}$ irreducible representations, each with
$2^{\frac{d}{2}-1}$ 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 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.
(7), (Ref. [10], 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, $\gamma^{a}$’s, when fermions are concerned ([10]
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 [7, 8, 9,
29]. What follows is a short overview of Subsect.3.2 of Ref. [10] and of
references cited in [10].

In Grassmann $d$-dimensional space there are $d$ anticommuting (operators)
$\theta^{a}$, and $d$ anticommuting operators which are derivatives with
respect to $\theta^{a}$, $\frac{\partial}{\partial\theta_{a}}$,

$\displaystyle\\{\theta^{a},\theta^{b}\\}_{+}=0\,,\,$
$\displaystyle\,\\{\frac{\partial}{\partial\theta_{a}},\frac{\partial}{\partial\theta_{b}}\\}_{+}=0\,,$
$\displaystyle\\{\theta_{a},\frac{\partial}{\partial\theta_{b}}\\}_{+}$
$\displaystyle=$ $\displaystyle\delta_{ab}\,,(a,b)=(0,1,2,3,5,\cdots,d)\,.$
(2)

Defining [30]

$\displaystyle(\theta^{a})^{\dagger}$ $\displaystyle=$
$\displaystyle\eta^{aa}\frac{\partial}{\partial\theta_{a}}\,,\quad{\rm
leads\,to}\quad(\frac{\partial}{\partial\theta_{a}})^{\dagger}=\eta^{aa}\theta^{a}\,,$
(3)

with $\eta^{ab}=diag\\{1,-1,-1,\cdots,-1\\}$.

$\theta^{a}$ and $\frac{\partial}{\partial\theta_{a}}$ 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 $\theta^{a}$
and correspondingly of $\frac{\partial}{\partial\theta_{a}}$ are made

$\displaystyle\\{\theta^{a}\\}^{*}$ $\displaystyle=$
$\displaystyle(\theta^{0},\theta^{1},-\theta^{2},\theta^{3},-\theta^{5},\theta^{6},...,-\theta^{d-1},\theta^{d})\,,$
$\displaystyle\\{\frac{\partial}{\partial\theta_{a}}\\}^{*}$ $\displaystyle=$
$\displaystyle(\frac{\partial}{\partial\theta_{0}},\frac{\partial}{\partial\theta_{1}},-\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial\theta_{3}},-\frac{\partial}{\partial\theta_{5}},\frac{\partial}{\partial\theta_{6}},...,-\frac{\partial}{\partial\theta_{d-1}},\frac{\partial}{\partial\theta_{d}})\,.$
(4)

The are $2^{d}$ superposition of products of $\theta^{a}$, the Hermitian
conjugated partners of which are the corresponding superposition of products
of $\frac{\partial}{\partial\theta_{a}}$.

There exist two kinds of the Clifford algebra elements (operators),
$\gamma^{a}$ and $\tilde{\gamma}^{a}$, expressible with $\theta^{a}$’s and
their conjugate momenta $p^{\theta a}=i\,\frac{\partial}{\partial\theta_{a}}$
[8], Eqs. (2, 3),

$\displaystyle\gamma^{a}$ $\displaystyle=$
$\displaystyle(\theta^{a}+\frac{\partial}{\partial\theta_{a}})\,,\quad\tilde{\gamma}^{a}=i\,(\theta^{a}-\frac{\partial}{\partial\theta_{a}})\,,$
$\displaystyle\theta^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,(\gamma^{a}-i\tilde{\gamma}^{a})\,,\quad\frac{\partial}{\partial\theta_{a}}=\frac{1}{2}\,(\gamma^{a}+i\tilde{\gamma}^{a})\,,$

offering together $2\cdot 2^{d}$ operators: $2^{d}$ are superposition of
products of $\gamma^{a}$ and $2^{d}$ of $\tilde{\gamma}^{a}$. It is easy to
prove, if taking into account Eqs. (3, 2), that they form two anticommuting
Clifford subalgebras, $\\{\gamma^{a},\tilde{\gamma}^{b}\\}_{+}=0$, Refs. ([10]
and references therein)

$\displaystyle\\{\gamma^{a},\gamma^{b}\\}_{+}$ $\displaystyle=$ $\displaystyle
2\eta^{ab}=\\{\tilde{\gamma}^{a},\tilde{\gamma}^{b}\\}_{+}\,,$
$\displaystyle\\{\gamma^{a},\tilde{\gamma}^{b}\\}_{+}$ $\displaystyle=$
$\displaystyle 0\,,\quad(a,b)=(0,1,2,3,5,\cdots,d)\,,$
$\displaystyle(\gamma^{a})^{\dagger}$ $\displaystyle=$
$\displaystyle\eta^{aa}\,\gamma^{a}\,,\quad(\tilde{\gamma}^{a})^{\dagger}=\eta^{aa}\,\tilde{\gamma}^{a}\,.$
(6)

While the Grassmann algebra offers the description of the ”anticommuting
integer spin second quantized fields” and of the ”commuting integer spin
second quantized fields” [33, 10], the Clifford algebras which are
superposition of odd products of either $\gamma^{a}$’s or
$\tilde{\gamma}^{a}$’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 $\gamma^{a}$’s or
$\tilde{\gamma}^{a}$’s offer the description of the commuting second quantized
boson fields with integer spins (as we can see in [12] 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 $\tilde{\gamma}^{a}$’s
operate on $\gamma^{a}$’s, reduces the two Clifford subalgebras,
$\gamma^{a}$’s and $\tilde{\gamma}^{a}$’s, to one, to the one described by
$\gamma^{a}$’s [20, 8, 27, 29, 30]

$\displaystyle\\{\tilde{\gamma}^{a}B$ $\displaystyle=$
$\displaystyle(-)^{B}\,i\,B\gamma^{a}\\}\,|\psi_{oc}>\,,$ (7)

with $(-)^{B}=-1$, if $B$ is (a function of) an odd products of
$\gamma^{a}$’s, otherwise $(-)^{B}=1$ [20], $|\psi_{oc}>$ is defined in Eq.
(16). of Subsect. 2.1.

After the postulate of Eq. (7) it follows:
a. The Clifford subalgebra described by $\tilde{\gamma}^{a}$’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 $\gamma^{a}$’s obey the postulates for the second quantization fields for
fermions or bosons, respectively, Sect.2.1.
c. It can be proven that the relations presented in Eq. (2) remain valid also
after the postulate of Eq. (7). The proof is presented in Ref. ([10], App. I,
Statement 3a.
d. Each irreducible representation of the Clifford odd ”basis vectors”
described by $\gamma^{a}$’s are equipped by the quantum numbers of the Cartan
subalgebra members of $\tilde{S}^{ab}$, chosen in Eq. (8), as follows

$\displaystyle{\cal{\bf S}}^{03},{\cal{\bf S}}^{12},{\cal{\bf
S}}^{56},\cdots,{\cal{\bf S}}^{d-1\;d}\,,$ $\displaystyle
S^{03},S^{12},S^{56},\cdots,S^{d-1\;d}\,,$
$\displaystyle\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56},\cdots,\tilde{S}^{d-1\;d}\,,$
$\displaystyle{\cal{\bf
S}}^{ab}=S^{ab}+\tilde{S}^{ab}=i\,(\theta^{a}\frac{\partial}{\partial\theta_{b}}-\theta^{b}\frac{\partial}{\partial\theta_{a}})\,.$
(8)

After the postulate of Eq. (7) no vector space of $\tilde{\gamma}^{a}$’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 Grassmann algebra is reduced to only one of the Clifford
subalgebras. The operators $\tilde{\gamma}^{a}$’s describe from now on
properties of fermion and boson ”basis vectors” determined by superposition of
products of odd or even numbers of $\gamma^{a}$’s, respectively.

$\tilde{\gamma}^{a}$’s equip each irreducible representation of the Lorentz
group (with the infinitesimal generators
$S^{ab}=\frac{i}{4}\\{\gamma^{a},\,\gamma^{b}\\}_{-}$) when applying on the
Clifford odd ”basis vectors” (which are superposition of odd products of
$\gamma^{a^{\prime}s}$) with the family quantum numbers (determined by
$\tilde{S}^{ab}=\frac{i}{4}\\{\tilde{\gamma}^{a},\,\tilde{\gamma}^{b}\\}_{-}$).

Correspondingly the Clifford odd ”basis vectors” (they are superposition of an
odd products of $\gamma^{a}$’s) form $2^{\frac{d}{2}-1}$ families, with the
quantum number $f$, each family have $2^{\frac{d}{2}-1}$ 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 $\gamma^{a}$’s) have no families as we shall see in what follows, but they
do carry both quantum numbers, $f$ and $m$. 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 ${\bf{\cal
S}}^{ab}=S^{ab}+\tilde{S}^{ab}$.

Properties of the Clifford odd and the Clifford even ”basis vectors” are
discussed in the next subsection.

###  2.1 ”Basis vectors” of fermions and bosons

After the reduction of the two Clifford subalgebras to only one, Eq. (7), 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
$\gamma^{a}$’s 333 In Ref. [10] 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. (8)..

Let us use the technique which makes ”basis vectors” products of nilpotents
and projectors [8, 9, 19, 20] which are eigenvectors of the (chosen) Cartan
subalgebra members, Eq. (8), of the Lorentz algebra in the space of
$\gamma^{a}$’s, either in the case of the Clifford odd or in the case of the
Clifford even products of $\gamma^{a}$’s .

There are $\frac{d}{2}$ members of the Cartan subalgebra, Eq. (8), in even
dimensional spaces.

One finds for any of the $\frac{d}{2}$ Cartan subalgebra member, $S^{ab}$ or
$\tilde{S}^{ab}$, both applying on a nilpotent $\stackrel{{\scriptstyle
ab}}{{(k)}}$ or on projector $\stackrel{{\scriptstyle ab}}{{[k]}}$

$\stackrel{{\scriptstyle
ab}}{{(k)}}:=\frac{1}{2}(\gamma^{a}+\frac{\eta^{aa}}{ik}\gamma^{b})\,,\;\;\;(\stackrel{{\scriptstyle
ab}}{{(k)}})^{2}=0,$ $\stackrel{{\scriptstyle
ab}}{{[k]}}:=\frac{1}{2}(1+\frac{i}{k}\gamma^{a}\gamma^{b})\,,\;\;\;(\stackrel{{\scriptstyle
ab}}{{[k]}})^{2}=\stackrel{{\scriptstyle ab}}{{[k]}}$

the relations

$\displaystyle S^{ab}\,\stackrel{{\scriptstyle
ab}}{{(k)}}=\frac{k}{2}\,\stackrel{{\scriptstyle ab}}{{(k)}}\,,\quad$
$\displaystyle\quad\tilde{S}^{ab}\,\stackrel{{\scriptstyle
ab}}{{(k)}}=\frac{k}{2}\,\stackrel{{\scriptstyle ab}}{{(k)}}\,,$
$\displaystyle S^{ab}\,\stackrel{{\scriptstyle
ab}}{{[k]}}=\frac{k}{2}\,\stackrel{{\scriptstyle ab}}{{[k]}}\,,\quad$
$\displaystyle\quad\tilde{S}^{ab}\,\stackrel{{\scriptstyle
ab}}{{[k]}}=-\frac{k}{2}\,\,\stackrel{{\scriptstyle ab}}{{[k]}}\,,$ (9)

with $k^{2}=\eta^{aa}\eta^{bb}$, demonstrating that the eigenvalues of
$S^{ab}$ on nilpotents and projectors expressed with $\gamma^{a}$’s differ
from the eigenvalues of $\tilde{S}^{ab}$ on nilpotents and projectors
expressed with $\gamma^{a}$’s, so that $\tilde{S}^{ab}$ can be used to equip
each irreducible representation of $S^{ab}$ with the ”family” quantum number.
444 The reader can find the proof of Eq. (2.1) in Ref. [10], App. (I).

We define in even $d$ the ”basis vectors” as algebraic, $*_{A}$, products of
nilpotents and projectors so that each product is eigenvector of all
$\frac{d}{2}$ Cartan subalgebra members.

