A New Way to Unify All Fermion and Boson Fields, Including Gravity

Abstract

The description of the internal spaces of fermion and boson fields with “basis vectors”, which are the superpositions of odd and even products of the operators ${\gamma }^a$, with the index a running in internal and external space-time, suggests in $d=2(2n+1)$-dimensions in internal space, such as $d=(13+1)$, and $d=(3+1)$ in external space-time, a unified picture of all so far observed fermions and bosons. Quarks, leptons, antiquarks, and antileptons appear in families - each family contains fermions and antifermions. Bosons—gravitons, photons, weak bosons, gluons, and scalars, which carry the spatial index $\alpha$ (for tensors and vectors $\alpha =n=$$(0,1,2,3)$ and for scalars $\alpha \ge 5$)—appear in two orthogonal groups. All fields are assumed to have non-zero momenta and angular momenta only in the $d=(3+1)$, $SO(3,1)$, of ordinary space-time. In any $d=2(2n+1)$-dimensional space, the number of internal states of fermions in all families and their Hermitian conjugate partners is equal to the number of internal states of bosons. The article presents general properties of massless fermion and boson fields and their mutual interactions in this theory, which determine the Lagrangian density of both fields and their interactions. It particularly illustrates “basis vectors” and their properties in $d=(13+1)$ and $d=(5+1)$. The article presents new results and discusses open problems with this theory.

1. Introduction

The author, with collaborators, succeeded in demonstrating in a long series of studies [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] that the model, named the “spin–charge-family” theory, suggests an elegant description of the second quantised fermion fields—appearing in families, written as the tensor products of the basis in ordinary space-time—and the basis, named “basis vectors”, in internal spaces, presented as superpositions of odd products of operators ${\gamma }^a$ (with a the index of the internal space) arranged in nilpotents and projectors, which are eigenvectors of the (chosen) Cartan subalgebra members [1,2,3,4,5,6,7,8,9,10].

Five years ago [18,19,20,21], the author started to use an equivalent description for boson fields to that so far used for fermion fields, recognising the possibility from thirty years ago [1,2,3]: the internal space of boson second quantised fields can be described by the “basis vectors”, presented as superpositions of even products of operators ${\gamma }^a$, arranged in nilpotents and projectors, which are eigenvectors of the Cartan subalgebra members.

The theory of massless fermions and bosons, with the non-zero momentum only in $d=(3+1)$ dimensions, describing internal spaces by an odd number of nilpotents, the rest of the projectors (for fermions) and an even number of nilpotents, the rest of the projectors (for bosons, which also carry the ordinary space-time index $n=0,1,2,3$), determines for the internal space-time dimension $d=(13+1)$ all the properties of fermion and boson fields and of their mutual interactions, realised by the algebraic multiplication as follows.

(a)

The theory explains the Dirac’s postulates for the second quantised fermion and boson fields.

(b)

The theory determines Lorentz (and correspondingly the gauge) symmetry of fermion and boson fields.

(c)

The theory determines couplings among fermion and boson fields and consequently the Lagrange densities.

(d)

The theory determines families of fermions, any family of which includes in $d=2(2n+1)$-dimensional internal space fermions and antifermions. Consequently, the vacuum does not have the negative energy of the Dirac vacuum; it is just the quantum vacuum.

(e)

The theory determines two orthogonal kinds of boson fields: one applies to fermions from the left-hand side and the other from the right-hand side. One transforms fermions within each family, the other transforms a member of a family into the same member of other families. Both are expressible as algebraic products of fermion fields and their Hermitian conjugate partners (knowing the “basic vectors” of fermion fields, one knows also the “basic vectors” of the boson fields, although the properties of fermion fields are very different from the properties of the boson fields).

(f)

Fermion states are algebraically orthogonal.

(g)

Although the internal spaces of fermions and bosons demonstrate numerous different properties (anticommuting fermions appear in families and have half-integer spins and charges in the fundamental representations; commuting bosons appear in two orthogonal groups, have no families, and have integer spins and charges in adjoint representations), the straightforward algebraic multiplication with the ${\gamma }^a$ relates both kinds of “basis vectors” (see Appendix B in Ref. [21]).

(h)

Properties listed in (d)–(g) influence the Feynman diagrams, which should reproduce the experimental data.

(i)

Second quantised fermion and boson fields are described as a tensor product of bases in ordinary space-time and of “basis vectors” describing the internal spaces of fields.

(j)

The analysis of the fermion and boson internal spaces with respect to the subgroups $SO(1,3),SU\left(2\right),$$SU\left(2\right),SU\left(3\right),U\left(1\right)$ of the group $SO(13,1)$ suggests the description of the observed families of quarks and leptons, appearing in families, and of tensor (gravitons), vector (photons, weak bosons, gluons), and scalar (Higgs) boson fields, explaining also other observed properties of fermions and bosons (like the appearance of the dark matter [13], the matter–antimatter asymmetry in the universe [14,15,16], and several other predictions [9,10,17,22]).

(k)

The Pauli matrices for spins and charges in any even d for either fundamental or adjoint representations can be straightforwardly found by applying the corresponding operators, $S^{ab}$ or ${\mathcal{S}}^{ab}$, on the “basis vectors” of fermions or of bosons.

(l)

There are—in internal space with $d=2(2n+1)$— $2^{{\displaystyle \frac{d}{2}}-1}$$\times 2^{{\displaystyle \frac{d}{2}}-1}$ “basis vectors” of fermion fields, arranged in an odd number of nilpotents, and the same number of their Hermitian conjugate partners. And there are twice $2^{{\displaystyle \frac{d}{2}}-1}$$\times 2^{{\displaystyle \frac{d}{2}}-1}$ “basis vectors” with an even number of nilpotents describing two orthogonal boson fields.

(m)

In odd-dimensional spaces, $d=(2n+1)$, the fermion and boson fields have exceptionally peculiar properties: half of the “basis vectors”, $2^{{\displaystyle \frac{2n}{2}}-1}$$\times 2^{{\displaystyle \frac{2n}{2}}-1}$, have the properties of fields in the $2n$-dimensional space (the anticommuting “basis vectors” appear in families and have their Hermitian conjugate partners in a separate group, the commuting “basis vectors” appear in two orthogonal groups). Among the rest of the “basis vectors”, that is, in $2^{{\displaystyle \frac{2n}{2}}-1}$$\times 2^{{\displaystyle \frac{2n}{2}}-1}$ cases, the anticommuting appear in two orthogonal groups, and the commuting appear in families and have their Hermitian conjugate partners in a separate group [19,20,21].

(n)

In this proposal, all fields, fermions and bosons (tensors, vectors, and scalars) are massless. This contribution does not discuss the breaking of symmetries and the appearance of massive fermion fields. There are condensates that make several scalar fields, as well as some of the fermion and vector boson fields, massive, as discussed in Refs. [9,10]. (The breaks of symmetries are expected to follow similarly to the case when we describe one kind of boson fields with ${{\omega }^{ab}}_{\alpha }$ and another kind with ${{\tilde{\omega }}^{ab}}_{\alpha }$, which are presented in Refs. [9,10].)

There are numerous studies using Clifford algebra for many different purposes, and many of the investigations do have some points in common with what current paper (and earlier studies by the author and collaborators cited herein) is proposing. However, it is not straightforward to comment on the differences—not even on the level when some studies [23,24,25,26,27,28] are aiming to understand the Dirac second quantisation postulates [29] in a novel way, see Refs. [23,24], as also this paper does, since different models are used.

Let us briefly review what this paper proposes with respect to the Dirac second quantisation postulates, which is built on the recognition that there are anticommuting fermions interacting with commuting bosons, governing the world, in order to comment on possible similarities with the other studies cited herein.

The current paper presents the equivalent description of fermion and boson fields in a novel way. The internal spaces of fermion and boson fields are described by superpositions of odd products of operators ${\gamma }^a$ if they are fermions and by the superpositions of even products of operators ${\gamma }^a$ if they are bosons. For more straightforward recognition, and understanding, the study takes care that the superposition of odd and even products of operators ${\gamma }^a$ are products of nilpotents and projectors, so that the states are eigenvectors of the chosen Cartan subalgebra members of the Lorentz algebra. Correspondingly, fermions have an odd number of nilpotents and the rest are projectors, while bosons have an even number of nilpotents and projectors.

All the rest follows.

(i)

All the eigenstates with an odd number of nilpotents anticommute and are mutually orthogonal; they are called odd “basis vectors” of $\psi$, $\psi {\ast }_A\psi =0$, where ${\ast }_A$ denotes the associative algebraic product.

(ii)

All their Hermitian conjugate partners, ${\psi }^{†}$, are mutually orthogonal, ${\psi }^{†}{\ast }_A{\psi }^{†}=0$.

(iii)

Products of one “basis vector” and one of the Hermitian conjugate partners $\psi {\ast }_A{\psi }^{†}$ are in general not zero and ${\psi }^{†}{\ast }_A\psi$ are not, either. It can be seen that internal spaces of boson fields, even “basis vectors”, of all bosons, can be written as even products of odd “basis vectors” (of fermions $\psi$). Even more, they have properties of the internal spaces of two kinds of mutually orthogonal bosons. Those are called even “basis vectors” of ${}^I\mathcal{A}_a$ and ${}^{II}\mathcal{A}_a$. One immediately sees that these two kinds of bosons must multiply fermions: one from the left-hand side, ${}^I\mathcal{A}_a$, $\left(\psi {\ast }_A{\psi }^{†}\right){\ast }_A\psi$, and the other from the right-hand side, ${}^{II}\mathcal{A}_a$, $\psi {\ast }_A\left({\psi }^{†}{\ast }_A\psi \right)$. Refs. [23,24,25,26,27,30] observe that there are two kinds of boson fields. (Multiplications of fermions and bosons are associative due to the properties of the odd and even“basis vectors”.)

(iv)

Since the way of building the second quantised fermion and boson fields here, fermions appear in families, there is the same number of the two kinds of the even “basis vectors” of boson fields, as it is the number of odd “basis vectors” appearing in families and of their Hermitian conjugated partners. In Refs. [31,32], the studies use triality to achieve families of fermions, while in Refs. [30,33], Clifford algebra, quaternions, and octonions are used to reproduce the Standard Model groups.

(v)

While in the here treated way of building second quantised fermion and boson fields, all the internal degrees of freedom, spins, and charges, of fermions and bosons (of all bosons with gravity included), are described by the odd and even “basis vectors” (spins in $SO(3,1)$ subgroup of $SO(13,1)$, charges in $SU\left(2\right)\times SU\left(2\right)\times SU\left(3\right)\times U\left(1\right)$ of $SO(13,1)$), and correspondingly, fermions anticommute and bosons commute [9,10,19], most of papers use Clifford algebra only to reproduce fermion fields [34].

In Section 2, the internal spaces of fermion and boson fields are presented as “basis vectors” which are algebraic products, ${\ast }_A$, of an odd number of nilpotents (for fermions) and an even number of nilpotents (for bosons). The rest are projectors. Nilpotents and projectors are chosen to be eigenvectors of the Cartan subalgebra members of the Lorentz algebra, as demonstrated in Section 2.1. Section 2.1 also presents algebraic relations among the “basis vectors” of fermion and two kinds of boson fields, which determines the Lagrange densities of interacting fermion and boson fields. The creation operators, presented in Section 2.2, are tensor products, ${\ast }_T$, of the “basis vectors” and the basis in ordinary space-time. In Section 2.3, the states active only in $d=(3+1)$-dimensions of ordinary space-time are discussed, while the internal space is active in $d=2(2n+1)$ dimensions. In Section 2.3, the definite algebraic relations among fermion and boson fields for the two cases that the internal space has $d=(5+1)$ and $d=(13+1)$, are presented.

In Section 3.1, the general algebraic structure of the second quantised fermion and boson fields, following from the properties presented in Section 2, is presented.

Section 4 presents what have been learned in the last three years. In this Section, the problems that remain to be solved in this theory, to find out whether the theory provides the correct description of the observed fermion and boson second quantised fields which determine the history (and the future) of the universe, are discussed.

In Appendix A, “basis vectors” of one family of quarks and leptons, and antiquarks and antileptons, are presented. In Appendix B, the relations among the Grassmann algebra and the two kinds of the Clifford algebras are discussed. In Appendix C, the relations are presented that are needed in all the sections. In Appendix D, odd and even “basis vectors” for $d=(5+1)$ cases are presented in detail, which are meant to be used like an exercise for $d=(13+1)$.

2. Internal Spaces of Second Quantised Fermion and Boson Fields

One shall need all of the requirements of the algebraic relations among the fermion and boson fields derived in this Section 3, which generalizes those requirements.

This Section (following several studies, see paper [21] and the references therein) overviews briefly the description of the internal spaces of the second quantised fermion and boson fields as algebraic products of nilpotents and projectors, which are the superposition of odd and even products of ${\gamma }^a$s. However, the novel recognitions, some of them been presented in Ref. [21] and the others that are clarified in what follows are presented.

As explained in Appendix B, Equation (A8), the Grassmann algebra provides two kinds of operators, ${\gamma }^a$s and ${\tilde{\gamma }}^a$s with the following properties

$$\begin{array}{ccc}\hfill {\{{\gamma }^a,{\gamma }^b\}}_{+}& =& 2{\eta }^{ab}={\{{\tilde{\gamma }}^a,{\tilde{\gamma }}^b\}}_{+},\hfill \\ \hfill {\{{\gamma }^a,{\tilde{\gamma }}^b\}}_{+}& =& 0,(a,b)=(0,1,2,3,5,\dots ,d),\hfill \\ \hfill {\left({\gamma }^a\right)}^{†}& =& {\eta }^{aa}{\gamma }^a,{\left({\tilde{\gamma }}^a\right)}^{†}={\eta }^{aa}{\tilde{\gamma }}^a,\hfill \end{array}$$

(1)

where ${\eta }^{ab}=\mathrm{diag}(1,-1,-1,\dots ,-1)$ and the dagger denotes Hermitian conjugate (in this paper, the definition of commutators (like $S^{ab}={\displaystyle \frac{i}{4}}{\{{\gamma }^a,{\gamma }^b\}}_{-}$) and anti-commutators (like ${\{{\gamma }^a,{\gamma }^b\}}_{+}=2{\eta }^{ab}={\{{\tilde{\gamma }}^a,{\tilde{\gamma }}^b\}}_{+}$) differ only in the signum; in standard definitions, commutators are presented as $S^{ab}={\displaystyle \frac{i}{4}}[{\gamma }^a,{\gamma }^b]$, while anti-commutators are presented as $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$).

