# QED based on eight-dimensional spinorial wave equation of the
electromagnetic field and the emergence of quantum gravity

Mikko Partanen  Photonics Group, Department of Electronics and
Nanoengineering, Aalto University, P.O. Box 13500, 00076 Aalto, Finland  Jukka
Tulkki  Engineered Nanosystems Group, School of Science, Aalto University,
P.O. Box 12200, 00076 Aalto, Finland

(October 17, 2023)

###### Abstract

Quantum electrodynamics (QED) is the most accurate of all experimentally
verified physical theories. How QED and other theories of fundamental
interactions couple to gravity through special unitary symmetries, on which
the standard model of particle physics is based, is, however, still unknown.
To enable a new kind of coupling between the electromagnetic field, Dirac
electron-positron field, and the gravitational field, we develop a covariant
spinorial photon equation with an eight-component spinorial representation of
photons that is analogous to the well-known representation of particles in the
Dirac theory. We reformulate QED using eight-component spinors. Furthermore,
we introduce the generating Lagrangian density of gravity based on the special
unitary symmetry of the eight-dimensional spinor space. The generating
Lagrangian density of gravity plays, in the definition of the gauge theory of
gravity and its symmetric stress-energy-momentum tensor source term, a similar
role as the conventional Lagrangian density of the free Dirac field plays in
the definition of the gauge theory of QED and its electric four-current
density source term. The fundamental consequence, the Yang-Mills gauge theory
of gravity, is studied in a separate work [arXiv:2310.01460], where the theory
is also extended to cover the other fundamental interactions of the standard
model. We devote ample space for details of the eight-spinor QED to provide
solid mathematical basis for the present work and the related work on the
Yang-Mills gauge theory of gravity.

##  I Introduction

Quantum electrodynamics (QED) agrees with experiments to an exceedingly high
accuracy Peskin2018 ; Fan2023 ; Parker2018 . In this regard, it can be
considered as the most successful theory of physics ever developed. The
present formulation of QED has, however, not enabled coupling of the
electromagnetic and Dirac electron-positron fields to the gravitational field
through the special unitary symmetries on which the standard model of particle
physics is based Schwartz2014 ; Peskin2018 . Therefore, alternative
approaches, such as string theory Becker2007 ; Green1987  and loop quantum
gravity Ashtekar1986 ; Jacobson1988 ; Rovelli1990 , are being developed. We
approach the problem from a different point of view: It is proposed that
identifying a special-unitary-symmetry-based coupling of the present standard
model to the gravitational field leads to the Yang-Mills gauge theory of
gravity, the unified theory of all known fundamental interactions of nature.
Thus, it is our interest to study whether QED, the simplest quantum field
theory of the standard model interactions Schwartz2014 ; Peskin2018 , can be
reformulated in alternative ways using novel mathematical structures that may
help us to identify the special-unitary-symmetry-based coupling to gravity.
This is the ultimate goal of the present work.

Advances in optics and photonics technologies enable detailed experimental
investigations of the spin and orbital angular momenta of light Shen2019 ;
Padgett2017 ; Devlin2017 ; Li2017  and the related physics of structured light
fields Forbes2021 ; Forbes2019 ; He2022 ; RubinszteinDunlop2017 ; Shen2022
and chiral quantum optics Aiello2022 ; Chen2023 ; Lininger2023 ; Lodahl2017 .
Classical Maxwell’s equations and QED provide solid basis for theoretical
understanding of the related physical phenomena Jackson1999 ; Landau1982 ;
Bliokh2015b ; Bliokh2015c ; Bliokh2017a ; WangM2018 ; Partanen2018a ;
Partanen2022b . However, in spite of overall success, theories of light are
also associated with well-known enigma that exist independently of the
question of the gravitational coupling.

