C}^{m}{}_{f\alpha}$.

The gravity fields, the vielbeins and the two kinds of spin connection fields,
$f^{a}{}_{\alpha}$, $\omega_{ab\alpha}$, $\tilde{\omega}_{ab\alpha}$,
respectively, are in the spin-charge-family theory (unifying spins, charges
and families of fermions and offering not only the explanation for all the
assumptions of the standard model but also for the increasing number of
phenomena observed so far) the only boson fields in $d=(13+1)$, observed in
$d=(3+1)$ besides as gravity also as all the other boson fields with the
Higgs’s scalars included [11].

We, therefore, need to relate:

$\displaystyle\\{\frac{1}{2}\sum_{ab}S^{ab}\,\omega_{ab\alpha}\\}\sum_{m}\beta^{mf}\,\hat{\bf
b}^{m\dagger}_{f}(\vec{p})$ $\displaystyle{\rm related\,\,to}$
$\displaystyle\\{\sum_{m^{\prime}f^{\prime}}{}^{I}{\hat{\cal
A}}^{m^{\prime}\dagger}_{f^{\prime}}\,{\cal
C}^{m^{\prime}f^{\prime}}_{\alpha}\\}\sum_{m}\beta^{mf}\,\hat{\bf
b}^{m\dagger}_{f}(\vec{p})\,,$ $\displaystyle\forall f\,{\rm
and}\,\forall\,\beta^{mf}\,,$ $\displaystyle{\bf\cal
S}^{cd}\,\sum_{ab}(c^{ab}{}_{mf}\,\omega_{ab\alpha})$ $\displaystyle{\rm
related\,\,to}$ $\displaystyle{\bf\cal S}^{cd}\,({}^{I}{\hat{\cal
A}}^{m\dagger}_{f}\,{\cal C}^{mf}_{\alpha})\,,$ (43)
$\displaystyle\forall\,(m,f),$ $\displaystyle\forall\,\,{\rm
Cartan\,\,subalgebra\,\,\,member}\,{\bf\cal S}^{cd}\,.$

Let be repeated that ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ are chosen to be
the eigenvectors of the Cartan subalgebra members, Eq. (8). Correspondingly we
can relate a particular ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}\,{}^{I}{\cal
C}^{m}{}_{f\alpha}$ with such a superposition of $\omega_{ab\alpha}$’s, which
is the eigenvector with the same values of the Cartan subalgebra members as
there is a particular ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}{\cal
C}^{mf}_{\alpha}$. We can do this in two ways:
i. Using the first relation in Eq. (2.4). On the left hand side of this
relation ${S}^{ab}$’s apply on $\hat{b}^{m\dagger}_{f}$ part of $\hat{\bf
b}^{m\dagger}_{f}(\vec{p})$. On the right hand side ${}^{I}{\hat{\cal
A}}^{m\dagger}_{f}$ apply as well on the same “basis vector”
$\hat{b}^{m\dagger}_{f}$.
ii. Using the second relation, in which ${\bf\cal S}^{cd}$ apply on the left
hand side on $\omega_{ab\alpha}$’s,

$\displaystyle\,{\bf\cal S}^{cd}\,\sum_{ab}\,c^{ab}{}_{mf}\,\omega_{ab\alpha}$
$\displaystyle=$
$\displaystyle\sum_{ab}\,c^{ab}{}_{mf}\,i\,(\omega_{cb\alpha}\eta^{ad}-\omega_{db\alpha}\eta^{ac}+\omega_{ac\alpha}\eta^{bd}-\omega_{ad\alpha}\eta^{bc}),$
(44)

on each $\omega_{ab\alpha}$ separately; $c^{ab}{}_{mf}$ are constants to be
determined from the second relation, where on the right-hand side of this
relation ${\bf\cal S}^{cd}(=S^{cd}+\tilde{S}^{cd})$ apply on the “basis
vector” ${}^{I}{\hat{\cal A}}^{m\dagger}_{f}$ of the corresponding gauge field
161616 The reader can find the relation of Eq. (2.4) demonstrated for the case
$d={3+1}$ in Ref. [15] at the end of Sect. 3..

We must treat equivalently also ${}^{II}{\hat{\cal
A}}^{m\dagger}_{f}\,{}^{II}{\cal C}^{m}{}_{f\alpha}$ and
$\tilde{\omega}_{ab\alpha}$.

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

##  3 Conclusions

In the spin-charge-family theory [6, 8, 11, 9, 22, 12, 14] the Clifford odd
algebra describes the internal space of fermion fields. The Clifford odd
“basis vectors” — the superposition of odd products of $\gamma^{a}$’s — in a
tensor product with the basis in ordinary space form the creation and
annihilation operators, in which the anti-commutativity of the “basis vectors”
is transferred to the creation and annihilation operators for fermions,
explaining the second quantization postulates for fermion fields.

The Clifford odd “basis vectors” have all the properties of fermions: Half
integer spins concerning the Cartan subalgebra members of the Lorentz algebra
in the internal space of fermions in even dimensional spaces ($d=2(2n+1)$ or
$d=4n$), as discussed in Subsects. (2.2, 2.4) (and in App A in a pedagogical
way). With respect to the subgroups of the $SO(d-1,1)$ group the Clifford odd
“basis vectors” appear in the fundamental representations, as illustrated in
Subsects. 2.3.

In this article, it is demonstrated that Clifford even algebra is offering the
description of the internal space of boson fields. The Clifford even “basis
vectors” — the superposition of even products of $\gamma^{a}$’s — in a tensor
product with the basis in ordinary space form the creation and annihilation
operators which manifest the commuting properties of the second quantized
boson fields, offering the explanation for the second quantization postulates
for boson fields [16, 15]. The Clifford even “basis vectors” have all the
properties of boson fields: Integer spins for the Cartan subalgebra members of
the Lorentz algebra in the internal space of bosons, as discussed in Subsects.
2.2.

With respect to the subgroups of the $SO(d-1,1)$ group the Clifford even
“basis vectors” manifest the adjoint representations, as illustrated in
Subsect. 2.3.

There are two kinds of anti-commuting algebras [6]: The Grassmann algebra,
offering in $d$-dimensional space $2\,.\,2^{d}$ operators ($2^{d}$
$\,\theta^{a}$’s and $2^{d}$ $\frac{\partial}{\partial\theta_{a}}$’s,
Hermitian conjugated to each other, Eq. (3)), and the two Clifford
subalgebras, each with $2^{d}$ operators named $\gamma^{a}$’s and
$\tilde{\gamma}^{a}$’s, respectively, [6, 10], Eqs. (2.1-2.1).

The operators in each of the two Clifford subalgebras appear in even-
dimensional spaces in two groups of $2^{\frac{d}{2}-1}\times$
$2^{\frac{d}{2}-1}$ of the Clifford odd operators (the odd products of either
$\gamma^{a}$’s in one subalgebra or of $\tilde{\gamma}^{a}$’s in the other
subalgebra), which are Hermitian conjugated to each other: In each Clifford
odd group of any of the two subalgebras, there appear $2^{\frac{d}{2}-1}$
irreducible representation each with the $2^{\frac{d}{2}-1}$ members and the
group of their Hermitian conjugated partners.

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

The Grassmann algebra operators are expressible with the operators of the two
Clifford subalgebras and opposite, Eq. (2.1). The two Clifford sub-algebras
are independent of each other, Eq. (2.1), forming two independent spaces.

Either the Grassmann algebra [12] or the two Clifford subalgebras can be used
to describe the internal space of anti-commuting objects, if the superposition
of odd products of operators ($\theta^{a}$’s or $\gamma^{a}$’s, or
$\tilde{\gamma}^{a}$’s) are used to describe the internal space of these
objects. The commuting objects must be a superposition of even products of
operators ($\theta^{a}$’s or $\gamma^{a}$’s or $\tilde{\gamma}^{a}$’s).

No integer spin anti-commuting objects have been observed so far, and to
describe the internal space of the so far observed fermions only one of the
two Clifford odd subalgebras are needed.