Here, one of the two kinds, ${\gamma }^a$s, is used to generate the “basis vectors” describing internal spaces of fermions and bosons. They are arranged in products of nilpotents and projectors:

$$\begin{array}{ccc}\hfill \stackrel{ab}{\left(k\right)}:& =& {\displaystyle \frac{1}{2}}({\gamma }^a+{\displaystyle \frac{{\eta }^{aa}}{ik}}{\gamma }^b),{\left(\stackrel{ab}{\left(k\right)}\right)}^2=0,\hfill \\ \hfill \stackrel{ab}{\left[k\right]}:& =& {\displaystyle \frac{1}{2}}(1+{\displaystyle \frac{i}{k}}{\gamma }^a{\gamma }^b),{\left(\stackrel{ab}{\left[k\right]}\right)}^2=\stackrel{ab}{\left[k\right]},\hfill \end{array}$$

(2)

so that each nilpotent and each projector is the eigenstate of one of the Cartan (chosen) subalgebras members of the Lorentz algebra

$$\begin{array}{ccc}& & S^{03},S^{12},S^{56},\cdots ,S^{d-1d},\hfill \\ & & {\tilde{S}}^{03},{\tilde{S}}^{12},{\tilde{S}}^{56},\cdots ,{\tilde{S}}^{d-1d},\hfill \\ & & {\mathbf{S}}^{ab}=S^{ab}+{\tilde{S}}^{ab},\hfill \end{array}$$

(3)

where $S^{ab}={\displaystyle \frac{i}{4}}{\{{\gamma }^a,{\gamma }^b\}}_{-}$, while ${\tilde{S}}^{ab}={\displaystyle \frac{i}{4}}{\{{\tilde{\gamma }}^a,{\tilde{\gamma }}^b\}}_{-}$ are used to determine additional quantum numbers; in the case of fermions, these are called the family quantum numbers.

Being eigenstates of both operators, of $S^{ab}$ and ${\tilde{S}}^{ab}$, nilpotents and projectors carry both quantum numbers $S^{ab}$ and ${\tilde{S}}^{ab}$:

$$\begin{array}{ccc}\hfill S^{ab}\stackrel{ab}{\left(k\right)}={\displaystyle \frac{k}{2}}\stackrel{ab}{\left(k\right)},& & {\tilde{S}}^{ab}\stackrel{ab}{\left(k\right)}={\displaystyle \frac{k}{2}}\stackrel{ab}{\left(k\right)},\hfill \\ \hfill S^{ab}\stackrel{ab}{\left[k\right]}={\displaystyle \frac{k}{2}}\stackrel{ab}{\left[k\right]},& & {\tilde{S}}^{ab}\stackrel{ab}{\left[k\right]}=-{\displaystyle \frac{k}{2}}\stackrel{ab}{\left[k\right]},\hfill \end{array}$$

(4)

with $k^2={\eta }^{aa}{\eta }^{bb}$.

In even-dimensional spaces, the states in internal spaces are defined by the “basis vectors” which are products of ${\displaystyle \frac{d}{2}}$ nilpotents and projectors, and they are the eigenstates of all the Cartan subalgebras members.

Fermions are products of an odd number of nilpotents (at least one), while the rest are projectors. Bosons are products of an even number of nilpotents (or none), while the rest are projectors. Those are called here odd and even “basis vectors”.

The odd “basis vectors” have the eigenvalues of the Cartan subalgebras members, as shown in Equations (3) and (4), either of $S^{ab}$ or ${\tilde{S}}^{ab}$, which are equal to a half integer, $\pm {\displaystyle \frac{i}{2}}$ or $\pm {\displaystyle \frac{1}{2}}$.

The even “basis vectors” have the eigenvalues of the Cartan subalgebras members, as shown in Equations (3) and (4), ${\mathcal{S}}^{ab}=$ $S^{ab}+{\tilde{S}}^{ab}$, which are equal to $\pm i$, $\pm 1$ or 0.

2.1. “Basis Vectors” Describing Internal Spaces of Fermion and Boson Fields

The odd products of nilpotents (at least one; the rest are projectors), called odd “basis vectors”, differ essentially from the even products of nilpotents (none or at least two), which are called even “basis vectors” (the rest are projectors). Either odd or even “basis vectors” are chosen to be eigenvectors of all the Cartan subalgebra members, as shown in Equation (3).

The algebraic product of any two members of the odd or even “basis vectors” can be straightforwardly calculated when taking into account the relations following from Equation (1). The most necessary relations are as follows:

$$\begin{array}{ccc}\hfill {\gamma }^a\stackrel{ab}{\left(k\right)}& =& {\eta }^{aa}\stackrel{ab}{[-k]},{\gamma }^b\stackrel{ab}{\left(k\right)}=-ik\stackrel{ab}{[-k]},{\gamma }^a\stackrel{ab}{\left[k\right]}=\stackrel{ab}{(-k)},{\gamma }^b\stackrel{ab}{\left[k\right]}=-ik{\eta }^{aa}\stackrel{ab}{(-k)},\hfill \\ \hfill \widetilde{{\gamma }^a}\stackrel{ab}{\left(k\right)}& =& -i{\eta }^{aa}\stackrel{ab}{\left[k\right]},\widetilde{{\gamma }^b}\stackrel{ab}{\left(k\right)}=-k\stackrel{ab}{\left[k\right]},\widetilde{{\gamma }^a}\stackrel{ab}{\left[k\right]}=i\stackrel{ab}{\left(k\right)},\widetilde{{\gamma }^b}\stackrel{ab}{\left[k\right]}=-k{\eta }^{aa}\stackrel{ab}{\left(k\right)},\hfill \\ \hfill \stackrel{ab}{\left(k\right)}\stackrel{ab}{(-k)}& =& {\eta }^{aa}\stackrel{ab}{\left[k\right]},\stackrel{ab}{(-k)}\stackrel{ab}{\left(k\right)}={\eta }^{aa}\stackrel{ab}{[-k]},\stackrel{ab}{\left(k\right)}\stackrel{ab}{\left[k\right]}=0,\stackrel{ab}{\left(k\right)}\stackrel{ab}{[-k]}=\stackrel{ab}{\left(k\right)},\hfill \\ \hfill \stackrel{ab}{(-k)}\stackrel{ab}{\left[k\right]}& =& \stackrel{ab}{(-k)},\stackrel{ab}{\left[k\right]}\stackrel{ab}{\left(k\right)}=\stackrel{ab}{\left(k\right)},\stackrel{ab}{\left[k\right]}\stackrel{ab}{(-k)}=0,\stackrel{ab}{\left[k\right]}\stackrel{ab}{[-k]}=0,\hfill \\ \hfill {\stackrel{ab}{\left(k\right)}}^{†}& =& {\eta }^{aa}\stackrel{ab}{(-k)},{\left(\stackrel{ab}{\left(k\right)}\right)}^2=0,\stackrel{ab}{\left(k\right)}\stackrel{ab}{(-k)}={\eta }^{aa}\stackrel{ab}{\left[k\right]},\hfill \\ \hfill {\stackrel{ab}{\left[k\right]}}^{†}& =& \stackrel{ab}{\left[k\right]},{\left(\stackrel{ab}{\left[k\right]}\right)}^2=\stackrel{ab}{\left[k\right]},\stackrel{ab}{\left[k\right]}\stackrel{ab}{[-k]}=0.\hfill \end{array}$$

(5)

More algebraic relations can be found in Appendix C.

The odd “basis vectors” are named ${\widehat{b}}_f^{m†}$, where m determines the family member quantum number, f determines the quantum number of a family, and they appear in $2^{{\displaystyle \frac{d}{2}}-1}$ irreducible representations—families, all with the same properties with respect to $S^{ab}$, while distinguishing with respect ${\tilde{S}}^{ab}$. Each family has $2^{{\displaystyle \frac{d}{2}}-1}$ members. Their Hermitian conjugate partners ${\left({\widehat{b}}_f^{m†}\right)}^{†}=$${\widehat{b}}_f^m$, appearing in a separate group, have $2^{{\displaystyle \frac{d}{2}}-1}\times 2^{{\displaystyle \frac{d}{2}}-1}$ members. As already said, the odd “basis vectors” have the eigenvalues of the Cartan subalgebra members, as shown in Equation (3), which are either of $S^{ab}$ or ${\tilde{S}}^{ab}$ half integer, $\pm {\displaystyle \frac{i}{2}}$ or $\pm {\displaystyle \frac{1}{2}}$.

The algebraic product of any two members of the odd “basis vectors” is equal to zero. The same is true for any two members of the Hermitian conjugate partner:

$$\begin{array}{ccc}\hfill {\widehat{b}}_f^{m†}{\ast }_A{\widehat{b}}_{f‘}^{m‘†}& =& 0,{\widehat{b}}_f^m{\ast }_A{\widehat{b}}_{f‘}^{m‘}=0,\hfill \\ \hfill \forall m,m^{\prime },& & f,f‘.\hfill \end{array}$$

(6)

Choosing the vacuum state equal to

$$\begin{array}{c}\hfill |{\psi }_{oc}⟩={\displaystyle \sum _{f=1}^{2^{\frac{d}{2}-1}}}{\widehat{b}}_f^m{}_{{\ast }_A}\widehat{b}_f^{m†}|1⟩\end{array}$$

(7)

(with the subscript “oc” standing to point to vacuum state denoted by “o” and to notify both—fermion and boson—vacua states then jointly denoted by “c”) for one of the members m, which can be any one of the odd irreducible representations f, it follows that the odd “basis vectors” obey the relations

$$\begin{array}{ccc}\hfill {\widehat{b}}_f^m{}_{{\ast }_A}|{\psi }_{oc}>& =& 0·|{\psi }_{oc}⟩,\hfill \\ \hfill {\widehat{b}}_f^{m†}{}_{{\ast }_A}|{\psi }_{oc}>& =& |{\psi }_f^m⟩,\hfill \\ \hfill {\{{\widehat{b}}_f^m,{\widehat{b}}_{f‘}^{m^{\prime }}\}}_{{\ast }_A+}|{\psi }_{oc}⟩& =& 0·|{\psi }_{oc}⟩,\hfill \\ \hfill {\{{\widehat{b}}_f^{m†},{\widehat{b}}_{f‘}^{m^{\prime }†}\}}_{{\ast }_A+}|{\psi }_{oc}⟩& =& 0·|{\psi }_{oc}⟩,\hfill \\ \hfill {\{{\widehat{b}}_f^m,{\widehat{b}}_{f‘}^{m^{\prime }†}\}}_{{\ast }_A+}|{\psi }_{oc}⟩& =& {\delta }^{m{m}^{\prime }}{\delta }_{ff‘}|{\psi }_{oc}⟩,\hfill \end{array}$$

(8)

as postulated by Dirac for the second quantised fermion fields. Here, the odd “basis vectors” anticommute, since the odd products of ${\gamma }^a$s anticommute (the algebraic product, ${\{·,·\}}_{{\ast }_A+}$, denotes, for example, ${\{{\widehat{b}}_f^m,{\widehat{b}}_{f‘}^{m^{\prime }†}\}}_{{\ast }_A+}|{\psi }_{oc}⟩=$ $({\widehat{b}}_f^m{\ast }_A{\widehat{b}}_{f‘}^{m^{\prime }†}+{\widehat{b}}_{f‘}^{m^{\prime }†}{\ast }_A{\widehat{b}}_f^m){\ast }_A|{\psi }_{oc}⟩$ $={\delta }_{ij}^{\left(lm\right)}$|${\psi }_{oc}⟩$). ${\delta }_{ij}^{\left(lm\right)}$ is the Kronecker delta.

The odd “basis vectors” and their Hermitian conjugate partners are normalised as follows:

$$\begin{array}{c}\hfill ⟨{\psi }_{oc}|{({\widehat{b}}_f^{m†})}^{†}{\ast }_A{\widehat{b}}_{f‘}^{m^{\prime }†}|{\psi }_{oc}⟩={\delta }^{m{m}^{\prime }}{\delta }_{ff‘}⟨{\psi }_{oc}|{\psi }_{oc}⟩,\end{array}$$

(9)

the vacuum state $⟨{\psi }_{oc}|{\psi }_{oc}⟩$ is normalised to identity. Let us, as an exercise, present the odd “basis vectors” and their Hermitian conjugate partners for $d=(3+1)$, which are taken from Ref. [20]. The odd “basis vectors” appear in two families; each family has two members:

$$\begin{array}{c}\hfill \begin{array}{cccc}f=1& f=2& & \\ {\tilde{S}}^{03}={\displaystyle \frac{i}{2}},{\tilde{S}}^{12}=-{\displaystyle \frac{1}{2}}& {\tilde{S}}^{03}=-{\displaystyle \frac{i}{2}},{\tilde{S}}^{12}={\displaystyle \frac{1}{2}}& \hfill S^{03}& \hfill S^{12}\\ {\widehat{b}}_1^{1†}=\stackrel{03}{(+i)}\stackrel{12}{[+]}& {\widehat{b}}_2^{1†}=\stackrel{03}{[+i]}\stackrel{12}{(+)}& \hfill {\displaystyle \frac{i}{2}}& \hfill {\displaystyle \frac{1}{2}}\\ {\widehat{b}}_1^{2†}=\stackrel{03}{[-i]}\stackrel{12}{(-)}& {\widehat{b}}_2^{2†}=\stackrel{03}{(-i)}\stackrel{12}{[-]}& \hfill -{\displaystyle \frac{i}{2}}& \hfill -{\displaystyle \frac{1}{2}}.\end{array}\end{array}$$

(10)

Their Hermitian conjugate partners have the properties

$$\begin{array}{c}\hfill \begin{array}{cccc}{S}^{03}=-{\displaystyle \frac{i}{2}},S^{12}={\displaystyle \frac{1}{2}}& S^{03}={\displaystyle \frac{i}{2}},S^{12}=-{\displaystyle \frac{1}{2}}& \hfill {\tilde{S}}^{03}& \hfill {\tilde{S}}^{12}\\ {\widehat{b}}_1^1=\stackrel{03}{(-i)}\stackrel{12}{[+]}& {\widehat{b}}_2^1=\stackrel{03}{[+i]}\stackrel{12}{(-)}& \hfill -{\displaystyle \frac{i}{2}}& \hfill -{\displaystyle \frac{1}{2}}\\ {\widehat{b}}_1^2=\stackrel{03}{[-i]}\stackrel{12}{(+)}& {\widehat{b}}_2^2=\stackrel{03}{(+i)}\stackrel{12}{[-]}& \hfill {\displaystyle \frac{i}{2}}& \hfill {\displaystyle \frac{1}{2}}.\end{array}\end{array}$$

(11)

The vacuum state $|{\psi }_{oc}⟩$, as shown in Equation (7), is equal to $|{\psi }_{oc}⟩={\displaystyle \frac{1}{\sqrt{2}}}(\stackrel{03}{[-i]}\stackrel{12}{[+]}+\stackrel{03}{[+i]}\stackrel{12}{[-]})$.

The even “basis vectors”, named ${}^I\widehat{\mathcal{A}}_f^{m†}$ and ${}^{II}\widehat{\mathcal{A}}_f^{m†}$, appear in two orthogonal groups

$$\begin{array}{ccc}\hfill {}^I\widehat{\mathcal{A}}_f^{m†}{\ast }_A{}^{II}\widehat{\mathcal{A}}_f^{m†}& =& 0={}^{II}\widehat{\mathcal{A}}^{m†}{.}_f{\ast }_A{}^I\widehat{\mathcal{A}}_f^{m†}.\hfill \end{array}$$

(12)

Each group has $2^{{\displaystyle \frac{d}{2}}-1}\times 2^{{\displaystyle \frac{d}{2}}-1}$ members with the Hermitian conjugate partners within the group.

