#  A unified field theory from a complexified quaternion-octonion Dirac
equation

Juan Camilo Vélez Quiñones<EMAIL_ADDRESS>Physics
Department, Universidad del Valle, Cali, Colombia.

(April 2022)

###### Abstract

It is set manifest an underlying algebraic structure of Dirac equation and
solutions, in terms of C$\ell_{2}$ Clifford algebra projectors and ladder
operators. From it, a scheme is proposed for constructing unified field
theories by enlarging the pointed algebra. A toy unified matter field model is
formulated, modifying the Dirac equation with complex quaternions and
octonions. The result describes a set of fermion fields with reminiscent
properties of one standard model particle generation, exhibiting U(1)
electromagnetism, SU(2) flavor, and SU(3) color symmetries, remarkably SU(2)
induces frame fields, though further explorations are needed.

##  I. INTRODUCTION

In particle physics, unified field theories (UFTs) try to describe fundamental
particles and forces of nature as different manifestations of a single
phenomenon aiming to elucidate underlying principles, behind seeming
complexity. Throughout the last decades, these theories have been applied as
an alternative for tackling the apparent structure arbitrariness of the
standard model, by merging elementary particles and interactions through
simple principles that could explain its rather ad hoc appearance [1, 2].

The roots of the unification pursuit lie in the successful unification of
electromagnetism and weak force into electroweak theory in the 60s. From there
the natural question was raised, if it is possible to encompass strong force
together with electroweak force into a grand unified theory (GUT). With the
aforementioned motivation, further attempts have been driven for constructing
theories meant to merge the standard model interactions into a single force.

The first GUTs were the four-color Pati Salam model [3], Georgi and Glashow
SU(5) [4] fusing all interactions in an electrocolor force, and the Georgi
[5]/Fritzsch and Minkowski [6] Spin(10), which go further not only unifying
bosons but fermions as well. These models give account in general for one
generation of standard model particles and their interactions. This comes at a
cost, like the intrusion of new interaction and proton decay predictions,
which remain experimentally unobserved up today. Despite these issues, a
plethora of theories have been formulated since then, but the problems from
initial works persist.

The usual approach for formulating GUTs is an extension of the methodology
used for the elaboration of electroweak theory: a master gauge group
containing U(1), SU(2), and SU(3) is proposed, and from such a bigger group,
an invariant Lagrangian under gauge transformations is crafted, requiring the
insertion of particle fields and interactions.

However, the mentioned GUT’s methodology is not the only one for conceiving
unified theories. An alternative is an algebraic approach, taking as a
starting point algebras, instead of master gauge groups. There, particles
correspond to algebra elements where inherent symmetries and hence charges,
emerge naturally in relation to actions of other algebra elements. This can be
done by elaborating minimal left ideals for representing particles [7, 8, 9,
10, 11, 12], but it is not strictly necessary [13]. The referred constructions
can describe different forces emerging from algebra symmetries, without
appealing to a master group as GUTs do, therefore avoiding a source of unknown
forces and unwanted effects [8]. Nevertheless, they could result in an
inaccurate description, not including all forces, or not getting standard
model symmetry groups.

Early examples of this algebraic approach were for instance the strong
interaction descriptions using octonions [14, 15, 16], later extended to other
forces using the four normed division algebras, which are the real
$\mathbb{R}$, complex $\mathbb{C}$, quaternions $\mathbb{H}$ and octonions
$\mathbb{O}$ numbers [17].

In the algebra approach, the usage of Clifford algebras is common. Among their
advantages is the fact that spinors are left or right ideals [18], making them
a suitable language for describing fermions, and for esthetic equations, which
can give algebraic structure inside [19, 20, 21]. Additionally, division
algebras are related to them through isomorphisms between quaternions,
octonions, and C$\ell_{2}$, C$\ell_{6}$ Clifford algebras respectively [22,
11]. Early examples of implementations of these algebras are the theories of
Chisholm and Farwell [23], or more updated models include C$\ell_{6}$ Stoica’s
[12], or Furey’s [10, 11] casted with octonions, but isomorphic to
C$\ell_{6}$.

The algebra approach it is sometimes not addressed the why and how the
algebraic nature of particles could interplay in Lagrangians, ending with a
lack of dynamical equations. However when it is tackled, it could yield that
Lagrangians allow being casted in modified versions, where the Dirac gamma
matrices are replaced by elements belonging to the algebra [8, 13, 23, 7, 16],
resulting in single expressions, which contain all fermion fields from one
generation. The fact that altered Lagrangians could describe several types of
particles in a single field [16, 13, 7, 24, 25], is not unexpected since
enlarging gamma matrices imply a wider dimension for the Dirac equation
solutions, a desirable feature for fermions unification.

Relative to what has been said, here is an exploration of an alternative
course of action for generating a UFT via modifying the Dirac equation (DE).
This is achieved through a concrete example where DE is reformulated with
complexified quaternions and octonions, resulting in a theory with a Dirac-
like equation, that simultaneously describes all fermions from one generation
with a unified matter field, and forces are included by Yang-Mills covariant
derivatives. All of these heavily rely on Furey’s [10, 11] previous results.
The procedure is generalized in a straightforward manner, being an alternative
scheme to GUTs master gauge groups, and serving as a complement to the
algebraic approach, providing a systematic method of dynamical equations
formulation for any algebra with a Clifford sub-algebra.

The paper is ordered as follows: in section II an underlying algebraic
structure of DE will be manifested, from there the schema for conceiving UFTs
is going to be formulated. In section III DE will be expressed using
quaternions, which allow finding simpler expressions for the usual terms in
Dirac theory. Section IV will build upon section III’s results and will apply
section II’s schema to the development of a toy theory that describes a set of
particles like those in one generation of standard model with U(1)
electromagnetism, SU(2) flavor and SU(3) symmetries, and a gravity-like
coupling shape.

##  II. DIRACS EQUATION ALGEBRA INSIGHTS

In this section, an underlying algebraic structure of DE is set explicit, and
it will serve as a starting point for further manipulations. This is going to
be done by noting that Dirac (gamma matrices) algebra is a C$\ell_{4}$
Clifford algebra equivalent to $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$.
Therefore it can be written in terms of two $\mathfrak{su}(2)$ copies, showing
an algebraic structure that relates the equation solutions to $C\ell_{2}$
ladder operators.

Setting conventions first, all calculations are going to be carried in natural
units $\hbar=c=1$. Einstein summation is adopted unless otherwise stated,
Greek indices sum over spacetime coordinates, while Latin indices run upon
space.

Starting with DE,

$\displaystyle(i\not\\!{\partial}-m)\psi=0,$ (1)

which explicitly in the Weyl representation is,

$\displaystyle\begin{bmatrix}\begin{pmatrix}0&\mathbb{1}\\\
\mathbb{1}&0\end{pmatrix}p_{0}+\begin{pmatrix}0&\bar{\sigma}^{j}\\\
\sigma^{j}&0\end{pmatrix}p_{j}-m\begin{pmatrix}\mathbb{1}&0\\\
0&\mathbb{1}\end{pmatrix}\end{bmatrix}\psi=0.$ (2)

This equation can be re-expressed in terms of tensor products between two
copies of Pauli matrices, a spin set of $\sigma^{i}$ and a chiral one of
$\chi_{i}$,

$\displaystyle\begin{bmatrix}(\chi_{1}\otimes\mathbb{1})p_{0}-i(\chi_{2}\otimes\sigma^{j})p_{j}-m(\mathbb{1}\otimes\mathbb{1})\end{bmatrix}\psi=0.$
(3)

The usual approach for tackling E.q(3) is to work column bi-spinor $\psi$, but
alternatively, matrices can also be used [7, 13]. Between the possibilities,
are a kind of square matrices known as algebraic spinors, which have the
particularity of being minimal left (right) ideals of some Clifford algebras
[18]. The C$\ell_{4}$ algebraic spinors will emerge naturally if one decides
to find solutions to the DE in terms of gamma matrices instead of bi-spinors.