The problem can be solved by reducing the two Clifford subalgebras to only
one, the one (chosen to be) determined by $\gamma^{a}$’s. The decision that
$\tilde{\gamma}^{a}$’s apply on $\gamma^{a}$ as follows:
$\\{\tilde{\gamma}^{a}B=(-)^{B}\,i\,B\gamma^{a}\\}\,|\psi_{oc}>$, Eq. (7),
(with $(-)^{B}=-1$, if $B$ is a function of an odd products of $\gamma^{a}$’s,
otherwise $(-)^{B}=1$) enables that $2^{\frac{d}{2}-1}$ irreducible
representations of $S^{ab}=\frac{i}{2}\,\\{\gamma^{a}\,,\,\gamma^{b}\\}_{-}$
(each with the $2^{\frac{d}{2}-1}$ members) obtain the family quantum numbers
determined by
$\tilde{S}^{ab}=\frac{i}{2}\,\\{\tilde{\gamma}^{a}\,,\,\tilde{\gamma}^{b}\\}_{-}$.

The decision to use in the spin-charge-family theory in $d=2(2n+1)$, $n\geq 3$
($d\geq(13+1)$ indeed), the superposition of the odd products of the Clifford
algebra elements $\gamma^{a}$’s to describe the internal space of fermions
which interact with gravity only (with the vielbeins, the gauge fields of
momenta, and the two kinds of the spin connection fields, the gauge fields of
$S^{ab}$ and $\tilde{S}^{ab}$, respectively), Eq. (1), offers not only the
explanation for all the assumed properties of fermions and bosons in the
standard model, with the appearance of the families of quarks and leptons and
antiquarks and antileptons ([14] and the references therein) and of the
corresponding vector gauge fields and the Higgs’s scalars included [11], but
also for the appearance of the dark matter [35] in the universe, for the
explanation of the matter/antimatter asymmetry in the universe [8], and for
several other observed phenomena, making several predictions [7, 33, 34, 36].

The recognition that the use of the superposition of the even products of the
Clifford algebra elements $\gamma^{a}$’s to describe the internal space of
boson fields, what appears to manifest all the properties of the observed
boson fields, as demonstrated in this article, makes clear that the Clifford
algebra offers not only the explanation for the postulates of the second
quantized anti-commuting fermion fields but also for the postulates of the
second quantized boson fields.

This recognition, however, offers the possibility to relate

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

and equivalently for ${}^{II}{\hat{\cal A}}^{m\dagger}_{f}\,{}^{II}{\cal
C}^{m}{}_{f\alpha}$ and $\tilde{\omega}_{ab\alpha}$, what offers the
possibility to replace the covariant derivative $p_{0\alpha}$

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

in Eq. (1) with

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

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

Although the properties of the Clifford odd and even “basis vectors” and
correspondingly of the creation and annihilation operators for fermion and
boson fields are, hopefully, demonstrated in this article, yet the proposed
way of the second quantization of fields, the fermion and the boson ones needs
further study to find out what new can the description of the internal space
of fermions and bosons bring into the understanding of the second quantized
fields.

This study showing up that the Clifford algebra can be used to describe the
internal spaces of fermion and boson fields equivalently, offering
correspondingly the explanation for the second quantization postulates for
fermion and boson fields is opening a new insight into the quantum field
theory, since studies of the interaction of fermion fields with boson fields
and of boson fields with boson fields so far looks very promising.

The study of properties of the second quantized boson fields, the internal
space of which is described by Clifford even algebra has just started and
needs further consideration.

##  Appendix A “Basis vectors” in $d=(3+1)$

This section, suggested by the referee, is to illustrate on a simple case of
$d=(3+1)$ the properties of “basis vectors” when describing internal spaces of
fermions and bosons by the Clifford algebra: i. The way of constructing
the“basis vectors” for fermions which appear in families and for bosons which
have no families. ii. The manifestation of anti-commutativity of the second
quantized fermion fields and commutativity of the second quantized boson
fields. iii. The creation and annihilation operators, described by a tensor
product, $*_{T}$, of the “basis vectors” and their Hermitian conjugated
partners with the basis in ordinary space-time.

This section is a short overview of some sections presented in the article
[18], equipped by concrete examples of “basis vectors” for fermions and bosons
in $d=(3+1)$.

“Basis vectors”

Let us start by arranging the “basis vectors” as a superposition of products
of (operators 171717We repeat that we treat $\gamma^{a}$ as operators, not as
matrices. We write “basis vectors” as the superposition of products of
$\gamma^{a}$. If we want to look for a matrix representation of any operator,
say $S^{ab}$, we arrange the “basis vectors” into a series and write a matrix
of transformations caused by the operator. However, we do not need to look for
the matrix representations of the operators since we can directly calculate
the application of any operators on “basis vectors”.) $\gamma^{a}$, each
“basis vector” is the eigenvector of all the Cartan subalgebra members, Eq.
(8). To achieve this, we arrange “basis vectors” to be products of nilpotents
and projectors, Eqs. (2.2, 2.2), so that every nilpotent and every projector
is the eigenvector of one of the Cartan subalgebra members.

Example 1.
Let us notice that, for example, two nilpotents anti-commute, while one
nilpotent and one projector (or two projectors) commute due to Eq. (2.1):
$\frac{1}{2}(\gamma^{0}-\gamma^{3})\frac{1}{2}(\gamma^{1}-i\gamma^{2})=-\frac{1}{2}(\gamma^{1}-i\gamma^{2})\frac{1}{2}(\gamma^{0}-\gamma^{3})$,
while
$\frac{1}{2}(\gamma^{0}-\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2})=\frac{1}{2}(1+i\gamma^{1}\gamma^{2})\frac{1}{2}(\gamma^{0}-\gamma^{3})$.

In $d=(3+1)$ there are $16\,(2^{d=4})$ “eigenvectors” of the Cartan subalgebra
members ($S^{03},S^{12}$) and (${\bf{\cal S}}^{03},{\bf{\cal S}}^{12}$) of the
Lorentz algebras $S^{ab}$ and ${\bf{\cal S}}^{ab}$ , Eq. (8).

Half of them are the Clifford odd “basis vectors” (and their Hermitian
conjugated partners in a separate group), appearing in two irreducible
representations, in two “families” ($2^{\frac{4}{2}-1}$, $f=(1,2)$), each with
two ($2^{\frac{4}{2}-1}$, $m=(1,2)$) members, $\hat{b}^{m\dagger}_{f}$, Eq.
(49).
There is $2^{\frac{4}{2}-1}\times$$2^{\frac{4}{2}-1}$ (Clifford odd) Hermitian
conjugated partners $\hat{b}^{m}_{f}=(\hat{b}^{m\dagger}_{f})^{\dagger}$
appearing in a separate group which is not reachable by $S^{ab}$, Eq. (53).

There are two separate groups of $2^{\frac{4}{2}-1}\times 2^{\frac{4}{2}-1}$
Clifford even ”basis vectors”, ${}^{i}{\bf{\cal A}}^{m\dagger}_{f},i=(I,II)$,
the $2^{\frac{4}{2}-1}$ members of each are self-adjoint, the rest have their
Hermitian conjugated partners within the same group, Eqs. (58, 62).
All the members of each group are reachable by $S^{ab}$ or $\tilde{S}^{ab}$
from any starting ”basis vector” ${}^{i}{\bf{\cal A}}^{1\dagger}_{1}$.