The even “basis vectors” have the eigenvalues of the Cartan subalgebra members, as shown in Equation (3), ${\mathcal{S}}^{ab}=$$S^{ab}+{\tilde{S}}^{ab}$, which are equal to $\pm i$, $\pm 1$ or 0.

The algebraic products, ${\ast }_A$, of two members of each of these two groups have the property

$$\begin{array}{c}\hfill {}^i\widehat{\mathcal{A}}_f^{m†}{\ast }_A{}^i\widehat{\mathcal{A}}_{f‘}^{m^{\prime }†}\to \left\{\begin{array}{c}\hfill {}^i\widehat{\mathcal{A}}_{f‘}^{m†},i=(I,II),\\ \hfill \mathrm{or}0.\end{array}\right.\end{array}$$

(13)

For a chosen ($m,f,f‘$), there is (out of $2^{{\displaystyle \frac{d}{2}}-1}$) only one $m^{\prime }$ giving a non-zero contribution. Let us, for an exercise, present the $2^{{\displaystyle \frac{4}{2}}-1}\times 2^{{\displaystyle \frac{4}{2}}-1}$ ”basis vectors” for $d=(3+1)$, the members of the group ${}^I\mathcal{A}_f^{m†}$, which are taken from Ref. [20]:

$$\begin{array}{c}\hfill \begin{array}{cccccc}& \hfill {\mathcal{S}}^{03}& \hfill {\mathcal{S}}^{12}& & \hfill {\mathcal{S}}^{03}& \hfill {\mathcal{S}}^{12}\\ {}^I\mathcal{A}_1^{1†}=\stackrel{03}{[+i]}\stackrel{12}{[+]}& \hfill 0& \hfill 0& ,{}^I\mathcal{A}_2^{1†}=\stackrel{03}{(+i)}\stackrel{12}{(+)}& \hfill i& \hfill 1\\ {}^I\mathcal{A}_1^{2†}=\stackrel{03}{(-i)}\stackrel{12}{(-)}& \hfill -i& \hfill -1& ,{}^I\mathcal{A}_2^{2†}=\stackrel{03}{[-i]}\stackrel{12}{[-]}& \hfill 0& \hfill 0,\end{array}\end{array}$$

(14)

and $2^{{\displaystyle \frac{4}{2}}-1}\times 2^{{\displaystyle \frac{4}{2}}-1}$ even ”basis vectors” ${}^{II}\mathcal{A}_f^{m†}$, $m=(1,2),f=(1,2)$,

$$\begin{array}{c}\hfill \begin{array}{cccccc}& \hfill {\mathcal{S}}^{03}& \hfill {\mathcal{S}}^{12}& & \hfill {\mathcal{S}}^{03}& \hfill {\mathcal{S}}^{12}\\ {}^{II}\mathcal{A}_1^{1†}=\stackrel{03}{[+i]}\stackrel{12}{[-]}& \hfill 0& \hfill 0& ,{}^{II}\mathcal{A}_2^{1†}=\stackrel{03}{(+i)}\stackrel{12}{(-)}& \hfill i& \hfill -1\\ {}^{II}\mathcal{A}_1^{2†}=\stackrel{03}{(-i)}\stackrel{12}{(+)}& \hfill -i& \hfill 1& ,{}^{II}\mathcal{A}_2^{2†}=\stackrel{03}{[-i]}\stackrel{12}{[+]}& \hfill 0& \hfill 0.\end{array}\end{array}$$

(15)

One can straightforwardly check the relations from Equations (10)–(15) if taking into account Equation (1) or Equation (5).

To be able to propose the action for fermion and boson second quantized fields, one needs to know the algebraic application, ${\ast }_A$, of boson fields on fermion fields and fermion fields on boson fields.

The algebraic application, ${\ast }_A$, of the even “basis vectors” ${}^I\widehat{\mathcal{A}}_f^{m†}$ on the odd “basis vectors” ${\widehat{b}}_{f‘}^{m^{\prime }†}$, called here “left multiplication”, gives

$$\begin{array}{c}\hfill {}^I\widehat{\mathcal{A}}_f^{m†}{\ast }_A{\widehat{b}}_{f‘}^{m^{\prime }†}\to \left\{\begin{array}{c}\hfill {\widehat{b}}_{f‘}^{m†},\\ \hfill \mathrm{or}0.\end{array}\right.\end{array}$$

(16)

Equation (16) demonstrates that ${}^I\widehat{\mathcal{A}}_f^{m†}$, applying on ${\widehat{b}}_{f‘}^{m^{\prime }†}$, transforms the odd “basis vector” into another odd “basis vector” of the same family, transferring to the odd “basis vector” integer spins or gives zero.

One finds for the second group of boson fields, ${}^{II}\widehat{\mathcal{A}}_f^{m†}$,

$$\begin{array}{c}\hfill {\widehat{b}}_f^{m†}{\ast }_A{}^{II}\widehat{\mathcal{A}}_{f‘}^{m^{\prime }†}\to \left\{\begin{array}{c}\hfill {\widehat{b}}_{f‘‘}^{m†},\\ \hfill \mathrm{or}0,\end{array}\right.\end{array}$$

(17)

demonstrating that the application of the odd “basis vector” ${\widehat{b}}_f^{m†}$ on ${}^{II}\widehat{\mathcal{A}}_{f‘}^{m^{\prime }†}$ leads to another odd “basis vector” ${\widehat{b}}_{f‘‘}^{m†}$ belonging to the same family member m of a different family $f‘‘$. This is called the “right multiplication” of ${}^{II}\widehat{\mathcal{A}}_f^{m†}$ on the odd “basis vector” ${\widehat{b}}_f^{m†}$.

The rest of the possibilities give zero:

$$\begin{array}{c}\hfill {\widehat{b}}_f^{m†}{\ast }_A{}^I\widehat{\mathcal{A}}_{f‘}^{m^{\prime }†}=0,{}^{II}\widehat{\mathcal{A}}_f^{m†}{\ast }_A{\widehat{b}}_{f‘}^{m^{\prime }†}=0,\forall (m,m‘,f,f‘).\end{array}$$

(18)

Let us add that the internal spaces of boson second-quantized fields can be written as the algebraic products of the odd “basis vectors” and their Hermitian conjugate partners: ${\widehat{b}}_f^{m†}$ and ${({\widehat{b}}_{f‘}^{m^{″}†})}^{†}$:

$$\begin{array}{ccc}\hfill {}^I\widehat{\mathcal{A}}_f^{m†}& =& {\widehat{b}}_{f‘}^{m^{\prime }†}{\ast }_A{({\widehat{b}}_{f‘}^{m^{″}†})}^{†}\hfill \end{array}$$

(19)

and

$$\begin{array}{ccc}\hfill {}^{II}\widehat{\mathcal{A}}_f^{m†}& =& {({\widehat{b}}_{f‘}^{m^{\prime }†})}^{†}{\ast }_A{\widehat{b}}_{f{‘}^{\prime }}^{m^{\prime }†}.\hfill \end{array}$$

(20)

Family members ${\widehat{b}}_{f‘}^{m^{\prime }†}$ of any family $f^{\prime }$ generates in the algebraic product ${\widehat{b}}_{f‘}^{m^{\prime }†}{\ast }_A$ $({\widehat{b}}_{f‘}^{m^{″}†}{)}^{†}$ the same $2^{{\displaystyle \frac{d}{2}}-1}\times$$2^{{\displaystyle \frac{d}{2}}-1}$ even “basis vectors” ${}^I\widehat{\mathcal{A}}_f^{m†}$; each family member $m^{\prime }$ generates in ${({\widehat{b}}_{f‘}^{m^{\prime }†})}^{†}{\ast }_A$ ${\widehat{b}}_{f{‘}^{\prime }}^{m^{\prime }†}$ the same $2^{{\displaystyle \frac{d}{2}}-1}\times$ $2^{{\displaystyle \frac{d}{2}}-1}$ even “basis vectors” ${}^{II}\widehat{\mathcal{A}}_f^{m†}$ (it follows that ${}^I\widehat{\mathcal{A}}_f^{m†}$, expressed by ${\widehat{b}}_{f‘}^{m^{\prime }†}{\ast }_A$ $({\widehat{b}}_{f‘}^{m^{″}†}{)}^{†}$, applying on ${\widehat{b}}_{f{‘}^{″}}^{m^{‴}†}$, obeys Equation (16), and ${\widehat{b}}_{f‘{‘}^{\prime }}^{m^{‴}†}$ applying on ${}^{II}\widehat{\mathcal{A}}_f^{m†}$, expressed by ${({\widehat{b}}_{f‘}^{m^{\prime }†})}^{†}$${\ast }_A$ ${\widehat{b}}_{f‘‘}^{m^{\prime }†}$, obeys Equation (17).).

2.2. Fermions and Bosons Creation Operators

The creation operators for either fermions or bosons must be defined as the tensor products, ${\ast }_T$, of both contributions, and the “basis vectors” are describing the internal space of fermions or bosons, and the basis in ordinary space-time is the momentum or coordinate representation (to the boson second quantised fields, one needs to add the space index $\alpha$. Since our space-time is flat, a shall be used instead of $\alpha$, and n instead of $\mu =(0,1,2,3)$ and for the scalar index, $a\ge 5$, (our fermions and bosons have non-zero momentum only in $d=(3+1)$ of ordinary space-time) index s).

Let us start with the definition of the single-particle states in ordinary space-time in momentum representation, briefly overviewing Ref. [21] (see also Section 3.3 and Appendix J in Ref. [9]):

$$\begin{array}{ccc}\hfill |\overrightarrow{p}⟩& =& {\widehat{b}}_{\overrightarrow{p}}^{†}|0_p⟩,⟨\overrightarrow{p}|=⟨0_p|{\widehat{b}}_{\overrightarrow{p}},\hfill \\ \hfill ⟨\overrightarrow{p}|{\overrightarrow{p}}^{\prime }⟩& =& \delta (\overrightarrow{p}-{\overrightarrow{p}}^{\prime })=⟨0_p|{\widehat{b}}_{\overrightarrow{p}}{\widehat{b}}_{{\overrightarrow{p}}^{\prime }}^{†}|0_p⟩,\hfill \\ \hfill ⟨0_p|{\widehat{b}}_{\overrightarrow{p^{\prime }}}{\widehat{b}}_{\overrightarrow{p}}^{†}|0_p⟩& =& \delta (\overrightarrow{p^{\prime }}-\overrightarrow{p}),\hfill \end{array}$$

(21)

with $⟨0_p|0_p⟩=1$. The operator ${\widehat{b}}_{\overrightarrow{p}}^{†}$ pushes a single particle state with zero momentum by an amount $\overrightarrow{p}$. Taking into account that ${\{{\widehat{p}}^i,{\widehat{p}}^j\}}_{-}=0$ and ${\{{\widehat{x}}^k,{\widehat{x}}^l\}}_{-}=0$, while ${\{{\widehat{p}}^i,{\widehat{x}}^j\}}_{-}=i{\eta }^{ij}$, it follows

$$\begin{array}{ccc}\hfill ⟨\overrightarrow{p}|\overrightarrow{x}⟩& =& ⟨0_{\overrightarrow{p}}|{\widehat{b}}_{\overrightarrow{p}}{\widehat{b}}_{\overrightarrow{x}}^{†}|0_{\overrightarrow{x}}⟩=(⟨0_{\overrightarrow{x}}|{\widehat{b}}_{\overrightarrow{x}}{\widehat{b}}_{\overrightarrow{p}}^{†}|0_{\overrightarrow{p}}{⟩)}^{†}\hfill \\ \hfill ⟨0_{\overrightarrow{p}}|{\{{\widehat{b}}_{\overrightarrow{p}}^{†},{\widehat{b}}_{{\overrightarrow{p}}^{\prime }}^{†}\}}_{-}|0_{\overrightarrow{p}}⟩& =& 0,⟨0_{\overrightarrow{p}}|{\{{\widehat{b}}_{\overrightarrow{p}},{\widehat{b}}_{{\overrightarrow{p}}^{\prime }}\}}_{-}|0_{\overrightarrow{p}}⟩=0,⟨0_{\overrightarrow{p}}|{\{{\widehat{b}}_{\overrightarrow{p}},{\widehat{b}}_{{\overrightarrow{p}}^{\prime }}^{†}\}}_{-}|0_{\overrightarrow{p}}⟩=0,\hfill \\ \hfill {⟨}_{\overrightarrow{x}}|{\{{\widehat{b}}_{\overrightarrow{x}}^{†},{\widehat{b}}_{{\overrightarrow{x}}^{\prime }}^{†}\}}_{-}|0_{\overrightarrow{x}}⟩& =& 0,⟨0_{\overrightarrow{x}}|{\{{\widehat{b}}_{\overrightarrow{x}},{\widehat{b}}_{{\overrightarrow{x}}^{\prime }}\}}_{-}|0_{\overrightarrow{x}}⟩=0,⟨0_{\overrightarrow{x}}|{\{{\widehat{b}}_{\overrightarrow{x}},{\widehat{b}}_{{\overrightarrow{x}}^{\prime }}^{†}\}}_{-}|0_{\overrightarrow{x}}⟩=0,\hfill \\ \hfill ⟨0_{\overrightarrow{p}}|{\{{\widehat{b}}_{\overrightarrow{p}},{\widehat{b}}_{\overrightarrow{x}}^{†}\}}_{-}|0_{\overrightarrow{x}}⟩& =& e^{i\overrightarrow{p}·\overrightarrow{x}}{\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^{d-1}}}},⟨0_{\overrightarrow{x}}|{\{{\widehat{b}}_{\overrightarrow{x}},{\widehat{b}}_{\overrightarrow{p}}^{†}\}}_{-}|0_{\overrightarrow{p}}⟩=e^{-i\overrightarrow{p}·\overrightarrow{x}}{\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^{d-1}}}}.\hfill \end{array}$$

(22)

The momentum basis is continuously infinite, while the internal space of either fermion or boson fields has a finite number of “basis vectors”—twice $2^{{\displaystyle \frac{d}{2}}-1}\times 2^{{\displaystyle \frac{d}{2}}-1}$ for fermions and twice $2^{{\displaystyle \frac{d}{2}}-1}\times 2^{{\displaystyle \frac{d}{2}}-1}$ for bosons, provided that the internal space includes d-dimensions. It is assumed that fermions and bosons are active only $d=(3+1)$.