Aiming to find solutions in terms of $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$,
a straight strategy is to solve E.q(3) at rest frame $p_{j}=0$, and then boost
it. From here and on, the solutions with the column bi-spinor will be denoted
by $\psi$, and the ones with algebraic elements $\Psi$. With ansatz solutions
of the form

$\displaystyle\Psi=\frac{1}{2}(\mathbb{1}\pm\chi_{1})e^{\mp imt},$ (4)

obviating the $\otimes$ for short writing, it is clear to see that E.q(4) is a
solution for E.q(3)

$\displaystyle[i\chi_{1}\partial_{0}-m\mathbb{1}](\chi_{1}\pm\mathbb{1})e^{\mp
imt}$ $\displaystyle=e^{\mp
imt}[\pm\chi_{1}m-m\mathbb{1}](\chi_{1}\pm\mathbb{1})$ $\displaystyle=\pm
me^{\mp imt}(\chi_{1}\mp\mathbb{1})(\chi_{1}\pm\mathbb{1})$ $\displaystyle=\mp
me^{\mp imt}[(\chi_{1})^{2}-\mathbb{1}]=0.$ (5)

Now, a generic boost can be found from the usual expression, by appealing to
the isomorphism between the Weyl and the $\sigma_{i},\leavevmode\nobreak\
\chi_{i}$ representations,

$\displaystyle\Lambda_{P}$
$\displaystyle=\begin{pmatrix}e^{-\frac{1}{2}\sigma^{\mu}p_{\mu}}&0\\\
0&e^{\frac{1}{2}\sigma^{\mu}p_{\mu}}\end{pmatrix}\rightarrow\frac{1}{2}(\mathbb{1}+\chi_{3})e^{-\frac{1}{2}\sigma^{\mu}p_{\mu}}+\frac{1}{2}(\mathbb{1}-\chi_{3})e^{\frac{1}{2}\sigma^{\mu}p_{\mu}},$
(6)

with four-vectos $P=(p_{0},p_{1},p_{2},p_{3})$ and
$\sigma^{\mu}=(\mathbb{1},\sigma^{1},\sigma^{2},\sigma^{3})$. Furthermore the
exponential factors can be worked as usual

$\displaystyle
e^{\pm\frac{1}{2}\sigma^{\mu}p_{\mu}}=\sqrt{\cosh|p_{0}|\mp\sinh|\vec{p}|\hat{n}\cdot\vec{\sigma}}=\sqrt{E/m\mp|\vec{p}|/m|}=\frac{1}{\sqrt{m}}\sqrt{p_{0}\mp\vec{p}\cdot\vec{\sigma}},$
(7)

and defining the chiral protectors $\chi_{L}=\frac{1}{2}(1+\chi_{3})$ and
$\chi_{R}=\frac{1}{2}(1-\chi_{3})$, the boost matrix is given by,

$\displaystyle\Lambda_{P}$
$\displaystyle=\frac{1}{\sqrt{m}}\left[\chi_{L}\sqrt{p_{\mu}\sigma^{\mu}}+\chi_{R}\sqrt{p_{\mu}\bar{\sigma}^{\mu}}\right],$
(8)

where $\bar{\sigma}^{\mu}=(\mathbb{1},-\sigma^{1},-\sigma^{2},-\sigma^{3})$.
For adding the spin degrees of freedom to the solution, in analogous way to
Lorentz boost the spin operator has the correspondence

$\displaystyle S=\frac{\hbar}{2}\begin{pmatrix}\sigma^{3}&0\\\
0&\sigma^{3}\end{pmatrix}\rightarrow\frac{\hbar}{2}\sigma^{3},$ (9)

with eigenvectors $s_{\uparrow\downarrow}=(1\pm\sigma^{3})/2$, since

$\displaystyle\sigma^{3}\frac{1}{2}(1\pm\sigma^{3})=\pm\frac{\hbar}{2}(1\pm\sigma^{3}).$
(10)

Hence some solutions of E.q(3) are of the form

$\displaystyle\Psi$
$\displaystyle=\Lambda_{P}s_{\uparrow\downarrow}(\mathbb{1}\pm\chi_{1})e^{\mp
ix_{\mu}p^{\mu}}.$ (11)

What is remarkable from E.q(11) is that
$\mathcal{P}_{\pm}=\frac{1}{2}(\mathbb{1}\mp\chi_{1})$ are actually
$\mathfrak{su}(2)$ projectors, and right multiplying them by $\chi_{2}$, one
gets

$\displaystyle\Psi$
$\displaystyle=\Lambda_{P}s_{\uparrow\downarrow}(\chi_{2}\pm i\chi_{3})e^{\mp
ix_{\mu}p^{\mu}},$ (12)

a form of solutions that explicitly have ladders operators from the chiral
algebra $\alpha_{\pm}=\frac{1}{2}(\mp\chi_{2}+i\chi_{3})$. Now these solutions
can be relate to the usual column spinors by tacking $u$’s as column
eigenvectors of $\alpha_{\pm}$ and $\mathcal{P}_{\mp}$, the conventional Dirac
wave functions are recovered just by left action of E.q(12) over some $u$.

In summary, DE solutions including the ones with algebraic spinors, can be
grouped as

$\displaystyle\Psi^{\mp}$
$\displaystyle=\Lambda_{P}s_{\uparrow\downarrow}\mathcal{P}_{\mp}e^{\mp
ix_{\mu}p^{\mu}}$ $\displaystyle\Psi_{f}^{\mp}$
$\displaystyle=\Lambda_{P}s_{\uparrow\downarrow}\alpha_{f}e^{\mp
ix_{\mu}p^{\mu}}$ $\displaystyle\psi_{f}^{\mp}$
$\displaystyle=\Lambda_{P}s_{\uparrow\downarrow}u_{f}e^{\mp ix_{\mu}p^{\mu}},$
(13)

taking $f$ as an arbitrary index for fermion solutions. In the column
solutions the structure including projectors and ladders is absent, but it is
explicit with algebraic spinor, the formers shedding light into underlying
algebraic architecture.

The E.q(II. DIRACS EQUATION ALGEBRA INSIGHTS) expressions serve as a ground
for the formulation of a unified matter field construction scheme, which
consists of assuming the algebra of gamma matrices as a sub-algebra of a
larger one. The logic behind this comes from E.q(II. DIRACS EQUATION ALGEBRA
INSIGHTS), one can see that enlarging the $\chi_{i}$’s chiral Clifford algebra
and hence widening the set of ladder operators, will yield a theory with more
fermion fields, described simultaneously by a single Dirac like equation. The
$\chi_{i}$’s belong to a Clifford algebra C$\ell_{2}$ then a natural extension
is to a C$\ell_{N}$, however, it is not a restriction since DE will conserve
its structure just by demanding that the equivalent gamma matrices follow the
Dirac C$\ell_{4}$ algebra.

##  III. DOUBLE QUATERNION DIRAC EQUATION

The present section is meant to show that changing the chiral $\chi_{i}$ and
spin $\sigma_{i}$ matrices by sets of quaternions, allows casting Dirac
solutions of E.q(II. DIRACS EQUATION ALGEBRA INSIGHTS) into expressions with
transparent algebraic simplicity that will serve as preliminary before using
octonions in the next sections.

Isomorphism between Pauli matrices and quaternions suggests a change of chiral
and spin matrices by two sets of quaternions. Though it will give a space of
solutions with the same dimension, they will be in terms of normed division
algebras, therefore with easier to handle properties that will allow
disclosing further algebraic simplicity.