We recognize in advance that the superposition of an odd products of
$\gamma^{a}$’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 $\gamma^{a}$”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

$\displaystyle\stackrel{{\scriptstyle ab}}{{(k)}}:$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\gamma^{a}+\frac{\eta^{aa}}{ik}\gamma^{b})\,,\quad\stackrel{{\scriptstyle
ab}}{{[k]}}:=\frac{1}{2}(1+\frac{i}{k}\gamma^{a}\gamma^{b})\,,$
$\displaystyle\stackrel{{\scriptstyle ab}}{{\tilde{(k)}}}:$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\tilde{\gamma}^{a}+\frac{\eta^{aa}}{ik}\tilde{\gamma}^{b})\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}:\frac{1}{2}(1+\frac{i}{k}\tilde{\gamma}^{a}\tilde{\gamma}^{b})\,,$
(10)

which can be derived after taking into account Eq. (2)

$\displaystyle\gamma^{a}\stackrel{{\scriptstyle ab}}{{(k)}}$ $\displaystyle=$
$\displaystyle\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[-k]}},\;\quad\gamma^{b}\stackrel{{\scriptstyle
ab}}{{(k)}}=-ik\stackrel{{\scriptstyle
ab}}{{[-k]}},\;\quad\gamma^{a}\stackrel{{\scriptstyle
ab}}{{[k]}}=\stackrel{{\scriptstyle
ab}}{{(-k)}},\;\quad\;\;\gamma^{b}\stackrel{{\scriptstyle
ab}}{{[k]}}=-ik\eta^{aa}\stackrel{{\scriptstyle ab}}{{(-k)}}\,,$
$\displaystyle\tilde{\gamma^{a}}\stackrel{{\scriptstyle ab}}{{(k)}}$
$\displaystyle=$ $\displaystyle-i\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[k]}},\quad\tilde{\gamma^{b}}\stackrel{{\scriptstyle
ab}}{{(k)}}=-k\stackrel{{\scriptstyle
ab}}{{[k]}},\;\qquad\,\tilde{\gamma^{a}}\stackrel{{\scriptstyle
ab}}{{[k]}}=\;\;i\stackrel{{\scriptstyle
ab}}{{(k)}},\;\quad\tilde{\gamma^{b}}\stackrel{{\scriptstyle
ab}}{{[k]}}=-k\eta^{aa}\stackrel{{\scriptstyle ab}}{{(k)}}\,,$
$\displaystyle\stackrel{{\scriptstyle ab}}{{(k)}}^{\dagger}$ $\displaystyle=$
$\displaystyle\eta^{aa}\stackrel{{\scriptstyle
ab}}{{(-k)}}\,,\quad(\stackrel{{\scriptstyle
ab}}{{(k)}})^{2}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{(k)}}\stackrel{{\scriptstyle
ab}}{{(-k)}}=\eta^{aa}\stackrel{{\scriptstyle ab}}{{[k]}}\,,$
$\displaystyle\stackrel{{\scriptstyle ab}}{{[k]}}^{\dagger}$ $\displaystyle=$
$\displaystyle\,\stackrel{{\scriptstyle
ab}}{{[k]}}\,,\quad\quad\quad\quad(\stackrel{{\scriptstyle
ab}}{{[k]}})^{2}=\stackrel{{\scriptstyle
ab}}{{[k]}}\,,\quad\stackrel{{\scriptstyle ab}}{{[k]}}\stackrel{{\scriptstyle
ab}}{{[-k]}}=0\,,$ $\displaystyle\stackrel{{\scriptstyle
ab}}{{(k)}}\stackrel{{\scriptstyle ab}}{{[k]}}$ $\displaystyle=$
$\displaystyle 0\,,\qquad\qquad\qquad\stackrel{{\scriptstyle
ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{(k)}}=\stackrel{{\scriptstyle
ab}}{{(k)}}\,,\quad\quad\quad\stackrel{{\scriptstyle
ab}}{{(k)}}\stackrel{{\scriptstyle ab}}{{[-k]}}=\stackrel{{\scriptstyle
ab}}{{(k)}}\,,\quad\,\stackrel{{\scriptstyle
ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{(-k)}}=0\,,$
$\displaystyle\stackrel{{\scriptstyle ab}}{{\tilde{(k)}}}^{\dagger}$
$\displaystyle=$ $\displaystyle\eta^{aa}\stackrel{{\scriptstyle
ab}}{{\tilde{(-k)}}}\,,\quad(\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}})^{2}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle
ab}}{{\tilde{(-k)}}}=\eta^{aa}\stackrel{{\scriptstyle ab}}{{\tilde{[k]}}}\,,$
$\displaystyle\stackrel{{\scriptstyle ab}}{{\tilde{[k]}}}^{\dagger}$
$\displaystyle=$ $\displaystyle\,\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\,,\quad\quad\quad\quad(\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}})^{2}=\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle ab}}{{\tilde{[-k]}}}=0\,,$
$\displaystyle\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle ab}}{{\tilde{[k]}}}$
$\displaystyle=$ $\displaystyle 0\,,\qquad\qquad\qquad\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}=\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\,,\quad\quad\quad\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle
ab}}{{\tilde{[-k]}}}=\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\,,\quad\,\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle ab}}{{\tilde{(-k)}}}=0\,.$ (11)

Looking at relations in Eq. (11) it is obvious that the properties of the
”basis vectors” which include odd number of nilpotents differer essentially
from the ”basis vectors” which include even number of nilpotents.

One namely recognizes:
i. Since the Hermitian conjugated partner of a nilpotent
$\stackrel{{\scriptstyle ab}}{{(k)}}^{\dagger}$ is
$\eta^{aa}\stackrel{{\scriptstyle ab}}{{(-k)}}$ and since neither $S^{ab}$ nor
$\tilde{S}^{ab}$ nor both can transform odd products of nilpotents to belong
to one of the $2^{\frac{d}{2}-1}$ members of one of $2^{\frac{d}{2}-1}$
irreducible representations (families), the Hermitian conjugated partners of
the Clifford odd ”basis vectors” must belong to a different group of
$2^{\frac{d}{2}-1}$ members of $2^{\frac{d}{2}-1}$ families.
Since $S^{ac}$ transforms $\stackrel{{\scriptstyle
ab}}{{(k)}}*_{A}\stackrel{{\scriptstyle cd}}{{(k^{\prime})}}$ into
$\stackrel{{\scriptstyle ab}}{{[-k]}}*_{A}\stackrel{{\scriptstyle
cd}}{{[-k^{\prime}]}}$, while $\tilde{S}^{ab}$ transforms
$\stackrel{{\scriptstyle ab}}{{[-k]}}*_{A}\stackrel{{\scriptstyle
cd}}{{[-k^{\prime}]}}$ into $\stackrel{{\scriptstyle
ab}}{{(-k)}}*_{A}\stackrel{{\scriptstyle cd}}{{(-k^{\prime})}}$ it is obvious
that the Hermitian conjugated partners of the Clifford odd ”basis vectors”
must belong to the same group of $2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$
members. Projectors are self adjoint.
ii. Since an odd products of $\gamma^{a}$’s anticommute with another group of
an odd product of $\gamma^{a}$, 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 postulated 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
subalgebra members equal to either $\pm\frac{1}{2}$ or to $\pm\frac{i}{2}$.
The Clifford even ”basis vectors” have all the eigenvalues of the Cartan
subalgebra members ${\bf{\cal S}}^{ab}$ equal to either $\pm 1$ and zero or to
$\pm i$ and zero.

Let us define odd an even ”basis vectors” as products of nilpotents and
projectors in even dimensional spaces.

a. Clifford odd ”basis vectors”

The Clifford odd ”basis vectors” must be products of an odd number of
nilpotents and the rest, up to $\frac{d}{2}$, of projectors, each nilpotent
and projector must be the ”eigenstates” of one of the members of the Cartan
subalgebra, Eq. (8), correspondingly are the ”basis vectors” eigenstates of
all the members of the Lorentz algebras: $S^{ab}$’s determine
$2^{\frac{d}{2}-1}$ members of one family, $\tilde{S}^{ab}$’s transform each
member of one family to the same same member of the rest of
$2^{\frac{d}{2}-1}$ families.

Let us name the Clifford odd ”basis vectors” $\hat{b}^{m\dagger}_{f}$, where
$m$ determines membership of ’basis vectors” in any family and $f$ determines
a particular family. The Hermitian conjugated partner of
$\hat{b}^{m\dagger}_{f}$ is named by
$\hat{b}^{m}_{f}=(\hat{b}^{m\dagger}_{f})^{\dagger}$.

Let us start in $d=2(2n+1)$ with the ”basis vector” $\hat{b}^{1\dagger}_{1}$
which is the product of only nilpotents, all the rest members belonging to the
$f=1$ family follow by the application of $S^{01}$, $S^{03}$,
$\dots,S^{0d},S^{15}$, $\dots,S^{1d},S^{5d}\dots,S^{d-2\,d}$. The algebraic
product mark $*_{A}$ is skipped.

$\displaystyle d=2(2n+1)\,,$
$\displaystyle\hat{b}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{(+)}}\,,$
$\displaystyle\hat{b}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{(+)}}\,,$
$\displaystyle\cdots$
$\displaystyle\hat{b}^{2^{\frac{d}{2}-1}\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[-]}}\,,$
$\displaystyle\cdots\,.$ (12)

The Hermitian conjugated partners of the Clifford odd ”basis vector”
$\hat{b}^{m\dagger}_{1}$, presented in Eq. (2.1), are

$\displaystyle d=2(2n+1)\,,$
$\displaystyle\hat{b}^{1}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(-)}}\,,$ $\displaystyle\hat{b}^{2}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(-)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{(-)}}\,,$
$\displaystyle\cdots$
$\displaystyle\hat{b}^{2^{\frac{d}{2}-1}\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(-)}}\stackrel{{\scriptstyle 78}}{{[-]}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[-]}}\,,$
$\displaystyle\cdots\,.$ (13)

In $d=4n$ the choice of the starting ”basis vector”with maximal number of
nilpotents must have one projector

$\displaystyle d=4n\,,$
$\displaystyle\hat{b}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,,$
$\displaystyle\hat{b}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle\cdots$
$\displaystyle\hat{b}^{2^{\frac{d}{2}-1}\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle\dots\,.$ (14)

The Hermitian conjugated partners of the Clifford odd ”basis vector”
$\hat{b}^{m\dagger}_{1}$, presented in Eq. (2.1), follow if all nilpotents
$\stackrel{{\scriptstyle ab}}{{(k)}}$ are transformed into
$\eta^{aa}\stackrel{{\scriptstyle ab}}{{(-k)}}$.

For either $d=2(2n+1)$ or for $d=4n$ all the $2^{\frac{d}{2}-1}$ families
follow by applying $\tilde{S}^{ab}$’s on all the members of the starting
family. (Or one can find the starting $\hat{b}^{1}_{f}$ for all families $f$
and then generate all the members $\hat{b}^{m}_{f}$ from $\hat{b}^{1}_{f}$ by
the application of $\tilde{S}^{ab}$ 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.(11).

