#  How Clifford algebra helps understand second quantized quarks and leptons
and corresponding vector and scalar boson fields, opening a new step beyond
the standard model
Nucl. Phys. B 994 (2023) 116326

N.S. Mankoč Borštnik1
1Department of Physics, University of Ljubljana
SI-1000 Ljubljana, Slovenia

###### Abstract

This article presents the description of the internal spaces of fermion and
boson fields in $d$-dimensional spaces, with the odd and even “basis vectors”
which are the superposition of odd and even products of operators
$\gamma^{a}$. While the Clifford odd “basis vectors” manifest properties of
fermion fields, appearing in families, the Clifford even “basis vectors”
demonstrate properties of the corresponding gauge fields. In $d\geq(13+1)$ the
corresponding creation operators manifest in $d=(3+1)$ the properties of all
the observed quarks and leptons, with the families included, and of their
gauge boson fields, with the scalar fields included, making several
predictions. The properties of the creation and annihilation operators for
fermion and boson fields are illustrated on the case $d=(5+1)$, when $SO(5,1)$
demonstrates the symmetry of $SU(3)\times U(1)$.

Keywords: Second quantization of fermion and boson fields with Clifford
algebra; Beyond the standard model; Kaluza-Klein-like theories in higher
dimensional spaces; Clifford algebra in odd dimensional spaces; Ghosts in
quantum field theories

##  1 Introduction

The standard model (corrected with the right-handed neutrinos) has been
experimentally confirmed without raising any severe doubts so far on its
assumptions, which, however, remain unexplained.

The standard model assumptions have several explanations in the literature,
mostly with several new, not explained assumptions. The most popular are the
grand unifying theories ([1, 2, 3, 4, 5] and many others).

In a long series of works ([6, 7, 8, 9],and the references there in) the
author has found, together with the collaborators ([10, 7, 11, 12, 13, 14] and
the references therein), the phenomenological success with the model named the
spin-charge-family theory with the properties:
a. The internal space of fermions are described by the “basis vectors” which
are superposition of odd products of anti-commuting objects (operators) 111
According to Eq. (2.1) $\\{\gamma^{a},\gamma^{b}\\}_{+}=2\eta^{ab}$ are
anticommuting unless $a=b$. $\gamma^{a}$ (in the sense
$\\{\gamma^{a},\gamma^{b}\\}_{+}=2\eta^{ab}$), Sect. 2.1, in
$d=(13+1)$-dimensional space [13, 14]. Correspondingly the “basis vectors” of
one Lorentz irreducible representation in internal space of fermions, together
with their Hermitian conjugated partners, anti-commute, fulfilling (on the
vacuum state) all the requirements for the second quantized fermion fields
([10, 14] and references therein).
a.i. The second kind of anti-commuting objects, $\tilde{\gamma}^{a}$, Sect.
2.1, equip each irreducible representation of odd “basis vectors” with the
family quantum number [13, 10].
a.ii. Creation operators for single fermion states — which are tensor
products, $*_{T}$, of a finite number of odd “basis vectors” appearing in
$2^{\frac{d}{2}-1}$ families, each family with $2^{\frac{d}{2}-1}$ members,
and the (continuously) infinite momentum/coordinate basis applying on the
vacuum state [13, 14] — inherit anti-commutativity of “basis vectors”.
Creation operators and their Hermitian conjugated partners correspondingly
anti-commute.
a.iii. The Hilbert space of second quantized fermion field is represented by
the tensor products, $*_{T_{H}}$, of all possible numbers of creation
operators, from zero to infinity [14], applying on a vacuum state.
a.iv. Spins from higher dimensions, $d>(3+1)$, described by the eigenvectors
of the superposition of the Cartan subalgebra $S^{ab}$, Table 4, manifest in
$d=(3+1)$ all the charges of the standard model quarks and leptons and
antiquarks and antileptons.
b. In a simple starting action, Eq. (1), massless fermions carry only spins
and interact with only gravity — with the vielbeins and the two kinds of spin
connection fields (the gauge fields of momenta, of
$S^{ab}=\frac{i}{4}(\gamma^{a}\gamma^{b}-\gamma^{b}\gamma^{a})$ and of
$\tilde{S}^{ab}=\frac{1}{4}(\tilde{\gamma}^{a}\tilde{\gamma}^{b}-\tilde{\gamma}^{b}\tilde{\gamma}^{a})$,
respectively 222 If no fermions are present, the two kinds of spin connection
fields are uniquely expressible by the vielbeins.). The starting action
includes only even products of $\gamma^{a}$’s and $\tilde{\gamma^{a}}$’s ([14]
and references therein).
b.i. Gravity — the gauge fields of $S^{ab}$, ($(a,b)=(5,6,....,d)$), with the
space index $m=(0,1,2,3)$ — manifest as the standard model vector gauge fields
[11], with the ordinary gravity included ($(a,b)=(0,1,2,3)$).
b.ii. The scalar gauge fields of $\tilde{S}^{ab}$, and of some of the
superposition of $S^{ab}$, with the space index $s=(7,8)$ manifest as the
scalar Higgs and Yukawa couplings [9, 14, 23], determining mass matrices (of
particular symmetry) and correspondingly the masses of quarks and leptons and
of the weak boson fields after (some of) the scalar fields with the space
index $(7,8)$ gain constant values.
b.iii. The scalar gauge fields of $\tilde{S}^{ab}$ and of $S^{ab}$ with the
space index $s=(9,10,...,14)$ and $(a,b)=(5,6,....,d)$ offer the explanation
for the observed matter/antimatter asymmetry [8, 9, 12, 14] in the universe.
c. The theory predicts at low energy two groups with four families. To the
lower group of four families the so far observed three belong [32, 33, 34, 36,
37], and the stable of the upper four families, the fifth family of (heavy)
quarks and leptons, offers the explanation for the appearance of dark matter.
Due to the heavy masses of the fifth family quarks, the nuclear interaction
among hadrons of the fifth family members is very different than the ones so
far observed [35, 38].
d. The theory offers a new understanding of the second quantized fermion
fields, as mentioned in a. and it is explained in Refs. [13, 23, 14], it also
enables a new understanding of the second quantization of boson fields which
is the main topics of this article [16, 15], both in even dimensional spaces.
d.i. The Clifford odd “basis vectors” appear in $2^{\frac{d}{2}-1}$ families,
each family having $2^{\frac{d}{2}-1}$ members. Their Hermitian conjugated
partners appear in a separate group, Sect. 2.
d.ii. The Clifford even “basis vectors” appear in two groups, each with
$2^{\frac{d}{2}-1}$ $\times 2^{\frac{d}{2}-1}$ members with their Hermitian
conjugated partners within the same group. One group of the Clifford even
“basis vectors” transform, when applying algebraically on the Clifford odd
“basis vector”, this Clifford odd “basis vector” into other members of the
same family. The other group of the Clifford even “basis vectors” transform,
when being applied algebraically by the Clifford odd “basis vector”, this
Clifford odd “basis vector” into the same member of another family; in
agreement with the action, Eq. (1).
d.iii. In odd dimensional spaces, $d=(2n+1)$, the properties of Clifford odd
and Clifford even “basis vectors” differ essentially from their properties in
even dimensional spaces, resembling the ghosts needed to make the
contributions of the Feynman diagrams finite [18].

The theory seems very promising to offer a new insight into the second
quantization of fermion and boson fields and to show the next step beyond the
standard model.

The more work is put into the theory, the more phenomena the theory can
explain.

Other references used a different approach by trying to make the next step
with Clifford algebra to the second quantized fermion, which might also be a
boson field [39, 40].

Let us present a simple starting action of the spin-charge-family theory ([14]
and the references therein) for massless fermions and anti-fermions which
interact with massless gravitational fields only; with vielbeins (the gauge
fields of momenta) and the two kinds of spin connection fields, the gauge
fields of the two kinds of the Lorentz transformations in the internal space
of fermions, of $S^{ab}$ and $\tilde{S}^{ab}$, in $d=2(2n+1)$-dimensional
space

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

Here 333$f^{\alpha}{}_{a}$ 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
$\eta^{ab}=diag\\{1,-1,-1,\cdots,-1\\}$. $f^{\alpha[a}f^{\beta b]}=f^{\alpha
a}f^{\beta b}-f^{\alpha b}f^{\beta a}$. The vielbeins, $f^{a}_{\alpha}$, 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 of the weak boson fields [11, 8, 9, 12] 444
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 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 $\gamma^{0}\gamma^{a}\hat{p}_{0a}$ or as
$\gamma^{0}\gamma^{c}\,S^{ab}\omega_{abc}$ and as
$\gamma^{0}\gamma^{c}\,\tilde{S}^{ab}\tilde{\omega}_{abc}$..

The action, Eq. (1), assumes two kinds of the spin connection gauge fields,
due to two kinds of the operators: $\gamma^{a}$ and $\tilde{\gamma}^{a}$. Let
be pointed out that the description of the internal space of bosons with the
Clifford even “basis vectors” offers as well two kinds of the Clifford even
“basis vectors”, as presented in d.ii..

In Sect. 2 the Grassmann and the Clifford algebras are explained, Subsect.2.1,
and creation and annihilation operators described as tensor products of the
“basis vectors” offering an 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. 2.2, the “basis vectors” are introduced and their properties
presented in even and odd-dimensional spaces, Subsects. 2.2.1, Subsect. 2.2.2,
respectively.

In Subsect. 2.3, the properties of the Clifford odd and even “basis vectors”
are demonstrated in the toy model in $d=(5+1)$.

In Subsect. 2.4, the properties of the creation and annihilation operators for
the second quantized fermion and boson fields in even dimensional spaces are
described.

Sect. 3 presents what the reader could learn new from this article.

In App. B, the answers of the spin-charge-family theory to some of the open
questions of the standard model are discussed.

In App. C, some useful formulas and relations are presented.

In App. D one irreducible representation (one family) of $SO(13,1)$, group,
analysed with respect to $SO(3,1)$, $SU(2)_{I}$, $SU(2)_{II}$, $SU(3)$, and
$U(1)$, representing “basis vectors” of quarks and leptons and antiquarks and
antilepons is discussed.

App. A, suggested by the referee, illustrates on the simplest case $d=(3+1)$
(and $d=(1+1)$; which offers only one “family” of fermions, $d=(3+1)$ has two
families) the properties of the Clifford odd and Clifford even“basis vectors”
describing the internal spaces of fermion and boson fields, explaining in a
pedagogical way in details their construction, manifestation of anti-
commutativity (in the fermion case) and commutativity (in the boson case) of
the tensor product of the “basis vectors” and the basis in ordinary space-
time.

The referee suggested also several footnotes.

##  2 Creation and annihilation operators for fermions and bosons in even and
odd dimensional spaces

Refs. [6, 10, 13, 8, 14] describe the internal space of fermion fields by the
superposition of odd products of $\gamma^{a}$ in even dimensional spaces
($d=2(2n+1)$, or $d=4n$). In any even dimensional space there appear
$2^{\frac{d}{2}-1}$ members of each irreducible representation of $S^{ab}$,
each irreducible representation representing one of $2^{\frac{d}{2}-1}$
families, carrying quantum numbers determined by $\tilde{S}^{ab}$. Their
Hermitian conjugated partners appear in a separate group (not reachable by
either $S^{ab}$ or $\tilde{S}^{ab}$). Since the tensor products, $*_{T}$, of
these Clifford odd “basis vectors” and basis in ordinary momentum or
coordinate space, applying on the vacuum state, fulfil the second quantization
postulates for fermions [19, 20, 21], it is obvious that the
$2^{\frac{d}{2}-1}$ $\times\,\,2^{\frac{d}{2}-1}$ anti-commuting Clifford odd
“basis vectors”, together with their Hermitian conjugated partners,
transferring their anti-commutativity to creation and annihilation operators,
explain the second quantization postulates of Dirac for fermions and their
families [13].

There are, however, the same number of the Clifford even “basis vectors”,
which obviously commute, transferring their commutativity to tensor products,
$*_{T}$, of the Clifford even “basis vectors” and basis in ordinary momentum
or coordinate space.