The creation operator for a free massless fermion field of the energy $p^0=\left|\overrightarrow{p}\right|$, belonging to the family f and to a superposition of family members m applying to the vacuum state ($|{\psi }_{oc}⟩{\ast }_T|0_{\overrightarrow{p}}⟩$) can be written as (following Section 3.3.2 of Ref. [9], and the references therein)

$$\begin{array}{ccc}\hfill {\widehat{\mathbf{b}}}_f^{s†}\left(\overrightarrow{p}\right)& =& \sum _m{c}^{sm}{}_f\left(\overrightarrow{p}\right){\widehat{b}}_{\overrightarrow{p}}^{†}{\ast }_T{\widehat{b}}_f^{m†}.\hfill \end{array}$$

(23)

The vacuum state for fermions, $|{\psi }_{oc}⟩{\ast }_T|0_{\overrightarrow{p}}⟩$, includes both spaces, the internal part, Equation (7), and the momentum part, Equation (21). The creation operators in the coordinate representation can be written as ${\widehat{\mathbf{b}}}_f^{s†}(\overrightarrow{x},x^0)={\sum }_m{\widehat{b}}_f^{m†}{\ast }_T{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d^{d-1}p}{{\left(\sqrt{2\pi }\right)}^{d-1}}}{c}^{sm}{}_f\left(\overrightarrow{p}\right){\widehat{b}}_{\overrightarrow{p}}^{†}{e}^{-i(p^0{x}^0-\overrightarrow{p}·\overrightarrow{x})}$, [19] (see also Section 3.3.2 in Ref. [9] and the references therein).

The creation operators, ${\widehat{\mathbf{b}}}_f^{s†}\left(\overrightarrow{p}\right)$, and their Hermitian conjugate partners annihilation operators, ${\left({\widehat{\mathbf{b}}}_f^{s†}\left(\overrightarrow{p}\right)\right)}^{†}$ $={\widehat{\mathbf{b}}}_f^s\left(\overrightarrow{p}\right)$, creating and annihilating the single fermion states, respectively, fulfil when applying the vacuum state, ($|{\psi }_{oc}>{\ast }_T|0_{\overrightarrow{p}}>$), the anti-commutation relations for the second-quantized fermions, postulated by Dirac [9,10], (Sections 3.3.1 and 5). The anticommuting properties of the creation operators for fermions are determined by the odd “basis vectors”, the basis in ordinary space-time, namely, commute:

$$\begin{array}{ccc}\hfill ⟨0_{\overrightarrow{p}}|⟨{\psi }_{oc}|{\{{\widehat{\mathbf{b}}}_{f‘}^{s^{\prime }}(\overrightarrow{p^{\prime }}),{\widehat{\mathbf{b}}}_f^{s†}(\overrightarrow{p})\}}_{+}|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩& =& {\delta }^{s{s}^{\prime }}{\delta }_{f{f}^{\prime }}\delta ({\overrightarrow{p}}^{\prime }-\overrightarrow{p}),\hfill \\ \hfill {\{{\widehat{\mathbf{b}}}_{f‘}^{s^{\prime }}(\overrightarrow{p^{\prime }}),{\widehat{\mathbf{b}}}_f^s(\overrightarrow{p})\}}_{+}|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩& =& 0·|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩,\hfill \\ \hfill {\{{\widehat{\mathbf{b}}}_{f^{\prime }}^{s^{\prime }†}(\overrightarrow{p^{\prime }}),{\widehat{\mathbf{b}}}_f^{s†}(\overrightarrow{p})\}}_{+}|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩& =& 0·|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩,\hfill \\ \hfill {\widehat{\mathbf{b}}}_f^{s†}(\overrightarrow{p})|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩& =& |{\psi }_f^s(\overrightarrow{p})⟩,\hfill \\ \hfill {\widehat{\mathbf{b}}}_f^s(\overrightarrow{p})|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩& =& 0·|{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩,\hfill \\ \hfill |p^0|& =& |\overrightarrow{p}|,\hfill \end{array}$$

(24)

since $⟨0_{\overrightarrow{p}}|⟨{\psi }_{oc}||{\psi }_{oc}⟩|0_{\overrightarrow{p}}⟩$ is normalised to 1.

The creation operator for a free massless boson field of the energy $p^0=\left|\overrightarrow{p}\right|$, with the “basis vectors” belonging to one of the two groups, ${}^i\widehat{\mathcal{A}}_f^{m†},i=(I,II)$, applying on the vacuum state, $|{\psi }_{oc}⟩{\ast }_T|0_{\overrightarrow{p}}⟩$, must carry the space index a, describing the a component of the boson field in the ordinary space-time (the vacuum state for bosons is chosen to be the same as the vacuum state for fermions). Therefore, the space index a is added, to boson fields, as well as the dependence on the momentum [21]:

$$\begin{array}{ccc}\hfill {}^i\widehat{\mathcal{A}}_{fa}^{m†}\left(\overrightarrow{p}\right)& =& {}^i\mathcal{C}^m{}_{fa}\left(\overrightarrow{p}\right){\ast }_T{}^i\widehat{\mathcal{A}}_f^{m†},i=(I,II),\hfill \end{array}$$

(25)

with ${}^i\mathcal{C}^m{}_{fa}\left(\overrightarrow{p}\right)={}^i\mathcal{C}^m{}_{fa}{\widehat{b}}_{\overrightarrow{p}}^{†}$, and ${\widehat{b}}_{\overrightarrow{p}}^{†}$ is defined in Equations (21) and (22) (in the general case, the energy eigenstates of bosons are in a superposition of ${}^i\widehat{\mathcal{A}}_f^{m†}$ for either $i=I$ or $i=II$).

The creation operators for boson fields in the coordinate representation one finds using Equations (21) and (22), ${}^i\widehat{\mathcal{A}}_{fa}^{m†}(\overrightarrow{x},x^0)={}^i\widehat{\mathcal{A}}_f^{m†}{\ast }_T{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d^{d-1}p}{{\left(\sqrt{2\pi }\right)}^{d-1}}}{}^i\mathcal{C}^m{}_{fa}{\widehat{b}}_{\overrightarrow{p}}^{†}{e}^{-i(p^0{x}^0-\epsilon \overrightarrow{p}·\overrightarrow{x})}{|}_{p^0=\left|\overrightarrow{p}\right|},$

Assuming that the internal space has $d=(13+1)$, while fermions and bosons have non-zero momenta only in $d=(3+1)$ of the ordinary space-time, both creation operators manifest (after analysing $SO(13,1)$ with respect to the subgroups $SO(3,1)$, $SU\left(2\right)\times SU\left(2\right)$, $SU\left(3\right)$ and $U\left(1\right)$ of the Lorentz group $SO(13,1)$) the properties of fermions and bosons Equation (13), as assumed by the Standard Model before the electroweak phase transitions: Clifford odd creation operators defining the fermion fields manifest all the properties of quarks and leptons, and antiquarks and antileptons, appearing in families, and the Clifford even boson creation operators, ${}^I\widehat{\mathcal{A}}_{fa}^{m†}$, defining the boson fields manifest for a equal to $n=(0,1,2,3)$ all the properties of gauge fields (photons, weak bosons, and gluons—predicting the second weak fields and explaining the gravitons).

For a equal to $s\ge 5$, the Clifford even boson creation operators, ${}^{II}\widehat{\mathcal{A}}_{fs}^{m†}$, manifest properties of the scalar Higgs, causing after the electroweak phase transition masses of quarks and leptons and antiquarks and antileptons as well as some of the gauge fields.

The assumption that the internal spaces of fermion and boson fields are describable by the odd and even “basis vectors”, respectively, leads to the conclusion that the internal spaces of all the boson fields—gravitons (the gauge fields of the spins $SO(3,1)$), photons (the gauge fields of $U\left(1\right)$), weak bosons (the gauge fields of one of the $SU\left(2\right)$) and gluons (the gauge fields of $SU\left(3\right)$, together with $U\left(1\right)$ origin in $SO\left(6\right)$)—must also be described by the even “basis vectors”; all must carry the index $a=$$n=(0,1,2,3)$.

Both groups of even “basis vectors” manifest as the gauge fields of the corresponding fermion fields: one concerning the family members quantum numbers, which are determined by $S^{ab}$, and the other concerning the family quantum numbers, which are determined by ${\tilde{S}}^{ab}$.

Let us point out that although it looks like that this theory postulates two kinds of boson fields, not yet observed so far, this is not the case. All the theories so far postulate the families of fermions and the scalar fields giving masses to fermions and weak bosons in addition to the internal spaces of fermions and bosons. In the case under consideration, the families are present without being postulated. Here, boson fields of the second kind have, in theories so far, realization in Higgs.

The proposed description of the internal spaces considers families of fermions, scalar fields, and gauge fields: ${}^I\widehat{\mathcal{A}}_f^{m†}$, transferring the integer quantum numbers to the odd “basis vectors”, ${\widehat{b}}_f^{m†}$, changes the family members’ quantum numbers, leaving the family quantum numbers unchanged, manifesting the properties of the gauge fields. The second group, ${}^{II}\widehat{\mathcal{A}}_f^{m†}$, transferring the integer quantum numbers to the “basis vector” ${\widehat{b}}_f^{m†}$, changes the family quantum numbers leaving the family members’ quantum numbers unchanged, manifesting properties of the scalar fields, which give masses to quarks and leptons as well as to the weak bosons.

The proposed description of the internal spaces of fermions and bosons predicts more families than observed so far and more gauge fields. It is believed that the breaks of the starting symmetry will show why those are not observed [9,19] (yet).

2.3. States of Fermions and Bosons Active Only in $d=(3+1)$

The states of fermion and boson fields are considered to have non-zero momentum only in $d=(3+1)$. This refers to the Poincaré group (with the infinitesimal generators $M^{ab}(=L^{ab}+{\mathcal{S}}^{ab}),p^c$) applying only in $d=(3+1)$, while in the internal space, the Lorentz group (with the infinitesimal generators ${\mathcal{S}}^{ab}$, ${\mathcal{S}}^{ab}=S^{ab}+{\tilde{S}}^{ab}$) applies to the whole internal space $d=2(2n+1)$. The algebraic relations among fermion and boson fields are discussed just below in the cases when the internal space is of $d=(5+1)$ (Section (a)) and $d=(13+1)$ (Section (b)) dimensions.

The odd and even “basis vectors” are presented in the case that $d=(5+1)$ in Appendix D in Table A2, Table A3, Table A4, Table A5 and Table A6. This toy model is discussed as an exercise.

In Table A1, the odd “basis vectors” are presented in the case that $d=(13+1)$ for one family of fermions—quarks and leptons and antiquarks and antileptons—as products of an odd number of nilpotents (at least one, up to seven). The “basis vectors” are eigenstates of all the Cartan subalgebra members, Equation (3), of the Lorentz algebra.

The creation and annihilation operators are for odd and even “basis vectors” the tensor products, ${\ast }_T$, of the basis in ordinary space-time in $d=(3+1)$, and the “basis vectors” in internal space, with $d=(5+1)$ or $d=(13+1)$: for anti-commuting creation operators, one has ${\widehat{\mathbf{b}}}_f^{s†}\left(\overrightarrow{p}\right)={\sum }_m{c}^{sm}{}_f\left(\overrightarrow{p}\right){\widehat{b}}_{\overrightarrow{p}}^{†}{\ast }_T{\widehat{b}}_f^{m†}$, as shown in Equation (23).

For the commuting creation operators with the “basis vectors” belonging to one of the two groups, ${}^i\widehat{\mathcal{A}}_f^{m†},i=(I,II)$, carrying the space index a, one has ${}^i\widehat{\mathcal{A}}_{fa}^{m†}\left(\overrightarrow{p}\right)={}^i\mathcal{C}^m{}_{fa}\left(\overrightarrow{p}\right){\ast }_T{}^i\widehat{\mathcal{A}}_f^{m†},i=(I,II)$, as shown in Equation (25).

• Internal Spaces of Fermions and Bosons in $d=(5+1)$ and $d=(13+1)$

(a)

Let us start with the toy model for electrons, positrons, photons and gravitons in the case that the internal space is for electrons, $d=(5+1)$, while fields have non-zero momenta in $d=(3+1)$.

The consideration here follows to some extent a similar part in Ref. ([21], and the references therein). This toy model is to show, in quite a simple model, what the new description of the internal spaces of fermion and boson fields suggests.

In Table A2, the odd “basis vectors”, ${\widehat{b}}_f^{m†}$, appearing in four ($2^{{\displaystyle \frac{d=6}{2}}-1}$) families, each family having four ($2^{{\displaystyle \frac{d=6}{2}}-1}$) family members, are presented in the first group as products of an odd number of nilpotents (one or three) and the remaining projectors. Their Hermitian conjugate partners are presented in the second group, again with members.

The even basis vectors appear in the third and the fourth group.

Table A2 presents the eigenvalues of all Cartan subalgebra members, as shown in Equation (3): $S^{ab}$ for a family members, and ${\tilde{S}}^{ab}$ for a family. ${\mathcal{S}}^{ab}=(S^{ab}+{\tilde{S}}^{ab})$ determine the Cartan eigenvalues of the even “basis vectors”, presenting internal spaces of boson fields.

One can check the relations among “basis vectors” of fermions and bosons appearing in Equations (6)–(20), for the case that the internal space has $d=(5+1)$ by taking into account Equations (5), (A12) and (A13).

The corresponding creation and annihilation operators for free massless fermion fields (${\widehat{\mathbf{b}}}_f^{m†}\left(\overrightarrow{p}\right)={\widehat{b}}_{\overrightarrow{p}}^{†}{\ast }_T{\widehat{b}}_f^{m†}$), and for free massless boson fields (${}^i\widehat{\mathcal{A}}_{fa}^{m†},i=(I,II)$, carrying the space index a, ${}^i\widehat{\mathcal{A}}_{fa}^{m†}\left(\overrightarrow{p}\right)={}^i\mathcal{C}^m{}_{fa}\left(\overrightarrow{p}\right){\ast }_T{}^i\widehat{\mathcal{A}}_f^{m†},i=(I,II)$), can be found in Equations (23) and (25).

Let us call the first ${\widehat{b}}_f^{m†}$ of the “basis vectors” in Table A2: ${\widehat{b}}_1^{1†}=$ $\stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}$, the “basis vector” of the “electron”, and the third “basis vector” ${\widehat{b}}_1^{3†}=$ $\stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)}$, both belong to the first family, the “basis vector” of the “positron”, choosing the quantum numbers of the “electron” equal to ($S^{03}={\displaystyle \frac{i}{2}}$, $S^{12}={\displaystyle \frac{1}{2}}$ and $S^{56}={\displaystyle \frac{1}{2}}$), and of the “positron” equal to ($S^{03}=-{\displaystyle \frac{i}{2}}$, $S^{12}={\displaystyle \frac{1}{2}}$ and $S^{56}={\displaystyle \frac{1}{2}}$). One can transform the “electron” to the “positron” by $S^{05}$.

The “basis vectors” of the “positron” and “electron” have fractional charges and both appear in four families, which are reachable from the first one by the application of ${\tilde{S}}^{ab}$.

For example, one generates the second family by applying ${\tilde{S}}^{05}$ on the first family.

The corresponding “photon” field, its “basis vector” indeed, describing the internal space of the “photon”, must be a product of projectors only, since the photon does not change the charge of the positron or electron.

There is only one even “basis vector”, that applying to the “basis vector” of the “electron” gives a non-zero contribution: the “basis vector” ${}^I\widehat{\mathcal{A}}_3^{1†}=$ $\stackrel{03}{[+i]}\stackrel{12}{[+]}\stackrel{56}{[+]}$. It is presented in Table A4.