$\displaystyle\hat{b}^{m\dagger}_{f}*_{A}\hat{b}^{m`\dagger}_{f`}$
$\displaystyle=$ $\displaystyle
0\,,\quad\hat{b}^{m}_{f}*_{A}\hat{b}^{m`}_{f`}=0\,,\quad\forall
m,m^{\prime},f,f`\,.$ (15)

If we require that each family of ”basis vectors”, determined by nilpotents
and projectors described by $\gamma^{a}$’s, carry the family quantum number
determined by $\tilde{S}^{ab}$ and define the vacuum state on which ”basis
vectors” apply as

$\displaystyle|\psi_{oc}>=\sum_{f=1}^{2^{\frac{d}{2}-1}}\,\hat{b}^{m}_{f}{}_{*_{A}}\hat{b}^{m\dagger}_{f}\,|\,1\,>\,,$
(16)

it follows that the Clifford odd ”basis vectors” obey the relations

$\displaystyle\hat{b}^{m}_{f}{}_{*_{A}}|\psi_{oc}>$ $\displaystyle=$
$\displaystyle 0.\,|\psi_{oc}>\,,$
$\displaystyle\hat{b}^{m\dagger}_{f}{}_{*_{A}}|\psi_{oc}>$ $\displaystyle=$
$\displaystyle|\psi^{m}_{f}>\,,$
$\displaystyle\\{\hat{b}^{m}_{f},\hat{b}^{m^{\prime}}_{f`}\\}_{*_{A}+}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle 0.\,|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m\dagger}_{f},\hat{b}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}+}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle 0.\,|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m}_{f},\hat{b}^{m^{\prime}\dagger}_{f}\\}_{*_{A}+}|\psi_{oc}>$
$\displaystyle=$
$\displaystyle\delta^{mm^{\prime}}\,\delta_{ff`}|\psi_{oc}>\,,$ (17)

while the normalization
$<\psi_{oc}|\hat{b}^{m^{\prime}\dagger}_{f^{\prime}}\,*_{A}\,\hat{b}^{m\dagger}_{f}*_{A}|\psi_{oc}>=\delta^{mm^{\prime}}\delta_{ff^{\prime}}\,$
is used and the anticommutation relation mean
$\\{\hat{b}^{m\dagger}_{f},\hat{b}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}+}=$
$\hat{b}^{m\dagger}_{f}\,*_{A}\,\hat{b}^{m^{\prime}\dagger}_{f`}+\hat{b}^{m^{\prime}\dagger}_{f`}\,*_{A}\,\hat{b}^{m\dagger}_{f}$.

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 1, presented in Subsect. 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 $\frac{d}{2}$, of projectors, each nilpotent
and projector in a product must be the ”eigenstate” of one of the members of
the Cartan subalgebra, Eq. (8), correspondingly are the ”basis vectors”
eigenstates of all the members of the Lorentz algebra: $S^{ab}$’s and
$\tilde{S}^{ab}$’s generate from the starting ”basis vector” all the
$2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$ members of one group which
includes as well the Hermitian conjugated partners of any member.
$2^{\frac{d}{2}-1}$ members of the group are products of projectors only. They
are self adjoint.

There are two groups with $2^{\frac{d}{2}-1}\times$$2^{\frac{d}{2}-1}$
members. The members of one group are not connected with the members of
another group by either by $S^{ab}$’s or $\tilde{S}^{ab}$’s or both.

Let us name the Clifford even ”basis vectors” ${}^{i}\hat{\cal
A}^{m\dagger}_{f}$, 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 ${}^{I}\hat{\cal A}^{m\dagger}_{f}$ or in the case of ${}^{II}\hat{\cal
A}^{m\dagger}_{f}$ 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

$\displaystyle d$ $\displaystyle=$ $\displaystyle 2(2n+1)$
$\displaystyle{}^{I}\hat{{\cal A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,,\qquad$ $\displaystyle{}^{II}\hat{{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle{}^{I}\hat{{\cal A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle{}^{I}\hat{{\cal A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{(-)}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{(-)}}\,,$
$\displaystyle\dots\qquad$ $\displaystyle\dots$ $\displaystyle d$
$\displaystyle=$ $\displaystyle 4n$ $\displaystyle{}^{I}\hat{{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{(+)}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(+)}}\,,$ $\displaystyle{}^{I}\hat{{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[-i]}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(+)}}\,,\qquad$ $\displaystyle{}^{II}\hat{{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[-i]}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(+)}}\,,$ $\displaystyle{}^{I}\hat{{\cal
A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[-]}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[-]}}\,$
$\displaystyle\dots\qquad$ $\displaystyle\dots$ (18)

There are $2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ Clifford even ”basis
vectors” of the kind ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ and there are
$2^{\frac{d}{2}-1}$ $\times 2^{\frac{d}{2}-1}$ Clifford even ”basis vectors”
of the kind ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}$.

Table 1, presented in Subsect. 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 two 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. (11).

$\displaystyle{}^{I}{\hat{\cal A}}^{m\dagger}_{f}*_{A}{}^{II}{\hat{\cal
A}}^{m\dagger}_{f}$ $\displaystyle=$ $\displaystyle 0={}^{II}{\hat{\cal
A}}^{m\dagger}_{f}*_{A}{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,.$ (19)

The members of each of this two groups have the property

$\displaystyle{}^{I,II}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,{}^{I,II}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\rightarrow\left\\{\begin{array}[]{r}{}^{I,II}{\hat{\cal
A}}^{m\dagger}_{f`}\,,{\rm only\;one\;for}\;\forall f`\,,\\\ {\rm
or\,zero}\,.\end{array}\right.$ (22)

Two ”basis vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ and
${}^{I}{\hat{\cal A}}^{m^{\prime}\dagger}_{f^{\prime}}$, the algebraic
product, $*_{A}$, of which gives nonzero contribution, ”scatter” into the
third one ${}^{I}{\hat{\cal A}}^{m\dagger}_{f`}$. The same is true also for
the ”basis vectors” ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}$.

Let us write the commutation relations for Clifford even ”basis vectors”
taking into account Eq. (22).

i. In the case that ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\rightarrow$ ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f`}$ and ${}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\,*_{A}\,{}^{I}{\hat{\cal A}}^{m\dagger}_{f}=0$ it
follows

$\displaystyle\\{{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,,\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}\,-}\rightarrow\left\\{\begin{array}[]{r}{}^{I}{\hat{\cal
A}}^{m\dagger}_{f`}\,,\;\;\;\;({\rm if\;}\quad\;{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\rightarrow{}^{I}{\hat{\cal A}}^{m\dagger}_{f`}\\\
{\rm and}\;\;{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}=0)\,,\end{array}\right.$ (25)

ii. In the case that ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\rightarrow$ ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f`}$ and ${}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\rightarrow$ ${}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f}\,$ it follows

$\displaystyle\\{{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,,\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}\,-}\rightarrow\left\\{\begin{array}[]{r}{}^{I}{\hat{\cal
A}}^{m\dagger}_{f`}-{}^{I}{\hat{\cal A}}^{m^{\prime}\dagger}_{f}\,,\;\;\;({\rm
if\;}\quad\;{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\rightarrow{}^{I}{\hat{\cal A}}^{m\dagger}_{f`}\\\
{\rm and}\;\;{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\rightarrow{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f})\,,\end{array}\right.$ (28)

iii. In all other cases we have

$\displaystyle\\{{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,,\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}\,-}=0\;.$ (29)

$\\{{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,,\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}\,-}$ means ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}-{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\,*_{A}\,{}^{I}{\hat{\cal A}}^{m\dagger}_{f}$.

It remains to evaluate the algebraic application, $*_{A}$, of the Clifford
even ”basis vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ on the Clifford odd
”basis vectors” $\hat{b}^{m^{\prime}\dagger}_{f`}$. One finds

$\displaystyle{}^{I}{\hat{\cal
A}}^{m\dagger}_{f`}\,*_{A}\,\hat{b}^{m^{\prime}\dagger}_{f}\rightarrow\left\\{\begin{array}[]{r}\hat{b}^{m\dagger}_{f}\,,\\\
{\rm or\,zero}\,.\end{array}\right.$ (32)

For each ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ there are among
$2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ members of the Clifford odd ”basis
vectors” (describing the internal space of fermion fields) $2^{\frac{d}{2}-1}$
members, $\hat{b}^{m^{\prime}\dagger}_{f`}$, fulfilling the relation of Eq.
(32). All the rest ($2^{\frac{d}{2}-1}\times(2^{\frac{d}{2}-1}-1)$, give zero
contributions.

Eq. (32) clearly demonstrates that ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$
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
description of the gauge fields to the corresponding fermion fields.

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.

###  2.2 Example demonstrating properties of Clifford odd and even ”basis
vectors” for $d=(5+1)$

‘

Subsect. 2.2 demonstrates properties of the Clifford odd and even ”basis
vectors” in a special case when $d=(5+1)$.

In Table 1 the $64\,(=2^{d=6})$ ”eigenvectors” of the Cartan subalgebra
members of the Lorentz algebra $S^{ab}$, Eq. (8), are presented. The Clifford
odd ”basis vectors”, they appear in $4\,(=2^{\frac{d=6}{2}-1})$ families, each
family has $4$ members, are products of an odd number of nilpotents, that is
either of three nilpotents or of one nilpotent. They appear in Table 1 in the
group named $odd\,I\,\hat{b}^{m\dagger}_{f}$. Their Hermitian conjugated
partners appear in the second group named $odd\,II\,\hat{b}^{m}_{f}$. Within
each of these two groups, the members are orthogonal, Eq. (15), which means
that the algebraic product of
$\hat{b}^{m\dagger}_{f}*_{A}\hat{b}^{m^{\prime}\dagger}_{f`}=0$ for all
$(m,m^{\prime},f,f`)$, what can be checked by using relations in Eq. (11).
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

$\displaystyle<\psi_{oc}|\hat{b}^{m}_{f}*_{A}\hat{b}^{m^{\prime}\dagger}_{f`}|\psi_{oc}>=\delta^{mm^{\prime}}\delta_{ff`}\,,$
(33)

since the vacuum state $|\psi_{oc}>=\frac{1}{\sqrt{2^{\frac{d=6}{2}-1}}}$
$(\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[-]}}\stackrel{{\scriptstyle 56}}{{[-]}}+\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}}+\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[-]}}\stackrel{{\scriptstyle 56}}{{[+]}}+\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[-]}})$ is normalized to one: $<\psi_{oc}|\psi_{oc}>=1$.

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. [10].

The Clifford even ”basis vectors” are products of an even number of
nilpotents, of either two or none in this case. They are presented in Table 1
in two groups, each with $16\,(=2^{\frac{d=6}{2}-1}\times
2^{\frac{d=6}{2}-1})$ members, as $even\,I\,{\bf{\cal A}}^{m\dagger}_{f}$ and
$even\,II\,{\bf{\cal A}}^{m\dagger}_{f}$. One can easily check, using Eq.
(11), that the algebraic product ${}^{I}{\bf{\cal A}}^{m\dagger}_{f}*_{A}$
${}^{II}{\bf{\cal
A}}^{m^{\prime}\dagger}_{f`}=0,\forall\,(m,m^{\prime},f.f`)$, Eq. (19). The
longer overview of the Clifford even ”basis vectors” and their Hermitian
conjugated partners for the case $d=(5+1)$\- can be found in Ref. [12].