Example 2.
$\hat{b}^{m=1\dagger}_{f=1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[+]}}(=\frac{1}{2}(\gamma^{0}-\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2}))$
is a Clifford odd “basis vector”, its Hermitian conjugated partner, Eq. (2.1),
is $\hat{b}^{m=1}_{f=1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[+]}}(=\frac{1}{2}(\gamma^{0}+\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2})$,
not reachable by either $S^{ab}$ or by $\tilde{S}^{ab}$ from any of two
members in any of two “families” of the group of $\hat{b}^{m\dagger}_{f}$,
presented in Eq. (49).
${}^{I}{\bf{\cal A}}^{m=1\dagger}_{f=1}(=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[+]}}=\frac{1}{2}(1+\gamma^{0}\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2})$
is self-adjoint, ${}^{I}{\bf{\cal
A}}^{m=2\dagger}_{f=1}(=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(-)}}=\frac{1}{2}(\gamma^{0}+\gamma^{3})(\gamma^{1}-i\gamma^{2})$. Its
Hermitian conjugated partner, belonging to the same group, is ${}^{I}{\bf{\cal
A}}^{m=1\dagger}_{f=2}$ and is reachable from ${}^{I}{\bf{\cal
A}}^{m=1\dagger}_{f=1}$ by the application of $\tilde{S}^{01}$, since
$\tilde{\gamma}^{0}*_{A}\stackrel{{\scriptstyle
03}}{{[+i]}}=i\stackrel{{\scriptstyle 03}}{{(+i)}}$ and
$\tilde{\gamma}^{1}*_{A}\stackrel{{\scriptstyle
12}}{{[+]}}=i\stackrel{{\scriptstyle 12}}{{(+)}}$.

Clifford odd “basis vectors”

Let us first present the Clifford odd anti-commuting “basis vectors”,
appearing in two “families” ${\hat{b}}^{m\dagger}_{f}$, and their Hermitian
conjugated partners $({\hat{b}}^{m\dagger}_{f})^{\dagger}$. Each member of the
two groups is a product of one nilpotent and one projector. We choose the
right-handed Clifford odd “basis vectors” 181818 We could choose the left-
handed Clifford odd “basis vectors” by exchanging the role of ‘basis vectors”
and their Hermitian conjugated partners.. Clifford odd “basis vectors” appear
in two families, each family has two members 191919 In the case of $d=(1+1)$,
we would have one family with one member only, which must be nilpotent.. Let
us notice that members of each of two families have the same quantum numbers
($S^{03}\,,S^{12}$). They distinguish in “family” quantum numbers
($\tilde{S}^{03}\,,\tilde{S}^{12}$).

$\displaystyle\begin{array}[]{ccrr}f=1&f=2&&\\\
\tilde{S}^{03}=\frac{i}{2},\tilde{S}^{12}=-\frac{1}{2}&\;\;\tilde{S}^{03}=-\frac{i}{2},\tilde{S}^{12}=\frac{1}{2}&S^{03}&S^{12}\\\
\hat{b}^{1\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[+]}}&\hat{b}^{1\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}}&\frac{i}{2}&\frac{1}{2}\\\
\hat{b}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(-)}}&\hat{b}^{2\dagger}_{2}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[-]}}&-\frac{i}{2}&-\frac{1}{2}\,.\end{array}$ (49)

We find for their Hermitian conjugated partners

$\displaystyle\begin{array}[]{ccrr}S^{03}=-\frac{i}{2},S^{12}=\frac{1}{2}&\;\;S^{03}=\frac{i}{2},S^{12}=-\frac{1}{2}&\tilde{S}^{03}&\tilde{S}^{12}\\\
\hat{b}^{1}_{1}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[+]}}&\hat{b}^{1}_{2}=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(-)}}&-\frac{i}{2}&-\frac{1}{2}\\\
\hat{b}^{2}_{1}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(+)}}&\hat{b}^{2}_{2}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{[-]}}&\frac{i}{2}&\frac{1}{2}\,.\end{array}$ (53)

The vacuum state $|\psi_{oc}>$, Eq. (67), on which the Clifford odd ”basis
vectors” apply is equal to:
$|\psi_{oc}>=\frac{1}{\sqrt{2}}(\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[+]}}+\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{[-]}})$.

Let us recognize that the Clifford odd ”basis vectors” anti-commute due to the
odd number of nilpotents, Example 1. And they are orthogonal according to Eqs.
(68, 69, C):
$\hat{b}^{m\dagger}_{f}*_{A}\hat{b}^{m^{\prime}\dagger}_{f^{\prime}}=0$.

Example 3.
According to the vacuum state presented above, one finds that, for example,
$\hat{b}^{1\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})|\psi_{oc}>$ is
$\hat{b}^{1\dagger}_{1}$ back, since $\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}*_{A}\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{[+]}}=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}}$, according to Eq. (68), while
$\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{[+]}}*_{A}\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[+]}}=0$ (due to $(\gamma^{0}+\gamma^{3})(1-\gamma^{0}\gamma^{3})=0$).
Let us apply $S^{01}$ and $\tilde{S}^{01}$ on some of the “basis vectors”
$\hat{b}^{m\dagger}_{f}$, say $\hat{b}^{1\dagger}_{1}$.
When applying $S^{01}=\frac{i}{2}\gamma^{0}\gamma^{1}$ on
$\frac{1}{2}(\gamma^{0}-\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2})(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})$ we get
$-\frac{i}{2}\frac{1}{2}(1-\gamma^{0}\gamma^{3})\frac{1}{2}(\gamma^{1}-i\gamma^{2})(\equiv(-\frac{i}{2}\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{(-)}})$.
When applying $\tilde{S}^{01}=\frac{i}{2}\tilde{\gamma}^{0}\tilde{\gamma}^{1}$
on
$\frac{1}{2}(\gamma^{0}-\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2})(\equiv\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})$ we get, according to Eq.
(7), or if using Eq. (2.2),
$-\frac{i}{2}\frac{1}{2}(1+\gamma^{0}\gamma^{3})\frac{1}{2}(\gamma^{1}+i\gamma^{2})(\equiv(-\frac{i}{2}\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}})$.

It then follows, after using Eqs. (2.2, 68, 69, C) or just the starting
relation, Eq. (2.1), and taking into account the above concrete evaluations,
the relations of Eq. (2.2.1) for our particular case

$\displaystyle\hat{b}^{m\dagger}_{f}*_{A}|\psi_{oc}>$ $\displaystyle=$
$\displaystyle|\psi^{m}_{f}>\,,$
$\displaystyle\hat{b}^{m}_{f}*_{A}|\psi_{oc}>$ $\displaystyle=$ $\displaystyle
0\cdot|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m\dagger}_{f},\hat{b}^{m^{\prime}\dagger}_{f^{\prime}}\\}_{-}*_{A}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle 0\cdot|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m}_{f},\hat{b}^{m^{\prime}}_{f^{\prime}}\\}_{-}*_{A}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle 0\cdot|\psi_{oc}>\,,$
$\displaystyle\\{\hat{b}^{m}_{f},\hat{b}^{m^{\prime}\dagger}_{f^{\prime}}\\}_{-}*_{A}|\psi_{oc}>$
$\displaystyle=$ $\displaystyle\delta^{mm^{\prime}}\delta_{ff`}|\psi_{oc}>\,.$
(54)

The last relation of Eq. (A) takes into account that each “basis vector”
carries the “family” quantum number, determined by $\tilde{S}^{ab}$ of the
Cartan subalgebra members, Eq. (8), and the appropriate normalization of
“basis vectors”, Eqs. (49, 53).

Clifford even “basis vectors”

Besides $2^{\frac{4}{2}-1}\times 2^{\frac{4}{2}-1}$ Clifford odd “basis
vectors” and the same number of their Hermitian conjugated partners, Eqs. (49,
53), the Clifford algebra objects offer two groups of $2^{\frac{4}{2}-1}\times
2^{\frac{4}{2}-1}$ Clifford even ”basis vectors”, the members of the group
${}^{I}{\bf{\cal A}}^{m\dagger}_{f}$ and ${}^{II}{\bf{\cal
A}}^{m\dagger}_{f}$, which have Hermitian conjugated partners within the same
group or are self-adjoint 202020 Let be repeated that ${\bf{\cal
S}}^{ab}=S^{ab}+\tilde{S}^{ab}$ [15].. We have the group ${}^{I}{\bf{\cal
A}}^{m\dagger}_{f}$, $m=(1,2),f=(1,2)$, the members of which are Hermitian
conjugated to each other or are self-adjoint,