We shall see in what follows that the Clifford even “basis vectors” appear in
two groups, each with $2^{\frac{d}{2}-1}$ $\times\,\,2^{\frac{d}{2}-1}$
members. The members of each group have their Hermitian conjugated partners
within the same group. As we shall see, one group transforms a particular
family member of a Clifford odd “basis vector” into other members of the same
family, keeping the family quantum number unchanged. The second group
transforms a particular family member of a Clifford odd “basis vector” into
the same member of another family [15]. We shall see that the Clifford even
“basis vectors” of each of the two groups has, in even dimensional spaces, the
properties of the gauge boson fields of the corresponding Clifford odd “basis
vectors”: One group with respect to $S^{ab}$, the other with respect to
$\tilde{S}^{ab}$.

The properties of the Clifford odd and the Clifford even “basis vectors” in
odd dimensional spaces, $d=(2n+1)$, differ essentially from their properties
in even dimensional spaces, as we shall review Ref. [18] in Subsect. 2.2.2.
Although anti-commuting, the Clifford odd “basis vectors” manifest properties
of the Clifford even “basis vectors” in even dimensional spaces. And the
Clifford even “basis vectors”, although commuting, manifest properties of the
Clifford odd “basis vectors” in even dimensional spaces.

###  2.1 Grassmann and Clifford algebras

This part is a short overview of several references, cited in Ref. ([14],
Subsects. 3.2,3.3), also appearing in Ref. [17, 13, 18].

The internal spaces of anti-commuting or commuting second quantized fields can
be described by using either the Grassmann or the Clifford algebras [6, 14]

In Grassmann $d$-dimensional space there are $d$ anti-commuting (operators)
$\theta^{a}$, and $d$ anti-commuting 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)

Making a choice [12]

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

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

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

There 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 $p^{\theta a}=i\,\frac{\partial}{\partial\theta_{a}}$
[6], Eqs. (2.1, 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, 2.1), that they form two anti-commuting
Clifford subalgebras, $\\{\gamma^{a},\tilde{\gamma}^{b}\\}_{+}=0$, Refs. ([14]
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 “anti-commuting
integer spin second quantized fields” and of the “commuting integer spin
second quantized fields” [13, 14], 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, Table 4.

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 [16, 15] 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 $\tilde{\gamma}^{a}$
operate on $\gamma^{a}$, reduces the two Clifford subalgebras, $\gamma^{a}$
and $\tilde{\gamma}^{a}$, to one, to the one described by $\gamma^{a}$ [10, 6,
9, 12]

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

with $(-)^{B}=-1$, if $B$ is (a function of) odd products of $\gamma^{a}$’s,
otherwise $(-)^{B}=1$ [10], the vacuum state $|\psi_{oc}>$ is defined in Eq.
(15) of Subsect. 2.2.

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 odd or even products of
$\gamma^{a}$’s obey the postulates for the second quantized fields for
fermions or bosons, respectively, Sect.2.2.
c. It can be proven that the relations presented in Eq. (2.1) remain valid
also after the postulate of Eq. (7). The proof is presented in Ref. ([14],
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

$\displaystyle S^{03},S^{12},S^{56},\cdots,S^{d-1\;d}\,,$
$\displaystyle\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56},\cdots,\tilde{S}^{d-1\;d}\,,$
$\displaystyle{\cal{\bf
S}}^{ab}=S^{ab}+\tilde{S}^{ab}=i\,(\theta^{a}\frac{\partial}{\partial\theta_{b}}-\theta^{b}\frac{\partial}{\partial\theta_{a}})\,.$
(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 operator $\tilde{\gamma}^{a}$ will from now on be used to
describe the properties of fermion “basis vectors”, determining by
$\tilde{S}^{ab}=\frac{i}{4}(\tilde{\gamma}^{a}\tilde{\gamma}^{b}-\tilde{\gamma}^{b}\tilde{\gamma}^{a})$
the “family” quantum numbers of the irreducible representations of the Lorentz
group in internal space of fermions, $S^{ab}$, and the properties of bosons
“basis vectors” determined by ${\cal S}^{ab}=S^{ab}+\tilde{S}^{ab}$. We shall
see that while the fermion “basis vectors” appear in “families”, the boson
“basis vectors” have no “families” and manifest properties of the gauge fields
of the corresponding fermion fields. In App. A the case of $d=(3+1)$ is
discussed.

$\tilde{\gamma}^{a}$’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 the superposition
of odd products of $\gamma^{a}$’s) form $2^{\frac{d}{2}-1}$ families, with the
quantum number $f$, each family has $2^{\frac{d}{2}-1}$ members, $m$. They
offer the description of the second quantized fermion fields.

The Clifford even “basis vectors” (they are the superposition of 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 $m$, offering 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 following subsection.

###  2.2 “Basis vectors” of fermions and bosons in even and odd dimensional
spaces

This subsection is a short overview of similar sections of several articles of
the author, like [17, 15, 18, 13].

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 555 In Ref. [14], 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 the eigenvectors 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 [6, 10] 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 in even-dimensional spaces $\frac{d}{2}$ members of the Cartan
subalgebra, Eq. (8). In odd-dimensional spaces there are $\frac{d-1}{2}$
members of the Cartan subalgebra.

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

$\displaystyle\stackrel{{\scriptstyle ab}}{{(k)}}:$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\gamma^{a}+\frac{\eta^{aa}}{ik}\gamma^{b})\,,\;\;\;(\stackrel{{\scriptstyle
ab}}{{(k)}})^{2}=0\,,$ $\displaystyle\stackrel{{\scriptstyle ab}}{{[k]}}:$
$\displaystyle=$
$\displaystyle\frac{1}{2}(1+\frac{i}{k}\gamma^{a}\gamma^{b})\,,\;\;\;(\stackrel{{\scriptstyle
ab}}{{[k]}})^{2}=\stackrel{{\scriptstyle ab}}{{[k]}},$ (9)

the relations

$\displaystyle S^{ab}\,\stackrel{{\scriptstyle
ab}}{{(k)}}=\frac{k}{2}\,\stackrel{{\scriptstyle ab}}{{(k)}}\,,\quad$
$\displaystyle\quad\tilde{S}^{ab}\,\stackrel{{\scriptstyle
ab}}{{(k)}}=\frac{k}{2}\,\stackrel{{\scriptstyle ab}}{{(k)}}\,,$
$\displaystyle S^{ab}\,\stackrel{{\scriptstyle
ab}}{{[k]}}=\frac{k}{2}\,\stackrel{{\scriptstyle ab}}{{[k]}}\,,\quad$
$\displaystyle\quad\tilde{S}^{ab}\,\stackrel{{\scriptstyle
ab}}{{[k]}}=-\frac{k}{2}\,\,\stackrel{{\scriptstyle ab}}{{[k]}}\,,$ (10)

with $k^{2}=\eta^{aa}\eta^{bb}$ 666Let us prove one of the relations in Eq.
(2.2): $S^{ab}\,\stackrel{{\scriptstyle
ab}}{{(k)}}=\frac{i}{2}\gamma^{a}\gamma^{b}\frac{1}{2}(\gamma^{a}+\frac{\eta^{aa}}{ik}\gamma^{b})=\frac{1}{2^{2}}\\{-i(\gamma^{a})^{2}\gamma^{b}+i(\gamma^{b})^{2}\gamma^{a}\frac{\eta^{aa}}{ik}\\}=\frac{1}{2}\frac{\eta^{aa}\eta^{bb}}{k}\frac{1}{2}\\{\gamma^{a}+\frac{k^{2}}{\eta^{bb}ik}\gamma^{b}\\}$.
For $k^{2}=\eta^{aa}\eta^{bb}$ the first relation follows., demonstrating that
the eigenvalues of $S^{ab}$ on nilpotents and projectors expressed with
$\gamma^{a}$ differ from the eigenvalues of $\tilde{S}^{ab}$ on nilpotents and
projectors expressed with $\gamma^{a}$, so that $\tilde{S}^{ab}$ can be used
to equip each irreducible representation of $S^{ab}$ with the ”family” quantum
number. 777 The reader can find the proof of Eq. (2.2) also in Ref. [14], App.
(I).

We define in even $d$ the “basis vectors” as algebraic, $*_{A}$, products of
nilpotents and projectors so that each product is an eigenvector of all
$\frac{d}{2}$ Cartan subalgebra members, Eq.(8). Fermion “basis vectors” are
(algebraic, $*_{A}$,) products of odd number of nilpotents; each of them is
the eigenvector of one of the Cartan subalgebra members, and the rest of the
projectors; again is each projector the eigenvector of one of the Cartan
subalgebra members. The boson “basis vectors” are (algebraic, $*_{A}$)
products of an even number of nilpotents and the rest of the projectors. (In
App. A, the reader can find concrete examples.)

It follows that the Clifford odd “basis vectors”, which are the superposition
of odd products of $\gamma^{a}$, must include an odd number of nilpotents, at
least one, while the superposition of an even products of $\gamma^{a}$, that
is Clifford even “basis vectors”, must include an even number of nilpotents or
only projectors.

We shall see that the Clifford odd “basis vectors” have properties appropriate
to describe the internal space of the second quantized fermion fields while
the Clifford even “basis vectors” have properties appropriate to describe the
internal space of the second quantized boson fields.

Taking into account Eq. (2.1) one finds

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

More relations are presented in App. C.

The relations in Eq. (2.2) demonstrate that the properties of “basis vectors”
which include an odd number of nilpotents, differ essentially from the “basis
vectors”, which include an 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 $\tilde{S}^{ac}$ 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 even “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 odd products of $\gamma^{a}$ anti-commute with another group of odd
products of $\gamma^{a}$, the Clifford odd “basis vectors” anti-commute,
manifesting in a tensor product, $*_{T}$, with the basis in ordinary space
(together with the corresponding Hermitian conjugated partners) properties of
the anti-commutation relations postulated by Dirac for the second quantized
fermion fields 888 So far, we multiply nilpotents and projectors, or products
of nilpotents and projectors forming “basis vectors”, among themselves. With
the tensor product, $*_{T}$, we include the basis in ordinary space.. The
creation and annihilation operators, which include the internal space of
fermions and bosons described by “basis vectors”, the anti-commutativity or
commutativity of which determine properties of the “basis vectors”, fulfil the
postulates of the second quantized fermion and boson fields. Basis of ordinary
space commute as presented in Eq.(31). App. (A) discuses the creation and
annihilation operators.
The Clifford even “basis vectors” correspondingly fulfil, in a tensor product,
$*_{T}$, with the basis in ordinary space, 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 ${\bf{\cal S}}^{ab}=S^{ab}+\tilde{S}^{ab}$ equal to either
$\pm 1$ and zero or to $\pm i$ and zero.

In odd-dimensional spaces the “basis vectors” can not be products of only
nilpotents and projections. As we shall see in Subsect. 2.2.2, half of “basis
vectors” can be chosen as products of nilpotents and projectors, the rest can
be obtained from the first half by the application of $S^{0d}$ on the first
half.

We shall demonstrate, shortly overviewing [18], that the second half of the
“basis vectors” have unusual properties: The Clifford odd “basis vectors have
properties of the Clifford even “basis vectors”, the Clifford even “basis
vectors” have properties of the Clifford odd “basis vectors”.

####  2.2.1 Clifford odd and even “basis vectors” in even $d$

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

a. Clifford odd “basis vectors”

This part overviews several papers with the same topic ([14, 18] and
references therein).

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 each projector 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 determine
$2^{\frac{d}{2}-1}$ members of one family, $\tilde{S}^{ab}$’s transform each
member of one family to the same member of the rest of $2^{\frac{d}{2}-1}$
families.

Let us call the Clifford odd “basis vectors” $\hat{b}^{m\dagger}_{f}$, if it
is the $m^{th}$ membership of the family $f$. The Hermitian conjugated partner
of $\hat{b}^{m\dagger}_{f}$ is called
$\hat{b}^{m}_{f}\,(=(\hat{b}^{m\dagger}_{f})^{\dagger}$.

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 $S^{01}$, $S^{03}$,
$\dots,S^{0d},S^{15}$, $\dots,S^{1d},S^{5d}\dots,S^{d-2\,d}$. They are
presented on the left-hand side. Their Hermitian conjugated partners are
presented on the right-hand side. The algebraic product mark $*_{A}$ among
nilpotents and projectors is skipped.