There is also only one even “basis vector”, which, applying to the “basis vector” of the “positron”, gives a non-zero contribution. Both even “basis vectors” have the properties of photons. It is presented in Table A4:

$$\begin{array}{c}{}^I\widehat{\mathcal{A}}_{3\mathrm{ph}}^{1†}(\equiv \stackrel{03}{[+i]}\stackrel{12}{[+]}\stackrel{56}{[+]}){\ast }_A{\widehat{b}}_f^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]})\to {\widehat{b}}_f^{1†},\hfill \\ {}^I\widehat{\mathcal{A}}_{2\mathrm{ph}}^{3†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{[-]}){\ast }_A{\widehat{b}}_f^{3†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)})\to {\widehat{b}}_f^{3†}.\hfill \end{array}$$

(26)

The same “photon” makes the same transformations on the corresponding “electron” (or “positron”) of all the families. Understandably, the Cartan subalgebra quantum numbers, Equation (3), ($S^{ab}+{\tilde{S}}^{ab}$), applying on any member of the “photon” is equal to zero: ($S^{03}+{\tilde{S}}^{03}=0$, $S^{12}+{\tilde{S}}^{12}=0$ and $S^{56}+{\tilde{S}}^{56}=0$) of either ${}^I\widehat{\mathcal{A}}_{3\mathrm{ph}}^{1†}$ or ${}^I\widehat{\mathcal{A}}_{2\mathrm{ph}}^{3†}$ are zero, since the projectors have properties that $S^{ab}$$=-{\tilde{S}}^{ab}$, as shown in Equation (4).

Let us check the relation of Equation (19), using Equations (5) and (A11).

$${}^I\widehat{\mathcal{A}}_3^{1†}(\equiv \stackrel{03}{[+i]}\stackrel{12}{[+]}\stackrel{56}{[+]})={\widehat{b}}_1^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}){\ast }_A{\left({\widehat{b}}_1^{1†}\right)}^{†}(\equiv {\left(\stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}\right)}^{†}).$$

$${}^I\widehat{\mathcal{A}}_2^{3†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{[-]})={\widehat{b}}_1^{3†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)}){\ast }_A{\left({\widehat{b}}_1^{3†}\right)}^{†}(\equiv {\left(\stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)}\right)}^{†}).$$

As it was demonstrated on one example that knowing the odd “basis vectors”, one can reproduce all the even “basis vectors”, ${}^I\widehat{\mathcal{A}}_f^{m†}$. (In Ref. [21], the relations among even “basis vectors” and the odd “basis vectors” are presented for the case $d=(5+1)$ in Tables 2–6 there. Tables 3 and 4 of Ref. [21] relate ${}^I\mathcal{A}_a$ with the algebraic products of odd “basis vectors” in the algebraic product with their Hermitian conjugated partners, while Tables 5 and 6 there relate ${}^{II}\mathcal{A}_a$ with the Hermitian conjugated partners in the algebraic product with the odd “basis vectors”. Table 1 of Ref. [21] presents the basis vectors of fermion fields appearing in families and their Hermitian conjugated partners as well as both kinds of boson fields, ${}^I\mathcal{A}_a$ and ${}^{II}\mathcal{A}_a$.)

Table A5 and Table A6 relate the odd “basis vectors” and their Hermitian conjugated partners.

One can repeat all the relations obtained for ${}^I\widehat{\mathcal{A}}_f^{m†}$ in this subsection also for ${}^{II}\widehat{\mathcal{A}}_f^{m†}$. Keepping in mind Equation (20), one straightforwardly see the essential difference between ${}^I\widehat{\mathcal{A}}_f^{m†}$ and ${}^{II}\widehat{\mathcal{A}}_f^{m†}$. While ${}^I\widehat{\mathcal{A}}_f^{m†}$ transform family members of odd “basis vectors” among themselves, keeping the family quantum number unchanged, ${}^{II}\widehat{\mathcal{A}}_f^{m†}$ transform a particular family member to the same family member of all the families, changing the family quantum numbers. Let us demonstrate the right application of ${}^{II}\widehat{\mathcal{A}}_f^{m†}$ on ${\widehat{b}}_{f‘}^{m^{\prime }†}$ for the case $d=(5+1)$.

The same ${}^{II}\widehat{\mathcal{A}}_f^{m†}$, applying from the right-hand side on ${\widehat{b}}_{f‘}^{m^{\prime }†}$, transforms any family member of one family to the same family member of another family.

$${\widehat{b}}_1^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}){\ast }_A{}^{II}\widehat{\mathcal{A}}_4^{1†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(+)}\stackrel{56}{[+]})\to {\widehat{b}}_3^{1†}(\equiv \stackrel{03}{[+i]}\stackrel{12}{(+)}\stackrel{56}{[+]}),$$

$${\widehat{b}}_1^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}){\ast }_A{}^{II}\widehat{\mathcal{A}}_1^{1†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(+)}\stackrel{56}{(+)})\to {\widehat{b}}_4^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{(+)}),$$

$${\widehat{b}}_1^{2†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(-)}\stackrel{56}{[+]}){\ast }_A{}^{II}\widehat{\mathcal{A}}_4^{1†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(+)}\stackrel{56}{[+]})\to {\widehat{b}}_3^{2†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{[-]}\stackrel{56}{[+]}),$$

$${\widehat{b}}_1^{2†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(-)}\stackrel{56}{[+]}){\ast }_A{}^{II}\widehat{\mathcal{A}}_1^{1†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(+)}\stackrel{56}{(+)})\to {\widehat{b}}_4^{2†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[-]}\stackrel{56}{(+)}).$$

Correspondingly, one sees in the second kind of bosons, when carrying the scalar space-time index $a\ge 5$, a scalar boson—a kind of Higgs.

Let us point out that the even “basis vectors”, determining the creation and annihilation operators in a tensor product with the basis in ordinary space-time, determine spins and charges of boson fields. Having non-zero momentum only in $d=(3+1)$, they carry space index $a=n=(0,1,2,3)$. They behave in the case that internal space has $(5+1)$ dimensions as a “photon”, as we just discussed. The “photon” can exchange the momentum in ordinary space-time with “electron” or “positron”, but it cannot influence any internal property, like there are the spins, $S^{03}$ and $S^{12}$, or the charge $S^{56}$.

Let us find out what represents the even “basis vectors”, ${}^I\widehat{\mathcal{A}}_4^{1†}$, with two nilpotents in the $SO(3,1)$ subgroup of the group $SO(5,1)$. The two spins, $S^{03}$ and $S^{12}$, enable the creation operators, which are the tensor product of the basis in ordinary space-time and the even “basis vectors” with two nilpotents, Equation (25), to form “gravitons”, as presented in Table A3. It presents two gravitons, ${}^I\widehat{\mathcal{A}}_{4n}^{1†}(\overrightarrow{p})$ and ${}^I\widehat{\mathcal{A}}_{3n}^{2†}(\overrightarrow{p})$,

$${}^I\widehat{\mathcal{A}}_{4n}^{1†}(\overrightarrow{p})={}^I\mathcal{C}^1{}_{4n}(\overrightarrow{p}){\ast }_T{}^I\widehat{\mathcal{A}}_4^{1†}(\equiv {\widehat{b}}_1^{1†}{\ast }_A{({\widehat{b}}_1^{2†})}^{†}=\stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]}),$$

$${}^I\widehat{\mathcal{A}}_{3n}^{2†}(\overrightarrow{p})={}^I\mathcal{C}^2{}_{3n}(\overrightarrow{p}){\ast }_T{}^I\widehat{\mathcal{A}}_3^{2†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(-)}\stackrel{56}{[+]}=({\widehat{b}}_1^{2†}{\ast }_A{({\widehat{b}}_1^{1†})}^{†}),$$

with the basis vectors ${}^I\widehat{\mathcal{A}}_4^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]})$, the first one, and with ${}^I\widehat{\mathcal{A}}_3^{2†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(-)}\stackrel{56}{[+]})$, the second one, which change the spins, $S^{03}$ and $S^{12}$, of fermions. When a boson ${}^I\widehat{\mathcal{A}}_{4n}^{1†}(\overrightarrow{p})$ scatters on a “electron” with the spin down, ${\widehat{\mathbf{b}}}_1^{2†}(\overrightarrow{p})(\equiv {\widehat{b}}_{\overrightarrow{p}}^{†}{\ast }_T{\widehat{b}}_1^{2†})$, as shown in Equation (23), changes its spin from ↓ (up) to ↑ (down), and transfers the momentum to the “electron”. This boson ${}^I\widehat{\mathcal{A}}_{4n}^{1†}(\overrightarrow{p})$, transferring the integer spin to the “electron” in addition to momentum of the space-time, is certainly “graviton” with ${\mathcal{S}}^{03}=i$ and ${\mathcal{S}}^{12}=1$, changing the quantum numbers $S^{03}=-{\displaystyle \frac{i}{2}}$ and $S^{12}=-{\displaystyle \frac{1}{2}}$ of ${\widehat{\mathbf{b}}}_1^{2†}(\overrightarrow{p})$ to $S^{03}={\displaystyle \frac{i}{2}}$ and $S^{12}={\displaystyle \frac{1}{2}}$ of ${\widehat{\mathbf{b}}}_1^{1†}(\overrightarrow{p})$.

Let us check, for two cases, how the “basis vectors” of “gravitons” behave when “gravitons” scatter:

$$\begin{array}{c}\hfill {}^I\widehat{\mathcal{A}}_{3\mathrm{gr}}^{2†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(-)}\stackrel{56}{[+]}){\ast }_A{}^I\widehat{\mathcal{A}}_{4\mathrm{gr}}^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]})\to {}^I\widehat{\mathcal{A}}_{4\mathrm{ph}}^{2†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[-]}\stackrel{56}{[+]}),\\ \hfill {}^I\widehat{\mathcal{A}}_{4\mathrm{gr}}^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]}){\ast }_A{}^I\widehat{\mathcal{A}}_{3\mathrm{gr}}^{2†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(-)}\stackrel{56}{[+]})\to {}^I\widehat{\mathcal{A}}_{3\mathrm{ph}}^{1†}(\equiv \stackrel{03}{[+i]}\stackrel{12}{[+]}\stackrel{56}{[+]}).\end{array}$$

(27)

There are also even “basis vectors” of the kind ${}^I\widehat{\mathcal{A}}_f^{m†}$ which change spin and charges, changing for example “positrons” into “electrons” (the corresponding bosons transform “positrons” into “electrons”, ${}^I\widehat{\mathcal{A}}_1^{2†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{[-]}\stackrel{56}{(+)}){\ast }_A{\widehat{b}}_1^{4†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(-)}\stackrel{56}{(-)})\to {\widehat{b}}_1^{2†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(-)}\stackrel{56}{[+]})$), changing at the same time the handedness.)

Looking at the even “basis vector” in this toy model, one fourth of ${}^I\widehat{\mathcal{A}}_f^{m†}$ are “photons” (like ${}^I\widehat{\mathcal{A}}_3^{1†}$ and ${}^I\widehat{\mathcal{A}}_4^{2†}$, not able to change the quantum numbers of the “electrons” and “positrons”, presented in Table A2) or “gravitons” (like ${}^I\widehat{\mathcal{A}}_3^{2†}$ and ${}^I\widehat{\mathcal{A}}_4^{1†}$, which change the spin of “electrons” and “positrons”).

The remaining eight ${}^I\widehat{\mathcal{A}}_f^{m†}$ relate “electrons” and “positrons”.

As already noticed, repeating the relations for ${}^I\widehat{\mathcal{A}}_f^{m†}$, Equations (26) and (27), also for ${}^{II}\widehat{\mathcal{A}}_f^{m†}$, one shall not obtain “photons” or “gravitons”, which both transform family members of odd “basis vectors” among themselves, keeping the family quantum number unchanged. Carrying the space index equal to $(5,6)$, the scalar bosons of the second kind, ${}^{II}\widehat{\mathcal{A}}_f^{m†}$, (“photons” and “gravitons”) cause, as a kind of “Higgs”, after breaking symmetries in this toy model, the masses of fermion fields.

(b)

The realistic case, which provides the “basis vectors” for all the so-far-observed fermion and boson fields, requires for internal space $d=(13+1)$, and for the space-time, in which fermions and bosons have non zero momenta, $d=(3+1)$, at least at observable energies.

In Table A1, Appendix A, the $2^{{\displaystyle \frac{14}{2}}-1}$ odd “basis vectors” present one irreducible representation, one family, of quarks and leptons and antiquarks and antileptons (both appearing in the same family), which are analysed with respect to the subgroups $SO(3,1),SU{\left(2\right)}_I,SU{\left(2\right)}_{II},$$SU\left(3\right),$$U\left(1\right)$ of the group $SO(13,1)$. One can notice in Table A1, that the content of the subgroup $SO(7,1)$ (including subgroups $SO(3,1),SU{\left(2\right)}_I,SU{\left(2\right)}_{II}$) is identical for quarks and leptons, and it is also identical for antiquarks and antileptons. Due to two $SU\left(2\right)$ subgroups $SU{\left(2\right)}_I,SU{\left(2\right)}_{II}$, first representing the weak charge, postulated by the Standard Model, the second $SU{\left(2\right)}_{II}$ group members are not (yet) observed at low energies. Quarks and leptons, and antiquarks and antileptons, distinguish only in the $SU\left(3\right)\times U\left(1\right)$ part of the group $SO(13,1)$.

From the first member, the odd “basis vector” $u_R^{c1}$ in Table A1 follows the remaining odd “basis vectors” by the application of the infinitesimal generators of the Lorentz group $S^{ab}$ as well as by the application of ${}^I\widehat{\mathcal{A}}_f^{m†}$. All the first members of the other families follow from the one presented in Table A1 by applying on $u_R^{c1}$ by ${\tilde{S}}^{ab}$, as well as by the application of ${}^{II}\widehat{\mathcal{A}}_f^{m†}$.

The corresponding creation and annihilation operators of fermions are tensor products of a “basis vector” and the basis in ordinary space-time, for example, ${\mathbf{u}}_R^{c1}\left(\overrightarrow{p}\right)=u_R^{c1}{\ast }_T{\widehat{b}}_{\overrightarrow{p}}$.

The even “basis vectors” can be obtained, according to Equations (19) and (20), as the algebraic products of the odd “basis vectors” and their Hermitian conjugate partners. In a tensor product with the basis in ordinary space-time, and with the space index $a=n(=0,1,2,3)$ added, ${}^I\widehat{\mathcal{A}}_{fa}^{m†}\left(\overrightarrow{p}\right)=$ ${}^I\mathcal{C}^m{}_{fa}\left(\overrightarrow{p}\right){\ast }_T$ ${}^I\widehat{\mathcal{A}}_f^{m†}$.

${}^I\widehat{\mathcal{A}}_{fa}^{m†}\left(\overrightarrow{p}\right)$ manifest the properties of the tensor ($a=n$), vector ($a=n$) and scalar ($a=s\ge 5$) gauge fields observed so far.