$\displaystyle\begin{array}[]{crrcrr}&{\bf{\cal S}}^{03}&{\bf{\cal
S}}^{12}&&{\bf{\cal S}}^{03}&{\bf{\cal S}}^{12}\\\ {}^{I}{\bf{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[+]}}&0&0&\,,\quad{}^{I}{\bf{\cal
A}}^{1\dagger}_{2}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(+)}}&i&1\\\ {}^{I}{\bf{\cal A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}}&-i&-1&\,,\quad{}^{I}{\bf{\cal
A}}^{2\dagger}_{2}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[-]}}&0&0\,,\end{array}$ (58)

and the group ${}^{II}{\bf{\cal A}}^{m\dagger}_{f}$, $m=(1,2),f=(1,2)$, the
members of which are either Hermitian conjugated to each other or are self
adjoint

$\displaystyle\begin{array}[]{crrcrr}&{\bf{\cal S}}^{03}&{\bf{\cal
S}}^{12}&&{\bf{\cal S}}^{03}&{\bf{\cal S}}^{12}\\\ {}^{II}{\bf{\cal
A}}^{1\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[-]}}&0&0&\,,\quad{}^{II}{\bf{\cal
A}}^{1\dagger}_{2}=\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle
12}}{{(-)}}&i&-1\\\ {}^{II}{\bf{\cal
A}}^{2\dagger}_{1}=\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(+)}}&-i&1&\,,\quad{}^{II}{\bf{\cal
A}}^{2\dagger}_{2}=\stackrel{{\scriptstyle 03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{[+]}}&0&0\,.\end{array}$ (62)

The Clifford even “basis vectors” have no families. The two groups,
${}^{I}{\bf{\cal A}}^{m\dagger}_{f}$ and ${}^{II}{\bf{\cal A}}^{m\dagger}_{f}$
(they are not reachable from one another by ${\bf{\cal S}}^{ab}$), are
orthogonal (which can easily be checked, since $\stackrel{{\scriptstyle
ab}}{{(\pm k)}}*_{A}\stackrel{{\scriptstyle ab}}{{(\pm k)}}=0$, and
$\stackrel{{\scriptstyle ab}}{{[\pm k]}}*_{A}\stackrel{{\scriptstyle
ab}}{{[\mp k]}}=0$).

$\displaystyle{}^{I}{\bf{\cal A}}^{m\dagger}_{f}*_{A}{}^{II}{\bf{\cal
A}}^{m^{\prime}\dagger}_{f`}=0,\quad{\rm for\;any}\;(m,m^{\prime},f,f`)\,.$
(63)

Application of ${}^{i}{\bf{\cal A}}^{m\dagger}_{f},i=(I,II)$ on
$\hat{b}^{m\dagger}_{f}$

Let us demonstrate the application of ${}^{i}{\bf{\cal
A}}^{m\dagger}_{f},i=(I,II)$, on the Clifford odd “basis vectors”
$\hat{b}^{m\dagger}_{f}$, Eqs. (24, 29), for our particular case $d=(3+1)$ and
compare the result with the result of application $S^{ab}$ and
$\tilde{S}^{ab}$ on $\hat{b}^{m\dagger}_{f}$ evaluated above in Example 3. We
found, for example, that
$S^{01}(=\frac{i}{2}\gamma^{0}\gamma^{1})*_{A}\hat{b}^{1\dagger}_{1}(=$
$\frac{1}{2}(\gamma^{0}-\gamma^{3})\frac{1}{2}(1+i\gamma^{1}\gamma^{2})(=\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})=$
$-\frac{i}{2}\frac{1}{2}(1-\gamma^{0}\gamma^{3})\frac{1}{2}(\gamma^{1}-i\gamma^{2})(=(-\frac{i}{2}\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle
12}}{{(-i)}})=-\frac{i}{2}\hat{b}^{2\dagger}_{1}$.

Applying ${}^{I}{\bf{\cal A}}^{2\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(-)}})*_{A}\,\hat{b}^{1\dagger}_{1}(=$ $\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})=-\stackrel{{\scriptstyle
03}}{{[-i]}}\stackrel{{\scriptstyle 12}}{{(-)}}$, which is
$-\hat{b}^{2\dagger}_{1}$, presented in Eq. (49). We obtain in both cases the
same result, up to the factor $\frac{i}{2}$ (in front of
$\gamma^{0}\gamma^{1}$ in $S^{01}$). In the second case one sees that
${}^{I}{\bf{\cal A}}^{2\dagger}_{1}$ (carrying ${\cal S}^{03}=-i,{\cal
S}^{12}=-1$) transfers these quantum numbers to $\hat{b}^{1\dagger}_{1}$
(carrying ${S}^{03}=\frac{i}{2},{S}^{12}=\frac{1}{2}$) what results in
$\hat{b}^{2\dagger}_{1}$ (carrying
${S}^{03}=\frac{-i}{2},{S}^{12}=\frac{-1}{2}$).

We can check what the application of the rest three ${}^{I}{\bf{\cal
A}}^{m\dagger}_{f}$, do when applying on $\hat{b}^{m\dagger}_{f}$. The self-
adjoint member carrying ${\cal S}^{03}=0,{\cal S}^{12}=0$, either gives
$\hat{b}^{m\dagger}_{f}$ back, or gives zero, according to Eq. (68). The
Clifford even “basis vectors”, carrying non zero ${\cal S}^{03}$ and ${\cal
S}^{12}$ transfer their internal values to $\hat{b}^{m\dagger}_{f}$ or give
zero. In all cases ${}^{I}{\bf{\cal A}}^{m\dagger}_{f}$ transform a “family”
member to another or the same “family” member of the same “family”.

Example 4.:
${}^{I}{\bf{\cal A}}^{1\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[+]}})*_{A}\,\hat{b}^{1\dagger}_{1}(=$ $\stackrel{{\scriptstyle
03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})=\hat{b}^{1\dagger}_{1}(=$
$\stackrel{{\scriptstyle 03}}{{(+i)}}\stackrel{{\scriptstyle 12}}{{[+]}})$ ,
${}^{I}{\bf{\cal A}}^{1\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{[+]}})*_{A}\,\hat{b}^{1\dagger}_{2}(=$ $\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}})=\hat{b}^{1\dagger}_{2}(=$
$\stackrel{{\scriptstyle 03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}})$ ,
${}^{I}{\bf{\cal A}}^{2\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle
12}}{{(-)}})*_{A}\,\hat{b}^{1\dagger}_{2}(=$ $\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle 12}}{{(+)}})=-\hat{b}^{2\dagger}_{2}(=$
$\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{[-]}})$ ,
${}^{I}{\bf{\cal A}}^{2\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(-)}})*_{A}\hat{b}^{2\dagger}_{2}(=$
$\stackrel{{\scriptstyle 03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{[-]}})=0$.

One easily sees that the application of ${}^{II}{\bf{\cal A}}^{m\dagger}_{f}$
on $\hat{b}^{m^{\prime}\dagger}_{f`}$ gives zero for all
$(m,m^{\prime},f,f^{\prime})$ (due to $\stackrel{{\scriptstyle ab}}{{[\pm
k]}}*_{A}\stackrel{{\scriptstyle ab}}{{[\mp k]}}=0$, $\stackrel{{\scriptstyle
ab}}{{[\pm k]}}*_{A}\stackrel{{\scriptstyle ab}}{{(\mp k)}}=0$, and similar
applications).

We realised in Example 3. that the application of
$\tilde{S}^{01}=\frac{i}{2}\tilde{\gamma}^{0}\tilde{\gamma}^{1}$ on
$\hat{b}^{1\dagger}_{1}$ gives $(-\frac{i}{2}\stackrel{{\scriptstyle
03}}{{[+i]}}\stackrel{{\scriptstyle
12}}{{(+i)}})=-\frac{i}{2}\hat{b}^{1\dagger}_{2}$.