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

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

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad d=4n\,,$
$\displaystyle\hat{b}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,,\qquad\qquad\qquad\qquad\quad\hat{b}^{1}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,$
$\displaystyle\hat{b}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,,\qquad\qquad\qquad\qquad\hat{b}^{2}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(-)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle\cdots\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\cdots\,,$
$\displaystyle\hat{b}^{2^{\frac{d}{2}-1}\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(-)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,,\qquad\hat{b}^{2^{\frac{d}{2}-1}}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(-)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle\cdots\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\cdots\,.$
(13)

The Hermitian conjugated partners of the Clifford odd “basis vectors”
$\hat{b}^{m\dagger}_{1}$, presented in Eq. (2.2.1) on the right-hand side,
follow if all nilpotents $\stackrel{{\scriptstyle ab}}{{(k)}}$ of
$\hat{b}^{m\dagger}_{1}$ 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 $\hat{b}^{1\dagger}_{f}$ for all
families $f$ and then generate all the members $\hat{b}^{m}_{f}$ from
$\hat{b}^{1\dagger}_{f}$ by the application of $S^{ab}$ 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. The same is true within their Hermitian conjugated
partners. Both can be proved by the algebraic multiplication using Eqs. (2.2,
68).

$\displaystyle\hat{b}^{m\dagger}_{f}*_{A}\hat{b}^{m`\dagger}_{f`}$
$\displaystyle=$ $\displaystyle
0\,,\quad\hat{b}^{m}_{f}*_{A}\hat{b}^{m`}_{f`}=0\,,\quad\forall
m,m^{\prime},f,f`\,.$ (14)

When we choose the vacuum state equal to

$\displaystyle|\psi_{oc}>=\sum_{f=1}^{2^{\frac{d}{2}-1}}\,\hat{b}^{m}_{f}{}_{*_{A}}\hat{b}^{m\dagger}_{f}\,|\,1\,>\,,$
(15)

for one of members $m$, which can be anyone of the odd irreducible
representations $f$ it follows that the Clifford odd “basis vectors” obey the
relations

$\displaystyle\hat{b}^{m}_{f}{}_{*_{A}}|\psi_{oc}>$ $\displaystyle=$
$\displaystyle 0.\,|\psi_{oc}>\,,$
$\displaystyle\hat{b}^{m\dagger}_{f}{}_{*_{A}}|\psi_{oc}>$ $\displaystyle=$
$\displaystyle|\psi^{m}_{f}>\,,$
$\displaystyle\\{\hat{b}^{m}_{f},\hat{b}^{m^{\prime}}_{f`}\\}_{*_{A}+}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle 0.\,|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m\dagger}_{f},\hat{b}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}+}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle 0.\,|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m}_{f},\hat{b}^{m^{\prime}\dagger}_{f`}\\}_{*_{A}+}|\psi_{oc}>$
$\displaystyle=$
$\displaystyle\delta^{mm^{\prime}}\,\delta_{ff`}|\psi_{oc}>\,,$ (16)

while the normalization
$<\psi_{oc}|\hat{b}^{m\dagger}_{f}\,*_{A}\,\hat{b}^{m\dagger}_{f}*_{A}|\psi_{oc}>=1$
is used and the anti-commutation 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 for fermions as the
tensor, $*_{T}$, products of “basis vectors” and the basis in ordinary space,
the creation and annihilation operators fulfil Dirac’s anti-commutation
postulates since the “basis vectors” transfer their anti-commutativity to
creation and annihilation operators; the ordinary basis namely commute as
presented in Eqs. (2.4, 2.4). Describing the internal space of fermions with
the Clifford odd “basis vectors”, makes creation operators fulfilling the
Dirac postulates for the second quantized fermion fields: No postulates are
needed. The creation and annihilation operators for fermions and bosons are
discussed in App. A, in the part with the title “Creation and annihilation
operators”.

It turns out, therefore, that not only the Clifford odd “basis vectors” offer
the description of the internal space of fermions, they explain the second
quantization postulates for fermions as well.

Table 1, presented in Subsect. 2.3, illustrates the properties of the Clifford
odd “basis vectors” on the case of $d=(5+1)$.

b. Clifford even “basis vectors”

This part proves that the Clifford even “basis vectors” are in even-
dimensional spaces offering the description of the internal spaces of boson
fields — the gauge fields of the corresponding Clifford odd “basis vectors”:
It is a new recognition, offering a new understanding of the second quantized
fermion and boson fields [15].

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 each projector is chosen to be the “eigenstate” of one of the members of
the Cartan subalgebra of the Lorentz algebra, ${\bf{\cal
S}}^{ab}=S^{ab}+\tilde{S}^{ab}$, Eq. (8). Correspondingly the “basis vectors”
are the eigenstates of all the members of the Cartan subalgebra of the Lorentz
algebra.

The Clifford even “basis vectors” appear in two groups, each group has
$2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$ members. The members of one
group can not be reached from the members of another group by either
$S^{ab}$’s or $\tilde{S}^{ab}$’s or both.

$S^{ab}$ and $\tilde{S}^{ab}$ generate from the starting “basis vector” of
each group all the $2^{\frac{d}{2}-1}\times$ $2^{\frac{d}{2}-1}$ members. Each
group contains the Hermitian conjugated partner of any member;
$2^{\frac{d}{2}-1}$ members of each group are products of only (self adjoint)
projectors.

Let us call the Clifford even “basis vectors” ${}^{i}\hat{\cal
A}^{m\dagger}_{f}$, where $i=(I,II)$ denotes the two groups of Clifford even
“basis vectors”, while $m$ and $f$ determine membership of “basis vectors” in
any of the two groups, $I$ or $II$.

$\displaystyle d$ $\displaystyle=$ $\displaystyle 2(2n+1)$
$\displaystyle{}^{I}\hat{{\cal A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{[+]}}\,,\qquad$ $\displaystyle{}^{II}\hat{{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle{}^{I}\hat{{\cal A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{[+]}}\,,$
$\displaystyle{}^{I}\hat{{\cal A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{(-)}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{(-)}}\,,$
$\displaystyle\dots\qquad$ $\displaystyle\dots$ $\displaystyle d$
$\displaystyle=$ $\displaystyle 4n$ $\displaystyle{}^{I}\hat{{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\cdots\stackrel{{\scriptstyle d-1\,d}}{{(+)}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(+)}}\,,$ $\displaystyle{}^{I}\hat{{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[-i]}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(+)}}\,,\qquad$ $\displaystyle{}^{II}\hat{{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[-i]}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-1\,d}}{{(+)}}\,,$ $\displaystyle{}^{I}\hat{{\cal
A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[-]}}\,,\qquad$
$\displaystyle{}^{II}\hat{{\cal A}}^{3\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\;\stackrel{{\scriptstyle d-1\,d}}{{[-]}}\,$
$\displaystyle\dots\qquad$ $\displaystyle\dots$ (17)

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. 2.3, illustrates properties of the Clifford odd
and Clifford even “basis vectors” on the case of $d=(5+1)$. Looking at this
case it is easy to evaluate properties of either even or odd “basis vectors”.
We shall discuss in this subsection the general case by carefully inspecting
properties of both kinds of “basis vectors”.

The Clifford even “basis vectors” belonging to two different groups are
orthogonal due to the fact that they differ in the sign of one nilpotent or
one projector, or the algebraic product of a member of one group with a member
of another group gives zero according to the first two lines of Eq. (68):
$\stackrel{{\scriptstyle ab}}{{(k)}}\stackrel{{\scriptstyle ab}}{{[k]}}=0$,
$\stackrel{{\scriptstyle ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{(-k)}}=0$,
$\stackrel{{\scriptstyle ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{[-k]}}=0$.

$\displaystyle{}^{I}{\hat{\cal A}}^{m\dagger}_{f}*_{A}{}^{II}{\hat{\cal
A}}^{m\dagger}_{f}$ $\displaystyle=$ $\displaystyle 0={}^{II}{\hat{\cal
A}}^{m\dagger}._{f}*_{A}{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,.$ (18)

The members of each of these two groups have the property

$\displaystyle{}^{i}{\hat{\cal A}}^{m\dagger}_{f}\,*_{A}\,{}^{i}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\rightarrow\left\\{\begin{array}[]{r}{}^{i}{\hat{\cal
A}}^{m\dagger}_{f`}\,,i=(I,II)\\\ {\rm or\,zero}\,.\end{array}\right.$ (21)

For a chosen ($m,f,f`$) there is only one $m^{\prime}$ (out of
$2^{\frac{d}{2}-1}$) which gives nonzero contribution.

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 non zero contribution, “scatter” into the
third one ${}^{i}{\hat{\cal A}}^{m\dagger}_{f`}$, for $i=(I,II)$.

Let us treat a particular case in $d=2(2n+1)$-dimensional internal space,
like:
${}^{I}{\hat{\cal A}}^{m\dagger}_{f}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(+)}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}}*_{A}$
$\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[-]}}\stackrel{{\scriptstyle 56}}{{(-)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(-)}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}}$
$\rightarrow\;\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{[+]}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[+]}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}}$, what follows if the
first two lines of Eq. (68) are taken into account. The eigenvalues of the
Cartan subalgebra members of $\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(+)}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}}$ are
($i,1,1,1,\dots,1,0$), of $\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(-)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(-)}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}}$ are
($0,0,-1,-1,\dots,-1,0$), and of $\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{[+]}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[+]}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}}$ are
($i,1,0,0,\dots,0,0$). The sum of the Cartan subalgebra eigenvalues of the two
scattered Clifford even “basis vectors” leads to the eigenvalues
($i,1,0,0,\dots,0,0$) of the third Clifford even “basis vector”.

It remains to evaluate the algebraic application, $*_{A}$, of the Clifford
even “basis vectors” ${}^{I,II}{\hat{\cal A}}^{m\dagger}_{f}$ on the Clifford
odd “basis vectors” $\hat{b}^{m^{\prime}\dagger}_{f`}$. One finds, taking into
account Eq. (68), for ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$

$\displaystyle{}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,\hat{b}^{m^{\prime}\dagger}_{f`}\rightarrow\left\\{\begin{array}[]{r}\hat{b}^{m\dagger}_{f`}\,,\\\
{\rm or\,zero}\,,\end{array}\right.$ (24)

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. (24). All the rest
($2^{\frac{d}{2}-1}\times\,(2^{\frac{d}{2}-1}-1)$) Clifford odd “basis
vectors” give zero contributions. Or equivalently, there are
$2^{\frac{d}{2}-1}$ pairs of quantum numbers $(f,m^{\prime})$ for which
$\hat{b}^{m\dagger}_{f`}\neq 0$.

Taking into account Eq. (68) one finds

$\displaystyle\hat{b}^{m\dagger}_{f}*_{A}{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}=0\,,\quad\forall(m,m`,f,f`)\,.$ (25)

Let us treat a particular case in $d=2(2n+1)$-dimensional space:
${}^{I}{\hat{\cal A}}^{m\dagger}_{f}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(+)}}\stackrel{{\scriptstyle d-1\,d}}{{[+]}})*_{A}$
$\hat{b}^{m^{\prime}\dagger}_{f`}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(-)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(-)}}\stackrel{{\scriptstyle
d-1\,d}}{{(+)}})\rightarrow\hat{b}^{m\dagger}_{f`}(\equiv\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[+]}}\stackrel{{\scriptstyle d-1\,d}}{{(+)}}$. The
${\cal{\bf{S}}}^{ab}$ (meaning
${\cal{\bf{S}}}^{03},{\cal{\bf{S}}}^{12},{\cal{\bf{S}}}^{56},\dots\cal{\bf{S}}^{d-1\,d}$)
say for the above case that the boson field with the quantum numbers
$(i,1,1,\dots,1,0)$ when “scattering” on the fermion field with the Cartan
subalgebra quantum numbers ($S^{03},S^{1,2},S^{56}\dots
S^{d-3\,d-2},S^{d-1\,d}$)
$=(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2},\frac{1}{2})$,
and the family quantum numbers
$(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2},\dots,$ $-\frac{1}{2},\frac{1}{2})$
transfers to the fermion field its quantum numbers $(i,1,1,\dots,1,0)$,
transforming fermion family members quantum numbers to
$(\frac{i}{2},\frac{1}{2},\frac{1}{2},\dots,\frac{1}{2},\frac{1}{2})$, leaving
family quantum numbers unchanged.

Eqs. (24, 25) demonstrates that ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$,
applying on $\hat{b}^{m^{\prime}\dagger}_{f`}$, transforms the Clifford odd
“basis vector” into another Clifford odd “basis vector” of the same family,
transferring to the Clifford odd “basis vector” integer spins, or gives zero.