In a tensor product with the basis in ordinary space-time, and with the space index $a=s\ge 5$ added, ${}^I\widehat{\mathcal{A}}_{fa}^{m†}\left(\overrightarrow{p}\right)$ manifest the properties of the scalar fields, like the Higgs and other scalar fields, bringing masses to quarks and leptons and antiquarks and antileptons and to weak bosons, for example.

Let us look at Table A1 for the $e_L^{-†}$, the twenty-ninth line. The photon ${}^I\widehat{\mathcal{A}}_{\mathrm{ph}{e}_L^{-†}\to e_L^{-†}}^{†}$ interacts with $e_L^{-†}$ as follows:

$$\begin{array}{c}{}^I\widehat{\mathcal{A}}_{\mathrm{ph}{e}_L^{-†}\to e_L^{-†}}^{†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{[-]}\stackrel{78}{[+]}\stackrel{910}{[+]}\stackrel{1112}{[+]}\stackrel{1314}{[+]}){\ast }_A{e}_L^{-†},(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)}\stackrel{78}{(+)}\stackrel{910}{(+)}\stackrel{1112}{(+)}\stackrel{1314}{(+)})\hfill \\ \to e_L^{-†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)}\stackrel{78}{(+)}\stackrel{910}{(+)}\stackrel{1112}{(+)}\stackrel{1314}{(+)}),{}^I\widehat{\mathcal{A}}_{\mathrm{ph}{e}_L^{-†}\to e_L^{-†}}^{†}=e_L^{-†}{\ast }_A{\left(e_L^{-†}\right)}^{†}.\hfill \end{array}$$

(28)

Let us look for the weak boson, transforming $e_L^{-†}$ from the twenty-ninth line into ${\nu }_L^{†}$ from the thirty-first line.

It follows

$$\begin{array}{c}{}^I\widehat{\mathcal{A}}_{\mathrm{w}1e_L\to {\nu }_L}^{†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(+)}\stackrel{78}{(-)}\stackrel{910}{[+]}\stackrel{1112}{[+]}\stackrel{1314}{[+]}){\ast }_A{e}_L^{-†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{(-)}\stackrel{78}{(+)}\stackrel{910}{(+)}\stackrel{1112}{(+)}\stackrel{1314}{(+)})\hfill \\ \to {\nu }_L^{†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{[+]}\stackrel{78}{[-]}\stackrel{910}{(+)}\stackrel{1112}{(+)}\stackrel{1314}{(+)}),{}^I\widehat{\mathcal{A}}_{\mathrm{w}1e_L\to {\nu }_L}^{†}={\nu }_L^{†}{\ast }_A{\left(e_L^{-†}\right)}^{†}.\hfill \end{array}$$

(29)

Knowing “basis vectors” of fermions, the “basis vectors” of all bosons fields can be found. Only few of them are among the so-far-observed boson fields. (We discuss in Refs. [9,21] that breaks of symmetries make that not all possibilities of the predicted vector and scalar boson fields are observable at relatively low energies. Also among the observed fermions and antifermions, the predicted right-handed neutrinos and left-handed antineutrinos wait to be observed. Our model predicts more families than observed, like the fourth family to the observed three [12,13] and the dark matter [14,15,16,17] as the (almost stable) fifth family belonging to the upper four family.)

However, studying all the boson fields might help to recognise why and how the properties of fermions and bosons change with breaking symmetries, if this theory, describing the internal spaces of fermion and boson fields with odd and even “basis vectors”, is what the universe obeys.

Demonstrating so many simple and considerably elegant descriptions of the second-quantized fields, explaining the assumptions of other theories, gives a hope that the theory might be what the universe obeys.

Since the graviton in this theory is understood in an equivalent way as all the gauge fields observed so far, let us at the end of this Section try to analyse the “basis vectors” of the gravitons if the internal space has $d=(13+1)$.

One must take into account that the “gravitons” do have the spin and handedness (non-zero ${\mathcal{S}}^{03}$ and ${\mathcal{S}}^{12}$, which means that this part must be presented by two nilpotents, $\stackrel{03}{(\pm i)}\stackrel{12}{(\pm )}$) in $d=(3+1)$, and they do not have weak, colour and $U\left(1\right)$ charges (which means that all the rest must be projectors), and they have, as all the vector gauge fields, the space index $n=(0,1,2,3)$.

One can then straightforwardly find the “basis vector” of the graviton, ${}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_R^{c1†}\to u_R^{c1†}}^{†}$, which applying on $u_R^{c1†}$ with spin up, appearing in the first line of Table A1, transforms it into $u_R^{c1†}$ with spin down, appearing in the second line of Table A1:

$$\begin{array}{c}{}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\uparrow }^{c1†}\to u_{R\downarrow }^{c1†}}^{†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(-)}\stackrel{56}{[+]}\stackrel{78}{[+]}\stackrel{910}{[+]}\stackrel{1112}{[-]}\stackrel{1314}{[-]}){\ast }_A{u}_{R\uparrow }^{c1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}\stackrel{78}{(+)}\stackrel{910}{(+)}\stackrel{1112}{[-]}\stackrel{1314}{[-]})\hfill \\ \to u_{R\downarrow }^{c1†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(-)}\stackrel{56}{[+]}\stackrel{78}{(+)}\stackrel{910}{(+)}\stackrel{1112}{[-]}\stackrel{1314}{[-]}),{}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\uparrow }^{c1†}\to u_{R\downarrow }^{c1†}}^{†}=u_{R\downarrow }^{c1†}{\ast }_A{\left(u_{R\uparrow }^{c1†}\right)}^{†}.\hfill \end{array}$$

(30)

Let us have a look at the “scattering” (that is the algebraic application, ${\ast }_A$) of the graviton with the “basis vector” ${}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\uparrow }^{c1†}\to u_{R\downarrow }^{c1†}}^{†}$ with the graviton with the “basis vector”

${}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\downarrow }^{c1†}\to u_{R\uparrow }^{c1†}}^{†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]}\stackrel{78}{[+]}\stackrel{910}{[+]}\stackrel{1112}{[-]}\stackrel{1314}{[-]})$, that is

$${}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\uparrow }^{c1†}\to u_{R\downarrow }^{c1†}}^{†}(\equiv \stackrel{03}{(-i)}\stackrel{12}{(-)}\stackrel{56}{[+]}\stackrel{78}{[+]}\stackrel{910}{[+]}\stackrel{1112}{[-]}\stackrel{1314}{[-]}){\ast }_A{}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\downarrow }^{c1†}\to u_{R\uparrow }^{c1†}}^{†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]}\stackrel{78}{[+]}\stackrel{910}{[+]}\stackrel{1112}{[-]}\stackrel{1314}{[-]})$$

$$\to (\equiv \stackrel{03}{[-i]}\stackrel{12}{[-]}\stackrel{56}{[+]}\stackrel{78}{[+]}\stackrel{910}{[+]}\stackrel{1112}{[-]}\stackrel{1314}{[-]})=u_{R\downarrow }^{c1†}{\ast }_A{\left(u_{R\downarrow }^{c1†}\right)}^{†}={}^I\widehat{\mathcal{A}}_{\mathrm{ph}{u}_{R\downarrow }^{c1†}\to u_{R\downarrow }^{c1†}}^{†},$$

to recognise how straightforwardly one finds the internal space of bosons.

The creation operators for gravitons must carry the space index $n=(0,1,2,3)$, like ${}^I\widehat{\mathcal{A}}_{\mathrm{gr}{u}_{R\uparrow }^{c1†}\to u_{R\downarrow }^{c1†}n}^{†}\left(\overrightarrow{p}\right)$.

3. General Algebraic Structure of Fermion and Boson Second Quantised Fields

In Section 2, the relations among second quantised fermion and boson fields were demonstrated when treating internal spaces of fermions and bosons in a unique way: fermions as a superposition of odd products of operators ${\gamma }^a$, and bosons as a superposition of even products of operators ${\gamma }^a$, carrying in addition the ordinary space-time index a, which are named odd and even “basis vectors”, respectively. Although the corresponding states of the creation and annihilation operators have, in the case that the internal space has $d=(13+1)$ while in the ordinary space-time fermions and bosons are active only in $d=(3+1)$, the desired properties as assumed by the other theories, yet there are differences, which might lead to different predictions and conclusions.

Let us point out the main differences.

(a)

All “basis vectors” of fermions are mutual orthogonal and so their Hermitian conjugate partners are mutually orthogonal as well and, correspondingly, their creation and annihilation operators.

(b)

Fermions appear in families; bosons appear in two orthogonal groups.

(c)

Bosons multiply fermions either from the left-hand side (when causing transformations within one family) or from the right-hand side (when causing transformations among families).

In this Section, the results of Section 2 are generalized by generalising the structure of massless fermion and boson second quantised fields with the non-zero momentum only in $d=(3+1)$ of the ordinary space-time, as presented in Refs. [19,20,21,35,36], and explained and discussed in Section 2 of this paper.

The decision that the internal spaces of the second quantised fermion and boson fields are described by a superposition of odd numbers of operators ${\gamma }^a$s for fermions and even numbers of ${\gamma }^a$s for bosons—bosons carry the ordinary space-time index a—determines all the properties of the fermion and boson fields, as well as their mutual interactions, which are realised by the algebraic multiplication.

This decision explains Dirac’s postulates for the second quantised fermion and boson fields. Section 3.2 below determines the statistics of fermions and bosons, determines families of fermions, each of which includes fermions and antifermions (consequently, the vacuum is not the negative-energy Dirac vacuum; it is just the quantum vacuum), and determines the Lorentz (and correspondingly gauge) symmetry of fermion and boson fields. Section 3.3 determines couplings among fermion and boson fields. Consequently, Section 3.4 and Section 3.6 determine the Lagrange densities. Section 3.5 and Section 3.6 determine two orthogonal kinds of boson fields. Finally, Section 3.7 determines the orthogonality of fermion states, which influences all Feynman diagrams (which should reproduce all the experimental data).

3.1. Algebraic Structure

The theory considers fermion fields; they are tensor products of “basis vectors” and basis in ordinary space-time, as shown in Equation (23),

$$\psi \left(x\right)\in C{l}^{\mathrm{odd}}(d-1,1),$$

(31)

as Clifford odd $2^{{\displaystyle \frac{d}{2}}-1}$ members in $2^{{\displaystyle \frac{d}{2}}-1}$ irreducible representations, families, together with the same number of their Clifford odd Hermitian conjugate partners. They satisfy the relations of Equation (6) (the oddness of “basis vectors” determines the orthogonality:

$$\psi {\ast }_A\psi =0,{\psi }^{†}{\ast }_A{\psi }^{†}=0,{\psi }^{†}{\ast }_A\psi \ne 0.,$$

(32)

The theory also contains two orthogonal bosonic fields, which are tensor products of “basis vectors” and basis in ordinary space-time, Equation (25), each with $2^{{\displaystyle \frac{d}{2}}-1}\times 2^{{\displaystyle \frac{d}{2}}-1}$ members,

$${}^I{\widehat{\mathcal{A}}}_a\left(x\right),{}^{II}{\widehat{\mathcal{A}}}_a\left(x\right),$$

with

$${}^I{\widehat{\mathcal{A}}}_a,{}^{II}{\widehat{\mathcal{A}}}_a\in C{l}^{\mathrm{even}}(d-1,1),$$

and a is the space-time index. The fields satisfy the orthogonality condition (12)

$${}^I{\widehat{\mathcal{A}}}_a{\ast }_A{}^{II}{\widehat{\mathcal{A}}}_b={}^{II}{\widehat{\mathcal{A}}}_a{\ast }_A{}^I{\widehat{\mathcal{A}}}_b=0.$$

(33)

The chosen algebraic structure leads to the relations of Equation (13):

$$\begin{array}{c}\hfill {}^i{\widehat{\mathcal{A}}}_a{\ast }_A{}^i{\widehat{\mathcal{A}}}_a\to {}^i{\widehat{\mathcal{A}}}_a,i=(I,II),\end{array}$$

(34)

and of Equation (16):

$$\begin{array}{c}\hfill {}^I{\widehat{\mathcal{A}}}_a{\ast }_A\psi \to \psi ,\end{array}$$

(35)

transforming fermion fields within an irreducible representation—family, and to Equation (17):

$$\begin{array}{c}\hfill \psi {\ast }_A{}^{II}{\widehat{\mathcal{A}}}_a\to \psi ,\end{array}$$

(36)

transforming fermion fields among irreducible representations—families.

3.2. Internal Lorentz Symmetry

Local internal Lorentz transformations in the internal space of fermion and boson fields are generated by, as seen in Ref. [19],

$${\mathcal{S}}^{ab}=(S^{ab}+{\tilde{S}}^{ab}).$$

The corresponding local transformation reads

$$\Lambda \left(x\right)=e^{i{\omega }_{ab}\left(x\right){\mathcal{S}}^{ab}},{\Lambda }^{†}\left(x\right){\ast }_A\Lambda \left(x\right)=1,$$

provided that ${\omega }_{0i}^{\ast }=-{\omega }_{0i}$ and ${\omega }_{ij}^{\ast }={\omega }_{ij}$.

The fermion field transforms as

$$\psi ⟶\Lambda {\ast }_A\psi .$$

Translations in ordinary space-time are generated by operators $p_a$. Rotations in both spaces, internal and ordinary space-time, are generated by $M^{ab}=(L^{ab}+S^{ab}+{\tilde{S}}^{ab})$ ($L^{ab}$ gives non-zero contributions only in $d=3+1$. Only in $d=3+1$ are the rotations in internal space and in ordinary space-time related).

3.3. Covariant Derivative

The covariant derivative corresponding to the algebraic properties of the fermion and boson fields acting on $\psi$ is defined by, as shown in Equations (16), (17), (35) and (36),

$$D_a\psi =p_a\psi -{}^I{\widehat{\mathcal{A}}}_a{\ast }_A\psi -\psi {\ast }_A{}^{II}{\widehat{\mathcal{A}}}_a.$$

(37)

Under a local Lorentz transformation in internal space (defined by $(S^{ab}+{\tilde{S}}^{ab})$, concerning the internal and space-time Lorentz transformations, one needs to take $M^{ab}=(L^{ab}+S^{ab}+{\tilde{S}}^{ab})$),

$$p_a⟶{\Lambda }^{†}{\ast }_A{p}_a{\ast }_A\Lambda =p_a+\left(p_a{\Lambda }^{†}\right){\ast }_A\Lambda .$$

The Lorentz covariance of $D_a\psi$ requires

$$D_a{\ast }_A(\Lambda {\ast }_A\psi )⟶({\Lambda }^{†}{\ast }_A{D}_a{\ast }_A\Lambda ){\ast }_A\psi ,$$

(38)

as one finds in Section 3.4 just below.