Let us algebraically, $*_{A}$, apply ${}^{II}{\bf{\cal
A}}^{2\dagger}_{1}(=\stackrel{{\scriptstyle
03}}{{(-i)}}\stackrel{{\scriptstyle 12}}{{(+)}}$), with quantum numbers
$({\cal S}^{03},{\cal S}^{12})=(-i,1)$, from the right hand side the Clifford
odd “basis vector” $\hat{b}^{1\dagger}_{1}$. This application causes the
transition of $\hat{b}^{1\dagger}_{1}$ (with quantum numbers
$(\tilde{S}^{03},\tilde{S}^{12})=(\frac{i}{2},-\frac{1}{2})$ (see Eq. (2.2))
into $\hat{b}^{1\dagger}_{2}$ (with quantum numbers
$(\tilde{S}^{03},\tilde{S}^{12})=(-\frac{i}{2},\frac{1}{2})$).
${}^{II}{\bf{\cal A}}^{2\dagger}_{1}$ obviously transfers its quantum numbers
to Clifford odd “basis vectors”, keeping $m$ unchanged, and changing the
“family” quantum number: $\hat{b}^{1\dagger}_{1}*_{A}{}^{II}{\bf{\cal
A}}^{2\dagger}_{1}=\hat{b}^{1\dagger}_{2}$.

We can conclude: The internal space of the Clifford even “basis vectors” has
properties of the gauge fields of the Clifford odd “basis vectors”;
${}^{I}{\bf{\cal A}}^{m\dagger}_{f}$ transform “family” members of the
Clifford odd “basis vectors” among themselves, keeping the “family” quantum
number unchanged, ${}^{II}{\bf{\cal A}}^{m\dagger}_{f}$ transform a particular
“family” member into the same “family” member of another “family”.

Creation and annihilation operators

To define creation and annihilation operators for fermion and boson fields, we
must include besides the internal space, the ordinary space, presented in Eq.
(2.4), which defines the momentum or coordinate part of fermion and boson
fields.

We define the creation operators for the single particle fermion states as a
tensor product, $*_{T}$, of the Clifford odd “basis vectors” and the basis in
ordinary space, Eq. (39):
${\bf\hat{b}}^{s\dagger}_{f}(\vec{p})=\sum_{m}c^{sm}{}_{f}(\vec{p})\,\hat{b}^{\dagger}_{\vec{p}}\,*_{T}\,\hat{b}^{m\dagger}_{f}$.
The annihilation operators are their Hermitian conjugated partners.

We have seen in Example 1. that Clifford odd “basis vectors” (having odd
products of nilpotents) anti-commute. The commuting objects
$\hat{b}^{\dagger}_{\vec{p}}$ (multiplying the “basis vectors”) do not change
the Clifford oddness of ${\bf\hat{b}}^{s\dagger}_{f}(\vec{p})$. The two
Clifford odd objects, ${\bf\hat{b}}^{s\dagger}_{f}(\vec{p})$ and
${\bf\hat{b}}^{s^{\prime}\dagger}_{f`}(\vec{p^{\prime}})$, keep their anti-
commutativity, fulfilling the anti-commutation relations as presented in Eq.
(40). Correspondingly we do not need to postulate anti-commutation relations
of Dirac. The Clifford odd “basis vectors” in a tensor product with the basis
in ordinary space explain the second quantized postulates for fermion fields.

The Clifford odd “basis vectors” contribute for each $\vec{p}$ a finite number
of ${\bf\hat{b}}^{s\dagger}_{f}(\vec{p})$, the ordinary basis offers infinite
possibilities 212121 An infinitesimally small difference between $\vec{p}$ and
$\vec{p^{\prime}}$ makes two creation operators
${\bf\hat{b}}^{s\dagger}_{f}(\vec{p})$ and
${\bf\hat{b}}^{s\dagger}_{f}(\vec{p^{\prime}})$ with the same “basis vector”
describing the internal space of fermion fields still fulfilling the anti-
commutation relations (as we learn from atomic physics; two electrons can
carry the same spin if they distinguish in the coordinate part of the state)..

Recognizing that internal spaces of fermion fields and their corresponding
boson gauge fields are describable in even dimensional spaces by the Clifford
odd and even “basis vectors”, respectively, it becomes evidently that when
including the basis in ordinary space, we must take into account that boson
gauge fields have the space index $\alpha$, which describes the $\alpha$
component of the boson fields in ordinary space.

We multiply, therefore, as presented in Eq. (41), the Clifford even “basis
vectors” with the coefficient ${}^{i}{\cal C}^{m}{}_{f\alpha}$ carrying the
space index $\alpha$ so that the creation operators ${\bf{}^{i}{\hat{\cal
A}}^{m\dagger}_{f\alpha}}(\vec{p})=\hat{b}^{\dagger}_{\vec{p}}\,*_{T}\,{}^{i}{\cal
C}^{m}{}_{f\alpha}\,{}^{i}{\hat{\cal A}}^{m\dagger}_{f}\,\,,i=(I,II)$ carry
the space index $\alpha$ 222222 Requiring the local phase symmetry for the
fermion part of the action, Eq. (1), would lead to the requirement of the
existence of the boson fields with the space index $\alpha$.. The self-adjoint
“basis vectors”, like (${}^{i}{\hat{\cal
A}}^{1\dagger}_{1\alpha},{}^{i}{\hat{\cal A}}^{2\dagger}_{2\alpha},i=(I,II)$),
do not change quantum numbers of the Clifford odd “basis vectors”, since they
have internal quantum numbers equal to zero.

In higher dimensional space, like in $d=(5+1)$, ${}^{I}{\hat{\cal
A}}^{1\dagger}_{3}$, presented in Table 3, could represent the internal space
of a photon field, which transfers to, for example, a fermion and anti-fermion
pair with the internal space described by ($\hat{b}^{1\dagger}_{1}$,
$\hat{b}^{3\dagger}_{1}$), presented in Table 2, the momentum in ordinary
space.

The subgroup structure of $SU(3)$ gauge fields can be recognized in Fig. 2.

Properties of the gauge fields ${}^{i}{\hat{\cal A}}^{m\dagger}_{f\alpha}$
need further studies.

In even dimensional spaces, the Clifford odd and even “basis vectors”,
describing internal spaces of fermion and boson fields, offer the explanation
for the second quantized postulates for fermion and boson fields [17].

##  Appendix B Discussion on the open questions of the standard model and
answers offered by the spin-charge-family theory

There are many suggestions in the literature for unifying charges in larger
groups, adding additional groups for describing families [1, 2, 3, 4, 5], or
by going to higher dimensional spaces of the Kaluza-Kline like theories [24,
25, 26, 27, 28, 29, 31, 30], what also the spin-charge-family is.

Let me present some open questions of the standard model and briefly tell the
answers offered by the spin-charge family theory.