Table 1:  $2^{d}=64$ ”eigenvectors” of the Cartan subalgebra of the Clifford odd and even algebras — the superposition of odd and even products of $\gamma^{a}$’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 ${\hat{b}}^{m\dagger}_{f}$, appearing in $2^{\frac{d}{2}-1}=4$ ”families” ($f=1,2,3,4$), each ”family” with $2^{\frac{d}{2}-1}=4$ ”family” members ($m=1,2,3,4$). The second group, $odd\,II$, contains Hermitian conjugated partners of the first group for each family separately, ${\hat{b}}^{m}_{f}=$ $({\hat{b}}^{m\dagger}_{f})^{\dagger}$. Either $odd\,I$ or $odd\,II$ are products of an odd number of nilpotents, the rest are projectors. The ”family” quantum numbers of ${\hat{b}}^{m\dagger}_{f}$, that is the eigenvalues of $(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$, are for the first odd I  group written above each ”family”, the quantum numbers of the members $(S^{03},S^{12},S^{56})$ are written in the last three columns. For the Hermitian conjugated partners of odd I, presented in the group odd II, the quantum numbers $(S^{03},S^{12},S^{56})$ are presented above each group of the Hermitian conjugated partners, the last three columns tell eigenvalues of $(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$. The two groups with the even number of $\gamma^{a}$’s, even I and even II, each has their Hermitian conjugated partners within its own group, have the quantum numbers $f$, that is the eigenvalues of $(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$, written above column of four members, the quantum numbers of the members, $(S^{03},S^{12},S^{56})$, are written in the last three columns.  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
${}^{\prime\prime}basis\,vectors^{\prime\prime}$ | $m$ | $f=1$ | $f=2$ | $f=3$ | $f=4$ |  |  |
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$ | $\rightarrow$ | $(\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $(\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $S^{03}$ | $S^{12}$ | $S^{56}$
|  |  |  |  |  |  |  |
$odd\,I\;{\hat{b}}^{m\dagger}_{f}$ | $1$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle 56}}{{[+]}}$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle 56}}{{(+)}}$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{[+]}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
| $2$ | $[-i](-)[+]$ | $(-i)(-)(+)$ | $(-i)[-][+]$ | $[-i][-](+)$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $3$ | $[-i][+](-)$ | $(-i)[+][-]$ | $(-i)(+)(-)$ | $[-i](+)[-]$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $4$ | $(+i)(-)(-)$ | $[+i](-)[-]$ | $[+i][-](-)$ | $(+i)[-][-]$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
|  |  |  |  |  |  |  |
$(S^{03},S^{12},S^{56})$ | $\rightarrow$ | $(-\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $(\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $(\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $\tilde{S}^{03}$ | $\tilde{S}^{12}$ | $\tilde{S}^{56}$
|  | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ |  |  |
|  |  |  |  |  |  |  |
$odd\,II\;{\hat{b}}^{m}_{f}$ | $1$ | $(-i)[+][+]$ | $[+i][+](-)$ | $[+i](-)[+]$ | $(-i)(-)(-)$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
| $2$ | $[-i](+)[+]$ | $(+i)(+)(-)$ | $(+i)[-][+]$ | $[-i][-](-)$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $3$ | $[-i][+](+)$ | $(+i)[+][-]$ | $(+i)(-)(+)$ | $[-i](-)[-]$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
5 | $-1$ | $-1$ |  |  |  |  |  |
| $4$ | $(-i)(+)(+)$ | $[+i](+)[-]$ | $[+i][-](+)$ | $(-i)[-][-]$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
|  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$ | $\rightarrow$ | $(-\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $(\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $(\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $S^{03}$ | $S^{12}$ | $S^{56}$
|  | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ |  |  |
|  |  |  |  |  |  |  |
$even\,I\;{}^{I}{\cal A}^{m}_{f}$ | $1$ | $[+i](+)(+)$ | $(+i)[+](+)$ | $[+i][+][+]$ | $(+i)(+)[+]$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
| $2$ | $(-i)[-](+)$ | $[-i](-)(+)$ | $(-i)(-)[+]$ | $[-i][-][+]$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $3$ | $(-i)(+)[-]$ | $[-i][+][-]$ | $(-i)[+](-)$ | $[-i](+)(-)$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $4$ | $[+i][-][-]$ | $(+i)(-)[-]$ | $[+i](-)(-)$ | $(+i)[-](-)$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
|  |  |  |  |  |  |  |
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$ | $\rightarrow$ | $(\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $(-\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $S^{03}$ | $S^{12}$ | $S^{56}$
|  | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ |  |  |
|  |  |  |  |  |  |  |
$even\,II\;{}^{II}{\cal A}^{m}_{f}$ | $1$ | $[-i](+)(+)$ | $(-i)[+](+)$ | $[-i][+][+]$ | $(-i)(+)[+]$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
| $2$ | $(+i)[-](+)$ | $[+i](-)(+)$ | $(+i)(-)[+]$ | $[+i][-][+]$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $3$ | $(+i)(+)[-]$ | $[+i][+][-]$ | $(+i)[+](-)$ | $[+i](+)(-)$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $4$ | $[-i][-][-]$ | $(-i)(-)[-]$ | $[-i](-)(-)$ | $(-i)[-](-)$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$

While the Clifford odd ”basis vectors” are (chosen to be) right handed,
$\Gamma^{(5+1)}=1$, have their Hermitian conjugated partners opposite
handedness 555The handedness $\Gamma^{(d)}$, one of the invariants of the
group $SO(d)$, with the infinitesimal generators of the Lorentz group
$S^{ab}$, is defined as $\Gamma^{(d)}=\alpha\,\varepsilon_{a_{1}a_{2}\dots
a_{d-1}a_{d}}\,S^{a_{1}a_{2}}\cdot S^{a_{3}a_{4}}\cdots S^{a_{d-1}a_{d}}$,
with $\alpha$ chosen so that $\Gamma^{(d)}=\pm 1$.

While the Clifford odd ”basis vectors” have half integer eigenvalues of the
Cartan subalgebra members, Eq.(8), that is of $S^{03},S^{12},S^{56}$ in this
particular case of $d=(5+1)$, the Clifford even ”basis vectors” have integer
spins, obtained by ${\bf{\cal S}}^{03}=S^{03}+\tilde{S}^{03}$, ${\bf{\cal
S}}^{12}=S^{12}+\tilde{S}^{12}$, ${\bf{\cal S}}^{56}=S^{56}+\tilde{S}^{56}$.

Let us check what does the algebraic application, $*_{A}$, of
${}^{I}{\hat{\cal A}}^{m\dagger}_{f=3},m=(1,2,3,4)$, presented in Table 1 in
the third column of $even\,I$, do on the Clifford odd ”basis vectors”
$\hat{b}^{m=1\dagger}_{f=1}$, presented as the first $odd\,I$ ”basis vector”
in Table 1. This can easily be evaluated by taking into account Eq. (2.1) for
any $m$.

$\displaystyle{}^{I}{\hat{\cal
A}}^{m\dagger}_{3}*_{A}\hat{b}^{1\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}}):$ $\displaystyle{}^{I}{\hat{\cal
A}}^{1\dagger}_{3}(\equiv\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}})*_{A}\hat{b}^{1\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}})\rightarrow\hat{b}^{1\dagger}_{1}\,,{\rm selfadjoint}$
$\displaystyle{}^{I}{\hat{\cal
A}}^{2\dagger}_{3}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{[+]}})*_{A}\hat{b}^{1\dagger}_{1}\rightarrow\hat{b}^{2\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{[+]}})\,,\quad$ $\displaystyle{}^{I}{\hat{\cal
A}}^{3\dagger}_{3}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{(-)}})*_{A}\hat{b}^{1\dagger}_{1}\rightarrow\hat{b}^{3\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{(-)}})\,,\quad$ $\displaystyle{}^{I}{\hat{\cal
A}}^{4\dagger}_{3}(\equiv\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(-)}})*_{A}\hat{b}^{1\dagger}_{1}\rightarrow\hat{b}^{4\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(-)}})\,.\;$ (34)

The sign $\rightarrow$ means that the relation is valid up to the constant.
${}^{I}{\hat{\cal A}}^{1\dagger}_{3}$ is self adjoint, the Hermitian
conjugated partner of ${}^{I}{\hat{\cal A}}^{2\dagger}_{3}$ is
${}^{I}{\hat{\cal A}}^{1\dagger}_{4}$, of ${}^{I}{\hat{\cal
A}}^{3\dagger}_{3}$ is ${}^{I}{\hat{\cal A}}^{1\dagger}_{2}$ and of
${}^{I}{\hat{\cal A}}^{4\dagger}_{3}$ is ${}^{I}{\hat{\cal
A}}^{1\dagger}_{1}$.

We can conclude that the algebraic, $*_{A}$, application of ${}^{I}{\hat{\cal
A}}^{m\dagger}_{3}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{(-)}})$ on $\hat{b}^{1\dagger}_{1}$ leads to the same or another family
member of the same family $f=1$, namely to $\hat{b}^{m\dagger}_{1}$,
$m=(1,2,3,4)$.

Calculating the eigenvalues of the Cartan subalgebra members, Eq. (8), before
and after the algebraic multiplication, $*_{A}$, one sees that
${}^{I}{\hat{\cal A}}^{m\dagger}_{3}$ carry the integer eigenvalues of the
Cartan subalgebra members, namely of ${\cal{\bf S}}^{ab}$
$=S^{ab}+\tilde{S}^{ab}$, 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 ${}^{I}{\hat{\cal
A}}^{m\dagger}_{3}$, $m=1,2,3,4$, on $\hat{b}^{1\dagger}_{1}$ transforms
$\hat{b}^{1\dagger}_{1}$ into $\hat{b}^{m\dagger}_{1}$, $m=(1,2,3,4)$.
Similarly we find that the algebraic application of ${}^{I}{\hat{\cal
A}}^{m}_{4},$ $m=(1,2,3,4)$ on $\hat{b}^{2\dagger}_{1}$ transforms
$\hat{b}^{2\dagger}_{1}$ into $\hat{b}^{m\dagger}_{1},m=(1,2,3,4)$. The
algebraic application of ${}^{I}{\hat{\cal A}}^{m}_{2},$ $m=(1,2,3,4)$ on
$\hat{b}^{3\dagger}_{1}$ transforms $\hat{b}^{3\dagger}_{1}$ into
$\hat{b}^{m\dagger}_{1},m=(1,2,3,4)$. And the algebraic application of
${}^{I}{\hat{\cal A}}^{m}_{1},$ $m=(1,2,3,4)$ on $\hat{b}^{4\dagger}_{1}$
transforms $\hat{b}^{4\dagger}_{1}$ into $\hat{b}^{m\dagger}_{1},m=(1,2,3,4)$.

The statement of Eq. (32) 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” ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ when they apply on each other. Let us denote the self
adjoint member in each group of ”basis vectors” of particular $f$ as
${}^{I}{\hat{\cal A}}^{m_{0}\dagger}_{f}$. We easily see that

$\displaystyle\\{{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,,{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f}\\}_{-}$ $\displaystyle=$ $\displaystyle
0\,,\quad{\rm if}\,(m,m^{\prime})\,\neq m_{0}\,{\rm
or}\,m=m_{0}=m^{\prime}\,,\forall\,f\,,$ $\displaystyle{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}*_{A}{}^{I}{\hat{\cal A}}^{m_{0}\dagger}_{f}$
$\displaystyle\rightarrow$ $\displaystyle{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,,\quad\forall\,m\,,\,\forall\,f\,.$ (35)

Since a projector to the second power is the projector back, and since a
projector is self adjoint, it means that the self adjoint Clifford even ”basis
vectors” is the product of projectors. In Table 1 we see that in each column
of either $even\,{}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ or of
$even{}^{II}{\hat{\cal A}}^{m\dagger}_{f}$ there is one self adjoint
${}^{I,II}{\hat{\cal A}}^{m_{0}\dagger}_{f}$. We also see that two ”basis
vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ and ${}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f}$ of the same $f$ and of $(m,m^{\prime})\neq m_{0}$
are orthogonal. We only have to take into account Eq. (11), which tells that

$\stackrel{{\scriptstyle ab}}{{(k)}}\stackrel{{\scriptstyle
ab}}{{[k]}}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{(k)}}=\stackrel{{\scriptstyle
ab}}{{(k)}}\,,\quad\stackrel{{\scriptstyle ab}}{{(k)}}\stackrel{{\scriptstyle
ab}}{{[-k]}}=\stackrel{{\scriptstyle
ab}}{{(k)}}\,,\quad\,\stackrel{{\scriptstyle
ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{(-k)}}=0.$

These relations tell us that ${}^{I}{\hat{\cal A}}^{1\dagger}_{4}$
$*_{A}{}^{I}{\hat{\cal A}}^{2\dagger}_{3}$ $={}^{I}{\hat{\cal
A}}^{1\dagger}_{3}$, what illustrates Eq. (29), while ${}^{I}{\hat{\cal
A}}^{2\dagger}_{3}$$*_{A}{}^{I}{\hat{\cal A}}^{1\dagger}_{4}$
$={}^{I}{\hat{\cal A}}^{2\dagger}_{4}$ illustrating Eq. (28), while
${}^{I}{\hat{\cal A}}^{1\dagger}_{3}$ $*_{A}{}^{I}{\hat{\cal
A}}^{2\dagger}_{4}=0$ illustrates Eq. (25).