For “scattering” the Clifford even “basis vectors” ${}^{II}{\hat{\cal
A}}^{m\dagger}_{f}$ on the Clifford odd “basis vectors”
$\hat{b}^{m^{\prime}\dagger}_{f`}$ it follows

$\displaystyle{}^{II}{\hat{\cal
A}}^{m\dagger}_{f}\,*_{A}\,\hat{b}^{m^{\prime}\dagger}_{f`}=0\,,\;\;\forall(m,m^{\prime},f,f`)\,,$
(26)

while we get

$\displaystyle\hat{b}^{m\dagger}_{f}*_{A}{}^{II}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}\,\rightarrow\left\\{\begin{array}[]{r}\hat{b}^{m\dagger}_{f``}\,,\\\
{\rm or\,zero}\,,\end{array}\right.$ (29)

For each $\hat{b}^{m\dagger}_{f}$ there are among $2^{\frac{d}{2}-1}$
$\times\;2^{\frac{d}{2}-1}$ members of the Clifford even “basis vectors”
(describing the internal space of boson fields) , ${}^{II}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f`}$, $2^{\frac{d}{2}-1}$ members (with appropriate
$f`$ and $m^{\prime}$) fulfilling the relation of Eq. (29) while $f``$ runs
over $(1-2^{\frac{d}{2}-1})$.

All the rest ($2^{\frac{d}{2}-1}\times\;(2^{\frac{d}{2}-1}-1)$) Clifford
even“basis vectors” give zero contributions.

Or equivalently, there are $2^{\frac{d}{2}-1}$ pairs of quantum numbers
$(f^{\prime},m^{\prime})$ for which $\hat{b}^{m\dagger}_{f}$ and
${}^{II}{\hat{\cal A}}^{m^{\prime}\dagger}_{f`}$ give non zero contribution.
2mm  Let us treat a particular case in $d=2(2n+1)$-dimensional space:
$\hat{b}^{m\dagger}_{f}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(-)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(-)}}\stackrel{{\scriptstyle d-1\,d}}{{(+)}})*_{A}$
${}^{II}{\hat{\cal A}}^{m`\dagger}_{f`}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{(+)}}\stackrel{{\scriptstyle d-1\,d}}{{[-]}})\rightarrow$
$\hat{b}^{m\dagger}_{f`^{\prime}}(\equiv\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}$ $\stackrel{{\scriptstyle
56}}{{[-]}}\dots\stackrel{{\scriptstyle
d-3\,d-2}}{{[-]}}\stackrel{{\scriptstyle d-1\,d}}{{(+)}})$ When the fermion
field with the Cartan subalgebra family members quantum numbers
($S^{03},S^{12},S^{56}\dots$ $S^{d-3\,d-2},S^{d-1\,d}$)
$=(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2},\frac{1}{2})$ and
family quantum numbers
($\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56}\dots\tilde{S}^{d-3\,d-2},\tilde{S}^{d-1\,d}$)
$(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2},\frac{1}{2})$
“absorbs” a boson field with the Cartan subalgebra quantum numbers
${\cal{\bf{S}}}^{ab}$ (meaning
${\cal{\bf{S}}}^{03},{\cal{\bf{S}}}^{12},{\cal{\bf{S}}}^{56},\dots\cal{\bf{S}}^{d-1\,d}$)
equal to $(i,1,1,\dots,1,0)$, the fermion field changes the family quantum
numbers
($\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56}\dots\tilde{S}^{d-3\,d-2},\tilde{S}^{d-1\,d}$)
to $(\frac{i}{2},\frac{1}{2},\frac{1}{2},\dots,\frac{1}{2},\frac{1}{2})$,
keeping family members quantum numbers unchanged.

Eqs. (26, 29) demonstrate that ${}^{II}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f^{\prime}}$, “absorbed” by $\hat{b}^{m\dagger}_{f}$,
transforms the Clifford odd “basis vector” into the Clifford odd “basis
vector” of the same family member and of another family, or gives zero.

The Clifford even “basis vectors” offer the description of the internal space
of the gauge fields of the corresponding fermion fields.

While the Clifford odd “basis vectors”, $\hat{b}^{m\dagger}_{f}$, offer the
description of the internal space of the second quantized anti-commuting
fermion fields, appearing in families, the Clifford even “basis vectors”,
${}^{I,II}{\hat{\cal A}}^{m\dagger}_{f}$, offer the description of the
internal space of the second quantized commuting boson fields, having no
families and appearing in two groups. One of the two groups, ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$, transferring their integer quantum numbers to the
Clifford odd “basis vectors”, $\hat{b}^{m\dagger}_{f}$, changes the family
members quantum numbers leaving the family quantum numbers unchanged. The
second group, transferring their integer quantum numbers to the Clifford odd
“basis vector”, changes the family quantum numbers leaving the family members
quantum numbers unchanged.

Both groups of Clifford even “basis vectors” manifest as the gauge fields of
the corresponding fermion fields: One concerning the family members quantum
numbers, the other concerning the family quantum numbers.

We shall discus properties of the Clifford even and odd “basis vectors” for
$d=(5+1)$\- dimensional internal spaces in Subsect. 2.3 in more details.

####  2.2.2 Clifford odd and even “basis vectors” in $d$ odd

Let us shortly overview properties of the fermion and boson “basis vectors” in
odd dimensional spaces, as presented in Ref. [18], Subsect. 2.2.

In even dimensional spaces the Clifford odd “basis vectors” fulfil the
postulates for the second quantized fermion fields, Eq. (2.2.1), and the
Clifford even ”basis vectors” have the properties of the internal spaces of
their corresponding gauge fields, Eqs. (21, 24, 29). In odd dimensional
spaces, the Clifford odd and even ”basis vectors” have unusual properties
resembling properties of the internal spaces of the Faddeev–Popov ghosts, as
we described in [18].

In $d=(2n+1)$-dimensional cases, $n=1,2,\dots$, half of the “basis vectors”,
$2^{\frac{2n}{2}-1}$ $\times\,2^{\frac{2n}{2}-1}$, can be taken from the
$2n$-dimensional part of space, presented in Eqs. (2.2.1, 2.2.1, 2.2.1, 21).

The rest of the “basis vectors” in odd dimensional spaces,
$2^{\frac{2n}{2}-1}$ $\times\,2^{\frac{2n}{2}-1}$, follow if $S^{0\,2n+1}$ is
applied on these half of the “basis vectors”. Since $S^{0\,2n+1}$ are Clifford
even operators, they do not change the oddness or evenness of the “basis
vectors”.

For the Clifford odd “basis vectors”, the $2^{\frac{d-1}{2}-1}$ members
appearing in $2^{\frac{d-1}{2}-1}$ families and representing the part which is
the same as in even, $d=2n$, dimensional space are present on the left-hand
side of Eq. (2.2.2), the part obtained by applying $S^{0\,2n+1}$ on the one of
the left-hand side is presented on the right hand side. Below the “basis
vectors” and their Hermitian conjugated partners are presented.

$\displaystyle d=$ $\displaystyle 2(2n+1)+1\,$
$\displaystyle\hat{b}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-2\,d-1}}{{(+)}}\,,\quad$
$\displaystyle\hat{b}^{1\dagger}_{2^{\frac{d-1}{2}-1}+1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-2\,d-1}}{{(+)}}\gamma^{d}\,,$
$\displaystyle\cdots\quad$ $\displaystyle\cdots$
$\displaystyle\hat{b}^{2^{\frac{d-1}{2}-1}\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle d-2\,d-1}}{{[-]}}\;,\quad$
$\displaystyle\hat{b}^{2^{\frac{d-1}{2}-1}\dagger}_{2^{\frac{d-1}{2}-1}+1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}}\dots\stackrel{{\scriptstyle d-2\,d-1}}{{[-]}}\;\gamma^{d}\,,$
$\displaystyle\cdots\quad$ $\displaystyle\cdots\,,$ $\displaystyle\cdots\,,$
$\displaystyle\hat{b}^{1}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(-)}}\cdots\stackrel{{\scriptstyle d-2\,d-1}}{{(-)}}\,,\quad$
$\displaystyle\hat{b}^{1}_{2^{\frac{d-1}{2}-1}+1}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(-)}}\cdots\stackrel{{\scriptstyle d-2\,d-1}}{{(-)}}\gamma^{d}\,,$
$\displaystyle\cdots\quad$ $\displaystyle\cdots\,.$ (30)

The application of $S^{0d}$ or $\tilde{S}^{0d}$ on the left-hand side of the
“basis vectors” (and the Hermitian conjugated partners of both) generate the
whole set of $2\times 2^{d-2}$ members of the Clifford odd “basis vectors” and
their Hermitian conjugated partners in $d=(2n+1)$\- dimensional space
appearing on the left-hand side and the right-hand sides of Eq. (2.2.2).

It is not difficult to see that $\hat{b}^{m\dagger}_{2^{\frac{d-1}{2}-1}+k}$
and $\hat{b}^{m^{\prime}}_{2^{\frac{d-1}{2}-1}+k^{\prime}}$ on the right-hand
side of Eq. (2.2.2) obtain properties of the two groups (they are orthogonal
to each other; the algebraic products, $*_{A}$, of a member from one group,
and any member of another group give zero) with the Hermitian conjugated
partners within the same group; they have properties of the Clifford even
“basis vectors” from the point of view of the Hermiticity property: The
operators $\gamma^{a}$ are up to a constant the self-adjoint operators, while
$S^{0d}$ transform one nilpotent into a projector.

$S^{ab}$ do not change the Clifford oddness of $\hat{b}^{m\dagger}_{f}$, and
$\hat{b}^{m}_{f}$; $\hat{b}^{m\dagger}_{f}$ remain to be Clifford odd objects,
however, with the properties of boson fields.

Let us find the Clifford even “basis vectors” in odd dimensional space
$d=2(2n+1)+1$.

$\displaystyle d=$ $\displaystyle 2(2n+1)+1\,$ $\displaystyle{}^{I}{\bf{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-2\,d-1}}{{[+]}}\,,\quad$ $\displaystyle{}^{I}{\bf{\cal
A}}^{1\dagger}_{2^{\frac{d-1}{2}-1}+1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-2\,d-1}}{{[+]}}\gamma^{d}\,,$
$\displaystyle\cdots\quad$ $\displaystyle\cdots$ $\displaystyle{}^{I}{\bf{\cal
A}}^{2^{\frac{d-1}{2}-1}\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{[-]}}\dots\stackrel{{\scriptstyle d-2\,d-1}}{{[+]}}\;,\quad$
$\displaystyle{}^{I}{\bf{\cal
A}}^{2^{\frac{d-1}{2}-1}\dagger}_{2^{{d-1}{2}-1}+1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{[-]}}\dots\stackrel{{\scriptstyle d-2\,d-1}}{{[+]}}\;\gamma^{d}\,,$
$\displaystyle\cdots\quad$ $\displaystyle\cdots\,,$ $\displaystyle\cdots\quad$
$\displaystyle\cdots$ $\displaystyle{}^{II}{\bf{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}\cdots\stackrel{{\scriptstyle
d-2\,d-1}}{{[+]}}\,,\quad$ $\displaystyle{}^{II}{\bf{\cal
A}}^{1\dagger}_{2^{\frac{d-1}{2}-1}+1}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(+)}}\cdots\stackrel{{\scriptstyle d-2\,d-1}}{{[+]}}\gamma^{d}\,,$
$\displaystyle\cdots\quad$ $\displaystyle\cdots\,.$ (31)

The right hand side of Eq. (2.2.2), although anti-commuting, is resembling the
properties of the Clifford even “basis vectors” on the left hand side of Eq.
(2.2.2), while the right-hand side of Eq. (2.2.2), although commuting,
resembles the properties of the Clifford odd “basis vectors”, from the left
hand side of Eq. (2.2.2): $\gamma^{a}$ are up to a constant the self adjoint
operators, while $S^{0d}$ transform one nilpotent into a projector (or one
projector into a nilpotent). However, $S^{ab}$ do not change Clifford evenness
of ${}^{I}{\bf{\cal A}}^{m\dagger}_{f},i=(I,II)$.

For illustration let me copy the special case for $d=(4+1)$ from
Subsect.3.2.2. of Ref. [18].