A. Where do fermions — quarks and leptons and antiquarks and antileptons — and
their families originate?
The answer offered by the spin-charge-family theory: In $d=(13+1)$ one
irreducible representation of $SO(13,1)$ analysed with respect to subgroups
$SO(7,1)$ (containing subgroups of $SO(3,1)\times SU(2)\times SU(2)$) and
$SO(6)$ (containing subgroups of $SU(3)\times U(1)$) offers the Clifford odd
“basis vectors”, describing the internal spaces of quarks and leptons and
antiquarks and antileptons, Table 4, as assumed by the standard model. The
Clifford odd “basis vectors” appear in families.
B. Why are charges of quarks so different from charges of leptons, and why
have left-handed family members so different charges from the right-handed
ones?
The answer offered by the spin-charge-family theory: The $SO(7,1)$ part of the
“basis vectors” is identical for quarks and leptons and identical for
antiquarks and antileptons, Table 4, they distinguish only in the $SU(3)$, the
colour or anticolour part, and in the fermion or antifermion $U(1)$ quantum
numbers. All families have the same content of $SO(7,1),SU(3)$ and $U(1)$ with
respect to $S^{ab}$. They distinguish only in the family quantum number,
determined by $\tilde{S}^{ab}$. The difference between left-handed and right-
handed members appears due to the difference in one quantum numbers of the two
$SU(2)$ groups, as seen in Table 4.
C. Why do family members — quarks and leptons — manifest such different masses
if they all start as massless, as (elegantly) assumed by the standard model?
The answer offered by the spin-charge-family theory: Masses of quarks and
leptons are in this theory determined by the spin connection fields
$\omega_{st\sigma}$, the gauge fields of $S^{ab}$ 232323 The three $U(1)$
singlets, the gauge fields of the “fermion” quantum number $\tau^{4}$, of the
hypercharge $Y$, and of the electromagnetic charge $Q$, determine the
difference in masses of quarks and leptons, presented in Table 4, Ref. ([14],
Sect, 6.2.2, Eq. (108)), and by $\tilde{\omega}_{st\sigma}$, the gauge fields
of $\tilde{S}^{ab}$, which are the same for quarks and leptons 242424 The two
times two $\widetilde{SU}(2)$ triplets are the same for quarks and leptons,
forming two groups of four families. Ref. ([14], Sect, 6.2.2, Eq. (108)..
Triplets and singlets are scalar gauge fields with the space index
$\sigma=(7,8)$. They have, with respect to the space index, the quantum
numbers of the Higgs scalars, Ref. ([14], Table 8, Eq. (110,111)).
D. What is the origin of boson fields, of vector fields which are the gauge
fields of fermions, and the Higgs’ scalars and the Yukawa couplings? Have all
boson fields, with gravity and scalar fields included a common origin?
The answer offered by the spin-charge-family theory: In a simple starting
action, Eq. (1), boson fields origin in gravity — in vielbeins and two kinds
of spin connection fields, $\omega_{ab\alpha}$ and
$\tilde{\omega}_{ab\alpha}$, in $d=(13+1)$ — and manifest in $d=(3+1)$ as
vector gauge fields, $\alpha=(0,1,2,3)$, or scalar gauge fields, $\alpha\geq
5$ [11], ([14], Sect. 6 and references therein). Boson gauge fields are
massless as there are fermion fields. The breaks of the starting symmetry
makes some gauge fields massive. This article describes the internal space of
boson fields by the Clifford even basis vectors, manifesting as the boson
gauge fields of the corresponding fermion fields described by the Clifford odd
“basis vectors”. The description of the boson fields with the Clifford even
“basis vectors” confirms the existence of two kinds of spin connection fields
as we see in Sects. 2.2 and2.3, but also open a door to a new understanding of
gravity. According to the starting action, Eq. (1), all gauge fields start in
$d\geq(13+1)$ as gravity.
E. How are scalar fields connected with the origin of families? How many
scalar fields determine properties of the so far (and others possibly be)
observed fermions and of weak bosons?
The answer offered by the spin-charge-family theory: The interaction between
quarks and leptons and the scalar gauge fields, which at the electroweak brake
obtain constant values, causes that quarks and leptons and the weak bosons
become massive. There are three singlets, they distinguish among quarks and
leptons, and two triplets, they do not distinguish among quarks and leptons,
which give masses to the lower four families 252525 There are the same three
singlets and two additional triplets, which determine the masses of the upper
four families- explaining the existence of the dark matter..
F. Where does the dark matter originate?
The answer offered by the spin-charge-family theory: The theory predicts two
groups of four families at low energy. The stable of the upper four groups are
candidates to form the dark matter [35].
G. Where does the “ordinary” matter-antimatter asymmetry originate?
The answer offered by the spin-charge-family theory: The theory predicts
scalars triplets and antitriplets with the space index
$\alpha=(9,10,11,12,13,14)$ [8].
H. How can we understand the second quantized fermion and boson fields?
The answer offered by the spin-charge-family theory: The main contribution of
this article, Sect. 2, is the description of the internal spaces of fermion
and boson fields with the superposition of odd (for fermions) and even (for
bosons) products of $\gamma^{a}$. The corresponding creation and annihilation
operators, which are tensor, $*_{T}$, products of (finite number) “basis
vectors” and (infinite) basis in ordinary space inherit anti-commutativity or
commutativity from the corresponding “basis vectors”, explaining the
postulates for the second quantized fermion and boson fields.
I. What is the dimension of space? $(3+1)?$, $((d-1)+1)?$, $\infty?$
The answer offered by the spin-charge-family theory: We observe
$(3+1)$-dimensional space. In order that one irreducible representation (one
family) of the Clifford odd “basis vectors”, analysed with respect to
subgroups $SO(3,1)\times$ $SO(4)$ $\times SU(3)$ $\times U(1)$ of the group
$SO(13,1)$ includes all quarks and leptons and antiquarks and antileptons, the
space must have $d\geq(13+1)$. (Since the only “elegantly” acceptable numbers
are $0$ and $\infty$, the space-time could be $\infty$.)
The $SO(10)$ theory [2], for example, unifies the charges of fermions and
bosons separately. Analysing $SO(10)$ with respect to the corresponding
subgroups, the charges of fermions appear in fundamental representations and
bosons in adjoint representations 262626The space-time is in unifying theories
$(3+1)$, consequently they have to relate handedness and charges “by hand”
[22], postulate the existence of antiparticles, and the existence of scalar
fields, as does the standard model..
There are additional open questions answers of which the spin-charge-family
the theory offers.

The spin-charge-family theory has to answer the question common to all the
Kaluza-Klein-like theories: How and why the space we observe has $d=(3+1)$
dimensions? The proposed description of the internal spaces of fermion and
boson fields might help.

##  Appendix C Some useful relations in Grassmann and Clifford algebras,
needed also in App. D

This appendix contains the helpful relations needed for the reader of this
paper. For more detailed explanations and for proofs, the reader is kindly
asked to read [14] and the references therein.

For fermions, the operator of handedness $\Gamma^{d}$ is determined as
follows:

$\displaystyle\Gamma^{(d)}=\prod_{a}(\sqrt{\eta^{aa}}\gamma^{a})\cdot\left\\{\begin{array}[]{l
l}(i)^{\frac{d}{2}}\,,&\rm{for\,d\,even}\,,\\\
(i)^{\frac{d-1}{2}}\,,&\rm{for\,d\,odd}\,,\end{array}\right.$ (66)

The vacuum state for the Clifford odd ”basis vectors”, $|\psi_{oc}>$, is
defined as

$\displaystyle|\psi_{oc}>=\sum_{f=1}^{2^{\frac{d}{2}-1}}\,\hat{b}^{m}_{f}{}_{*_{A}}\hat{b}^{m\dagger}_{f}\,|\,1\,>\,.$
(67)

Taking into account that the Clifford objects $\gamma^{a}$ and
$\tilde{\gamma}^{a}$ fulfil relations of Eq. 2.1, one obtains beside the
relations presented in Eq. (2.2) the following ones

$\displaystyle\stackrel{{\scriptstyle ab}}{{(k)}}\stackrel{{\scriptstyle
ab}}{{(-k)}}$ $\displaystyle=$ $\displaystyle\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[k]}}\,,\quad\stackrel{{\scriptstyle ab}}{{(-k)}}\stackrel{{\scriptstyle
ab}}{{(k)}}=\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[-k]}}\,,\quad\stackrel{{\scriptstyle ab}}{{(k)}}\stackrel{{\scriptstyle
ab}}{{[k]}}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{(k)}}\stackrel{{\scriptstyle ab}}{{[-k]}}=\stackrel{{\scriptstyle
ab}}{{(k)}}\,,\quad$ $\displaystyle\stackrel{{\scriptstyle
ab}}{{(-k)}}\stackrel{{\scriptstyle ab}}{{[k]}}$ $\displaystyle=$
$\displaystyle\stackrel{{\scriptstyle
ab}}{{(-k)}}\,,\quad\stackrel{{\scriptstyle ab}}{{[k]}}\stackrel{{\scriptstyle
ab}}{{(k)}}=\stackrel{{\scriptstyle ab}}{{(k)}}\,,\quad\stackrel{{\scriptstyle
ab}}{{[k]}}\stackrel{{\scriptstyle
ab}}{{(-k)}}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{[k]}}\stackrel{{\scriptstyle ab}}{{[-k]}}=0\,,\quad$
$\displaystyle\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle ab}}{{(k)}}$ $\displaystyle=$
$\displaystyle 0\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle
ab}}{{(-k)}}=-i\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[-k]}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\widetilde{(-k)}}}\stackrel{{\scriptstyle
ab}}{{(k)}}=-i\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[k]}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle
ab}}{{[k]}}=i\stackrel{{\scriptstyle ab}}{{(k)}}\,,\quad$
$\displaystyle\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle ab}}{{[-k]}}$ $\displaystyle=$
$\displaystyle 0\,,\quad\stackrel{{\scriptstyle
ab}}{{\widetilde{(-k)}}}\stackrel{{\scriptstyle
ab}}{{[k]}}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{\widetilde{(-k)}}}\stackrel{{\scriptstyle
ab}}{{[-k]}}=i\stackrel{{\scriptstyle
ab}}{{(-k)}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle ab}}{{(k)}}=\stackrel{{\scriptstyle
ab}}{{(k)}}\,,$ $\displaystyle\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle ab}}{{(-k)}}$ $\displaystyle=$
$\displaystyle 0\,,\quad\stackrel{{\scriptstyle
ab}}{{\widetilde{[k]}}}\stackrel{{\scriptstyle
ab}}{{[k]}}=0\,,\quad\stackrel{{\scriptstyle
ab}}{{\widetilde{[-k]}}}\stackrel{{\scriptstyle
ab}}{{[k]}}=\stackrel{{\scriptstyle ab}}{{[k]}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle
ab}}{{[-k]}}=\stackrel{{\scriptstyle ab}}{{[-k]}}\,,$ (68)