Table 2 presents the Clifford even ”basis vectors” ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ for $d=(5+1)$ with the properties: i.They are products of
an even number of nilpotents, $\stackrel{{\scriptstyle ab}}{{(k)}}$, with the
rest up to $\frac{d}{2}$ of projectors, $\stackrel{{\scriptstyle ab}}{{[k]}}$.
ii. Nilpotents and projectors are eigenvectors of the Cartan subalgebra
members ${\bf{\cal S}}^{ab}$ $=S^{ab}+\tilde{S}^{ab}$, Eq. (8), carrying the
integer eigenvalues of the Cartan subalgebra members.
iii. They have their Hetmitian conjugated partners within the same group of
${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ with $2^{\frac{d}{2}-1}$ $\times$
$2^{\frac{d}{2}-1}$ 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”, transferring 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 $SU(3)$ $\times U(1)$ subgroups of the Clifford odd and
Clifford even ”basis vectors”.

We present in Eqs. (36, 2.2) the superposition of members of Cartan
subalgebra, Eq. (8) for $S^{ab}$ for the Clifford odd ”basis vectors”, for the
subgroup $SO(3,1)\times U(1)$ ($N^{3}_{\pm}\,,\tau$) and for the subgroup
$SU(3)$ $\times U(1)$: ($\tau^{\prime},\tau^{3},\tau^{8}$). The same relations
can be used also for the corresponding operators determining the ”family”
quantum numbers ($\tilde{N}^{3}_{\pm}\,,\tilde{\tau}$) of the Clifford odd
”basis vectors’, if $S^{ab}$’s are replaced by $\tilde{S}^{ab}$’s. For the
Clifford even objects ${\cal{\bf S}}^{ab}(=S^{ab}+\tilde{S}^{ab})$ must
replace $S^{ab}$.

$\displaystyle N^{3}_{\pm}(=N^{3}_{(L,R)}):=\,\frac{1}{2}(S^{12}\pm
iS^{03})\,,\quad\tau=S^{56}\,,\quad$ (36) $\displaystyle\tau^{3}:=$
$\displaystyle\frac{1}{2}\,(-S^{1\,2}-iS^{0\,3})\,,\qquad\tau^{8}=\frac{1}{2\sqrt{3}}(-iS^{0\,3}+S^{1\,2}-2S^{5\;6})\,,$
$\displaystyle\tau^{\prime}=$
$\displaystyle-\frac{1}{3}(-iS^{0\,3}+S^{1\,2}+S^{5\,6})\,.$ (37)

Let us, for example, algebraically apply ${}^{I}{\hat{\cal A}}^{2}_{3}$
($\equiv\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle
12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$), denoted by $\odot\odot$ on
Table 2, carrying
$(\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{2}{3})$,
represented also on Fig. 2 by $\odot\odot$, on the Clifford odd ”basis vector”
$\hat{b}^{1\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}})$, presented on Table 1, with
$(\tau^{3}=0,\tau^{8}=0,\tau^{\prime}=-\frac{1}{2})$, as we can calculate
using Eq. (2.2) and which is represented on Fig. 1 by a square as a singlet.
${}^{I}{\hat{\cal A}}^{2}_{3}$ transforms $\hat{b}^{1\dagger}_{1}$ (by
transferring to $\hat{b}^{1\dagger}_{1}$
$(\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{2}{3})$) to
$\hat{b}^{1\dagger}_{2}$ with
$(\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{1}{6})$,
belonging on Fig. 1 to the triplet, denoted by $\bigcirc$. The corresponding
gauge fields, presented on Fig. 2, if belonging to the sextet, would transform
the triplet of quarks among themselves.

Figure 1:  Representations of the subgroups $SU(3)$ and $U(1)$ of the group
$SO(5,1)$, the properties of which appear in Table 1, are presented.
($\tau^{3},\tau^{8}$ and $\tau^{\prime}$) can be calculated if using Eqs.(36,
2.2). On the abscissa axis, on the ordinate axis and on the third axis the
eigenvalues of the superposition of the three Cartan subalgebra members,
$\tau^{3}$, $\tau^{8}$, $\tau^{\prime}$ are presented. One notices one
triplet, denoted by ${\bf\bigcirc}$ with the values
$\tau^{\prime}=\frac{1}{6}$,
($\tau^{3}=-\frac{1}{2},\tau^{8}=\frac{1}{2\sqrt{3}},\tau^{\prime}=\frac{1}{6})$,
($\tau^{3}=\frac{1}{2},\tau^{8}=\frac{1}{2\sqrt{3}},\tau^{\prime}=\frac{1}{6}$),
($\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{1}{6}$),
respectively, and one singlet denoted by the square.
($\tau^{3}=0,\tau^{8}=0,\tau^{\prime}=-\frac{1}{2}$). The triplet and the
singlet appear in four families.

In the case of the group $SO(6)$, manifesting as $SU(3)\times U(1)$ 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
leptons. The corresponding gauge of the fields, presented on Fig. 2, if
belonging to the sextet, would transform the triplet of quarks among
themselves, changing the colour and leaving the ”fermion” quantum number equal
to $\frac{1}{6}$.

Figure 2:  The Clifford even ”basis vectors” ${}^{I}{\hat{\cal A}}^{m}_{f}$,
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
$SO(5,1)$: $\tau^{3}=\frac{1}{2}$ $(-{\cal S}^{12}-i{\cal S}^{03})$,
$\tau^{8}=\frac{1}{2\sqrt{3}}({\cal S}^{12}-i{\cal S}^{03}-2{\cal S}^{56})$,
$\tau^{\prime}=-\frac{1}{3}({\cal S}^{12}-i{\cal S}^{03}+{\cal S}^{56})$.
Their properties appear in Table 2. The abscissa axis carries the eigenvalues
of $\tau^{3}$, the ordinate axis of $\tau^{8}$ and the third axis the
eigenvalues of $\tau^{\prime}$, One notices four singlets with
($\tau^{3}=0,\tau^{8}=0,\tau^{\prime}=0$), denoted by $\bigcirc$, representing
four self adjoint Clifford even ”basis vectors” ${}^{I}{\hat{\cal
A}}^{m}_{f}$, one sextet of three pairs with $\tau^{\prime}=0$, Hermitian
conjugated to each other, denoted by $\bigtriangleup$ (with
($\tau^{\prime}=0,\tau^{3}=-\frac{1}{2},\tau^{8}=-\frac{3}{2\sqrt{3}}$) and
($\tau^{\prime}=0,\tau^{3}=\frac{1}{2},\tau^{8}=\frac{3}{2\sqrt{3}}$) ),
respectively, by $\ddagger$ (with ($\tau^{\prime}=0,\tau^{3}=-1,\tau^{8}=0$)
and ($\tau^{\prime}=0,\tau^{3}=1,\tau^{8}=0$), respectively, and by $\otimes$
(with ($\tau^{\prime}=0,\tau^{3}=\frac{1}{2},\tau^{8}=-\frac{3}{2\sqrt{3}}$)
and ($\tau^{\prime}=0,\tau^{3}=-\frac{1}{2},\tau^{8}=\frac{3}{2\sqrt{3}}$) ),
respectively, and one triplet, denoted by $\star\star$ with
($\tau^{\prime}=\frac{2}{3},\tau^{3}=\frac{1}{2},\tau^{8}=\frac{1}{2\sqrt{3}}$),
by $\bullet$ with
($\tau^{\prime}=\frac{2}{3},\tau^{3}=-\frac{1}{2},\tau^{8}=\frac{1}{2\sqrt{3}}$),
and by $\odot\odot$ with
($\tau^{\prime}=\frac{2}{3},\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}}$), as well
as one antitriplet, Hermitian conjugated to the triplet, denoted by
$\star\star$ with
($\tau^{\prime}=-\frac{2}{3},\tau^{3}=-\frac{1}{2},\tau^{8}=-\frac{1}{2\sqrt{3}}$),
by $\bullet$ with
($\tau^{\prime}=-\frac{2}{3},\tau^{3}=\frac{1}{2},\tau^{8}=-\frac{1}{2\sqrt{3}}$),
and by $\odot\odot$ with
($\tau^{\prime}=-\frac{2}{3},\tau^{3}=0,\tau^{8}=\frac{1}{\sqrt{3}}$).

We can see that ${}^{I}{\hat{\cal A}}^{m\dagger}_{3}$ with $(m=2,3,4)$, if
applied on the $SU(3)$ singlet $\hat{b}^{1\dagger}_{1}$ with
($\tau^{\prime}=-\frac{1}{2},\tau^{3}=0,\tau^{8}=0$), transforms it to
$\hat{b}^{m=2,3,4)\dagger}_{1}$, respectively, which are members of the
$SU(3)$ triplet. All these Clifford even ”basis vectors” have $\tau^{\prime}$
equal to $\frac{2}{3}$, changing correspondingly $\tau^{\prime}=-\frac{1}{2}$
into $\tau^{\prime}=\frac{1}{6}$ and bringing the needed values of $\tau^{3}$
and $\tau^{8}$.

In Table 2 we find $(6+4)$ Clifford even ”basis vectors” ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ with $\tau`=0$. 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 ($\tau^{\prime}=0,\tau^{3}=0,\tau^{8}=0$), denoted in the
third column of Table 2 by $\bigcirc$. The rest $6$ Clifford even ”basis
vectors” belong to one triplet with $\tau^{\prime}=\frac{2}{3}$ and
$(\tau^{3},\tau^{8})$ equal to [$(0,-\frac{1}{\sqrt{3}})$,
$(-\frac{1}{2},\frac{1}{2\sqrt{3}})$, $(\frac{1}{2},\frac{1}{2\sqrt{3}})$] and
one antitriplet with $\tau^{\prime}=-\frac{2}{3}$ and ($(\tau^{3},\tau^{8})$
equal to [$(-\frac{1}{2},-\frac{1}{2\sqrt{3}})$,
$(\frac{1}{2},-\frac{1}{2\sqrt{3}})$, $(0,\frac{1}{\sqrt{3}})$].

Each triplet has Hermitian conjugated partners in antitriplet and opposite. In
Table 2 the Hermitian conjugated partners of the triplet and antitriplet are
denoted by the same signum: (${}^{I}{\hat{\cal A}}^{1\dagger}_{1}$,
${}^{I}{\hat{\cal A}}^{4\dagger}_{3}$) by $\star\star$, (${}^{I}{\hat{\cal
A}}^{1\dagger}_{2}$, ${}^{I}{\hat{\cal A}}^{3\dagger}_{3}$) by $\bullet$, and
(${}^{I}{\hat{\cal A}}^{2\dagger}_{3}$, ${}^{I}{\hat{\cal A}}^{1\dagger}_{4}$)
by $\odot\odot$.

The octet and the two triplets are presented in Fig. 2.