$\displaystyle d=$ $\displaystyle 4+1\,$ $\displaystyle{\rm Clifford\;odd}$
$\displaystyle\hat{b}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[+]}}\,,\;\,\hat{b}^{1\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\,,$
$\displaystyle\hat{b}^{1\dagger}_{3}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[+]}}\gamma^{5}\,,\;\,\hat{b}^{1\dagger}_{4}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\gamma^{5}\,,$
$\displaystyle\hat{b}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(-)}}\,,\;\,\hat{b}^{2\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{[-]}}\,,$
$\displaystyle\hat{b}^{2\dagger}_{3}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(-)}}\gamma^{5}\,,\;\,\hat{b}^{2\dagger}_{4}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\gamma^{5}\,,$
$\displaystyle\hat{b}^{1}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[+]}}\,,\;\,\hat{b}^{1}_{2}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(-)}}\,,$
$\displaystyle\hat{b}^{1}_{3}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[+]}}\gamma^{5}\,,\;\,\hat{b}^{1}_{4}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\gamma^{5}\,,$
$\displaystyle\hat{b}^{2}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(+)}}\,,\;\,\hat{b}^{2}_{2}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[-]}}\,,$
$\displaystyle\hat{b}^{2}_{3}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}\gamma^{5}\,,\;\,\hat{b}^{2}_{4}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\gamma^{5}\,,$
$\displaystyle{\rm Clifford\;even}$ $\displaystyle{}^{I}{\bf{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[+]}}\,,\;\,{}^{I}{\bf{\cal A}}^{1\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\,,$
$\displaystyle{}^{I}{\bf{\cal A}}^{1}_{3}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[+]}}\gamma^{5}\,,\;\,{}^{I}{\bf{\cal
A}}^{1}_{4}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(+)}}\gamma^{5}\,,$ $\displaystyle{}^{I}{\bf{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(-i)}}\,,\;\,{}^{I}{\bf{\cal A}}^{2\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\,,$
$\displaystyle{}^{I}{\bf{\cal A}}^{2}_{3}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{(-)}}\gamma^{5}\,,\;\,{}^{I}{\bf{\cal
A}}^{2}_{4}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[-]}}\,\gamma^{5}\,,$ $\displaystyle{}^{II}{\bf{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[+]}}\,,\;\,{}^{II}{\bf{\cal A}}^{1\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\,,$
$\displaystyle{}^{II}{\bf{\cal A}}^{1\dagger}_{3}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[+]}}\gamma^{5}\,,\;\,{}^{II}{\bf{\cal
A}}^{1\dagger}_{4}=\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{(+)}}\gamma^{5}\,,$ $\displaystyle{}^{II}{\bf{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(-)}}\,,\;\,{}^{II}{\bf{\cal A}}^{2\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\,,$
$\displaystyle{}^{II}{\bf{\cal A}}^{2\dagger}_{3}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(-)}}\gamma^{5}\,,\;\,{}^{II}{\bf{\cal
A}}^{2\dagger}_{4}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[-]}}\,\gamma^{5}\,.$ (32)

It can clearly be seen that the left-hand side of the Clifford odd “basis
vectors” and the right-hand side of the Clifford even “basis vectors”,
although the former are the Clifford odd objects and the latter are Clifford
even objects, have similar properties [18].

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

Subsect. 2.3 demonstrates the properties of the Clifford odd and even “basis
vectors” in the special case when $d=(5+1)$ to clear up the relations of the
Clifford odd and even “basis vectors” to fermion and boson fields,
respectively.

Table 1 presents the $64\,(=2^{d=6})$ “eigenvectors” of the Cartan subalgebra
members of the Lorentz algebra, $S^{ab}$ and ${\bf\cal{S}}^{ab}$, Eq. (8).

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, either of three or one. They appear 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 mutually orthogonal (which can be checked by using
Eq. (68)); $\hat{b}^{m\dagger}_{f}*_{A}\hat{b}^{m^{\prime}\dagger}_{f`}=0$ for
all $(m,m^{\prime},f,f`)$. Equivalently,
$\hat{b}^{m}_{f}*_{A}\hat{b}^{m^{\prime}}_{f`}=0$ for all
$(m,m^{\prime},f,f`)$. The “basis vectors” and their Hermitian conjugated
partners are normalized as

$\displaystyle<\psi_{oc}|\hat{b}^{m}_{f}*_{A}\hat{b}^{m^{\prime}\dagger}_{f`}|\psi_{oc}>=\delta^{mm^{\prime}}\delta_{ff`}\,,$
(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 more extended 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. [14].

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. (68), that the algebraic
product ${}^{I}{\bf{\cal A}}^{m\dagger}_{f}*_{A}$ ${}^{II}{\bf{\cal
A}}^{m^{\prime}\dagger}_{f`}=0={}^{II}{\bf{\cal A}}^{m\dagger}_{f}*_{A}$
${}^{I}{\bf{\cal A}}^{m^{\prime}\dagger}_{f`},\forall\,(m,m^{\prime},f.f`)$,
Eq. (18). An overview of the Clifford even “basis vectors” and their Hermitian
conjugated partners for the case $d=(5+1)$ can be found in Ref. [15].

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 (one or three) and projectors (two or none). 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 appearing above each “family”, the quantum numbers of the family 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 group has their Hermitian conjugated partners within its 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. To find the quantum numbers of $({\cal{\bf S}}^{03},{\cal{\bf S}}^{12},{\cal{\bf S}}^{56})$ one has to take into account that ${\cal{\bf S}}^{ab}$ $=S^{ab}+\tilde{S}^{ab}$.  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
${}^{\prime\prime}basis\,vectors^{\prime\prime}$ | $m$ | $f=1$ | $f=2$ | $f=3$ | $f=4$ |  |  |
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$ | $\rightarrow$ | $(\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $(\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $S^{03}$ | $S^{12}$ | $S^{56}$
|  |  |  |  |  |  |  |
$odd\,I\;{\hat{b}}^{m\dagger}_{f}$ | $1$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle 56}}{{[+]}}$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle 56}}{{(+)}}$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{[+]}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
| $2$ | $[-i](-)[+]$ | $(-i)(-)(+)$ | $(-i)[-][+]$ | $[-i][-](+)$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $3$ | $[-i][+](-)$ | $(-i)[+][-]$ | $(-i)(+)(-)$ | $[-i](+)[-]$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $4$ | $(+i)(-)(-)$ | $[+i](-)[-]$ | $[+i][-](-)$ | $(+i)[-][-]$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
|  |  |  |  |  |  |  |
$(S^{03},S^{12},S^{56})$ | $\rightarrow$ | $(-\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $(\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $(\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $\tilde{S}^{03}$ | $\tilde{S}^{12}$ | $\tilde{S}^{56}$
|  | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ |  |  |
|  |  |  |  |  |  |  |
$odd\,II\;{\hat{b}}^{m}_{f}$ | $1$ | $(-i)[+][+]$ | $[+i][+](-)$ | $[+i](-)[+]$ | $(-i)(-)(-)$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
| $2$ | $[-i](+)[+]$ | $(+i)(+)(-)$ | $(+i)[-][+]$ | $[-i][-](-)$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $3$ | $[-i][+](+)$ | $(+i)[+][-]$ | $(+i)(-)(+)$ | $[-i](-)[-]$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $4$ | $(-i)(+)(+)$ | $[+i](+)[-]$ | $[+i][-](+)$ | $(-i)[-][-]$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
|  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$ | $\rightarrow$ | $(-\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $(\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $(\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $S^{03}$ | $S^{12}$ | $S^{56}$
|  | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ |  |  |
|  |  |  |  |  |  |  |
$even\,I\;{}^{I}{\cal A}^{m}_{f}$ | $1$ | $[+i](+)(+)$ | $(+i)[+](+)$ | $[+i][+][+]$ | $(+i)(+)[+]$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
| $2$ | $(-i)[-](+)$ | $[-i](-)(+)$ | $(-i)(-)[+]$ | $[-i][-][+]$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $3$ | $(-i)(+)[-]$ | $[-i][+][-]$ | $(-i)[+](-)$ | $[-i](+)(-)$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $4$ | $[+i][-][-]$ | $(+i)(-)[-]$ | $[+i](-)(-)$ | $(+i)[-](-)$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$
|  |  |  |  |  |  |  |
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})$ | $\rightarrow$ | $(\frac{i}{2},\frac{1}{2},\frac{1}{2})$ | $(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ | $(\frac{i}{2},-\frac{1}{2},-\frac{1}{2})$ | $(-\frac{i}{2},\frac{1}{2},-\frac{1}{2})$ | $S^{03}$ | $S^{12}$ | $S^{56}$
|  | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ | $\stackrel{{\scriptstyle 03}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 12}}{{\;\,}}\;\;\,\stackrel{{\scriptstyle 56}}{{\;\,}}$ |  |  |
|  |  |  |  |  |  |  |
$even\,II\;{}^{II}{\cal A}^{m}_{f}$ | $1$ | $[-i](+)(+)$ | $(-i)[+](+)$ | $[-i][+][+]$ | $(-i)(+)[+]$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$
| $2$ | $(+i)[-](+)$ | $[+i](-)(+)$ | $(+i)(-)[+]$ | $[+i][-][+]$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$
| $3$ | $(+i)(+)[-]$ | $[+i][+][-]$ | $(+i)[+](-)$ | $[+i](+)(-)$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$
| $4$ | $[-i][-][-]$ | $(-i)(-)[-]$ | $[-i](-)(-)$ | $(-i)[-](-)$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$

While the Clifford odd “basis vectors” are (chosen to be) left handed,
$\Gamma^{(5+1)}=-1$, their Hermitian conjugated partners have opposite
handedness, Eq. 66 in App. C 999 Let us check the hadedness of the chosen
representation:
$\Gamma^{5+1}\hat{b}^{1\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}})=\sqrt{(-1)^{5}}i^{3}(\frac{2}{i})^{3}S^{03}S^{12}S^{56}(\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}})=\frac{i^{4}2^{3}}{i^{3}}\frac{i}{2}\frac{1}{2}\frac{1}{2}(\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}\stackrel{{\scriptstyle
56}}{{[+]}})=-1(\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[+]}}\stackrel{{\scriptstyle 56}}{{[+]}})$)..

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
${}^{I}{\hat{\cal A}}^{m=1\dagger}_{f=4}$, for example, presented in Table 1
in the first line of the fourth column of $even\,I$, do on the Clifford odd
“basis vector” $\hat{b}^{m=2\dagger}_{f=2}$, presented in $odd\,I$ as the
second member of the second column. (This can easily be evaluated by taking
into account Eq. (68) for any $m$.)

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

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

Let us check the Cartan subalgebra quantum numbers of this “scattering”:
${}^{I}{\hat{\cal A}}^{1\dagger}_{4}$ has
$({\bf\cal{S}}^{03},{\bf\cal{S}}^{12},{\bf\cal{S}}^{56})=(i,1,0)$,
$\hat{b}^{2\dagger}_{2}$ has
$(S^{03},S^{12},S^{56})=(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$ and
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})=(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$,
and $\hat{b}^{1\dagger}_{2}$ has
$(S^{03},S^{12},S^{56})=(\frac{i}{2},\frac{1}{2},\frac{1}{2})$ and
$(\tilde{S}^{03},\tilde{S}^{12},\tilde{S}^{56})=(-\frac{i}{2},-\frac{1}{2},\frac{1}{2})$.
This means that Clifford even “basis vector” changes the family members
quantum numbers of the Clifford odd “basis vector”, leaving the family quantum
numbers unchanged.

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

Calculating the eigenvalues of the Cartan subalgebra members, Eq. (8), before
and after the algebraic multiplication, $*_{A}$, assures us 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 to the Clifford odd “basis
vector” integer eigenvalues of the Cartan subalgebra members, changing the
Clifford odd “basis vector” into another Clifford odd “basis vector” of the
same family.

We, therefore, confirm 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 $\hat{b}^{m\dagger}_{1},m=(1,2,3,4)$.

One easily checks Eq. (25) if taking into account Eq. (68); like:
$\hat{b}^{1\dagger}_{1}*_{A}{}^{I}{\hat{\cal A}}^{m}_{4}=0$, $(m=(1,2,3,4))$,
since either $(\stackrel{{\scriptstyle 03}}{{(+i)}})^{2}=0$ or
$\stackrel{{\scriptstyle 12}}{{[+]}}*_{A}\stackrel{{\scriptstyle
12}}{{[-]}}=0$ or$\stackrel{{\scriptstyle 56}}{{[+]}}\stackrel{{\scriptstyle
56}}{{(-)}}=0$.