The algebraic multiplication among $\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}$ and $\stackrel{{\scriptstyle ab}}{{\tilde{[k]}}}$ goes as
in the case of $\stackrel{{\scriptstyle ab}}{{(k)}}$ and
$\stackrel{{\scriptstyle ab}}{{[k]}}$

$\displaystyle\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle ab}}{{\tilde{[k]}}}$
$\displaystyle=$ $\displaystyle 0\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}=\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\stackrel{{\scriptstyle
ab}}{{\tilde{[-k]}}}=\stackrel{{\scriptstyle
ab}}{{\tilde{(k)}}}\,,\quad\,\stackrel{{\scriptstyle
ab}}{{\tilde{[k]}}}\stackrel{{\scriptstyle ab}}{{\tilde{(-k)}}}=0\,,$
$\displaystyle\stackrel{{\scriptstyle
ab}}{{\widetilde{(-k)}}}\stackrel{{\scriptstyle ab}}{{\tilde{(k)}}}$
$\displaystyle=$ $\displaystyle\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[-k]}}\,,\quad\stackrel{{\scriptstyle
ab}}{{\widetilde{(-k)}}}\stackrel{{\scriptstyle
ab}}{{\widetilde{[-k]}}}=0\,,\quad$ (69)

One can further find that

$\displaystyle S^{ac}\stackrel{{\scriptstyle
ab}}{{(k)}}\stackrel{{\scriptstyle cd}}{{(k)}}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\eta^{aa}\eta^{cc}\stackrel{{\scriptstyle
ab}}{{[-k]}}\stackrel{{\scriptstyle cd}}{{[-k]}}\,,\quad
S^{ac}\stackrel{{\scriptstyle ab}}{{[k]}}\stackrel{{\scriptstyle
cd}}{{[k]}}=\frac{i}{2}\stackrel{{\scriptstyle
ab}}{{(-k)}}\stackrel{{\scriptstyle cd}}{{(-k)}}\,,$ $\displaystyle
S^{ac}\stackrel{{\scriptstyle ab}}{{(k)}}\stackrel{{\scriptstyle cd}}{{[k]}}$
$\displaystyle=$ $\displaystyle-\frac{i}{2}\eta^{aa}\stackrel{{\scriptstyle
ab}}{{[-k]}}\stackrel{{\scriptstyle cd}}{{(-k)}}\,,\quad
S^{ac}\stackrel{{\scriptstyle ab}}{{[k]}}\stackrel{{\scriptstyle
cd}}{{(k)}}=\frac{i}{2}\eta^{cc}\stackrel{{\scriptstyle
ab}}{{(-k)}}\stackrel{{\scriptstyle cd}}{{[-k]}}\,.$ (70)

##  Appendix D One family representation of Clifford odd “basis vectors” in
$d=(13+1)$

This appendix, is following App. D of Ref. [18], with a short comment on the
corresponding gauge vector and scalar fields and fermion and boson
representations in $d=(14+1)$-dimensional space included.

In even dimensional space $d=(13+1)$ ([15], App. A), one irreducible
representation of the Clifford odd “basis vectors”, analysed from the point of
view of the subgroups $SO(3,1)\times SO(4)$ (included in $SO(7,1)$) and
$SO(7,1)\times SO(6)$ (included in $SO(13,1)$, while $SO(6)$ breaks into
$SU(3)\times U(1)$), contains the Clifford odd “basis vectors” describing
internal spaces of quarks and leptons and antiquarks, and antileptons with the
quantum numbers assumed by the standard model before the electroweak break.
Since $SO(4)$ contains two $SU(2)$ groups, $Y=\tau^{23}+\tau^{4}$, one
irreducible representation includes the right-handed neutrinos and the left-
handed antineutrinos, which are not in the standard model scheme.

The Clifford even “basis vectors”, analysed to the same subgroups, offer the
description of the internal spaces of the corresponding vector and scalar
fields, appearing in the standard model before the electroweak break [16, 15];
as explained in Subsect. 2.2.1.

For an overview of the properties of the vector and scalar gauge fields in the
spin-charge-family theory, the reader is invited to see Refs. ([14, 11] and
the references therein). The vector gauge fields, expressed as the
superposition of spin connections and vielbeins, carrying the space index
$m=(0,1,2,3)$, manifest properties of the observed boson fields. The scalar
gauge fields, causing the electroweak break, carry the space index $s=(7,8)$
and determine the symmetry of mass matrices of quarks and leptons. [23,
NH2006, nh2008]).

In this Table 4, one can check the quantum numbers of the Clifford odd “basis
vectors” representing quarks and leptons and antiquarks and antileptons if
taking into account that all the nilpotents and projectors are eigenvectors of
one of the Cartan subalgebra members, ($S^{03},S^{12},S^{56},\dots,$
$S^{13\,14}$), with the eigenvalues $\pm\frac{i}{2}$ for
$\stackrel{{\scriptstyle ab}}{{(\pm i)}}$ and $\stackrel{{\scriptstyle
ab}}{{[\pm i]}}$, and with the eigenvalues $\pm\frac{1}{2}$ for
$\stackrel{{\scriptstyle ab}}{{(\pm 1)}}$ and $\stackrel{{\scriptstyle
ab}}{{[\pm 1]}}$.

Taking into account that the third component of the weak charge,
$\tau^{13}=\frac{1}{2}(S^{56}-S^{78})$, for the second $SU(2)$ charge,
$\tau^{23}=\frac{1}{2}(S^{56}+S^{78})$, for the colour charge
[$\tau^{33}=\frac{1}{2}(S^{9\,10}-S^{11\,12})$ and
$\tau^{38}=\frac{1}{2\sqrt{3}}(S^{9\,10}+S^{11\,12}-2S^{13\,14})$], for the
“fermion charge” $\tau^{4}=-\frac{1}{3}(S^{9\,10}+S^{11\,12}+S^{13\,14})$, for
the hyper charge $Y=\tau^{23}+\tau^{4}$, and electromagnetic charge
$Q=Y+\tau^{13}$, one reproduces all the quantum numbers of quarks, leptons,
and antiquarks, and antileptons. One notices that the $SO(7,1)$ part is the
same for quarks and leptons and the same for antiquarks and antileptons.
Quarks distinguish from leptons only in the colour and “fermion” quantum
numbers and antiquarks distinguish from antileptons only in the anti-colour
and “anti-fermion” quantum numbers.

In odd dimensional space, $d=(14+1)$, the eigenstates of handedness are the
superposition of one irreducible representation of $SO(13,1)$, presented in
Table 4, and the one obtained if on each “basis vector” appearing in
$SO(13,1)$ the operator $S^{0\,(14+1)}$ applies, Subsect. 2.2.2, Ref. [18].

Let me point out that in addition to the electroweak break of the standard
model the break at $\geq 10^{16}$ GeV is needed ([14], and references
therein). The condensate of the two right-handed neutrinos causes this break
(Ref. [14], Table 6); it interacts with all the scalar and vector gauge
fields, except the weak, $U(1),SU(3)$ and the gravitational field in
$d=(3+1)$, leaving these gauge fields massless up to the electroweak break,
when the scalar fields, leaving massless only the electromagnetic, colour and
gravitational fields, cause masses of fermions and weak bosons.