Table 2: The Clifford even ”basis vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$, 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, ${\cal S}^{03}$, ${\cal S}^{12}$, ${\cal S}^{56}$, Eq. (8), are presented for $d=(5+1)$-dimensional case. Indexes $m$ and $f$ determine $2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ different members ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$. In the third column the ”basis vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ which are Hermitian conjugated partners to each other (and can therefore annihilate each other) are pointed out with the same symbol. For example, with $\star\star$ are equipped the first member with $m=1$ and $f=1$ and the last member of $f=3$ with $m=4$. The sign $\bigcirc$ denotes the Clifford even ”basis vectors” which are self adjoint $({}^{I}{\hat{\cal A}}^{m\dagger}_{f})^{\dagger}$ $={}^{I}{\hat{\cal A}}^{m^{\prime}\dagger}_{f`}$. It is obvious that † has no meaning, since ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ are self adjoint or are Hermitian conjugated partner to another ${}^{I}{\hat{\cal A}}^{m^{\prime}\dagger}_{f`}$. This table represents also the eigenvalues of the three commuting operators ${\cal N}^{3}_{L,R}$ and ${\cal S}^{56}$ of the subgroups $SU(2)\times SU(2)\times U(1)$ of the group $SO(5,1)$ and the eigenvalues of the three commuting operators ${\tau}^{3},{\tau}^{8}$ and ${\tau^{\prime}}$ of the subgroups $SU(3)\times U(1)$.  $\,f$ | $m$ | $*$ | ${}^{I}\hat{\cal A}^{m\dagger}_{f}$ | ${\cal S}^{03}$ | ${\cal S}^{12}$ | ${\cal S}^{56}$ | ${\cal N}^{3}_{L}$ | ${\cal N}^{3}_{R}$ | ${\cal\tau}^{3}$ | ${\cal\tau}^{8}$ | ${\cal\tau}^{\prime}$
---|---|---|---|---|---|---|---|---|---|---|---
$I$ | $1$ | $\star\star$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}$ | $0$ | $1$ |  $1$. | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\sqrt{3}}$ | $-\frac{2}{3}$
| $2$ | $\bigtriangleup$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{(+)}}$ | $-i$ | $0$ | $1$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{3}{2\sqrt{3}}$ | $0$
| $3$ | $\ddagger$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $-i$ | $1$ | $0$ | $1$ | $0$ | $-1$ | $0$ | $0$
| $4$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$II$ | $1$ | $\bullet$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{(+)}}$ | $i$ | $0$ | $1$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\sqrt{3}}$ | $-\frac{2}{3}$
| $2$ | $\otimes$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{(+)}}$ | $0$ | $-1$ | $1$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{3}{2\sqrt{3}}$ | $0$
| $3$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4$ | $\ddagger$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $i$ | $-1$ | $0$ | $-1$ | $0$ | $1$ | $0$ | $0$
$III$ | $1$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2$ | $\odot\odot$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $-i$ | $-1$ | $0$ | $0$ | $-1$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{2}{3}$
| $3$ | $\bullet$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $-i$ | $0$ | $-1$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{2}{3}$
| $4$ | $\star\star$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $0$ | $-1$ | $-1$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{2}{3}$
$IV$ | $1$ | $\odot\odot$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $i$ | $1$ | $0$ | $0$ | $1$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{2}{3}$
| $2$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $3$ | $\otimes$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $0$ | $1$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{3}{2\sqrt{3}}$ | $0$
| $4$ | $\bigtriangleup$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $i$ | $0$ | $-1$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{3}{2\sqrt{3}}$ | $0$

Fig. 2 represents the $2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ members
${}^{I}{\hat{\cal A}}^{m}_{f}$ of the Clifford even ”basis vectors” for the
case that $d=(5+1)$. The properties of ${}^{I}{\hat{\cal A}}^{m}_{f}$ are
presented also in Table 2. There are in this case again $16$ members.
Manifesting the structure of subgroups $SU(3)\times U(1)$ of the group
$SO(5,1)$ they are represented as eigenvectors of the superposition of the
Cartan subalgebra members (${\bf{\cal S}}^{03},{\bf{\cal S}}^{12},{\bf{\cal
S}}^{56}$), that is with $\tau^{3}=\frac{1}{2}(-{\bf{\cal S}}^{12}-i{\bf{\cal
S}}^{03})$, $\tau^{8}=\frac{1}{2\sqrt{3}}({\bf{\cal S}}^{12}-i{\bf{\cal
S}}^{03}-2{\bf{\cal S}}^{56})$, and $\tau^{\prime}=-\frac{1}{3}({\bf{\cal
S}}^{12}-i{\bf{\cal S}}^{03}+{\bf{\cal S}}^{56})$. There are four self adjoint
Clifford even ”basis vectors” with ($\tau^{3}=0,\tau^{8}=0,\tau^{\prime}=0$),
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” ${}^{I}{\hat{\cal A}}^{m}_{f}$ are the boson
”partners” of the Clifford odd ”basis vectors” $\hat{b}^{m\dagger}_{f}$,
presented in Fig. 1 for one of four families, anyone. The reader can check
that the algebraic application of ${}^{I}{\hat{\cal A}}^{m}_{f}$, belonging to
the triplet, transforms the Clifford odd singlet, denoted on Fig. 1 by a
square, to one of the members of the triplet, denoted on Fig. 1 by the circle
${\bf\bigcirc}$.

Looking at the boson fields ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ from the
point of view of subgroups $SU(3)\times U(1)$ 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.

###  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 sub section 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 $\gamma^{a}$’s, respectively, offer the description of the
internal spaces of fermion and boson fields.

The Clifford odd algebra offers $2^{\frac{d}{2}-1}$ ”basis vectors”
$\hat{b}^{m\dagger}_{f}$, appearing in $2^{\frac{d}{2}-1}$ families (with the
family quantum numbers determined by
$\tilde{S}^{ab}=\frac{i}{2}\\{\tilde{\gamma}^{a},\tilde{\gamma}^{b}\\}_{-}$),
which together with their $2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$
Hermitian conjugated partners $\hat{b}^{m}_{f}$ fulfil the postulates for the
second quantized fermion fields, Eq. (2.1) in this paper, Eq.(26) in Ref.
[10], explaining the second quantization postulates of Dirac.

The Clifford even algebra offers $2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$
”basis vectors” of ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ (and the same number
of ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}$) with the properties of the second
quantized boson fields manifesting as the gauge fields of fermion fields
described by the Clifford odd ”basis vectors” $\hat{b}^{m\dagger}_{f}$.

The Clifford odd and the Clifford even ”basis vectors” are chosen to be
products of nilpotents, $\stackrel{{\scriptstyle ab}}{{(k)}}$ (with the odd
number of nilpotents if describing fermions and the even number of nilpotents
if describing bosons), and projectors, $\stackrel{{\scriptstyle ab}}{{[k]}}$.
Nilpotents and projectors are (chosen to be) eigenvectors of the Cartan
subalgebra members of the Lorentz algebra in the internal space of $S^{ab}$
for the Clifford odd ”basis vectors” and of ${\bf{\cal
S}}^{ab}(=S^{ab}+\tilde{S}^{ab}$) 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 ordinary space in momentum or coordinate representation is
needed. Here Ref. [10], 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. [10] in Subsect. 3.3 and in App. J.

$\displaystyle|\vec{p}>$ $\displaystyle=$
$\displaystyle\hat{b}^{\dagger}_{\vec{p}}\,|\,0_{p}\,>\,,\quad<\vec{p}\,|=<\,0_{p}\,|\,\hat{b}_{\vec{p}}\,,$
$\displaystyle<\vec{p}\,|\,\vec{p}^{\prime}>$ $\displaystyle=$
$\displaystyle\delta(\vec{p}-\vec{p}^{\prime})=<\,0_{p}\,|\hat{b}_{\vec{p}}\;\hat{b}^{\dagger}_{\vec{p}^{\prime}}|\,0_{p}\,>\,,$
$\displaystyle{\rm leading\;to\;}$
$\displaystyle\hat{b}_{\vec{p^{\prime}}}\,\hat{b}^{\dagger}_{\vec{p}}$
$\displaystyle=$ $\displaystyle\delta(\vec{p^{\prime}}-\vec{p})\,,$ (38)

with the normalization $<\,0_{p}\,|\,0_{p}\,>=1$. While the quantized
operators $\hat{\vec{p}}$ and $\hat{\vec{x}}$ commute
$\\{\hat{p}^{i}\,,\hat{p}^{j}\\}_{-}=0$ and
$\\{\hat{x}^{k}\,,\hat{x}^{l}\\}_{-}=0$, it follows for
$\\{\hat{p}^{i}\,,\hat{x}^{j}\\}_{-}=i\eta^{ij}$. One correspondingly finds

$\displaystyle<\vec{p}\,|\,\vec{x}>$ $\displaystyle=$
$\displaystyle<0_{\vec{p}}\,|\,\hat{b}_{\vec{p}}\;\hat{b}^{\dagger}_{\vec{x}}|0_{\vec{x}}\,>=(<0_{\vec{x}}\,|\,\hat{b}_{\vec{x}}\;\hat{b}^{\dagger}_{\vec{p}}\,|0_{\vec{p}}\,>)^{\dagger}\,$
$\displaystyle\\{\hat{b}^{\dagger}_{\vec{p}}\,,\,\hat{b}^{\dagger}_{\vec{p}\,^{\prime}}\\}_{-}$
$\displaystyle=$ $\displaystyle
0\,,\qquad\\{\hat{b}_{\vec{p}},\,\hat{b}_{\vec{p}\,^{\prime}}\\}_{-}=0\,,\qquad\\{\hat{b}_{\vec{p}},\,\hat{b}^{\dagger}_{\vec{p}\,^{\prime}}\\}_{-}=0\,,$
$\displaystyle\\{\hat{b}^{\dagger}_{\vec{x}},\,\hat{b}^{\dagger}_{\vec{x}\,^{\prime}}\\}_{-}$
$\displaystyle=$ $\displaystyle
0\,,\qquad\\{\hat{b}_{\vec{x}},\,\hat{b}_{\vec{x}\,^{\prime}}\\}_{-}=0\,,\qquad\\{\hat{b}_{\vec{x}},\,\hat{b}^{\dagger}_{\vec{x}\,^{\prime}}\\}_{-}=0\,,$
$\displaystyle\\{\hat{b}_{\vec{p}},\,\hat{b}^{\dagger}_{\vec{x}}\\}_{-}$
$\displaystyle=$ $\displaystyle
e^{i\vec{p}\cdot\vec{x}}\frac{1}{\sqrt{(2\pi)^{d-1}}}\,,\qquad,\\{\hat{b}_{\vec{x}},\,\hat{b}^{\dagger}_{\vec{p}}\\}_{-}=e^{-i\vec{p}\cdot\vec{x}}\frac{1}{\sqrt{(2\pi)^{d-1}}}\,,$
(39)

.

The internal space of either fermion or boson fields has the finite number of
”basis vectors”, $2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$, 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
$p^{0}=|\vec{p}|$, belonging to a family $f$ and to a superposition of family
members $m$ applying on the vacuum state $|\psi_{oc}>\,*_{T}\,|0_{\vec{p}}>$
can be written as ([10], Subsect.3.3.2, and the references therein)

$\displaystyle{\bf\hat{b}}^{s\dagger}_{f}(\vec{p})\,$ $\displaystyle=$
$\displaystyle\,\sum_{m}c^{sm}{}_{f}(\vec{p})\,\hat{b}^{\dagger}_{\vec{p}}\,*_{T}\,\hat{b}^{m\dagger}_{f}\,\,,$
(40)

where the vacuum state for fermions $|\psi_{oc}>\,*_{T}\,|0_{\vec{p}}>$
includes both spaces, the internal part, Eq.(16), and the momentum part, Eq.
(2.3) (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 $\hat{b}^{\dagger}_{\vec{p}}$ which pushes the
momentum by an amount $\vec{p}$ 666 The creation operators and their Hermitian
conjugated partners annihilation operators in the coordinate representation
can be read in [10] and the references therein: $\hat{\bf
b}^{s\dagger}_{f}(\vec{x},x^{0})=\sum_{m}\,\hat{b}^{m\dagger}_{f}\,\int_{-\infty}^{+\infty}\,\frac{d^{d-1}p}{(\sqrt{2\pi})^{d-1}}\,c^{ms}{}_{f}\;(\vec{p})\;\hat{b}^{\dagger}_{\vec{p}}\;e^{-i(p^{0}x^{0}-\varepsilon\vec{p}\cdot\vec{x})}$
([10], subsect. 3.3.2., Eqs. (55,57,64) and the references therein).).
The creation operators fulfil the anticommutation relations for the second
quantized fermion fields

$\displaystyle\\{\hat{\bf b}^{s^{\prime}}_{f`}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle\delta^{ss^{\prime}}\delta_{ff^{\prime}}\,\delta(\vec{p}^{\prime}-\vec{p})\,|\psi_{oc}>|0_{\vec{p}}>\,,$
$\displaystyle\\{\hat{\bf b}^{s^{\prime}}_{f`}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle 0\,.\,|\psi_{oc}>|0_{\vec{p}}>\,,$ $\displaystyle\\{\hat{\bf
b}^{s^{\prime}\dagger}_{f^{\prime}}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle 0\,.\,\,|\psi_{oc}>|0_{\vec{p}}>\,,$ $\displaystyle\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle|\psi^{s}_{f}(\vec{p})>\,$ $\displaystyle\hat{\bf
b}^{s}_{f}(\vec{p})\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$ $\displaystyle
0\,.\,\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle|p^{0}|$ $\displaystyle=$
$\displaystyle|\vec{p}|\,.$ (41)

The creation operators $\hat{\bf b}^{s\dagger}_{f}(\vec{p}))$ and their
Hermitian conjugated partners annihilation operators $\hat{\bf
b}^{s}_{f}(\vec{p})$, creating and annihilating the single fermion states,
respectively, fulfil when applying on the vacuum state,
$|\psi_{oc}>*_{T}|0_{\vec{p}}>$, the anticommutation relations for the second
quantized fermions, postulated by Dirac (Ref. [10], Subsect. 3.3.1, Sect. 5).
777 The anticommutation relations of Eq. (41) are valid also if we replace the
vacuum state, $|\psi_{oc}>|0_{\vec{p}}>$, by the Hilbert space of Clifford
fermions generated by the tensor product multiplication, $*_{T_{H}}$, 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 $\hat{\bf
b}^{s\,\dagger}_{f}(\vec{p})$ of any $(s,f,\vec{p})$), Ref. ([10], Sect. 5.).