Similarly, one can check Eq. (26) by evaluating, for example,
${}^{II}{\hat{\cal A}}^{m}_{4}*_{A}\hat{b}^{1\dagger}_{1}$, since either
$\stackrel{{\scriptstyle 12}}{{(+)}}*_{A}\stackrel{{\scriptstyle
12}}{{[+]}}=0$ or $\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{[+]}}=0$.

Let us check the validity of Eq. (29) on the case:
$\hat{b}^{4\dagger}_{1}*_{A}{}^{II}{\hat{\cal A}}^{m}_{4}=$
$\hat{b}^{4\dagger}_{3}$ for $m=1$, and zero for $m=(2,3,4)$, while
$\hat{b}^{4\dagger}_{1}*_{A}{}^{II}{\hat{\cal A}}^{1}_{f}=$
($\hat{b}^{4\dagger}_{4},\hat{b}^{4\dagger}_{2},\hat{b}^{4\dagger}_{1},\hat{b}^{4\dagger}_{3}$)
for $f=(1,2,3,4)$. All ${}^{II}{\hat{\cal A}}^{m}_{f}$ giving non zero
contributions, keep the family member quantum numbers of the Clifford odd
“basis vectors” unchanged, changing the family quantum number. All the rest
give zero contribution.

The statements of Eq. (24, 25, 26, 29), are, therefore, demonstrated on the
case of $d=(5+1)$.

The Cartan subalgebra has in $d=(5+1)$-dimensional space $3$ members. To
illustrate that the Clifford even “basis vectors” have the properties of the
gauge fields of the corresponding Clifford odd “basis vectors” let us study
properties of the $SU(3)$ $\times U(1)$ subgroups of the Clifford odd and
Clifford even “basis vectors”. We need the relations between $S^{ab}$ and
($\tau^{3},\tau^{8},\tau^{`}$)

$\displaystyle\tau^{3}:=$
$\displaystyle\frac{1}{2}\,(-S^{1\,2}-iS^{0\,3})\,,\qquad\tau^{8}=\frac{1}{2\sqrt{3}}(-iS^{0\,3}+S^{1\,2}-2S^{5\;6})\,,$
$\displaystyle\tau^{\prime}=$
$\displaystyle-\frac{1}{3}(-iS^{0\,3}+S^{1\,2}+S^{5\,6})\,.$ (35)

The corresponding relations for ($\tilde{\tau}^{3},\tilde{\tau}^{8}$,
$\tilde{\tau}^{\prime}$) can be read from Eq. (2.3), if replacing $S^{ab}$ by
$\tilde{S}^{ab}$.

The corresponding relations for superposition of the Cartan subalgebra
elements ($\tau^{\prime},\tau^{3},\tau^{8}$) for ${\bf{\cal
S}}^{ab}=S^{ab}+\tilde{S}^{ab}$ follow if in Eq. (2.3) $S^{ab}$ is replaced by
${\bf{\cal S}}^{ab}$.

In Tables (2, 3) the Clifford odd and even “basis vectors”
($\hat{b}^{m\dagger}_{f}$ and ${}^{I}{\hat{\cal A}}^{m}_{f}$, respectively)
are presented as products of nilpotents (odd number of nilpotents for
$\hat{b}^{m\dagger}_{f}$ and even number of nilpotents for ${}^{I}{\hat{\cal
A}}^{m}_{f}$) and projectors: Like in Table 1. Besides the eigenvalues of the
Cartan subalgebra members of Eq. (8) also ($\tau^{3},\tau^{8},\tau^{`}$) are
added on both tables. In Table 2 also
($\tilde{\tau}^{3},\tilde{\tau}^{8},\tilde{\tau}^{`}$) are written. In Fig.
(1) only one family is presented; all four families have the same
($\tau^{3},\tau^{8},\tau^{`}$), they only distinguish in
($\tilde{\tau}^{3},\tilde{\tau}^{8},\tilde{\tau}^{`}$) .

The corresponding table for the Clifford even “basis vectors”
${}^{II}{\hat{\cal A}}^{m}_{f}$ are not presented. ${}^{II}{\hat{\cal
A}}^{m}_{f}$ carry, namely, the same quantum numbers
($\tau^{3},\tau^{8},\tau^{`}$) as ${}^{I}{\hat{\cal A}}^{m}_{f}$. There are
only products of nilpotents and projectors which distinguish among
${}^{I}{\hat{\cal A}}^{m}_{f}$ and ${}^{II}{\hat{\cal A}}^{m}_{f}$, causing
differences in properties with respect to the Clifford odd “basis vectors”;
${}^{II}{\hat{\cal A}}^{m^{\prime}}_{f`}$ transform $\hat{b}^{m\dagger}_{f}$
with a family member $m$ of particular family $f$ into
$\hat{b}^{m\dagger}_{f^{\prime\prime}}$ of the same family member $m$ of
another family $f^{\prime\prime}$. ${}^{I}{\hat{\cal A}}^{m}_{f}$ transform a
family member of particular family $\hat{b}^{m^{\prime}\dagger}_{f`}$ into
another family member $m$ of the same family $\hat{b}^{m\dagger}_{f`}$. (Let
us remind the reader that the $SO(5,1)$ group and the $SU(3),U(1)$ subgroups
have the same number of commuting operators, but different number of
generators; $SO(5,1)$ has $15$ generators, $SU(3)$ and $U(1)$ have together
$9$ generators.)

Table 2:  The ”basis vectors” $\hat{b}^{m\dagger}_{f}$ are presented for $d=(5+1)$-dimensional case. Each $\hat{b}^{m\dagger}_{f}$ is a product of projectors and of an odd number of nilpotents and is the ”eigenvector” of all the Cartan subalgebra members, ($S^{03}$, $S^{12}$, $S^{56}$) and ($\tilde{S}^{03}$, $\tilde{S}^{12}$, $\tilde{S}^{56}$), Eq. (8), $m$ counts the members of each family, while $f$ determines the family quantum numbers (the eigenvalues of ($\tilde{S}^{03}$, $\tilde{S}^{12}$, $\tilde{S}^{56}$)). This table also presents in the columns ($8^{th},9^{th},10^{th}$) the eigenvalues of the three commuting operators ($\tau^{3},\tau^{8}$ and $\tau^{\prime}$) of the subgroups $SU(3)\times U(1)$, Eq. (2.3), as well as (in the last three columns) the corresponding ($\tilde{\tau}^{3},\tilde{\tau}^{8},\tilde{\tau}^{\prime}$). $\Gamma^{(3+1)}$ $=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ is written in the $7^{th}$ column. $\Gamma^{(5+1)}=-1$ ($=-\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{5}\gamma^{6}$). Operators $\hat{b}^{m\dagger}_{f}$ and $\hat{b}^{m}_{f}$ fulfil the anti-commutation relations of Eq. (2.2.1).  $\,f$ | $m$ | $\hat{b}^{m\dagger}_{f}$ | $S^{03}$ | $S^{12}$ | $S^{56}$ | $\Gamma^{3+1}$ | $\tau^{3}$ | $\tau^{8}$ | $\tau^{\prime}$ | $\tilde{S}^{03}$ | $\tilde{S}^{12}$ | $\tilde{S}^{56}$ | $\tilde{\tau}^{3}$ | $\tilde{\tau}^{8}$ | $\tilde{\tau}^{`}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$I$ | $1$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $0$ | $-\frac{1}{2}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
| $2$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
| $3$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
| $4$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
$II$ | $1$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(+)}}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$
| $2$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(+)}}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$
| $3$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[-]}}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$
| $4$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[-]}}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$
$III$ | $1$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{(+)}}|\stackrel{{\scriptstyle 56}}{{[+]}}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
| $2$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{[-]}}|\stackrel{{\scriptstyle 56}}{{[+]}}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
| $3$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{(+)}}|\stackrel{{\scriptstyle 56}}{{(-)}}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
| $4$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[-]}}|\stackrel{{\scriptstyle 56}}{{(-)}}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$
$IV$ | $1$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(+)}}|\stackrel{{\scriptstyle 56}}{{(+)}}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $0$ | $-\frac{1}{2}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$
| $2$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[-]}}|\stackrel{{\scriptstyle 56}}{{(+)}}$ | $-\frac{i}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$
| $3$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(+)}}|\stackrel{{\scriptstyle 56}}{{[-]}}$ | $-\frac{i}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$
| $4$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[-]}}|\stackrel{{\scriptstyle 56}}{{[-]}}$ | $\frac{i}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{i}{2}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$

Figure 1:  The representations of the subgroups $SU(3)$ and $U(1)$ of the
group $SO(5,1)$, the properties of which appear in Tables (1, 2) for the
Clifford odd “basis vectors”, are presented. ($\tau^{3},\tau^{8}$,
$\tau^{\prime}$) can be calculated if using Eq. (2.3). 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, with the family quantum numbers presented in
the last three columns of Table 2.

.

In the case that the group $SO(5,1)$ — manifesting as $SU(3)\times U(1)$ and
representing the colour group with quantum numbers ($\tau^{3}$, $\tau^{8}$)
and the “fermion” group with the quantum number $\tau^{,}$ — is embedded into
$SO(13,1)$ the triplet would represent quarks (and antiquarks), and the
singlet leptons (and antileptons).

The corresponding gauge fields, presented in Table 3 and 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
$\frac{1}{6}$.

Table 3 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 Hermitian 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 the Clifford even
“basis vectors”, ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$, apply 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” of the same family, 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 subgroups.)

If, for example, ${}^{I}{\hat{\cal A}}^{1\dagger}_{3}$ applies on a singlet
$\hat{b}^{1\dagger}_{1}$ keeps the internal space of $\hat{b}^{1\dagger}_{1}$
unchanged (it can change only momentum), while if ${}^{I}{\hat{\cal
A}}^{2\dagger}_{3}$ applies on $\hat{b}^{1\dagger}_{1}$ transforms it to a
member of a triplet, to $\hat{b}^{2\dagger}_{1}$.