Following the same idea of Rowlands nilpotent DE [26], two idependent
quaternions sets are defined

$\displaystyle i\sigma^{1}=\bm{i}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
i\sigma^{2}=\bm{j}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ i\sigma^{3}=\bm{k}$ $\displaystyle
i\chi^{1}={\text{{i}}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
i\chi^{2}={\text{{j}}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ i\chi^{3}={\text{{k}}},$ (14)

also two kinds of quaternions conjugates are set, for a generic element
$\bm{a}=a_{0}+a_{1}\bm{i}+a_{2}\bm{j}+a_{3}\bm{k}$ the spin quaternion
conjugate is defined as
$\bar{\bm{a}}=a_{0}-a_{1}\bm{i}-a_{2}\bm{j}-a_{3}\bm{k}$, and for an object of
the type $\bm{b}=b_{0}+b_{1}{\text{{i}}}+b_{2}{\text{{j}}}+b_{3}{\text{{k}}}$
the chiral conjugate is
$\tilde{\bm{b}}=b_{0}-b_{1}{\text{{i}}}-b_{2}{\text{{j}}}-b_{3}{\text{{k}}}$,
where all elements satisfy
$\overline{\bm{a}\bm{a}^{\prime}}=\bar{\bm{a}}^{\prime}\bar{\bm{a}}$ and
$\widetilde{\bm{b}\bm{b}^{\prime}}=\tilde{\bm{b}}^{\prime}\tilde{\bm{b}}$.
Finally for completeness the hermitian of a generic object $X$, is its spin,
chiral and complex conjugate, $X^{\dagger}=\tilde{{\overline{X}}}^{*}$.

Using the quaternions of E.q(III. DOUBLE QUATERNION DIRAC EQUATION), and
defining the spacial gradient as
$\bm{\partial}=\partial_{1}\bm{i}+\partial_{2}\bm{j}+\partial_{3}\bm{k}$, the
DE can be written as,

$\displaystyle\begin{bmatrix}{\text{{i}}}\partial_{0}+{\text{{j}}}\bm{\partial}-m\end{bmatrix}\psi$
$\displaystyle=0,$ (15)

which is an equation equivalent to the one obtained by Rowlands [26]. However
similarities end here since Rowlands solutions are DE in momentum and
frequency domain. Here instead the E.q(II. DIRACS EQUATION ALGEBRA INSIGHTS)
expression will be taken as solutions, just substituting Pauli matrices by
their correspondent quaternions, where the spin factor becomes
$s_{\uparrow\downarrow}=\frac{1}{2}(1\pm\bm{k})$, the projector
$\mathcal{P}_{\pm}=\frac{1}{2}(1\pm i{\text{{i}}})$ and the ladder operators
can be taken as $\alpha_{f}=\frac{1}{2}({\text{{j}}}\mp i{\text{{i}}})$.

For rewriting Lorentz boost, a spin four-momentum can be defined as
$\bm{P}=p_{0}-ip_{1}\bm{i}-ip_{2}\bm{j}-ip_{3}\bm{k}$, which matches
Schuricht’s four-vectors shape [27] in his conplexified quaternion description
of flat space-time. Then Lorentz boost of E.q(II. DIRACS EQUATION ALGEBRA
INSIGHTS) can be casted as

$\displaystyle\Lambda_{P}$
$\displaystyle=\frac{1}{2\sqrt{m}}\left[(1-i{\text{{k}}})\sqrt{\bm{P}}+(1+i{\text{{k}}}){\overline{\sqrt{\bm{P}}}}\right].$
(16)

At this point quaternions pay out, by defining the left chiral projector
$\chi=\frac{1}{2}(1-i{\text{{k}}})$, where in E.q(16) are recognizable the
left and right Lorentz boosts $\Lambda_{L}=\chi\sqrt{\bm{P}}$ and
$\Lambda_{R}=\tilde{\chi}\sqrt{\bm{P}}^{*}$, adopting the simple form

$\displaystyle\Lambda_{P}$
$\displaystyle={\small\frac{1}{\sqrt{m}}}[\chi\sqrt{\bm{P}}+\tilde{\chi}\sqrt{\bm{P}}^{*}]$
(17)
$\displaystyle={\small\frac{1}{\sqrt{m}}}\left[\Lambda_{L}+\Lambda_{L}^{*}\right],$
(18)

and the algebraic Dirac solutions of E.q(II. DIRACS EQUATION ALGEBRA
INSIGHTS), transforming to each other just by

$\displaystyle\Psi^{+}=\Psi^{-*}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Psi_{f}^{+}=\Psi_{f}^{-*},$ (19)

breaking down complexity of expression, to clear relations in terms of
conjugates, projectors and ladder operators.

##  IV. MODIFIED DIRAC EQUATION

This section will address an example of unified field theory, constructed from
the altered DE scheme presented at the end of section II. This will be done by
changing chiral quaternions of III by octonions. The properties
characterization for the spawned fields lays heavily on Furey’s results [10,
11].

### A. OCTONIONS AS CHIRAL ALGEBRA

For an example of a unified theory constructed with section’s II methodology,
it is enough to choose a random algebra with a $\mathfrak{su}(2)$ sub-algebra
to play the $\chi_{i}$’s chiral roll, enriching its inner structure by
enlarging solutions’ dimensions, without altering the E.q(3) DE form. However,
the simplicity of section’s III expression is desirable, induced by the normed
division algebra as a chiral component. Octonions are the next bigger division
algebra, additionally, it has been proven that they can give account for
electric and color symmetries, therefore using them for writing a DE could
yields solutions that inherit those properties. Hence it is plausible to ask
what if the chiral algebra is not quaternionic but octonionic. Remarkably this
theory will encompass all possible division algebras
$\mathbb{R}\oplus\mathbb{C}\oplus\mathbb{H}\oplus\mathbb{O}$, known as Dixon
algebra.

Octonions are not associative, a feature that is overcome in the following
steps by considering only the left action of $\mathbb{C}\oplus\mathbb{O}$ on
itself, since an isomorphism to $C\ell_{6}$ exists [11]. Ergo, the choosing of
octonions could be argued as merely cosmetic, since non-associativity is
disregarded. Nevertheless, it is a matter of taste, rather than a semantic
discussion.

Figure 1: Octonion multiplication rules. Each trio of unities $\mathbf{e}_{j}$
on a single segment, conforms a set of quaternions. The $\mathbf{o}_{j}$
objects are the nearest octonions unities in the diagram, under unitary
transformation $U\mathbf{e}_{j}U^{-1}$.