To write the creation operators for boson fields we must take into account
that boson gauge fields have the space index $\alpha$, describing the $\alpha$
component of the boson field in the ordinary space 888 In the spin-charge-
family theory the Higgs’s scalars origin in the boson gauge fields with the
vector index $(7,8)$, Ref. ([10], Sect. 7.4.1, and the references therein)..
We therefore add the space index $\alpha$ as follows

$\displaystyle{\bf{}^{I}{\hat{\cal A}}^{m\dagger}_{f\alpha}}(\vec{p})\,$
$\displaystyle=$ $\displaystyle\hat{b}^{\dagger}_{\vec{p}}\,*_{T}\,{\cal
C}^{m}{}_{f\alpha}\,{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,\,.$ (42)

We treat free massless bosons of momentum $\vec{p}$ and energy
$p^{0}=|\vec{p}|$ and of particular ”basis vectors” ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$’s which are eigenvectors of all the Cartan subalgebra
members 999 In general the energy eigenstates of bosons are in superposition
of ${\bf{}^{I}{\hat{\cal A}}^{m\dagger}_{f}}$. One example, which uses the
superposition of the Cartan subalgebra eigenstates manifesting the
$SU(3)\times U(1)$ subgroups of the group $SO(6)$, is presented in Fig. 2.,
${\cal C}^{m}{}_{f\alpha}$ carry the space index $\alpha$ of the boson field.
Creation operators operate on the vacuum state
$|\psi_{oc_{ev}}>\,*_{T}\,|0_{\vec{p}}>$ with the internal space part just a
constant, $|\psi_{oc_{ev}}>=$ $|\,1>$, and for a starting single boson state
with a zero momentum from which one obtains the other single boson states with
the same ”basis vector” by the operators $\hat{b}^{\dagger}_{\vec{p}}$ which
push the momentum by an amount $\vec{p}$, making also ${\cal
C}^{m}{}_{f\alpha}$ depending on $\vec{p}$.

For the creation operators for boson fields in a coordinate representation we
find using Eqs. (2.3, 2.3)

$\displaystyle{\bf{}^{I}{\hat{\cal A}}^{m\dagger}_{f\alpha}}(\vec{x},x^{0})$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\,\frac{d^{d-1}p}{(\sqrt{2\pi})^{d-1}}\,{}^{I}{\hat{\cal
A}}^{m\dagger}_{f\alpha}(\vec{p})\,e^{-i(p^{0}x^{0}-\varepsilon\vec{p}\cdot\vec{x})}|_{p^{0}=|\vec{p}|}\,.$
(43)

To understand what new does the Clifford algebra description of the internal
space of fermion and boson fields, Eqs. (42, 43, 40), 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
$\sum_{ab}c^{ab}\omega_{ab\alpha}$ and $\sum_{mf}{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}{\cal C}^{mf}_{\alpha}$, recognizing that ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}{\cal C}^{mf}_{\alpha}$ are eigenstates of the Cartan
subalgebra members, while $\omega_{ab\alpha}$ are not.

The gravity fields, the vielbeins and the two kinds of the spin connection
fields, $f^{a}{}_{\alpha}$, $\omega_{ab\alpha}$, $\tilde{\omega}_{ab\alpha}$,
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 [25].

We therefore need to relate

$\displaystyle\\{\frac{1}{2}\sum_{ab}S^{ab}\,\omega_{ab\alpha}\\}\sum_{m}\beta^{mf}\,\hat{\bf
b}^{m\dagger}_{f}(\vec{p})$ $\displaystyle{\rm relate\,\,to}$
$\displaystyle\\{\sum_{m^{\prime}f^{\prime}}{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f^{\prime}}\,{\cal
C}^{m^{\prime}f^{\prime}}_{\alpha}\\}\sum_{m}\beta^{mf}\,\hat{\bf
b}^{m\dagger}_{f}(\vec{p})\,,$ $\displaystyle\forall f\,{\rm
and}\,\forall\,\beta^{mf}\,,$ $\displaystyle{\bf\cal
S}^{cd}\,\sum_{ab}(c^{ab}{}_{mf}\,\omega_{ab\alpha})$ $\displaystyle{\rm
relate\,\,to}$ $\displaystyle{\bf\cal S}^{cd}\,({}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,{\cal C}^{mf}_{\alpha})\,,$ (44)
$\displaystyle\forall\,(m,f),$ $\displaystyle\forall\,\,{\rm
Cartan\,\,subalgebra\,\,\,member}\,{\bf\cal S}^{cd}\,.$

Let be repeated that ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ are chosen to be
the eigenvectors of the Cartan subalgebra members, Eq. (8). Correspondingly we
can relate a particular ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}{\cal
C}^{mf}_{\alpha}$ with such a superposition of $\omega_{ab\alpha}$’s which is
the eigenvector with the same values of the Cartan subalgebra members as there
is a particular ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}{\cal C}^{mf}_{\alpha}$.
We can do this in two ways:
i. Using the first relation in Eq. (2.3). On the left hand side of this
relation ${S}^{ab}$’s apply on $\hat{b}^{m\dagger}_{f}$ part of $\hat{\bf
b}^{m\dagger}_{f}(\vec{p})$. On the right hand side ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ apply as well on the same ”basis vector”
$\hat{b}^{m\dagger}_{f}$.
ii. Using the second relation, in which ${\bf\cal S}^{cd}$ apply on the left
hand side on $\omega_{ab\alpha}$’s

$\displaystyle\,{\bf\cal S}^{cd}\,\sum_{ab}\,c^{ab}{}_{mf}\,\omega_{ab\alpha}$
$\displaystyle=$
$\displaystyle\sum_{ab}\,c^{ab}{}_{mf}\,i\,(\omega_{cb\alpha}\eta^{ad}-\omega_{db\alpha}\eta^{ac}+\omega_{ac\alpha}\eta^{bd}-\omega_{ad\alpha}\eta^{bc}),$
(45)

on each $\omega_{ab\alpha}$ separately; $c^{ab}{}_{mf}$ are constants to be
determined from the second relation, where on the right hand side of this
relation ${\bf\cal S}^{cd}(=S^{cd}+\tilde{S}^{cd})$ apply on the ”basis
vector” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ of the corresponding gauge
field.

Let us conclude this section by pointing out that either the Clifford odd
”basis vectors” $\hat{b}^{m\dagger}_{f}$ or the Clifford even ”basis vectors”
${}^{i}{\hat{\cal A}}^{m\dagger}_{f},i=(I,II)$ have in any even $d$
$2^{\frac{d}{2}-1}$ $\times 2^{\frac{d}{2}-1}$ members, while
$\omega_{ab\alpha}$ as well as $\tilde{\omega}_{ab\alpha}$ have each for each
$\alpha$ $\frac{d}{2}(d-1)$ members. It is needed to find out what new does
this difference bring into the - unifying theories of the Kaluza-Klein
theories are.

##  3 Short overview and achievements of spin-charge-family theory

The spin-chare-family theory [7, 8, 21, 23, 25, 26, 27, 29, 28, 30] is a kind
of the Kaluza-Klein theories [36, 37, 38, 39, 40, 41, 42, 43, 25] since it is
built on the assumption that the dimension of space-time is $\geq(13+1)$
101010 $d=(13+1)$ 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.,
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 $S^{ab}$ and of $\tilde{S}^{ab}\;$111111If there
are no fermions present both spin connection fields are expressible with
vielbeins ([10], Eq. (103)).).

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 121212 Fermions and bosons internal spaces are assumed to be
superposition of odd products of $\gamma^{a}$’s (fermion fields) or of even
products of $\gamma^{a}$’s (boson fields) what offers the explanation for the
second quantized postulates of Dirac [14]. The ”basis vectors” of the internal
spaces namely determine anticommutativity or commutativity of the
corresponding creation and annihilation operators..

The theory is offering the explanation for all the assumptions of the standard
model, unifying not only charges, but also spins, charges and families, and
consequently offering the explanation for the appearance of families of quarks
and leptons and antiquarks and antileptons, of vector gauge fields, of Higgs’s
scalar field and the Yukawa couplings, for the differences in masses among
quarks and leptons for the matter-antimatter asymmetry in the universe, for
the dark matter, 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 131313The 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 $\gamma^{a}$’s seems very promising
in looking for a new way to second quantization of all fields, with gravity
included, as discussed in this talk.. b. The spontaneous brake of the starting
symmetry which would at low energies manifest the observed almost massless
fermions [37]. 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 ([17], Table III), ([10], Table 6) bringing masses of the scale
$\propto 10^{16}$ GeV or higher to all the vector and scalar gauge fields,
which interact with the condensate [23] is promissing to show the right way
[32].

The scalar fields (scalar fields are the spin connection fields with the space
index $\alpha$ 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. [25, 22] it is
pointed out that the spin connection 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. (1).

The fermion part of the action, Eq. (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 $S^{ab}$ and
$\tilde{S}^{ab}$, Eqs. (8, 2.2, 36). ii. Couplings of fermions to the vector
gauge fields, which are the superposition of gauge fields
$\omega^{st}{}_{\alpha}$, Sect. 6.2 in Ref. [10], with the space index
$\alpha=(0,1,2,3)$ and with the charges determined by the Cartan subalgebra of
$S^{ab}$ and $\tilde{S}^{ab}$ manifesting the symmetry of space $(d-4)$, and
to the scalar gauge fields [21, 24, 27, 29, 34, 35, 7, 8, 46, 47, 48, 22] with
the space index $\alpha\geq 5$ and the charges determined by the Cartan
subalgebra of $S^{ab}$ and $\tilde{S}^{ab}$ (as explained in the case of the
vector gauge fields), and which are superposition of either
$\omega^{st}{}_{\alpha}$ or $\tilde{\omega}^{ab}{}_{\alpha}$,

$\displaystyle{\mathcal{L}}_{f}$ $\displaystyle=$
$\displaystyle\bar{\psi}\gamma^{m}(p_{m}-\sum_{A,i}\;g^{Ai}\tau^{Ai}A^{Ai}_{m})\psi+$
(46) $\displaystyle\\{\sum_{s=7,8}\;\bar{\psi}\gamma^{s}p_{0s}\;\psi\\}+$
$\displaystyle\\{\sum_{t=5,6,9,\dots,14}\;\bar{\psi}\gamma^{t}p_{0t}\;\psi\\}\,,$

where
$p_{0s}=p_{s}-\frac{1}{2}S^{s^{\prime}s"}\omega_{s^{\prime}s"s}-\frac{1}{2}\tilde{S}^{ab}\tilde{\omega}_{abs}$,
$p_{0t}=p_{t}-\frac{1}{2}S^{t^{\prime}t"}\omega_{t^{\prime}t"t}-\frac{1}{2}\tilde{S}^{ab}\tilde{\omega}_{abt}$,
with $p_{0s}=e_{s}^{\alpha}p_{0\alpha}$, $m\in(0,1,2,3)$,
$s\in(7,8),\,(s^{\prime},s")\in(5,6,7,8)$, $(a,b)$ (appearing in
$\tilde{S}^{ab}$) run within either $(0,1,2,3)$ or $(5,6,7,8)$, $t$ runs
$\in(5,\dots,14)$, $(t^{\prime},t")$ run either $\in(5,6,7,8)$ or
$\in(9,10,\dots,14)$. The spinor function $\psi$ represents all family members
of all the $2^{\frac{7+1}{2}-1}=8$ families.