Figure 2:  The Clifford even ”basis vectors” ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ in the case that $d=(5+1)$ are presented concerning the
eigenvalues of the commuting operators of the subgroups $SU(3)$ and $U(1)$ of
the group $SO(5,1)$, Eq. (2.3): (${\cal\bf{\tau}}^{3}$, ${\cal\bf{\tau}}^{8}$,
${\cal\bf{\tau}}^{\prime}$). Their properties appear in Table 3. The abscissa
axis carries the eigenvalues of ${\cal\bf{\tau}}^{3}$, the ordinate axis
carries the eigenvalues of ${\cal\bf{\tau}}^{8}$ and the third axis carries
the eigenvalues of ${\cal\bf{\tau}}^{\prime}$. One notices four singlets with
(${\cal\bf{\tau}}^{3}=0,{\cal\bf{\tau}}^{8}=0,{\cal\bf{\tau}}^{\prime}=0$),
denoted by $\bigcirc$, representing four self adjoint Clifford even ”basis
vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$, with ($f=1,m=4$), ($f=2,m=3$),
($f=3,m=1$), ($f=4,m=2$) , one sextet of three pairs, Hermitian conjugated to
each other, with ${\cal\bf{\tau}}^{\prime}=0$, denoted by $\bigtriangleup$
(${}^{I}{\hat{\cal A}}^{2\dagger}_{1}$ with
(${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=-\frac{1}{2},{\cal\bf{\tau}}^{8}=-\frac{3}{2\sqrt{3}}$)
and ${}^{I}{\hat{\cal A}}^{4\dagger}_{4}$ with
(${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=\frac{1}{2},{\cal\bf{\tau}}^{8}=\frac{3}{2\sqrt{3}}$)),
by $\ddagger$ (${}^{I}{\hat{\cal A}}^{3\dagger}_{1}$ with
(${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=-1,{\cal\bf{\tau}}^{8}=0$)
and ${}^{I}{\hat{\cal A}}^{4\dagger}_{2}$ with
${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=1,{\cal\bf{\tau}}^{8}=0$)),
and by $\otimes$ (${}^{I}{\hat{\cal A}}^{2\dagger}_{2}$ with
(${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=\frac{1}{2},{\cal\bf{\tau}}^{8}=-\frac{3}{2\sqrt{3}}$)
and ${}^{I}{\hat{\cal A}}^{3\dagger}_{4}$ with
(${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=-\frac{1}{2},{\cal\bf{\tau}}^{8}=\frac{3}{2\sqrt{3}}$)),
and one triplet, denoted by $\star\star$ (${}^{I}{\hat{\cal
A}}^{4\dagger}_{3}$ with
(${\cal\bf{\tau}}^{\prime}=\frac{2}{3},{\cal\bf{\tau}}^{3}=\frac{1}{2},{\cal\bf{\tau}}^{8}=\frac{1}{2\sqrt{3}}$)),
by $\bullet$ (${}^{I}{\hat{\cal A}}^{3\dagger}_{3}$ with
(${\cal\bf{\tau}}^{\prime}=\frac{2}{3},{\cal\bf{\tau}}^{3}=-\frac{1}{2},{\cal\bf{\tau}}^{8}=\frac{1}{2\sqrt{3}}$)),
and by $\odot\odot$ (${}^{I}{\hat{\cal A}}^{2\dagger}_{3}$ with
(${\cal\bf{\tau}}^{\prime}=\frac{2}{3},{\cal\bf{\tau}}^{3}=0,{\cal\bf{\tau}}^{8}=-\frac{1}{\sqrt{3}}$)),
as well as one antitriplet, Hermitian conjugated to triplet, denoted by
$\star\star$ (${}^{I}{\hat{\cal A}}^{1\dagger}_{1}$ with
(${\cal\bf{\tau}}^{\prime}=-\frac{2}{3},{\cal\bf{\tau}}^{3}=-\frac{1}{2},{\cal\bf{\tau}}^{8}=-\frac{1}{2\sqrt{3}}$)),
by $\bullet$ (${}^{I}{\hat{\cal A}}^{1\dagger}_{2}$ with
(${\cal\bf{\tau}}^{\prime}=-\frac{2}{3},{\cal\bf{\tau}}^{3}=\frac{1}{2},{\cal\bf{\tau}}^{8}=-\frac{1}{2\sqrt{3}}$)),
and by $\odot\odot$ (${}^{I}{\hat{\cal A}}^{4\dagger}_{1}$ with
(${\cal\bf{\tau}}^{\prime}=-\frac{2}{3},{\cal\bf{\tau}}^{3}=0,{\cal\bf{\tau}}^{8}=\frac{1}{\sqrt{3}}$)).

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

In Table 3 we find $(6+4)$ Clifford even “basis vectors” ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ with ${\cal\bf{\tau}}^{\prime}=0$. 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
(${\cal\bf{\tau}}^{\prime}=0,{\cal\bf{\tau}}^{3}=0,{\cal\bf{\tau}}^{8}=0$),
denoted in the third column of Table 3 by $\bigcirc$. The rest $6$ Clifford
even “basis vectors” belong to one triplet with
${\cal\bf{\tau}}^{\prime}=\frac{2}{3}$ and
$({\cal\bf{\tau}}^{3},{\cal\bf{\tau}}^{8})$ 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
${\cal\bf{\tau}}^{\prime}=-\frac{2}{3}$ and
($({\cal\bf{\tau}}^{3},{\cal\bf{\tau}}^{8})$ 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 anti-triplet and opposite.
In Table 3 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, two triplets and four singlets are presented in Fig. 2.

Table 3: 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 also represents the eigenvalues of the three commuting operators ${\cal{\bf\tau}}^{3},{\cal{\bf\tau}}^{8}$ and ${\cal{\bf\tau}}^{\prime}$ of the subgroups $SU(3)\times U(1)$.  $\,f$ | $m$ | $*$ | ${}^{I}\hat{\cal A}^{m\dagger}_{f}$ | ${\cal S}^{03}$ | ${\cal S}^{12}$ | ${\cal S}^{56}$ | ${\cal{\bf\tau}}^{3}$ | ${\cal{\bf\tau}}^{8}$ | ${\cal{\bf\tau}}^{\prime}$
---|---|---|---|---|---|---|---|---|---
$I$ | $1$ | $\star\star$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle 56}}{{(+)}}$ | $0$ | $1$ | $1$ | $-\frac{1}{2}$ | $-\frac{1}{2\sqrt{3}}$ | $-\frac{2}{3}$
| $2$ | $\bigtriangleup$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{(+)}}$ | $-i$ | $0$ | $1$ | $-\frac{1}{2}$ | $-\frac{3}{2\sqrt{3}}$ | $0$
| $3$ | $\ddagger$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $-i$ | $1$ | $0$ | $-1$ | $0$ | $0$
| $4$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$II$ | $1$ | $\bullet$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{(+)}}$ | $i$ | $0$ | $1$ | $\frac{1}{2}$ | $-\frac{1}{2\sqrt{3}}$ | $-\frac{2}{3}$
| $2$ | $\otimes$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{(+)}}$ | $0$ | $-1$ | $1$ | $\frac{1}{2}$ | $-\frac{3}{2\sqrt{3}}$ | $0$
| $3$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $4$ | $\ddagger$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{[-]}}$ | $i$ | $-1$ | $0$ | $1$ | $0$ | $0$
$III$ | $1$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $2$ | $\odot\odot$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $-i$ | $-1$ | $0$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{2}{3}$
| $3$ | $\bullet$ | $\stackrel{{\scriptstyle 03}}{{(-i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $-i$ | $0$ | $-1$ | $-\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{2}{3}$
| $4$ | $\star\star$ | $\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(-)}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $0$ | $-1$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2\sqrt{3}}$ | $\frac{2}{3}$
$IV$ | $1$ | $\odot\odot$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $i$ | $1$ | $0$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{2}{3}$
| $2$ | $\bigcirc$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{[+]}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| $3$ | $\otimes$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(+)}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $0$ | $1$ | $-1$ | $-\frac{1}{2}$ | $\frac{3}{2\sqrt{3}}$ | $0$
| $4$ | $\bigtriangleup$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[-]}}\,\stackrel{{\scriptstyle 56}}{{(-)}}$ | $i$ | $0$ | $-1$ | $\frac{1}{2}$ | $\frac{3}{2\sqrt{3}}$ | $0$

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 3. 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 gauge fields “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 applying on the Clifford odd singlet, denoted in Fig. 1 by a
square, this singlet to one of the members of the triplet, denoted in 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
recognize in the part of fields forming the octet the colour gauge fields of
quarks and leptons and antiquarks and antileptons. The Clifford even “basis
vectors” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ transform when applying on the
Clifford odd “basis vectors” $\hat{b}^{m^{\prime}\dagger}_{f`}$ to another (or
the same) member, keeping the family member unchanged.

We can check that although ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}$ have
different structure of an even number of nilpotents, and the rest of the
projectors than ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$, having correspondingly
different properties with respect to the Clifford odd “basis vectors”:
${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ transform
$\hat{b}^{m^{\prime}\dagger}_{f`}$ among the family members, keeping the
family quantum numbers unchanged, ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}$
transform $\hat{b}^{m\dagger}_{f}$ into the same member of another family,
keeping the family member’s quantum number unchanged, both, ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ and ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}$ do have the
equivalent figure and equivalent ${\cal{\bf S}}^{ab}$ and correspondingly also
$({\cal{\bf\tau}}^{3},{\cal{\bf\tau}}^{8},{\cal{\bf\tau}}^{\prime})$ content.

Let us anyhow demonstrate properties of “scattering” of
$\hat{b}^{m\dagger}_{f}$ on ${}^{II}{\hat{\cal A}}^{m^{\prime}\dagger}_{f`}$,
paying attention on $SU(3)$ and $U(1)$ substructure of $SO(5,1)$.

Let us look at the “scattering” of the kind of Eq. (34)

$\displaystyle\hat{b}^{2\dagger}_{2}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(+)}})*_{A}{}^{II}{\hat{\cal
A}}^{3\dagger}_{1}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{[-]}})\rightarrow\hat{b}^{2\dagger}_{4}(\equiv\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}})\,,$ (36)

$\hat{b}^{2\dagger}_{2}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}\stackrel{{\scriptstyle
56}}{{(+)}})$ has
$(\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{1}{6}$) and
$(\tilde{\tau}^{3}=0,\tilde{\tau}^{8}=-\frac{1}{\sqrt{3}},\tilde{\tau}^{\prime}=\frac{1}{6}$).
$\hat{b}^{2\dagger}_{4}(\equiv\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{(+)}})$ has
$(\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{1}{6}$) and
$(\tilde{\tau}^{3}=0,\tilde{\tau}^{8}=0,\tilde{\tau}^{\prime}=-\frac{1}{2}$).
${}^{II}{\hat{\cal A}}^{1\dagger}_{4}(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{[-]}}$) has
$({\cal{\bf\tau}}^{3}=0,{\cal{\bf\tau}}^{8}=\frac{1}{\sqrt{3}},{\cal{\bf\tau}}^{\prime}=-\frac{2}{3}).$

If $\hat{b}^{2\dagger}_{2}$ absorbs ${}^{II}{\hat{\cal
A}}^{3\dagger}_{4}(\equiv\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}}\stackrel{{\scriptstyle
56}}{{(-)}})$ with
$({\cal{\bf\tau}}^{3}=-\frac{1}{2},{\cal{\bf\tau}}^{8}=\frac{3}{2\sqrt{3}},{\cal{\bf\tau}}^{\prime}=0$)
becomes $\hat{b}^{2\dagger}_{3}(\equiv\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{[-]}}\stackrel{{\scriptstyle
56}}{{[+]}})$ with quantum numbers
$(\tau^{3}=0,\tau^{8}=-\frac{1}{\sqrt{3}},\tau^{\prime}=\frac{1}{6}$) and
$(\tilde{\tau}^{3}=-\frac{1}{2},\tilde{\tau}^{8}=\frac{1}{2\sqrt{3}},\tilde{\tau}^{\prime}=\frac{1}{6}$).

${}^{II}{\hat{\cal A}}^{3\dagger}_{4}$ transfers its quantum numbers to
$\hat{b}^{2\dagger}_{2}$, changing family and leaving the family member $m$
unchanged.

###  2.4 Second quantized fermion and boson fields with internal spaces
described by Clifford “basis vectors” in even dimensional spaces

We learned in the previous Subsects. (2.2, 2.3) 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. (2.2.1) in this paper, Eq.(26) in Ref.
[14], explaining the second quantization postulate 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
commutation relations of ${}^{i}{\hat{\cal A}}^{m\dagger}_{f},i=(I,II),$ are
commented in the last paragraph of App. A on a simple case of $d=(3+1)$. The
subgroup structure of $SU(3)$ can be recognized on Fig. 2, leading to the
commutation relations of the observed colour boson gauge fields. However,
further studies are needed to recognize what new this way of describing
internal spaces of fermion and boson fields with the Clifford algebra is
offering.

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, for fermions or bosons, besides the “basis
vectors” defining the internal space of fermions and bosons, the basis in
ordinary space in momentum or coordinate representation is needed. Here Ref.
[14], Subsect. 3.3 and App. J is overviewed.

Let us introduce the momentum part of the single-particle states. (The
extended version is presented in Ref. [14] in Subsect. 3.3 and App. J.)