For a brief introduction octonions have seven imaginary units $\bm{e}_{j}$
with multiplication rules shown in Fig.(1). All unities anti-conmute
$\bm{e}_{j}\bm{e}_{i}=-\bm{e}_{i}\bm{e}_{j}$, and not associative for unities
belonging to different segments of Fig.(1), existing the possibility of
$(\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}))\neq((\bm{e}_{i}\bm{e}_{j})\bm{e}_{k})$.
For the simplicity, only left multiplication is going to be considered, taking
as equivalents
$\bm{e}_{i}\bm{e}_{j}\bm{e}_{k}=(\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}))$ [17].
Therefore a generic element of octonions is
$f=f_{0}+\sum_{j=1}^{7}f_{j}\bm{e}_{j}$, and the chiral conjugate is now
adopted as $\tilde{f}=f_{0}-\sum_{j=1}^{7}f_{j}\bm{e}_{j}$. And additional to
the seven unities $\bm{e}_{j}$, are defined the plus/minos elements as
$\bm{e}_{+}=\bm{e}_{4}\bm{e}_{2}\bm{e}_{1}$ and
$\bm{e}_{-}=\bm{e}_{3}\bm{e}_{6}\bm{e}_{5}$. Finally, from Fig.(1), it can be
infer a representation, for left multiplication of two generic octonions, $f$
and $g$,

$\displaystyle
fg=\footnotesize{\left(\begin{array}[]{cccccccc}f_{0}&-f_{1}&-f_{2}&-f_{3}&-f_{4}&-f_{5}&-f_{6}&-f_{7}\\\
f_{1}&f_{0}&-f_{4}&-f_{7}&f_{2}&-f_{6}&f_{5}&f_{3}\\\
f_{2}&f_{4}&f_{0}&-f_{5}&-f_{1}&f_{3}&-f_{7}&f_{6}\\\
f_{3}&f_{7}&f_{5}&f_{0}&-f_{6}&-f_{2}&f_{4}&-f_{1}\\\
f_{4}&-f_{2}&f_{1}&f_{6}&f_{0}&-f_{7}&-f_{3}&f_{5}\\\
f_{5}&f_{6}&-f_{3}&f_{2}&f_{7}&f_{0}&-f_{1}&-f_{4}\\\
f_{6}&-f_{5}&f_{7}&-f_{4}&f_{3}&f_{1}&f_{0}&-f_{2}\\\
f_{7}&-f_{3}&-f_{6}&f_{1}&-f_{5}&f_{4}&f_{2}&f_{0}\end{array}\right)\left(\begin{array}[]{c}g_{0}\\\
g_{1}\\\ g_{2}\\\ g_{3}\\\ g_{4}\\\ g_{5}\\\ g_{6}\\\
g_{7}\end{array}\right)}.$ (36)

Now the question is which elements to select as chiral components of E.q(11).
For such selection and clarity for further expression manipulation E.q( 36) is
going to be transformed into a basis where $\bm{e}_{+}$ is diagonal. This is
done through the unitary transformation

$U=\footnotesize{\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-i\\\
0&0&0&0&0&0&-i&0\\\ 0&0&0&0&0&-i&0&0\\\ 0&0&0&-i&0&0&0&0\end{array}\right)},$
(37)

such that $\bm{o}_{\ell}=U\mathbf{e}_{j}U^{-1}$ as is indicated in Fig.(1),
yields

$UfU^{-1}=\footnotesize{\left(\begin{array}[]{cccccccc}f_{0}&-f_{2}&-f_{4}&-f_{1}&-if_{7}&-if_{6}&-if_{5}&-if_{3}\\\
f_{2}&f_{0}&-f_{1}&f_{4}&if_{6}&-if_{7}&if_{3}&-if_{5}\\\
f_{4}&f_{1}&f_{0}&-f_{2}&if_{5}&-if_{3}&-if_{7}&if_{6}\\\
f_{1}&-f_{4}&f_{2}&f_{0}&if_{3}&if_{5}&-if_{6}&-if_{7}\\\
-if_{7}&if_{6}&if_{5}&if_{3}&f_{0}&f_{2}&f_{4}&f_{1}\\\
-if_{6}&-if_{7}&-if_{3}&if_{5}&-f_{2}&f_{0}&f_{1}&-f_{4}\\\
-if_{5}&if_{3}&-if_{7}&-if_{6}&-f_{4}&-f_{1}&f_{0}&f_{2}\\\
-if_{3}&-if_{5}&if_{6}&-if_{7}&-f_{1}&f_{4}&-f_{2}&f_{0}\\\
\end{array}\right).}$ (38)

Among this objects, it is noteworthy that
$\\{i\bm{o}_{0},-\bm{o}_{-},\bm{o}_{+}\\}$ behaves as Pauli matrices,
explicitly

$\displaystyle\bm{o}_{0}=-i\footnotesize{\left(\begin{array}[]{cccc}0&0&\mathbb{1}&0\\\
0&0&0&\mathbb{1}\\\ \mathbb{1}&0&0&0\\\ 0&\mathbb{1}&0&0\\\
\end{array}\right)}$ (43)
$\displaystyle\bm{o}_{+}=\footnotesize{\left(\begin{array}[]{cccc}\mathbb{1}&0&0&0\\\
0&\mathbb{1}&0&0\\\ 0&0&-\mathbb{1}&0\\\ 0&0&0&-\mathbb{1}\\\
\end{array}\right)}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\bm{o}_{-}=\footnotesize{i\left(\begin{array}[]{cccc}0&0&\mathbb{1}&0\\\
0&0&0&\mathbb{1}\\\ -\mathbb{1}&0&0&0\\\ 0&-\mathbb{1}&0&0\\\
\end{array}\right)},$ (52)

which gives a straightforward identification for the chiral matrices

$\displaystyle\chi_{1}\rightarrow i\bm{o}_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \chi_{2}\rightarrow-\bm{o}_{-}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \chi_{3}\rightarrow\bm{o}_{+}.$ (53)

From E.q(53), the E.q(15) DE, with octonions (chiral), and quaternions (spin)
becomes in

$\displaystyle\begin{bmatrix}\bm{o}_{0}\partial_{0}+i\bm{o}_{-}\bm{\partial}+m\end{bmatrix}\psi=0.$
(54)

This form yields the direct identification for gamma matrices,

$\displaystyle\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}=i\bm{o}_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{1}=\bm{o}_{-}\bm{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{2}=\bm{o}_{-}\bm{j}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{3}=\bm{o}_{-}\bm{k}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}=\bm{o}_{+},$ (55)

in consequence permits to write Dirac Lagrangian in its usual form with a
thick-bar notation for adjoint spinors
$\accentset{\rule{2.79996pt}{1.2pt}}{\psi}=\psi^{\dagger}\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}$,
as

$\displaystyle\mathcal{L}=\accentset{\rule{2.79996pt}{1.2pt}}{\psi}\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}\partial_{\mu}-m\right]\psi=0.$
(56)

It is worth highlighting that the Lagrangian conserves its form from original
Dirac theory, which make the treatment analogous to the unmodified one. With
that said, it is expected that discrete symmetries of original theory hold.
Now, even though the form is alike, the
$\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}$ are $16\times 16$ complex matrices
($\mathbb{C}\oplus\mathbb{H}\oplus\mathbb{O}$), providing the theory with a
richer structure for internal degrees of freedom, hence for describing more
kinds of fermions.

### B. SOLUTIONS

With the Lagrangian set, the next step is finding the fields it describes.
Starting from the projector solution $\Psi^{\mp}=\Lambda
s_{\uparrow\downarrow}\mathcal{P}^{\mp}e^{\mp ix_{\mu}p^{\mu}}$, with
$\mathcal{P}_{\mp}=\frac{1}{2}(1\mp i\bm{o}_{0})$. Spin component remains
unchanged, and left chiral projector $\chi=\frac{1}{2}(1-\bm{o}_{+})$, E.q(17)
expression holds, enabling to write the Lorentz boost,

$\displaystyle\Lambda_{P}$
$\displaystyle=\frac{1}{\sqrt{m}}\left[\chi\sqrt{\bm{P}}+\widetilde{(\chi\sqrt{\bm{P}})^{*}}\right].$
(57)

The following stage is to find the $\mathcal{P}_{\mp}$ algebraic spinors
eigenvectors, the ladders operators or arches are defined for this

$\displaystyle\alpha_{\ell}^{\dagger}=\frac{1}{2}(-\bm{o}_{\ell}+i\bm{o}_{\bar{\ell}})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\alpha_{\ell}=\frac{1}{2}(\bm{o}_{\ell}+i\bm{o}_{\bar{\ell}}),$ (58)

given a $C\ell_{6}$ fermion algebra,

$\displaystyle\\{\alpha_{\ell}^{\dagger},\alpha_{j}\\}=\delta_{\ell j}.$ (59)