3.4. Fermion Lagrangian and Local Lorentz Transformations in Internal Space

Following the properties of the fermion and boson fields, repeated above in Section 2 and in this Section up to here, one is able to consider the fermionic Lagrangian density in a Dirac way, and with the covariant derivative $D_a$:

$${\mathcal{L}}_F\left(\psi \right)={\displaystyle \frac{1}{2}}\left[{\psi }^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A{D}_a\psi +{\left(D_a\psi \right)}^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A\psi \right].$$

(39)

The covariant derivative acting on the fermion operator $\psi$ is

$$D_a{\ast }_A\psi =(p_a-{}^I\mathcal{A}_a){\ast }_A\psi -\psi {\ast }_A{}^{II}\mathcal{A}_a.$$

(40)

Let us consider a local Lorentz transformation in internal space

$$\psi \to {\psi }^{\prime }=\Lambda \psi ,$$

(41)

with

$$\Lambda \left(x\right)=e^{i{\omega }_{ab}\left(x\right){\mathcal{S}}^{ab}}.$$

Using this in the Lagrange density, it follows from looking at transformations in the first order in the way

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\{{({\gamma }^0{\gamma }^a{p}_a(1+i{\omega }_{ij}{S}^{ij}+i{\omega }_{0i}{S}^{0i}){\psi }^{\prime })}^{†})(1+i{\omega }_{ij}{S}^{ij}+i{\omega }_{0i}{S}^{0i}){\psi }^{\prime }+\\ \hfill {((1+i{\omega }_{ij}{S}^{ij}+i{\omega }_{0i}{S}^{0i}){\psi }^{\prime })}^{†}{\gamma }^0{\gamma }^a{p}_a(1+i{\omega }_{ij}{S}^{ij}+i{\omega }_{0i}{S}^{0i}){\psi }^{\prime }\},\\ \hfill ={\displaystyle \frac{1}{2}}\{{(p_a{\psi }^{\prime })}^{†}{\gamma }^0{\gamma }^a{\psi }^{\prime }+{({\psi }^{\prime })}^{†}{\gamma }^0{\gamma }^a{p}_a{\psi }^{\prime }\},\end{array}$$

(42)

after taking into account that ${\omega }_{ij}^{\ast }={\omega }_{ij}$, while ${\omega }_{0i}^{\ast }=-{\omega }_{0i}$, and that ${\left(S^{ij}\right)}^{†}=S^{ij},$${\left(S^{0i}\right)}^{†}=-S^{0i}$,

$${\mathcal{L}}_F(\Lambda \psi )={\displaystyle \frac{1}{2}}\left[{(\Lambda \psi )}^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A{D}_a{\ast }_A(\Lambda \psi )+{(D_a{\ast }_A\Lambda \psi )}^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A\Lambda \psi \right],$$

which leads to

$${\mathcal{L}}_F(\Lambda \psi )={\displaystyle \frac{1}{2}}\left[{\left(\psi \right)}^{†}{\Lambda }^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A\Lambda {\Lambda }^{†}{\ast }_A{D}_a{\ast }_A\Lambda \psi +{(\Lambda {\Lambda }^{†}{\ast }_A{D}_a{\ast }_A\Lambda \psi )}^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A\Lambda \psi \right].$$

Taking into account that ${\Lambda }^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A\Lambda ={\gamma }^{‘0}{\gamma }^{‘a}$ is the Lorentz transformed object, and ${\Lambda }^{†}{\ast }_A{D}_a{\ast }_A\Lambda =D_a^{\prime }$ is the Lorentz transformed object, while ${(\Lambda {\ast }_A{D}_a^{\prime }\psi )}^{†}={\left(D_a^{\prime }{\ast }_A\psi \right)}^{†}{\Lambda }^{†}$, the new, transformed, fermion Lagrange density is

$${\mathcal{L}}_F(\Lambda \psi )={\displaystyle \frac{1}{2}}\left[{\psi }^{†}{\ast }_A{\gamma }^{‘0}{\gamma }^{‘a}{\ast }_A{D}_a^{\prime }{\ast }_A\psi +{\left(D_a^{\prime }\psi \right)}^{†}{\ast }_A{\gamma }^{‘0}{\gamma }^{‘a}{\ast }_A\psi \right].$$

(43)

Expanding the derivative term yields

$$p_a(\Lambda \psi )=(p_a\Lambda )\psi +\Lambda \left(p_a\psi \right).$$

The new covariant derivative is, if one recalls that $-{\Lambda }^{†}{\ast }_A{}^I\mathcal{A}_a{\ast }_A\Lambda +{\Lambda }^{†}{p}_a\Lambda =-{}^I{\mathcal{A}}^{\prime }_a$ and $-{\Lambda }^{†}{\ast }_A{}^{II}\mathcal{A}_a{\ast }_A\Lambda =-{}^{II}{\mathcal{A}}^{\prime }_a$, equal to:

$${\Lambda }^{†}{D}_a(\Lambda \psi )=D_a^{\prime }\psi =\{p_a-{}^I{\mathcal{A}}^{\prime }_a\}{\ast }_A\psi -\psi {\ast }_A{}^{II}{\mathcal{A}}^{\prime }_a.$$

(44)

This determines the transformation laws of the bosonic fields

$$\begin{array}{ccc}\hfill {}^I\mathcal{A}_a^{\prime }& =& {\Lambda }^{†}{}^I\mathcal{A}_a\Lambda -{\Lambda }^{†}(p_a\Lambda ),\hfill \\ \hfill {}^{II}\mathcal{A}_a^{\prime }& =& {\Lambda }^{†}{}^{II}\mathcal{A}_a\Lambda \hfill \end{array}$$

(45)

(one finds for ${}^I{\mathcal{A}}^{\prime }_a={\Lambda }^{†}\psi {\ast }_A{\psi }^{†}\Lambda -{\Lambda }^{†}(p_a\Lambda )$ and ${}^{II}{\mathcal{A}}^{\prime }_a={\psi }^{†}{\ast }_A\psi ={}^{II}\mathcal{A}_a$, while ${\Lambda }^{†}\psi {\ast }_A{\psi }^{†}\Lambda$ is not $\psi {\ast }_A{\psi }^{†}$, as demonstrated in Equation (50) below). With the transformation rules (45), one obtains

$${\gamma }^0{\gamma }^a{D}_a(\Lambda \psi )={\gamma }^{‘0}{\gamma }^{‘a}{\ast }_A{D}_a^{\prime }{\ast }_A\psi ,$$

(46)

and the fermion Lagrange density preserves its form under local Lorentz transformations as noticedin Section 3.3. (The gauge transformations of boson fields coincide with (some of) the Lorentz transformations concerning only the internal space with $d>3$ if the boson fields carry the space index $n=(0,1,2,3)$.)

It may be as well assumed the quadratic form of the Lagrange density for fermions, again with the covariant derivative $(D_a\psi$ presented in Equation (44))

$${\mathcal{L}}_F={\left(D_a\psi \right)}^{†}{\ast }_A\left(D^a\psi \right).$$

(47)

This Lagrange density, certainy Hermitian, transforming under Lorentz rotations as required by Equations (44) and (45), is waiting to be studied.

Equation of Motion for Fermion Fields

Varying the fermion Lagrangian density (39) with respect to the Hermitian conjugate field ${\psi }^{†}$ gives

$$\delta {\mathcal{L}}_F={\displaystyle \frac{1}{2}}\left[\delta {\psi }^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A{D}_a\psi +{\left(D_a\delta \psi \right)}^{†}{\ast }_A{\gamma }^0{\gamma }^a{\ast }_A\psi \right].$$

After integration by parts, one obtains the equation of motion

$${\gamma }^0{\gamma }^a{\ast }_A{D}_a\psi =0.$$

(48)

Using the definition of the covariant derivative

$$D_a\psi =p_a\psi -{}^I{\widehat{\mathcal{A}}}_a{\ast }_A\psi -\psi {\ast }_A{}^{II}{\widehat{\mathcal{A}}}_a,$$

the explicit form of the fermion equation becomes

$${\gamma }^0{\gamma }^a\left(p_a\psi -{}^I{\widehat{\mathcal{A}}}_a{\ast }_A\psi -\psi {\ast }_A{}^{II}{\widehat{\mathcal{A}}}_a\right)=0.$$

(49)

One sees that the theory, assuming that internal spaces of fermion and boson fields are described by the odd number of nilpotents (fermions) and the even number of nilpotents (bosons), the rest of the projectors, requires that fermions can interact only by exchanging bosons, either with left or right multiplication.

3.5. Algebraic Structure of Boson Fields

Boson fields in the present approach carry a space-time index

$$a=0,1,2,3,5,6,\dots ,d.$$

(Here, the indices $(0,1,2,3)$ correspond to the ordinary $(3+1)$-dimensional space-time when representing vectors and tensors (only in this space-time bosons, as well as fermions, are active), while $a\ge 5$ determines scalars. The internal space is active in all $d=2(2n+1)$.)

For $a=0,1,2,3$, the fields

$${\mathcal{A}}_a\left(x\right)$$

transform as vectors in space-time. For $a\ge 5$, the corresponding fields appear as scalars from the $(3+1)$–dimensional point of view. Boson fields therefore appear either as space-time vectors or as space-time scalars.

The internal structure of bosons can be expressed in terms of fermion operators $\psi$ and ${\psi }^{†}$, Equations (19) and (20). The algebra satisfies the nilpotent relations (6):

$$\psi {\ast }_A\psi =0,{\psi }^{†}{\ast }_A{\psi }^{†}=0,$$

while mixed products do not vanish. The internal structure of bosonic operators can be expressed through the bilinears, as shown in Equations (19) and (20):

$$\psi {\ast }_A{\psi }^{†},{\psi }^{†}{\ast }_A\psi .$$

The algebra distinguishes two types of bosons, as shown in Equations (19) and (20):

$${}^I\mathcal{A}_a\sim \psi {\ast }_A{\psi }^{†},{}^{II}\mathcal{A}_a\sim {\psi }^{†}{\ast }_A\psi .$$

Both transform under Lorentz transformation as presented in Equation (45). (Let us see what Equation (45) gives for the transformation of bosons ${}^{II}\mathcal{A}_a\sim {\psi }^{†}{\ast }_A\psi$ and ${}^I\mathcal{A}_a\sim \psi {\ast }_A{\psi }^{†}$:

$$\begin{array}{ccc}\hfill {}^I\mathcal{A}_a& =& \psi {\ast }_A{\psi }^{†},{}^{II}\mathcal{A}_a={\psi }^{†}{\ast }_A\psi ,\hfill \\ \hfill {}^I{\mathcal{A}}^{\prime }_a& =& \left({\Lambda }^{†}\psi \right){\ast }_A{\left({\Lambda }^{†}\psi \right)}^{†}-{\Lambda }^{†}(p_a\Lambda )={\Lambda }^{†}\psi {\ast }_A{\psi }^{†}\Lambda -{\Lambda }^{†}(p_a\Lambda )={}^I{\mathcal{A}}^{\prime }_a\hfill \\ \hfill {}^{II}{\mathcal{A}}^{\prime }_a& =& {\left({\Lambda }^{†}\psi \right)}^{†}{\ast }_A\left({\Lambda }^{†}\psi \right)={\psi }^{†}\Lambda {\ast }_A{\Lambda }^{†}\psi ={}^{II}\mathcal{A}_a\hfill \\ \hfill \mathrm{leading}\mathrm{to}\\ \hfill {}^I{\mathcal{A}}^{\prime }_a& =& {\Lambda }^{†}\psi {\ast }_A{\psi }^{†}\Lambda -{\Lambda }^{†}(p_a\Lambda ),{}^{II}{\mathcal{A}}^{\prime }_a={\psi }^{†}{\ast }_A\psi ={}^{II}\mathcal{A}_a.\hfill \end{array}$$

(50)

While ${}^I\mathcal{A}_a$ transforms, as was used to seeing the transformation properties of vector gauge fields, the second kind does not transform at all.)

Because of the nilpotent relations—their mutual algebraic products are zero—these two sectors do not interact directly. They can interact only in the presence of a fermion. (Let us demonstrate the relation in Equation (56) on the case that $d=(5+1)$, with the fermion and boson “basis vectors” presented in Table A2. Let the “basis vector” of ${}^I\mathcal{A}_a$ be equal to ${}^I\widehat{\mathcal{A}}_4^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{(+)}\stackrel{56}{[+]}))={\widehat{b}}_1^{1†}(\equiv \stackrel{03}{(+i)}\stackrel{12}{[+]}\stackrel{56}{[+]}){\ast }_A{({\widehat{b}}_1^{2†})}^{†}(\equiv {(\stackrel{03}{[-i]}\stackrel{12}{(-)}\stackrel{56}{[+]})}^{†})$,and let it applies on a fermion $\psi$ with the “basis vector” equal to ${\widehat{b}}_1^{2†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{(-)}\stackrel{56}{[+]})$, $\psi$ with the “basis vector” ${\widehat{b}}_1^{2†}$ scatters to ${\widehat{b}}_1^{1†}$ and ${}^{II}\mathcal{A}_a$ with the “basis vector” equal to ${}^{II}\widehat{\mathcal{A}}_3^{1†}(\equiv \stackrel{03}{[-i]}\stackrel{12}{[+]}\stackrel{56}{[+]}$). $\psi$ and ${}^{II}\mathcal{A}_a$ either share the momentum and energy, or ${}^{II}\mathcal{A}_a$ remains momentumless in the vacuum, obeying Equation (16).) One finds schematically

$${}^I\mathcal{A}_a{\ast }_A\psi =\left(\psi {\ast }_A{\psi }^{†}\right){\ast }_A\psi =\psi {\ast }_A\left({\psi }^{†}{\ast }_A\psi \right)\sim \psi {\ast }_A{}^{II}\mathcal{A}_a.$$

(51)

Thus, bosons of type I and type II are connected only through fermions, what must be recognized in Feynman diagrams, which must, certainly, reproduce the experimental data.

When ${}^I\mathcal{A}_a{\ast }_A\psi \to \psi$, it is indeed ${}^I\mathcal{A}_a{\ast }_A\psi \to \psi {\ast }_A{}^{II}\mathcal{A}_a$. In this case, ${}^{II}\mathcal{A}_a$ can either remain without energy and momentum, taken by $\psi$, or both - ${}^{II}\mathcal{A}_a$ and $\psi$ - share energy and momentum. When applying either ${}^I\mathcal{A}_a\left(x\right)$ on $\psi$ or $\psi$ on ${}^{II}\mathcal{A}_a\left(x\right)$, one always obtains two fields back. In the first case, when ${}^I\mathcal{A}_a\left(x\right)$ applies on $\psi$, one obtains (another) $\psi$ and ${}^{II}\mathcal{A}_a\left(x\right)$. The boson field may stay in the vacuum without energy or appear as well as a propagating field. In the second case, when $\psi$ applies on ${}^{II}\mathcal{A}_a\left(x\right)$, one obtains $\psi {\ast }_A\left({\psi }^{†}{\ast }_A\psi \right)=\left(\psi {\ast }_A{\psi }^{†}\right){\ast }_A\psi$, that is ${}^I\mathcal{A}_a\left(x\right)$ and another $\psi$. The boson fields may stay in the vacuum without energy or appear as well as a propagating field. This becomes of high importance when looking at the Feynman diagrams.)