The first line of Eq. (46) determines in $d=(3+1)$ the kinematics and dynamics
of fermion fields coupled to the vector gauge fields [25, 29, 21]. The vector
gauge fields are the superposition of the spin connection fields
$\omega_{stm}$, $m=(0,1,2,3)$, $(s,t)=(5,6,\cdots,13,14)$, and are the gauge
fields of $S^{st}$, Subsect. (6.2.1) of Ref. [10].

The reader can find in Sect. 6 of Ref. [10] a quite detailed overview of the
properties which the massless fermion and boson fields appearing in the simple
starting action, Eq. (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.

Table 5 and Eq. (110) of Ref. [10] explain that the scalar fields with the
space index $\alpha=(7,8)$ carry the weak charge $\tau^{13}=\pm\frac{1}{2}$
and the hyper charge $Y=\mp\frac{1}{2}$, just as assumed by the standard
model.

Masses of families of quarks and leptons are determined by the superposition
of the scalar fields, Eq. (108-120) of Ref. [10], appearing in two groups,
each of them manifesting the symmetry $SU(2)$$\times SU(2)$ $\times U(1)$
141414 The assumption that the symmetry $SO(13,1)$ first breaks into
$SU(3)\times$ $U(1)\times SO(7,1)$ makes that quarks and leptons distinguish
only in the part $SU(3)\times U(1)$, while the $SO(7,1)$ part is identical
separately for quarks and leptons and separately for antiquarks and
antileptons. Table 7 of Ref. [10], presenting one family, which includes
quarks and leptons and antiquarks and antileptons, manifests these properties.
The $\omega_{ab\alpha}$, with the space index $(7,8)$ carry with respect to
the flat index $ab$ only quantum numbers $Q,Y,\tau^{4}$, ($Q$
$(=\tau^{13}+Y)$,$\tau^{13}$ $(=\frac{1}{2}({\cal S}^{56}-{\cal S}^{78})$, $Y$
$(=\tau^{4}+\tau^{23})$ and $\tau^{4}=-\frac{1}{3}({\cal S}^{9\,10}+{\cal
S}^{11\,12}+{\cal S}^{13\,14})$, the flat index $(ab)$ of
$\tilde{\omega}_{ab\alpha}$, with the space index $(7,8)$, includes all
($0,1,\dots,8$) correspondingly forming the symmetry $SU(2)$$\times SU(2)$
$\times U(1)$..

The scalar gauge fields with the space index $(7,8)$ determine correspondingly
the symmetry of mass matrices of quarks and leptons ([10], Eq. (111)) which
appear in two groups as the scalar fields do. In Table 5 in Ref. [10]) the
symmetry $SU(2)\times SU(2)\times U(1)$ 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. [10], enables
to determine parameters of mass matrices from the measured data of the
$3\times 3$ sub mixing matrices and the masses of the measured three families
of quarks and leptons.

Although the known $3\times 3$ submatrix of the unitary $4\times 4$ matrix
enables to determine $4\times 4$ matrix, the measured $3\times 3$ mixing sub
matrix is even for quarks far accurately enough measured, so that we only can
predict the matrix elements of the $4\times 4$ mixing matrix for quarks if
assuming that masses (times $c^{2}$) of the fourth family quarks are heavy
enough, that is above one TeV [44, 47]. The new measurements 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. ([48, 49]) and the overview in Ref.
([10], 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. [47] and a short overview in Subsect. 7.3.1
of [10], 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 [47, 50].

Besides the scalar fields with the space index $\alpha=(7,8)$, which manifest
in $d=(3+1)$ as scalar gauge fields with the weak and hyper charge
$\pm\frac{1}{2}$ and $\mp\frac{1}{2}$, respectively, and which gaining at low
energies constant values make families of quarks and leptons and the weak
gauge field massive, there are in the starting action, Eqs. (1), additional
scalar gauge fields with the space index $\alpha=(9,10,11,12,13,14)$. They are
with respect to the space index $\alpha$ either triplets or antitriplets
causing transitions from antileptons into quarks and from antiquarks into
quarks and back.

Their properties are presented in Ref. [23] and briefly in Table 9 and Fig. 1
of Ref. [10].

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 [32].

In $d=(3+1)$-dimensional space — at low energies — the gauge gravitational
fields manifest as the observed vector gauge fields [25], 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. (1) — reproduce in $d=(3+1)$ the observed properties of quarks and leptons
[21, 23, 25, 26, 27, 29, 28, 30], the observed vector gauge fields, the scalar
field and thge 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.

##  4 Conclusions

In the spin-charge-family theory [7, 8, 21, 23, 25, 26, 27, 29, 28, 30, 10]
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 $\gamma^{a}$’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. (2.1, 2.3).
With respect to the subgroups of the $SO(d-1,1)$ group the Clifford odd ”basis
vectors” appear in the fundamental representations, as illustrated in
Subsects. 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 $\gamma^{a}$’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 [12]. 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. 2.1.

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. 2.2.

There are two kinds of anticommuting algebras [8]: The Grassmann algebra,
offering in $d$-dimensional space $2\,.\,2^{d}$ operators ($2^{d}$
$\,\theta^{a}$’s and $2^{d}$ $\frac{\partial}{\partial\theta_{a}}$’s,
Hermitian conjugated to each other, Eq. (3)), and the two Clifford
subalgebras, each with $2^{d}$ operators named $\gamma^{a}$’s and
$\tilde{\gamma}^{a}$’s, respectively, [8, 19, 20], Eqs. (2-2).

The operators in each of the two Clifford subalgebras appear in two groups of
$2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$ of the Clifford odd operators
(the odd products of either $\gamma^{a}$’s in one subalgebra or of
$\tilde{\gamma}^{a}$’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 $2^{\frac{d}{2}-1}$ irreducible representation each
with the $2^{\frac{d}{2}-1}$ members and the group of their Hermitian
conjugated partners.

There are as well the Clifford even operators (the even products of either
$\gamma^{a}$’s in one subalgebra or of $\tilde{\gamma}^{a}$’s in another
subalgebra) which again appear in two groups of $2^{\frac{d}{2}-1}\times$
$2^{\frac{d}{2}-1}$ members each. In the case of the Clifford even objects the
members of each group of $2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$ members
have the Hermitian conjugated partners within the same group, Subsect. 2.1,
Table 1.

The Grassmann algebra operators are expressible with the operators of the two
Clifford subalgebras and opposite, Eq. (2). The two Clifford subalgebras are
independent of each other, Eq. (2), forming two independent spaces.

Either the Grassmann algebra [18, 13] or the two Clifford subalgebras can be
used to describe the internal space of anticommuting objects, if the
superposition of odd products of operators ($\theta^{a}$’s or $\gamma^{a}$’s,
or $\tilde{\gamma}^{a}$’s) are used to describe the internal space of these
objects. The commuting objects must be superposition of even products of
operators ($\theta^{a}$’s or $\gamma^{a}$’s or $\tilde{\gamma}^{a}$’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 $\gamma^{ab}$’s. The decision that
$\tilde{\gamma}^{a}$’s apply on $\gamma^{a}$ as follows:
$\\{\tilde{\gamma}^{a}B=(-)^{B}\,i\,B\gamma^{a}\\}\,|\psi_{oc}>$, Eq. (7),
(with $(-)^{B}=-1$, if $B$ is a function of an odd products of $\gamma^{a}$’s,
otherwise $(-)^{B}=1$) enables that $2^{\frac{d}{2}-1}$ irreducible
representations of $S^{ab}=\frac{i}{2}\,\\{\gamma^{a}\,,\,\gamma^{b}\\}_{-}$
(each with the $2^{\frac{d}{2}-1}$ members) obtain the family quantum numbers
determined by
$\tilde{S}^{ab}=\frac{i}{2}\,\\{\tilde{\gamma}^{a}\,,\,\tilde{\gamma}^{b}\\}_{-}$.

The decision to use in the spin-charge-family theory in $d=2(2n+1)$, $n\geq 3$
($d\geq(13+1)$ indeed), the superposition of the odd products of the Clifford
algebra elements $\gamma^{a}$’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
$S^{ab}$ and $\tilde{S}^{ab}$, respectively), Eq. (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 ([10] and the references therein) and of the
corresponding vector gauge fields and the Higgs’s scalars included [25], but
also for the appearance of the dark matter [47] in the universe, for the
explanation of the matter/antimatter asymmetry in the universe [23], and for
several other observed phenomena, making several predictions [35, 45, 46, 48].

Recognition that the use of the superposition of the even products of the
Clifford algebra elements $\gamma^{a}$’s to describe the internal space of
boson fields, what appear 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. (2.3)

$\displaystyle\\{\frac{1}{2}\sum_{ab}S^{ab}\,\omega_{ab\alpha}\\}\sum_{m}\beta^{mf}\,\hat{\bf
b}^{m\dagger}_{f}(\vec{p})$ $\displaystyle{\rm relate\,\,to}$
$\displaystyle\\{\sum_{m^{\prime}f^{\prime}}{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f^{\prime}}\,{\cal
C}^{m^{\prime}f^{\prime}}_{\alpha}\\}\sum_{m}\beta^{mf}\,\hat{\bf
b}^{m\dagger}_{f}(\vec{p})\,,$ $\displaystyle\forall f\,{\rm
and}\,\forall\,\beta^{mf}\,,$ $\displaystyle{\bf\cal
S}^{cd}\,\sum_{ab}(c^{ab}{}_{mf}\,\omega_{ab\alpha})$ $\displaystyle{\rm
relate\,\,to}$ $\displaystyle{\bf\cal S}^{cd}\,({}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,{\cal C}^{mf}_{\alpha})\,,$ $\displaystyle\forall\,(m,f),$
$\displaystyle\forall\,\,{\rm Cartan\,\,subalgebra\,\,\,member}\,{\bf\cal
S}^{cd}\,,$

offers the possibility to replace the covariant derivative $p_{0\alpha}$

$p_{0\alpha}=p_{\alpha}-\frac{1}{2}S^{ab}\omega_{ab\alpha}-\frac{1}{2}\tilde{S}^{ab}\tilde{\omega}_{ab\alpha}\quad\quad\quad\quad\;$

in Eq. (1) with

$p_{0\alpha}=p_{\alpha}-\sum_{mf}{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}{}^{I}{\cal
C}^{m}_{f\alpha}-\sum_{mf}{}^{I}{\hat{\widetilde{\cal
A}}}^{m\dagger}_{f}\,{}^{I}{\widetilde{\cal C}}^{m}_{f\alpha}\,,$

where the relation among ${}^{I}{\hat{\widetilde{\cal
A}}}^{m\dagger}_{f}{}^{I}{\widetilde{\cal C}}^{m}_{f\alpha}$ and
${}^{II}{\hat{\widetilde{\cal A}}}^{m\dagger}_{f}\,{}^{II}{\widetilde{\cal
C}}^{m}_{f\alpha}$ with respect to $\omega_{ab\alpha}$ and
$\tilde{\omega}_{ab\alpha}$, not discussed directly in this article, needs
additional study and explanation.

Although the properties of the Clifford odd and even ”basis vectors” and
correspondingly 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.

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 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.

###  4.1 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
facilities 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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