An orthogonal set of eigenvectors for the projector $\mathcal{P}^{\pm}$ from
these operators can be constructed. This can be done first by realizing that
eigenvectors are the nilpotent operators

$\displaystyle\omega^{\dagger}=\alpha_{1}^{\dagger}\alpha_{2}^{\dagger}\alpha_{3}^{\dagger}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \omega=\alpha_{3}\alpha_{2}\alpha_{1}.$ (60)

Upon them other objects can be built via successive application of E.q(59)
ladder operators over the idempotent $\omega\omega^{\dagger}$, this leads to
linear combinations of algebraic spinors (ideals) [11]

$\displaystyle\begin{array}[]{c c}\Psi=&\ell_{1}\omega\omega^{\dagger}\\\
&+\bar{q}^{r}_{1}\alpha_{1}^{\dagger}\omega\omega^{\dagger}+\bar{q}^{g}_{1}\alpha_{2}^{\dagger}\omega\omega^{\dagger}+\bar{q}^{b}_{1}\alpha_{3}^{\dagger}\omega\omega^{\dagger}\\\
&+q^{r}_{2}\alpha_{3}^{\dagger}\alpha_{2}^{\dagger}\omega\omega^{\dagger}+q^{g}_{2}\alpha_{1}^{\dagger}\alpha_{3}^{\dagger}\omega\omega^{\dagger}+q^{b}\alpha_{2}^{\dagger}\alpha_{1}^{\dagger}\omega\omega^{\dagger}\\\
&+\bar{\ell}_{2}\alpha_{3}^{\dagger}\alpha_{2}^{\dagger}\alpha_{1}^{\dagger}\omega\omega^{\dagger},\end{array}$
(65)

where, for coming reasons, $\ell$ is for leptons, $q$ for quarks, and the
small over-bar is for antiparticles. For constructing column spinorial
solutions it is enough to find the eigenvector $u$ of
$\omega\omega^{\dagger}$, with eigenvalue different to zero, and from it
finding the other ones by analogue construction to the algebraic spinor case.
The corresponding eigenvector is $u=(1,0,0,0,1,0,0,0)^{\top},$ hence by
successive application of the ladder operators,

$\begin{array}[]{c}\big{(}\ell_{1}+\bar{q}_{1}^{r}\alpha_{1}^{\dagger}+\bar{q}_{1}^{g}\alpha_{2}^{\dagger}+\bar{q}_{1}^{b}\alpha_{3}^{\dagger}\\\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\alpha_{2}^{\dagger}\alpha_{3}^{\dagger}q_{2}^{r}+\alpha_{3}^{\dagger}\alpha_{1}^{\dagger}q_{2}^{g}+\alpha_{1}^{\dagger}\alpha_{2}^{\dagger}q_{2}^{b}+\bar{\ell}_{2}\alpha_{3}^{\dagger}\alpha_{2}^{\dagger}\alpha_{1}^{\dagger}\big{)}u=\end{array}\footnotesize{\left(\begin{array}[]{c}\ell_{1}+\bar{\ell}_{2}\\\
q_{2}^{g}+\bar{q}_{1}^{g}\\\ q_{2}^{b}+\bar{q}_{1}^{b}\\\
q_{2}^{r}+\bar{q}_{1}^{r}\\\ \ell_{1}-\bar{\ell}_{2}\\\
q_{2}^{g}-\bar{q}_{1}^{g}\\\ q_{2}^{b}-\bar{q}_{1}^{b}\\\
q_{2}^{r}-\bar{q}_{1}^{r}\end{array}\right)}.$ (66)

From E.q(66) and E.q(II. DIRACS EQUATION ALGEBRA INSIGHTS), a generic column
solution can be written as $\psi=(e^{\pm
ip_{\mu}x^{\mu}}\Lambda_{P}s_{\uparrow\downarrow})(\alpha u)$ where $\alpha$
is any product of $\alpha_{i}$’s, letting to classified the eight solutions,
into four kinds

$\displaystyle\ell_{1}$
$\displaystyle=(e^{-ip_{\mu}x^{\mu}}\Lambda_{P}s_{\uparrow\downarrow})(\omega\omega^{\dagger}u)$
$\displaystyle\bar{\ell}_{2}$
$\displaystyle=(e^{ip_{\mu}x^{\mu}}\Lambda_{P}s_{\uparrow\downarrow})(\alpha_{3}^{\dagger}\alpha_{2}^{\dagger}\alpha_{1}^{\dagger}\omega\omega^{\dagger}u)$
$\displaystyle\bar{q}_{1}^{c}$
$\displaystyle=(e^{ip_{\mu}x^{\mu}}\Lambda_{P}s_{\uparrow\downarrow})(\alpha_{i}^{\dagger}\omega\omega^{\dagger}u)$
$\displaystyle q_{2}^{c}$
$\displaystyle=(e^{-ip_{\mu}x^{\mu}}\Lambda_{P}s_{\uparrow\downarrow})(\alpha_{i}^{\dagger}\alpha_{j}^{\dagger}\omega\omega^{\dagger}u),$
(67)

showing an evident structure, with separable spacetime/spin components, from
the $\alpha_{j}$’s.

As it is shown in subsection C, each E.q(67) solution type describes a
particle class, where the main distinguishable feature is the $\alpha_{j}$’s
factors, endowing them with charges and a sort of flavor notion, from there
the suitable label of arches. Then the theory solutions meets a speculated
correspondence of ideals$\sim$particles,
$\mathbb{C}\oplus\mathbb{H}\sim$Lorentz and
$\mathbb{C}\oplus\mathbb{O}\sim$internal degrees of freedom [11].

### C. SOLUTION CHARACTERIZATION AND SYMMETRIES

The next step is to identify each solution with a kind of particle, for such
task charge operators can be formulated from the arches.

An $U(1)$ generator that behaves as an electric charge can be constructed as

$Q^{u}=\frac{1}{3}\sum_{i}\alpha_{i}^{\dagger}\alpha_{i},$ (68)

$SU(3)$ generators are given by,

$\begin{array}[]{ll}\Lambda_{1}^{u}=-\alpha_{2}^{\dagger}\alpha_{1}-\alpha_{1}^{\dagger}\alpha_{2}&\Lambda_{2}^{u}=i\alpha_{2}^{\dagger}\alpha_{1}-i\alpha_{1}^{\dagger}\alpha_{2}\\\
\Lambda_{3}^{u}=\alpha_{2}^{\dagger}\alpha_{2}-\alpha_{1}^{\dagger}\alpha_{1}&\Lambda_{4}^{u}=-\alpha_{1}^{\dagger}\alpha_{3}-\alpha_{3}^{\dagger}\alpha_{1}\\\
\Lambda_{5}^{u}=-i\alpha_{1}^{\dagger}\alpha_{3}+i\alpha_{3}^{\dagger}\alpha_{1}&\Lambda_{6}^{u}=-\alpha_{3}^{\dagger}\alpha_{2}-\alpha_{2}^{\dagger}\alpha_{3}\\\
\Lambda_{7}^{u}=i\alpha_{3}^{\dagger}\alpha_{2}-i\alpha_{2}^{\dagger}\alpha_{3}&\Lambda_{8}^{u}=-\frac{1}{\sqrt{3}}\left[\alpha_{1}^{\dagger}\alpha_{1}+\alpha_{2}^{\dagger}\alpha_{2}-2\alpha_{3}^{\dagger}\alpha_{3}\right],\end{array}$
(69)

and the complex conjugate results in other set of independent generators,

$Q^{d}=Q^{u*}=\frac{1}{3}\sum_{i}\alpha_{i}\alpha_{i}^{\dagger}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Lambda_{i}^{d}=\Lambda_{i}^{u*}.$ (70)

The confluence of all those quadratic terms, is the general linear
combination:

$\sum_{\mathcal{H}}\mathcal{H}=a^{u}Q^{u}+\sum_{i=1}^{8}g_{i}^{u}\Lambda_{i}^{u}+a^{d}Q^{d}+\sum_{i=1}^{8}g_{i}^{d}\Lambda_{i}^{d},$
(71)

with $a_{j}^{u/d}$’s and $g_{i}^{u/d}$’s some $\mathbb{R}$ coefficients.