$\displaystyle|\vec{p}>$ $\displaystyle=$
$\displaystyle\hat{b}^{\dagger}_{\vec{p}}\,|\,0_{p}\,>\,,\quad<\vec{p}\,|=<\,0_{p}\,|\,\hat{b}_{\vec{p}}\,,$
$\displaystyle<\vec{p}\,|\,\vec{p}^{\prime}>$ $\displaystyle=$
$\displaystyle\delta(\vec{p}-\vec{p}^{\prime})=<\,0_{p}\,|\hat{b}_{\vec{p}}\;\hat{b}^{\dagger}_{\vec{p}^{\prime}}|\,0_{p}\,>\,,$
$\displaystyle{\rm pointing\;out\;}$
$\displaystyle<\,0_{p}\,|\hat{b}_{\vec{p^{\prime}}}\,\hat{b}^{\dagger}_{\vec{p}}\,|\,0_{p}\,>$
$\displaystyle=$ $\displaystyle\delta(\vec{p^{\prime}}-\vec{p})\,,$ (37)

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

$\displaystyle<\vec{p}\,|\,\vec{x}>$ $\displaystyle=$
$\displaystyle<0_{\vec{p}}\,|\,\hat{b}_{\vec{p}}\;\hat{b}^{\dagger}_{\vec{x}}|0_{\vec{x}}\,>=(<0_{\vec{x}}\,|\,\hat{b}_{\vec{x}}\;\hat{b}^{\dagger}_{\vec{p}}\,\,|0_{\vec{p}}\,>)^{\dagger}\,$
$\displaystyle<0_{\vec{p}}\,|\\{\hat{b}^{\dagger}_{\vec{p}}\,,\,\hat{b}^{\dagger}_{\vec{p}\,^{\prime}}\\}_{-}|0_{\vec{p}}\,>$
$\displaystyle=$ $\displaystyle
0\,,\qquad<0_{\vec{p}}\,|\\{\hat{b}_{\vec{p}},\,\hat{b}_{\vec{p}\,^{\prime}}\\}_{-}|0_{\vec{p}}\,>=0\,,\qquad<0_{\vec{p}}\,|\\{\hat{b}_{\vec{p}},\,\hat{b}^{\dagger}_{\vec{p}\,^{\prime}}\\}_{-}|0_{\vec{p}}\,>=0\,,$
$\displaystyle<0_{\vec{x}}\,|\\{\hat{b}^{\dagger}_{\vec{x}},\,\hat{b}^{\dagger}_{\vec{x}\,^{\prime}}\\}_{-}|0_{\vec{x}}\,>$
$\displaystyle=$ $\displaystyle
0\,,\qquad<0_{\vec{x}}\,|\\{\hat{b}_{\vec{x}},\,\hat{b}_{\vec{x}\,^{\prime}}\\}_{-}|0_{\vec{x}}\,>=0\,,\qquad<0_{\vec{x}}\,|\\{\hat{b}_{\vec{x}},\,\hat{b}^{\dagger}_{\vec{x}\,^{\prime}}\\}_{-}|0_{\vec{x}}\,>=0\,,$
$\displaystyle<0_{\vec{p}}\,|\\{\hat{b}_{\vec{p}},\,\hat{b}^{\dagger}_{\vec{x}}\\}_{-}|0_{\vec{x}}\,>$
$\displaystyle=$ $\displaystyle
e^{i\vec{p}\cdot\vec{x}}\frac{1}{\sqrt{(2\pi)^{d-1}}}\,,\quad<0_{\vec{x}}\,|\\{\hat{b}_{\vec{x}},\,\hat{b}^{\dagger}_{\vec{p}}\\}_{-}|0_{\vec{p}}\,>=e^{-i\vec{p}\cdot\vec{x}}\frac{1}{\sqrt{(2\pi)^{d-1}}}\,.$
(38)

.  The internal space of either fermion or boson fields has the finite number
of “basis vectors”, $2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ for fermions
(and the same number of their Hermitian conjugated partners), and twice
$2^{\frac{d}{2}-1}\times 2^{\frac{d}{2}-1}$ for bosons, the momentum basis is
continuously infinite.

The creation operators for either fermions or bosons must be tensor products,
$*_{T}$, 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 ([14], Subsect.3.3.2, and the references therein)

$\displaystyle{\bf\hat{b}}^{s\dagger}_{f}(\vec{p})\,$ $\displaystyle=$
$\displaystyle\,\sum_{m}c^{sm}{}_{f}(\vec{p})\,\hat{b}^{\dagger}_{\vec{p}}\,*_{T}\,\hat{b}^{m\dagger}_{f}\,\,,$
(39)

where the vacuum state for fermions $|\psi_{oc}>\,*_{T}\,|0_{\vec{p}}>$
includes both spaces, the internal part, Eq.(15), and the momentum part, Eq.
(2.4) (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 $\vec{p}$ 101010 The creation operators and their
Hermitian conjugated annihilation operators in the coordinate representation
can be read in [14] and the references therein: $\hat{\bf
b}^{s\dagger}_{f}(\vec{x},x^{0})=\sum_{m}\,\hat{b}^{m\dagger}_{f}\,*_{T}\,\int_{-\infty}^{+\infty}\,\frac{d^{d-1}p}{(\sqrt{2\pi})^{d-1}}\,c^{sm}{}_{f}\;(\vec{p})\;\hat{b}^{\dagger}_{\vec{p}}\;e^{-i(p^{0}x^{0}-\varepsilon\vec{p}\cdot\vec{x})}$
([14], subsect. 3.3.2., Eqs. (55,57,64) and the references therein).).
The creation operators and annihilation operators for fermion fields fulfil
the anti-commutation relations for the second quantized fermion fields 111111
Let us evaluate: $<0_{\vec{p}}\,|\\{\hat{\bf
b}^{s^{\prime}}_{f`}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>=\delta^{ss^{\prime}}\delta_{ff^{\prime}}\,\delta(\vec{p}^{\prime}-\vec{p})\,\cdot|\psi_{oc}>=$
$<0_{\vec{p}}\,|\hat{\bf b}^{s^{\prime}}_{f`}\,\hat{\bf
b}^{s\dagger}_{f}\,\hat{b}_{\vec{p}^{\prime}}\hat{b}^{\dagger}_{\vec{p}}+\,\hat{b}^{\dagger}_{\vec{p}}\hat{b}_{\vec{p}^{\prime}}\,\hat{\bf
b}^{s\dagger}_{f}\,\hat{\bf b}^{s^{\prime}}_{f`}\,|\psi_{oc}>|0_{\vec{p}}>=$
$<0_{\vec{p}}\,|\hat{\bf b}^{s^{\prime}}_{f`}\,\hat{\bf
b}^{s\dagger}_{f}\,\hat{b}_{\vec{p}^{\prime}}\hat{b}^{\dagger}_{\vec{p}}\,|\psi_{oc}>|0_{\vec{p}}>$,
since, according to Eq. (2.2.1), $\hat{\bf
b}^{s^{\prime}}_{f`}\,|\psi_{oc}>=0.$ Let us demonstrate for free fields
$|\vec{p}>=e^{-i\vec{p}\cdot\vec{x}}|0_{p}>=\hat{b}^{\dagger}_{\vec{p}}\,|0_{p}>$
, $\,<\vec{p}|=<0_{p}|e^{i\vec{p}\cdot\vec{x}}=<0_{p}|\,\hat{b}_{\vec{p}}\,$
$<\vec{p^{\prime}}|\vec{p}>=<0_{p}|\,\hat{b}_{\vec{p^{\prime}}}\,\hat{b}^{\dagger}_{\vec{p}}\,|0_{p}>=\delta{(\vec{p^{\prime}}-\vec{p})}\,,$
$<\vec{-p^{\prime}}|\vec{-p}>=<0_{p}|\,\hat{b}^{\dagger}_{\vec{p^{\prime}}}\,\hat{b}_{\vec{p}}\,|0_{p}>=\delta{(\vec{-p^{\prime}}-(\vec{-p}))}=\delta{(\vec{p}-\vec{p^{\prime}})}\,,$
consequently
$<0_{p}|\\{\,\hat{b}_{\vec{p}}\,,\hat{b}^{\dagger}_{\vec{p^{\prime}}}\\}_{-}\,|0_{p}>=0$
.  121212 Two fermion states (formed from two creation operators applying on
the vacuum state) with the orthogonal basis part in ordinary space (with two
different momenta in ordinary space in the case of free fields) ”do not meet”;
correspondingly, each can carry the same “basis vector”. They must differ in
the internal basis if they have the identical ordinary part of the basis.
(Otherwise, the tensor product, $*_{T_{H}}$, of such two fermion states is
zero.) Illustration: Let us treat an atom with many electrons. Each electron
has a spin of either $1/2$ or $-1/2.$ Their orthogonal basis in ordinary space
allows them to have the internal spin $\pm 1/2$ (leading to total angular
momentum either $\pm 1/2$ or larger due to the angular momentum in ordinary
space). As mentioned in the introduction section in a.iii. the Hilbert space
of the second quantized fermion states is represented by the tensor products,
$*_{T_{H}}$, of all possible members of creation operators from zero to
infinity applying on the simple vacuum state. For any of these members the
scalar product is obtained by multiplying from the left hand side by their
Hermitian conjugated partner..

$\displaystyle<0_{\vec{p}}\,|\\{\hat{\bf
b}^{s^{\prime}}_{f`}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle\delta^{ss^{\prime}}\delta_{ff^{\prime}}\,\delta(\vec{p}^{\prime}-\vec{p})\,\cdot|\psi_{oc}>\,,$
$\displaystyle\\{\hat{\bf b}^{s^{\prime}}_{f`}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle 0\,\cdot\,|\psi_{oc}>|0_{\vec{p}}>\,,$
$\displaystyle\\{\hat{\bf
b}^{s^{\prime}\dagger}_{f^{\prime}}(\vec{p^{\prime}})\,,\,\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\\}_{+}\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle 0\,\cdot\,|\psi_{oc}>|0_{\vec{p}}>\,,$ $\displaystyle\hat{\bf
b}^{s\dagger}_{f}(\vec{p})\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$
$\displaystyle|\psi^{s}_{f}(\vec{p})>\,,$ $\displaystyle\hat{\bf
b}^{s}_{f}(\vec{p})\,|\psi_{oc}>|0_{\vec{p}}>$ $\displaystyle=$ $\displaystyle
0\,\cdot\,\,|\psi_{oc}>|0_{\vec{p}}>\,,$ $\displaystyle|p^{0}|$
$\displaystyle=$ $\displaystyle|\vec{p}|\,.$ (40)

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 the vacuum state,
$|\psi_{oc}>*_{T}|0_{\vec{p}}>$, the anti-commutation relations for the second
quantized fermions, postulated by Dirac (Ref. [14], Subsect. 3.3.1, Sect. 5).
131313 The anti-commutation relations of Eq. (40) are valid also if we replace
the vacuum state, $|\psi_{oc}>|0_{\vec{p}}>$, by the Hilbert space of the
Clifford fermions generated by the tensor products 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. ([14],
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 141414 In the spin-charge-
family theory the Higgs’s scalars origin in the boson gauge fields with the
vector index $(7,8)$, Ref. ([14], Sect. 7.4.1, and the references therein)..
We, therefore, add the space index $\alpha$ as follows.

$\displaystyle{\bf{}^{i}{\hat{\cal A}}^{m\dagger}_{f\alpha}}(\vec{p})\,$
$\displaystyle=$ $\displaystyle\hat{b}^{\dagger}_{\vec{p}}\,*_{T}\,{}^{i}{\cal
C}^{m}{}_{f\alpha}\,{}^{i}{\hat{\cal A}}^{m\dagger}_{f}\,\,,i=(I,II)\,.$ (41)

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 151515 In the general case, the energy eigenstates of bosons are in a
superposition of ${\bf{}^{i}{\hat{\cal A}}^{m\dagger}_{f}}$, for either $i=I$
or $i=II$. One example, which uses the superposition of the Cartan subalgebra
eigenstates manifesting the $SU(3)\times U(1)$ subgroups of the group
$SO(5,1)$, is presented in Fig. 2., ${}^{i}{\cal C}^{m}{}_{f\alpha}$ 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 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 ${}^{i}{\cal C}^{m}{}_{f\alpha}$ depending on $\vec{p}$.

For the creation operators for boson fields in a coordinate representation one
finds using Eqs. (2.4, 2.4)

$\displaystyle{\bf{}^{i}{\hat{\cal A}}^{m\dagger}_{f\alpha}}(\vec{x},x^{0})$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\,\frac{d^{d-1}p}{(\sqrt{2\pi})^{d-1}}\,{}^{i}{\hat{\cal
A}}^{m\dagger}_{f\alpha}(\vec{p})\,e^{-i(p^{0}x^{0}-\varepsilon\vec{p}\cdot\vec{x})}|_{p^{0}=|\vec{p}|}\,,i=(I,II)\,.$
(42)

To understand what new the Clifford algebra description of the internal space
of fermion and boson fields, Eqs. (41, 42, 39), 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
$\sum_{mf}{}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,{}^{I}{\cal
C}^{m}{}_{f\alpha}$, recognizing that ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,{}^{I}{\cal C}^{m}{}_{f\alpha}$ are eigenstates of the
Cartan subalgebra members, while $\omega_{ab\alpha}$ are not. And,
equivalently, we need to relate
$\sum_{ab}\tilde{c}^{ab}\tilde{\omega}_{ab\alpha}$ and
$\sum_{mf}{}^{II}{\hat{\cal A}}^{m\dagger}_{f}\,{}^{II}{\cal