The theory predicts two groups of four families: To the lower group of four
families, the three so far observed contribute. The theory predicts the
symmetry of both groups to be $SU(2)\times SU(2)\times U(1)$, Ref. ([14],
Sect. 7.3), which enable to calculate mixing matrices of quarks and leptons
for the accurately enough measured $3\times 3$ sub-matrix of the $4\times 4$
unitary matrix. No sterile neutrinos are needed, and no symmetry of the mass
matrices must be guessed [36].

In the literature, one finds a lot of papers trying to reproduce mass matrices
and measured mixing matrices for quarks and leptons [41, 42, 43, 44, 45, 47,
46].

The stable of the upper four families predicted by the spin-charge-family
theory is a candidate for the dark matter, as discussed in Refs. [35, 14]. In
the literature, there are several works suggesting candidates for the dark
matter and also for matter/antimatter asymmetry [49, 48].

i |  | $|^{a}\psi_{i}>$ | $\Gamma^{(3,1)}$ | $S^{12}$ | $\tau^{13}$ | $\tau^{23}$ | $\tau^{33}$ | $\tau^{38}$ | $\tau^{4}$ | $Y$ | $Q$
---|---|---|---|---|---|---|---|---|---|---|---
|  | ${\rm(Anti)octet},\,\Gamma^{(7,1)}=(-1)\,1\,,\,\Gamma^{(6)}=(1)\,-1$ |  |  |  |  |  |  |  |  |
|  | ${\rm of\;(anti)quarks\;and\;(anti)leptons}$ |  |  |  |  |  |  |  |  |
1 | $u_{R}^{c1}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
2 | $u_{R}^{c1}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
3 | $d_{R}^{c1}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
4 | $d_{R}^{c1}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
5 | $d_{L}^{c1}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
6 | $d_{L}^{c1}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
7 | $u_{L}^{c1}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
8 | $u_{L}^{c1}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
9 | $u_{R}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
10 | $u_{R}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
11 | $d_{R}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
12 | $d_{R}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
13 | $d_{L}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
14 | $d_{L}^{c2}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
15 | $u_{L}^{c2}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
16 | $u_{L}^{c2}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2\,\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
17 | $u_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
18 | $u_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $\frac{2}{3}$
19 | $d_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
20 | $d_{R}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $-\frac{1}{3}$ | $-\frac{1}{3}$
21 | $d_{L}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
22 | $d_{L}^{c3}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{3}$
23 | $u_{L}^{c3}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
24 | $u_{L}^{c3}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{2}{3}$
25 | $\nu_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $0$ | $0$
26 | $\nu_{R}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $0$ | $0$
27 | $e_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $-1$ | $-1$
28 | $e_{R}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $-\frac{1}{2}$ | $-1$ | $-1$
29 | $e_{L}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$
30 | $e_{L}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$
31 | $\nu_{L}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $0$
32 | $\nu_{L}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $0$
33 | $\bar{d}_{L}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
34 | $\bar{d}_{L}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
35 | $\bar{u}_{L}^{\bar{c1}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
36 | $\bar{u}_{L}^{\bar{c1}}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
37 | $\bar{d}_{R}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
38 | $\bar{d}_{R}^{\bar{c1}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
39 | $\bar{u}_{R}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
40 | $\bar{u}_{R}^{\bar{c1}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
41 | $\bar{d}_{L}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
42 | $\bar{d}_{L}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
43 | $\bar{u}_{L}^{\bar{c2}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
44 | $\bar{u}_{L}^{\bar{c2}}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
45 | $\bar{d}_{R}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
46 | $\bar{d}_{R}^{\bar{c2}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
47 | $\bar{u}_{R}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
48 | $\bar{u}_{R}^{\bar{c2}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{(+)}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2\,\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
49 | $\bar{d}_{L}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
50 | $\bar{d}_{L}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $\frac{1}{3}$ | $\frac{1}{3}$
51 | $\bar{u}_{L}^{\bar{c3}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
52 | $\bar{u}_{L}^{\bar{c3}}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$
53 | $\bar{d}_{R}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
54 | $\bar{d}_{R}^{\bar{c3}}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $\frac{1}{3}$
55 | $\bar{u}_{R}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
56 | $\bar{u}_{R}^{\bar{c3}}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{(+)}}\;\;\stackrel{{\scriptstyle 11\;12}}{{(+)}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{2}{3}$
57 | $\bar{e}_{L}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $1$ | $1$
58 | $\bar{e}_{L}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $1$ | $1$
59 | $\bar{\nu}_{L}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $0$ | $0$
60 | $\bar{\nu}_{L}$ | $-\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | -1 | $-\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $0$ | $0$
61 | $\bar{\nu}_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$
62 | $\bar{\nu}_{R}$ | $-\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{(-)}}\,\stackrel{{\scriptstyle 78}}{{(+)}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $-\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$
63 | $\bar{e}_{R}$ | $\stackrel{{\scriptstyle 03}}{{(+i)}}\,\stackrel{{\scriptstyle 12}}{{[+]}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$
64 | $\bar{e}_{R}$ | $\stackrel{{\scriptstyle 03}}{{[-i]}}\,\stackrel{{\scriptstyle 12}}{{(-)}}|\stackrel{{\scriptstyle 56}}{{[+]}}\,\stackrel{{\scriptstyle 78}}{{[-]}}||\stackrel{{\scriptstyle 9\;10}}{{[-]}}\;\;\stackrel{{\scriptstyle 11\;12}}{{[-]}}\;\;\stackrel{{\scriptstyle 13\;14}}{{[-]}}$ | 1 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $0$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$
Table 4:  The left-handed ($\Gamma^{(13,1)}=-1$, Eq. (66)) irreducible
representation of one family of spinors — the product of the odd number of
nilpotents and of projectors, which are eigenvectors of the Cartan subalgebra
of the $SO(13,1)$ group [8, 10], manifesting the subgroup $SO(7,1)$ of the
colour charged quarks and antiquarks and the colourless leptons and
antileptons — is presented. It contains the left-handed ($\Gamma^{(3,1)}=-1$)
weak ($SU(2)_{I}$) charged ($\tau^{13}=\pm\frac{1}{2}$), and $SU(2)_{II}$
chargeless ($\tau^{23}=0$) quarks and leptons, and the right-handed
($\Gamma^{(3,1)}=1$) weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$ charged
($\tau^{23}=\pm\frac{1}{2}$) quarks and leptons, both with the spin $S^{12}$
up and down ($\pm\frac{1}{2}$, respectively). Quarks distinguish from leptons
only in the $SU(3)\times U(1)$ part: Quarks are triplets of three colours
($c^{i}$ $=(\tau^{33},\tau^{38})$
$=[(\frac{1}{2},\frac{1}{2\sqrt{3}}),(-\frac{1}{2},\frac{1}{2\sqrt{3}}),(0,-\frac{1}{\sqrt{3}})$,
carrying the ”fermion charge” ($\tau^{4}=\frac{1}{6}$). The colourless leptons
carry the ”fermion charge” ($\tau^{4}=-\frac{1}{2}$). The same multiplet
contains also the left handed weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$
charged antiquarks and antileptons and the right handed weak ($SU(2)_{I}$)
charged and $SU(2)_{II}$ chargeless antiquarks and antileptons. Antiquarks
distinguish from antileptons again only in the $SU(3)\times U(1)$ part:
Antiquarks are anti-triplets carrying the ”fermion charge”
($\tau^{4}=-\frac{1}{6}$). The anti-colourless antileptons carry the ”fermion
charge” ($\tau^{4}=\frac{1}{2}$). $Y=(\tau^{23}+\tau^{4})$ is the hyper
charge, the electromagnetic charge is $Q=(\tau^{13}+Y$).

## Acknowledgment

The author thanks Department of Physics, FMF, University of Ljubljana, Society
of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the
research on the spin-charge-family theory by offering the room and computer
facilities and Matjaž Breskvar of Beyond Semiconductor for donations, in
particular for the annual workshops entitled ”What comes beyond the standard
models”.

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