The algebraic spinors of E.q(65) under the action of the above generators,
behaves as representations of standard model particles, assigning the ID of
electron $\mathcal{E}$, neutrino $\mathcal{V}$, antiquark up
$\bar{\mathcal{U}}$, antidown $\bar{\mathcal{D}}$, and so on [10]. However,
they are identified as up $S^{u}$ or down $S^{d}$ weak-isospin particles,
depending on which generators act on them as it is consigned in Tab.(1). As
can be seen, electrons and quarks are excitations from what can be identified
as a neutrino, which plays as a sort of vacuum [10].

$\underline{Q}^{u}$ | $\underline{\Lambda}^{u}$ | $\underline{S^{u}\leavevmode\nobreak\ ID}$ | $-\underline{Q}^{d}$ | $-\underline{\Lambda}^{d}$ | $\underline{S^{d}\leavevmode\nobreak\ ID}$ | $\underline{\alpha}$
---|---|---|---|---|---|---
$0$ | $1$ | $\mathcal{V}$ | $-1$ | $1$ | $\mathcal{E}$ | $\omega\omega^{\dagger}$
$1/3$ | $\bar{3}$ | $\bar{\mathcal{D}}$ | $-2/3$ | $\bar{3}$ | $\bar{\mathcal{U}}$ | $\alpha_{i}^{\dagger}\omega\omega^{\dagger}$
$2/3$ | $3$ | $\mathcal{U}$ | $-1/3$ | $3$ | $\mathcal{D}$ | $\alpha_{i}^{\dagger}\alpha_{j}^{\dagger}\omega\omega^{\dagger}$
$1$ | $\bar{1}$ | $\mathcal{E}^{+}$ | $0$ | $\bar{1}$ | $\bar{\mathcal{V}}$ | $\alpha_{i}^{\dagger}\alpha_{j}^{\dagger}\alpha_{k}^{\dagger}\omega\omega^{\dagger}$
Table 1:  Particle properties under generators action. $Q^{u/d}$ is electric
charge operator for isospin up/down. For $\Lambda$ generators $\bar{3}$, means
that the corresponding $\alpha$ transform as color anti-triplet, $1$ for color
singlet, et cetera.

It will be clear later how to include both (weak-)isospins subspaces in a
single description, for now it is illustrative enough to go ahead with up
isospin generators. For incorporating interactions one can start with

$\sum_{\mathcal{H}^{u}}\mathcal{H}^{u}=aQ^{u}+\sum_{i=1}^{8}g^{i}\Lambda_{i}^{u}.$
(72)

From here a gauge covariant derivative can be constructed, making the equation
invariant under local phase transformation in the usual Yang-Mills form.
Promoting position dependency for $a=a(x)$ and $g=g_{i}(x)$, for a
transformation

$\displaystyle\Lambda_{P}s_{\uparrow\downarrow}\alpha e^{\pm
ip_{\mu}x^{\mu}}\rightarrow\Lambda_{P}s_{\uparrow\downarrow}\left(e^{-i\mathcal{H}^{u}}\right)\alpha
e^{\pm ip_{\mu}x^{\mu}}.$ (73)

DE is not invariant under such transformation,

$\displaystyle\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}\partial_{\mu}-m\right]\Lambda
s_{\uparrow\downarrow}\left(e^{i\mathcal{H}^{u}}\right)\alpha e^{\pm
ip_{\mu}x^{\mu}}=0$ (74)
$\displaystyle\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}\left(\partial_{\mu}+i\\{\Lambda\partial_{\mu}\mathcal{H}^{u}\Lambda^{-1}\\}\right)-m\right]\Lambda
s_{\uparrow\downarrow}\alpha e^{\pm ip_{\mu}x^{\mu}}\neq 0,$ (75)

to overcome this, is added the counterterm,

$\displaystyle\partial_{\mu}\mathcal{H}^{u}=A_{\mu}^{u}Q^{u}+\sum_{i=1}^{8}G_{\mu}^{u}\Lambda_{i}^{u},$
(76)

with $A_{\mu}=\partial_{\mu}a$ and $G_{\mu}^{i}=\partial_{\mu}g^{i}$. Hence
the covariant derivative acting on the particles with isospin up should be
defined like

$\displaystyle
D_{\mu}^{u}=\partial_{\mu}-i\Lambda_{P}\left(A_{\mu}^{u}Q^{u}+\sum_{i=1}^{8}G^{u}\Lambda_{i}^{u}\right)\Lambda_{P}^{-1},$
(77)

where the $\Lambda_{P}$ has been added, regarding that $Q^{u}$ and
$\Lambda_{i}^{u}$ are now matrices which do not commute with Lorentz boost.

Now E.q(56) produces solutions with same masses, to give a different mass to
each kind of particle, an ad hoc isospin up mass matrix $M^{u}$ can be
constructed as

$\displaystyle\begin{array}[]{c
c}M^{u}=&m_{\mathcal{V}}\omega\omega^{\dagger}\\\
&+m_{\mathcal{D}}(\alpha_{1}^{\dagger}\omega\omega^{\dagger}\alpha_{1}+\alpha_{2}^{\dagger}\omega\omega^{\dagger}\alpha_{2}+\alpha_{3}^{\dagger}\omega\omega^{\dagger}\alpha_{3})\\\
&+m_{\mathcal{U}}(\alpha_{3}^{\dagger}\alpha_{2}^{\dagger}\omega\omega^{\dagger}\alpha_{3}\alpha_{2}+\alpha_{1}^{\dagger}\alpha_{3}^{\dagger}\omega\omega^{\dagger}\alpha_{1}\alpha_{3}+\alpha_{2}^{\dagger}\alpha_{1}^{\dagger}\omega\omega^{\dagger}\alpha_{2}\alpha_{1})\\\
&+m_{\mathcal{E}}\omega^{\dagger}\omega,\end{array}$ (82)

and also the $1/\sqrt{M^{u}}$ matrix, as $M^{u}$ with all $m_{f}\rightarrow
m_{f}^{-1/2}$. In a similar way for $Q^{u}$ and $\Lambda_{i}^{u}$ generators,
the downs isospin analogue is given by the complex conjugate $M^{d}=M^{u*}$
and $1/\sqrt{M^{d}}=1/\sqrt{M^{u}}^{*}$. Introduction of these elements could
be seen as enlarging the algebra, if products between algebra elements and $M$
are taking into account, however this will not be explored in this paper.

Putting together E.q(54), the gauge covariant derivative E.q(91) and the mass
matrix E.q(82), arrive to an equation that contains the particle spectrum of
isospin up subspace of one standard model generation, and the corresponding
equation for isospin down is gotten by changing $u\rightarrow d$. In summary

$\displaystyle\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}D_{\mu}^{u/d}-M^{u/d}\right]\psi^{u/d}=0,$
(83)

and the solutions only change by $1/\sqrt{m}\rightarrow 1/\sqrt{M^{u/d}}$ in
E.q(57) Lorentz boost. E.q(83) describes a unified matter field theory with
electromagnetic and strong like forces, but just for a half of one standard
model particle generation isospin up or down at a time.