In general, a boson field can be written as an expansion over Clifford algebra elements, as shown in Equation (25),

$${\mathcal{A}}_a\left(x\right)=\sum _k{A}_a^{\left(k\right)}\left(x\right){\Gamma }_k,$$

(52)

where ${\Gamma }_k$ denotes elements of the Clifford algebra built from nilpotents and projectors. A boson field is a tensor product, ${\ast }_T$, of a field in the ordinary space-time in the coordinate representation and one of the two kinds of Clifford even “basis vectors”: ${\mathcal{C}}^m{}_{fa}\left(x\right)$ ${\ast }_T{\widehat{\mathcal{A}}}_f^{m†}$ (see Equation (25)). In Equation (52), ${\Gamma }_k$ replaces ${\widehat{\mathcal{A}}}_f^{m†}$ and $A_a^{\left(k\right)}\left(x\right)$ replaces ${\mathcal{C}}^m{}_{fa}\left(x\right)$.

The space-time index a determines whether the field behaves as a vector or scalar in space-time, while the Clifford algebra structure determines the physical properties such as spin and charges.

3.6. Bosonic Lagrange Density

Since the algebra distinguishes two kinds of non-interacting bosons,

$${}^I\mathcal{A}_a\mathrm{and}{}^{II}\mathcal{A}_a,$$

the bosonic Lagrange density separates naturally into two parts,

$${\mathcal{L}}_B={\mathcal{L}}_B^I+{\mathcal{L}}_B^{II}.$$

Taking into account Equations (13) and (37), and requiring that each kind contains the kinetic term for the corresponding boson fields, and the self-interacting term following Equations (13) and (37), one ends up with the field strengths

$$\begin{array}{ccc}\hfill {}^{I}F_{ab}& =& -i(p_a{}^I\mathcal{A}_b-p_b{}^I\mathcal{A}_a-{\{{}^I\mathcal{A}_a,{}^I\mathcal{A}_b\}}_{-}),\hfill \\ \hfill {}^{II}F_{ab}& =& -i(p_a{}^{II}\mathcal{A}_b-p_b{}^{II}\mathcal{A}_a-{\{{}^{II}\mathcal{A}_a,{}^{II}\mathcal{A}_b\}}_{-},\hfill \end{array}$$

(53)

and correspondingly with the bosonic Lagrange density

$$\begin{array}{c}\hfill {\mathcal{L}}_B={\displaystyle \frac{1}{4}}{}^{I}F_{ab}{}^{I}F^{ab}+{\displaystyle \frac{1}{4}}{}^{II}F_{ab}{}^{II}F^{ab}.\end{array}$$

(54)

Because the two bosonic sectors correspond to different Clifford structures, orthogonal to each other, they do not interact in the bosonic part of the Lagrangian.

The interaction between the two sectors occurs only through fermions. The fermion interaction term can schematically be, taking Equation (39) into account, written as

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{int}}& =& {\displaystyle \frac{1}{2}}\left\{\right({\psi }^{†}{\gamma }^0{\gamma }^a{}^I\mathcal{A}_a\psi +{\psi }^{†}{\gamma }^0{\gamma }^a\psi {}^{II}\mathcal{A}_a+\hfill \\ & & {[{}^I\mathcal{A}_a\psi +\psi {}^{II}\mathcal{A}_a]}^{†}{\gamma }^0{\gamma }^a\psi \}.\hfill \end{array}$$

(55)

In this way, bosons of type I and type II meet only in the presence of fermionic fields.

3.7. Feynman Rules and Interaction Structure

The algebraic structure discussed above has essential consequences for the interaction vertices and the corresponding Feynman rules.

The proposed description of the internal space of bosonic operators, having the even number of nilpotents, leads to the relations among fermion operators and boson operators

$$\psi {\ast }_A{\psi }^{†}\mathrm{and}{\psi }^{†}{\ast }_A\psi ,$$

describing two kinds of bosons,

$${}^I\mathcal{A}_a\mathrm{and}{}^{II}\mathcal{A}_a,$$

respectively. Because of the nilpotent relations

$$\psi {\ast }_A\psi =0\mathrm{and}{\psi }^{†}{\ast }_A{\psi }^{†}=0,$$

the two kinds of bosons do not interact directly. They can interact only through fermionic operators. Schematically, one finds relations of the form

$${}^I\mathcal{A}_a{\ast }_A\psi =\left(\psi {\ast }_A{\psi }^{†}\right){\ast }_A\psi =\psi {\ast }_A\left({\psi }^{†}{\ast }_A\psi \right)\sim \psi {\ast }_A{}^{II}\mathcal{A}_a.$$

The above implies that bosons of type I and type II can be connected only through fermion lines. Correspondingly, it is the fermionic Lagrange density which is responsible for the interaction among fermions and bosons.

$${\mathcal{L}}_{int}={\psi }^{†}{\ast }_A{\gamma }^0{\gamma }^a({}^I\mathcal{A}_a{\ast }_A\psi +\psi {\ast }_A{}^{II}\mathcal{A}).$$

4. Conclusions

The proposed theory, built on the assumption that the internal spaces of fermion and boson fields are describable by odd (for fermions) and even (for bosons) products of operators ${\gamma }^a$, provides the unique description of spins and charges of fermion and boson second quantised fields, as one can notice in Tables (Table A1 for the observed quarks and leptons and Table A2 for a toy model) as well as in the covariant derivatives of the Lagrange densities for fermion and boson fields in interaction; see Equations (39), (40), (54) and (55).

The theory gives the explanation for the appearance of all the so-far-observed fermions (quarks and leptons and antiquarks and antileptons; as well as for their families) and bosons ($SO(3,1)$ gravitons [19,21], two $SU\left(2\right)$ weak fields [9,10,19,21], $SU\left(3\right)$ gluons [9,10,19,21] $U\left(1\right)$ photons, and new predicted fermions (right-handed neutrinos and left-handed antineutrinos ([9,10] and references therein), the fourth family to the observed three [12,17], a new group of four families of quarks and leptons, the stable one at low energies explaining the Dark matter [13]). (Let us point out that the graviton in this theory is explained in an equivalent way as all the gauge fields observed so far.)

Both fields, fermions and bosons, are assumed to be massless and appear in a flat space-time.

Without breaking symmetries, there would also exist boson fields carrying more than one charge at the same time, like the weak and colour charge, or the spin, weak charge and colour charge, which have not (yet) been observed.

There is the break in the symmetries that make fermion fields and some of the boson fields massive, as studied in Refs. [9,10]. (The breaks of symmetries were studied when the boson fields were described by $S^{ab}{\omega }_{ab\alpha }$ and ${\tilde{S}}^{ab}{\tilde{\omega }}_{ab\alpha }$ (see Section 6.2 in Ref. [9] and references therein), instead of by ${}^I\widehat{\mathcal{A}}_{f\alpha }^{m†}$ and ${}^{II}\widehat{\mathcal{A}}_{f\alpha }^{m†}$.)

The Lorentz transformations in internal space, presented in Section 3.2, manifest the gauge transformations of other theories when the internal space with $d\ge 5$ is considered. The discussed description of the internal spaces of fermion and boson fields requires that all bosons are treated equivalently with gravitons (with the space index $a=(0,1,2,3)$) and scalars included (scalars have the space index $a\ge 5$, while they can have all the internal space properties as vectors and tensors).

The choice that the internal spaces of fermions and bosons are described by a superposition of odd and even products of operators ${\gamma }^a$, respectively, explains Dirac’s postulates of the second quantised fermion and boson fields.

It is assumed that the Dirac-like action for fermion fields but with the covariant derivative required by the description of the internal spaces of fermion and boson fields, as presented in Equations (39) and (40). The fermion Lagrangian density, preserving its form under the local Lorentz transformations as shown in Section 3.4, explains the gauge invariance of the existing theories if our internal space concerns $d\ge 5$. (There are gravitons with two nilpotents in $SO(3,1)$, the rest are projectors; see Equation (30), suggesting that gravitons should be treated according to the standard gauge fields.) There is no negative energy Dirac sea for fermions. Fermions have only ordinary quantum vacuum. Fermions and antifermions appear in the same family.

The quadratic Lagrange density for fermion fields, preserving its form under local Lorentz transformations, presented in Equation (47), remains to be studied.

Also, the boson Lagrangian density preserves its form under the local Lorentz transformations, as shown in Equation (50).

The proposed description of the internal spaces of fermion and boson fields requires two kinds of mutually non-interacting boson fields; see Section 3.5 for ${}^I\mathcal{A}_a$ and ${}^{II}\mathcal{A}_a$, ${}^I\mathcal{A}_a{\ast }_A{}^{II}\mathcal{A}_a=0$. Their Lorentz transformation properties are presented in Equations (45) and (50).

The Lagrange density for bosons, generated as in ordinary theories, is, correspondingly, the sum of two independent Lagrange densities, as presented in Section 3.6, Equations (53) and (54), for ${}^I\mathcal{A}_a$ and ${}^{II}\mathcal{A}_a$, which interact only through fermions.

The boson fields either manifest the left multiplication to fermions, ${}^I\mathcal{A}_a$, or the right one, ${}^{II}\mathcal{A}_a$, as shown in Equation (40).

Their mutual interaction term in the Lagrange densities, presented in Equation (55), determined by our description of the internal spaces of the fermion and boson fields, requires two kinds of bosons.

The choice of the algebra used to describe the internal properties of fermion and boson fields determines also the Lagrange densities for fermions and bosons.

In any $d=2(2n+1)$ dimensional internal space, the fermion and boson states are arranged to be eigenvectors of all the members of the Cartan subalgebra, as shown in Equation (3); eigenstates were called the “basis vectors”. The “basis vectors” for fermion fields have an odd number of nilpotents, and for the boson fields, there are an even number of nilpotents, and the rest are projectors; see Equation (4).

The fermion “basis vectors” appear in $2^{{\displaystyle \frac{d}{2}}-1}$ families, each family having $2^{{\displaystyle \frac{d}{2}}-1}$ members; and there are $2^{{\displaystyle \frac{d}{2}}-1}\times$$2^{{\displaystyle \frac{d}{2}}-1}$ of their Hermitian conjugate partners, appearing in a separate group.

The boson “basis vectors” appear in two orthogonal groups, each with $2^{{\displaystyle \frac{d}{2}}-1}\times$$2^{{\displaystyle \frac{d}{2}}-1}$ members, and they have their Hermitian conjugate partners within the same group.

The “basis vectors” for bosons are expressible as the algebraic products of fermion “basis vectors” and their Hermitian conjugate partners, as shown in Equations (19) and (20).

The second quantised fermion fields are tensor products of the “basis vectors” and basis in ordinary space time; see Equation (23).

The second quantised boson fields are tensor products of the “basis vectors” and basis in ordinary space time, and they carry the space-time index, as shown in Equation (25).

The properties of fermion and boson fields presented in this study are outlined as follows.

• The algebraic product of two fermions is zero:

  $$\psi {\ast }_A\psi =0\mathrm{and}{\psi }^{†}{\ast }_A{\psi }^{†}=0,$$
• The algebra distinguishes two types of bosons, as shown in Equations (19) and (20):

  $${}^I\mathcal{A}_a\sim \psi {\ast }_A{\psi }^{†}\mathrm{and}{}^{II}\mathcal{A}_a\sim {\psi }^{†}{\ast }_A\psi ,$$
• Bosons can interact only in the presence of a fermion:

  $${}^I\mathcal{A}_a{\ast }_A\psi =\left(\psi {\ast }_A{\psi }^{†}\right){\ast }_A\psi =\psi {\ast }_A\left({\psi }^{†}{\ast }_A\psi \right)\sim \psi {\ast }_A{}^{II}\mathcal{A}_a,$$

  influencing the Feynman diagrams. The Feynman rules are studied in Refs. [35,36].

Demonstrating quite a number of simple and elegant enough descriptions of the second quantised fields, explaining the assumptions of existing theories, gives a hope that the theory considered here might be what the universe obeys.

What Should One Understand

If the proposed description of the internal degrees of freedom of the second quantised fermion and boson fields is what nature uses in the case that the space-time is flat, and all the second quantised fields are massless, while all the fields are active only in $d=(3+1)$ in ordinary space-time, the proposed theory is able not only to explain the Standard Model assumptions before the electroweak break but also find the solution for all the open questions connected with the elementary fermion and boson fields.

There are two kinds of boson fields in this theory. One kind describes the observed gauge fields and the graviton field on an equal level, unifying all the boson fields. This kind, causing transformations within members of each family of fermions and antifermions, predicts the existence of the second kind of the weak force and, correspondingly, the existence of the right-handed neutrino and the left-handed anti-neutrino, as seen in Table A1.

The second kind of boson fields transforms a family member of one family to the same family member of all the families, giving the explanation for the appearance of the Higgs scalars and predicting new scalar fields [14,15,16].

The internal spaces of all the observed boson fields are describable with “basis vectors” having only two nilpotents (gravitons in $SO(3,1)$ part of $SO(13,1)$, two kinds of weak bosons in $SO\left(4\right)$ part of $SO(13,1)$, gluons in $SU\left(3\right)$ part of $SO\left(6\right)$, part of $SO(13,1)$), and the remaining being projectors and photons with only projectors in all $SO(13,1)$.

There are, however, many more boson fields. Their “basis vectors” have more than two nilpotents (they can have four or six nilpotents), carrying correspondingly, more than one kind of the so-far-observed charges. There are many more families in this theory than the observed three. The theory predicts that the three observed families are the members of the group of four families [17]. The theory predicts also the second group of four families, contributing to the dark matter [13].

To be able to explain why “nature has decided” to break symmetries, one should know the properties this theory has with respect to the following.

(a)

The renormalisability and anomalies in even and odd dimensional spaces.

(b)

How the second kind of bosons contributes to the breaking of symmetries, while the first kind of the boson “basis vectors” seems to mainly determine the properties of all the observed boson fields with the gravity included.

(c)

The differences in odd, $d=(2n+1)$, and even, $d=\left(2\right(2n+1\left)\right)$, dimensional spaces. In even dimensional spaces, $d=2(2n+1)$, the odd “basis vectors” anticommute and have their Hermitian conjugated “basis vectors” in a separate group, and the even “basis vectors” commute and appear in two orthogonal groups, having the “basis vectors” in $d=2(2n+1)+1$ strange properties. Half of the odd and even “basis vectors” behave like in $d=2(2n+1)$; in the second half, the commuting odd “basis vectors” appear in two orthogonal groups, while the commuting even “basis vectors” appear in families and have the Hermitian conjugate partners in a separate group [20].

(d)

The differences in even-dimensional internal spaces, when $d=2(2n+1)$ and $d=4n$. While in $d=2(2n+1)$ the “basis vectors” for fermions and antifermions appear in the same family, in $d=4n$, the “basis vectors” of a family do not include antifermions. Correspondingly, the vacuum in $d=2(2n+1)$ is just the quantum vacuum, while in $d=4n$, the Dirac sea with the negative energies must be invented.

(e)

How to present and interpret the Feynman diagrams in this theory in comparison with the Feynman diagrams so far presented and interpreted [35,36].

(f)

It might be needed to extend the second quantised fermion and boson fields to strings with the first step already completed in Refs. [35,36].