Additionally, it is pointed out that it is possible to write the DE in terms
of electric charge operator, considering

$\displaystyle\bm{o}_{0}=\bm{o}_{3}\bm{o}_{2}\bm{o}_{1}\bm{o}_{\bar{1}}\bm{o}_{\bar{2}}\bm{o}_{\bar{3}},$
(84)

and using $\bm{o}_{j}=\alpha_{j}-\alpha_{j}^{\dagger}$,
$i\bm{o}_{\bar{j}}=\alpha_{j}^{\dagger}+\alpha_{j}$, it is possible to express
$\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}$ in terms of the electric charge
operator $Q^{u}$

$\displaystyle\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}=\left(-1+2\left(10Q^{u}-9Q^{u2}+2Q^{u3}\right)\right),$
(85)

which quaintly looks like first orders of an expansion.

###  D. ONE GENERATION UNIFIED FIELD LAGRANGIAN

Although the octonionic DE presented in E.q(83) succeeded in reproducing
particles that can be identified as two leptons and two quarks under
electromagnetic and strong force, it describes only a isospin up or down
subspace of solutions, meaning that they only have a particle or antiparticle
of each type. A clue for a possible way of overcoming this is CPT symmetry,
which should transform E.q(83) between particles up and down isospin versions
exchanging particles and antiparticles.

From the usual CPT transformation it is given the form

$\psi(t,\vec{x})\overset{\mathrm{CPT}}{\rightarrow}i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}\psi(-t,-\vec{x}),$
(86)

but in the normal DE the $Q$ and $\Lambda_{i}$ generators commute with
$\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}=\bm{o}_{+}$, in this case instead, the
formers belong to the same algebra having

$\displaystyle
Q^{u}\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}=\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}Q^{d}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Lambda^{u}\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}=\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}\Lambda^{d}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
M^{u}\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}=\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}M^{d}.$
(87)

Taking into account that the CPT transformation also transform between isospin
operators, the correspondence between particles and antiparticles remains as

$\displaystyle\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}D_{\mu}^{u}-M^{u}\right]\psi^{u}\overset{\mathrm{CPT}}{\leftrightarrow}\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}D_{\mu}^{d}-M^{d}\right]\psi^{d}.$
(88)

Then an invariant Lagrangian (density) under CPT transformation, can be
formulated containing both isospin version of E.q(83),

$\displaystyle\mathcal{L}=$
$\displaystyle\accentset{\rule{2.79996pt}{1.2pt}}{\psi}^{u}\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}D_{\mu}^{u}-M^{u}\right]\psi^{u}+\accentset{\rule{2.79996pt}{1.2pt}}{\psi}^{d}\left[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}D_{\mu}^{d}-M^{d}\right]\psi^{d},$
(89)

For short writing from here and on it is going to be suppressed the super-
index $u$ on all up isospin operators, and Dirac operator is going to be
denoted by

$\displaystyle\mathcal{D}=-i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}D_{\mu}+M=\bm{o}_{0}D_{0}+i\bm{o}_{-}\bm{D}+M.$
(90)

The E.q(90) Lagrangian can be reformulated in a simpler form. It is point out
that for this from E.q(58) for any arche
$\alpha_{j}^{*}=-\alpha_{j}^{\dagger}$. This is the reason why for all
generators in gauge derivatives and mass matrix the conjugate complex
transform between isospins subspaces. Also, applied to the solutions of
E.q(67), complex conjugate relates wave functions in both isospin subspaces.
However, E.q(57) Lorentz boosts are not invariant under complex conjugate,
this is fixed noting that they are invariant under simultaneous complex and
quaternion conjugate, with quaternion conjugate not affecting the arches. From
here it is straightforward to write E.q(88) into the succinct formula

$\displaystyle\mathcal{L}=$
$\displaystyle\accentset{\rule{2.79996pt}{1.2pt}}{\psi}\mathcal{D}\psi+\overline{\left(\accentset{\rule{2.79996pt}{1.2pt}}{\psi}\mathcal{D}\psi\right)^{*}}.$
(91)

In this expression the relation between isospin sub-spaces is explicit. But as
it is known Lagrangians are not unique, a further form can be ideated which
allows an additional interpretation. Summing 0 terms as Dirac operator powers

$\displaystyle\sum_{n=1}^{\infty}\frac{(i\mathcal{D})^{n}}{n!}\psi=$
$\displaystyle 0,$ (92)

equivalent to

$\displaystyle e^{i\mathcal{D}}\psi=$ $\displaystyle\psi,$ (93)

and it corresponds to the alternative Lagrangian

$\displaystyle\mathcal{L}$
$\displaystyle=\accentset{\rule{2.79996pt}{1.2pt}}{\psi}e^{i\mathcal{D}}\psi-\accentset{\rule{2.79996pt}{1.2pt}}{\psi}\psi+q.c.c,$
(94)

where $q.c.c$ is for quaternion-complex conjugate. This Lagrangian describes a
unified matter field theory with endogenous gauge groups $U(1)$
electromagnetism and $SU(3)$ color, alike to a full generation of standard
model particles, and CPT symmetry is equivalent to a quaternion-complex
conjugation. It is noteworthy, that the matter fields are the ones unified
while the interactions arise from the symmetry properties of the fermion
fields, rather than the imposition of a gauge master group. Additionally, the
form of equation E.q(93) and E.q(94) Lagrangian exhibits the feature that the
system symmetries and dynamics emanate from demanding invariance under phase
transformation $\psi\rightarrow\psi^{\prime}=e^{i\mathcal{D}}\psi$. This will
come in handy when introducing flavor interactions.

###  E. ON FLAVOR INTERACTION AND FRAME FIELDS

For the theory developed so far, is natural to ask if it can be extended to
include a flavor changing interaction by using elements of the algebra. A path
for this is insinuated by E.q(89) being conveniently arranged separating
isospin subspaces, which suggest to group $\psi^{u}$ and $\psi^{d}$ in an
isospin doublet,

$\displaystyle\mathcal{L}=\footnotesize{\begin{pmatrix}\accentset{\rule{2.23996pt}{1.2pt}}{\psi}&\overline{\accentset{\rule{2.23996pt}{1.2pt}}{\psi}^{*}}\end{pmatrix}\begin{pmatrix}\mathcal{D}&0\\\
0&\overline{\mathcal{D}^{*}}&\end{pmatrix}\begin{pmatrix}\psi\\\
\overline{\psi^{*}}\end{pmatrix}},$ (95)

at first glance, it seems that for a SU(2) flavor change elements outside of
the algebra are needed, though this can be achieved with elements inside it as
will be subsequently proven. A hint is given by
$\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}=\bm{o}_{+}$ in the CPT transformation
which connects isospin doublets, allowing the speculation of a relation
between flavor force and space-time.

Exploring the action of E.q(58) chiral matrices
$i\bm{o}_{0},-\bm{o}_{-},\bm{o}_{+}$ on algebraic solutions of E.q(65), it is
identifiable that they act as $\mathfrak{su}(2)$ flavor changing matrices

$\displaystyle\footnotesize{\tau_{1}=-\bm{o}_{+}=-\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}\equiv\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tau_{2}=\bm{o}_{-}=i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{5}\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}\equiv\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tau_{3}=i\bm{o}_{0}=\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}\equiv\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}},$ (96)

on doublets,

$\displaystyle\begin{pmatrix}\ell_{1}\\\
\ell_{2}\end{pmatrix}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\begin{pmatrix}q_{2}\\\ q_{1}\end{pmatrix}.$ (97)

In contrast to the standard model left weak force, here the generators act on
left and right particles equally, and they are also in terms of gamma
matrices. An attempt to add left chiral projectors to mimic the usual weak
force is reasonable, however the $\chi\tau_{i}$’s do not form a
$\mathfrak{su}(2)$ algebra neither over algebraic or column spinors.

From E.q(96) and E.q(96) the tentative next step is to use Yang-Mills scheme,
for constructing an invariant theory under the $\tau_{i}$’s induced SU(2)
transformation

$\begin{pmatrix}\psi\\\ \overline{{\psi}^{*}}\end{pmatrix}\rightarrow
e^{\frac{i}{2}\text{w}_{i}\tau_{i}}\begin{pmatrix}\psi\\\
\overline{{\psi}^{*}}\end{pmatrix}.$ (98)

However, this will not succeed since the Lagrangian is not globally invariant
under the transformation due to $\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{\mu}$’s
and $\tau_{i}$’s do not commute being one and the same from E.q(96).

Now if one insists on exploring the inclusion of an interaction with E.q(58)
chiral generators, the Lagrangian can be made invariant under certain
conditions, which produces curious expressions to be derived.

To find how a flavor changing interaction can be introduced into the theory,
one can insert a new $Y$ matrix to discompose the mass matrix
$-M=\text{b}Y+\text{w}_{1}\tau_{1}$, with b,$\text{w}_{1}\in\mathbb{R}$.
Regarding the equivalence given in E.q(96) Dirac operator can be recast as

$\displaystyle\mathcal{D}=-i\tau_{3}D_{0}+i\tau_{2}\bm{D}-\text{w}_{1}\tau_{1}-\text{b}Y,$
(99)

in conjunction with E.q(93)

$\displaystyle
e^{-i(\text{w}_{1}\tau_{1}-i\tau_{2}\bm{D}+i\tau_{3}D_{0}+\text{b}Y)}\psi=$
$\displaystyle\psi.$ (100)

A form that tempts the unconventional eigenvalues associations of $iD_{0}$ and
$i\bm{D}$, to $E\equiv\text{w}_{3}$ and $-\bm{P}\equiv\text{w}_{2}$
respectively. They lead to see DE as an spinor invariance down flavor phase
transformations

$\displaystyle e^{-i(\text{b}Y+\vec{\tau}\cdot\vec{\text{w}})}\psi=$
$\displaystyle\psi,$ (101)

constrained by the equivalent Einstein energy-momentum relation

$\text{w}_{3}=\pm\sqrt{|\text{w}_{2}|^{2}+(\text{w}_{1}+\text{b})^{2}}.$ (102)

Just by the form of E.q(102) $Y$ could be identified as a sort of (weak)
hypercharge, and its relation to mass $M=\text{w}_{1}\tau_{1}-\text{b}Y$ looks
more or less like a Gell-Mann–Nishijima formula. However, its nature and
relation to the already ad hoc mass matrix remain unexplored. On the other
hand, $Y_{\text{w}}$ hypercharge can be defined by the usual Gell-
Mann–Nishijima relation equivalent to $Y_{\text{w}}=\tau_{3}-2Q$, nonetheless
$Y_{\text{w}}$ values for right-handed particles will not correspond to those
of standard model since $\tau_{i}$’s acts on left and right indistinguishably.

At this point, flavor force can be included by gauge covariant derivatives,
making the $\text{w}_{i}$’s phases spacetime functions

$\sum_{\mathcal{H}}\mathcal{H}=Qa(x)+\vec{\tau}\cdot\vec{\text{w}}(x)+\sum_{i=1}^{8}\Lambda_{i}g^{i}_{\mu}(x),$
(103)

changing the gauge covariant derivative to

$D_{\mu}=\partial_{\mu}-i\Lambda_{P}\left(A_{\mu}Q+i\vec{\tau}\cdot\vec{W}_{\mu}+\sum_{i=1}^{8}G\Lambda_{i}\right)\Lambda_{P}^{-1},$
(104)

with $\vec{W}_{\mu}=\partial_{\mu}\vec{\text{w}}$, meeting E.q(102) constrain.
Promoting the flavor phases from global to local, carries the abnormal
consequence of energy and momentum induced spacetime dependency since
$\text{w}_{3}(x)\equiv E(x)$ and $\text{w}_{2}(x)\equiv-\bm{P}(x)$, explicitly
yielding into DE

$(i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}E(x)+\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{j}p_{j}(x)-M)\psi=0.$
(105)

Now the question is what the spacetime dependency could mean. A possible
interpretation came from general relativity. Gravity-like effects are
introduced into DE, by the hand of $e_{a}^{\mu}(x)$ (vierbeins) frame fields,
which are directly coupled to gamma matrices, giving the curve spacetime DE

$[i(\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{a}e_{a}^{0})D_{0}+i(\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{a}e_{a}^{j})D_{j}-M]\psi=0,$
(106)

where $D_{\mu}$ is a covariant derivative including also spin connections, and
Latin indices of $e_{a}^{\mu}$, are for flat Lorentzian spacetime, while Greek
are for curve relating Minkowski metric $\eta^{ab}$ to a curve metric
$g^{\mu\nu}$ by $g^{\mu\nu}(x)=e_{a}^{\mu}(x)e_{b}^{\nu}(x)\eta^{ab}$.
Reordering E.q(106)

$[i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{0}(e_{0}^{\mu}D_{\mu})+i\bm{\gamma}\\!\\!\\!\\!\bm{\gamma}^{j}(e_{j}^{\mu}D_{\mu})-M]\psi=0.$
(107)

Comparing term by term between E.q(105) and E.q(107), suggests the straight
forward correspondence of $E(x)$ and $p_{j}(x)$, as eigenvalues of
$e_{0}^{\mu}D_{\mu}$ and $e_{j}^{\mu}D_{\mu}$ respectively, setting in this
theory a direct relation between flavor bosons and framefields. Further
studies in how gravity could behave are needed.

##  V. CONCLUSIONS

By breaking down the Dirac equation in terms of chiral and spin
$\mathfrak{su}(2)$ matrices, it was possible to express its solutions using
projectors and ladder operators. A construction scheme for formulating unified
field theories was proposed from the obtained forms. This consists in taking
an algebra where Dirac gamma matrices are sub-algebra elements, done by
enlarging the chiral matrices, introducing new degrees of freedom that allow
the resultant Dirac-like equation to describes several fermions in a unified
matter field. The fields could have symmetries endowed by inheritance of the
algebra, and forces are introduced by Yang-Mills covariant derivatives.

As an illustrative example, a unified matter field theory was constructed
using quaternions as spin matrices and left multiply octonions as chiral
components. These elections were motivated by the fact that they are normed
division algebras, therefore have nice properties to work out, yielding
concise expression by use of conjugates, projectors, and ladder operators. The
resulting theory describes a full generation of standard model particles,
under U(1) electromagnetism, SU(2) flavor and SU(3) color gauge groups, and
remarkably SU(2) flavor enters with abnormal behavior, coupling its bosons to
gamma matrices as frame fields, opening connections to gravity. It is
noteworthy that what is unified are the fermion fields, and forces emerge from
inherent symmetries, rather than from a master gauge group.

Demanding Dirac equation to describe particles with different masses, an ad
hoc mass matrix had to be included, which later is shown to be equivalent to
introducing a matrix that looks like a weak hypercharge analog. However, the
model does not give account for electroweak force, and the Gell-Mann–Nishijima
formula does not yield the correct weak isospin charges.

Based on what has been developed through this paper future paths of
investigation could be exploration of the dynamics induced by frame fields,
algebra enlargement by mass ($Y$) matrix, and further algebras for the
proposed scheme to emulate standard model particles more accurately.

### ACKNOWLEDGMENTS

The author is deeply indebted to P. Arevalo for correcting the manuscript and
would like to thank W. E. Salazar for valuable discussions and feedback.

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