In QED, the usual approach to quantize the electromagnetic field Glauber2006 ;
Klauder1968  is based on the transverse-vector-potential eigenstates
describing the transverse photons Loudon2000 ; Landau1982 ; Sakurai1967 . In
optical spectroscopy Parson2015 , only radiative processes, which can be
associated with transverse photons, are discovered. One enigmatic feature of
the transverse-vector-potential eigenstates is that their Lorentz
transformation between inertial frames leads to a nonzero scalar potential and
a nonzero longitudinal component in the transformed inertial frame Gupta1950 .
Thus, the transformed state does not belong to the set of transverse-vector-
potential eigenstates, which excludes scalar and longitudinal photon states
Feynman1961 . This is associated with the vector potential being gauge
dependent Landau1982 ; Feynman1961 . The related enigma of the definition of
the photon wavefunction has been a subject of a continuing dialogue
BialynickiBirula1994 ; BialynickiBirula1996 ; Sipe1995 ; Newton1949 ;
Kiessling2018 ; Dressel2015 ; Landau1982 ; Kidd1989 ; Hawton2019 ; Hawton1999
; SmithB2007 ; Sebens2019 . The same applies to the definition of the
relativistic quantum spin structure of light Yang2022 ; YangL2021 .

For comparison, the wavefunctions of spin-$\frac{1}{2}$ particles and their
antiparticles are described by the Dirac equation whose solutions are four-
component spinors Feynman1961 ; Landau1982 . The Dirac spinors transform
between inertial frames by their own spinorial Lorentz transformation, which
is not equivalent to the Lorentz transformation of four-vectors. The Lorentz-
transformed spinors belong to the set of eigenstates in contrast to the case
of the transverse-four-potential eigenstates of photons as discussed above.

Furthermore, the spin emerges from the Dirac theory through the generators of
Lorentz transformations on the Dirac spinors Peskin2018 . The origin of the
spin in the Dirac theory suggests that a similar Lorentz-transformation-based
origin may be found for the spin of photons. A strict relativistic description
of the spin of photons does not emerge within the principles of classical
physics from Maxwell’s equations, from the electromagnetic field tensor, or
from the electromagnetic four-potential. The existing formulations of the
quantum field theory of photons neither lead to a natural emergence of the
covariant and gauge-invariant spin structure of light Barnett2016 ;
Andrews2013 .

The initial goal of the present work was to shed light on the wavefunction and
spin properties of light by constructing a covariant and gauge-invariant
spinorial wavefunction and its Lorentz transformation for photons with a
natural emergence of the spin structure of light. However, we discovered later
that the theory of light based on spinorial wavefunctions leads to much more
far-reaching consequences on the gravitational coupling as discussed below.
First, we introduce an eight-component _spinorial photon equation_ , the
solutions of which are called _spinorial photon states_. This approach leads
to the appearance of $8\times 8$ matrices in the spinorial photon equation and
to the spinorial photon states having eight components. The relativistic
description of the quantum spin of light emerges from this theory through the
generators of Lorentz transformations on photon spinors in analogy to the
emergence of the spin in the Dirac theory. Since the spinorial photon states
are gauge independent, they also avoid the gauge dependence problem of the
four-potential eigenstates.

The spinorial representation of the electromagnetic field enables rewriting
the conventional QED and its Lagrangian density without changes in the
physical predictions of the theory. However, as a fundamental consequence, the
eight-spinor formulation of the theory enables the description of _the
generating Lagrangian density of gravity_ based on the special unitary
symmetry of the eight-dimensional spinor space. The generating Lagrangian
density of gravity leads to an elegant derivation of the _symmetric_ stress-
energy-momentum (SEM) tensors of the electromagnetic and Dirac electron-
positron fields in accordance with Noether’s theorem Noether1918 ; Zee2010 .
Previously, the SEM tensors have been derived only by utilizing the external
space-time symmetry. Furthermore, the canonical SEM tensors of the
conventional QED are asymmetric, being as such incompatible with _general
relativity_. This suggests that the generating Lagrangian density of gravity
provides the basis for the derivation of the quantum field theory of gravity.
Accordingly, the fundamental consequence of the present eight-spinor theory,
the Yang-Mills gauge theory of gravity, is investigated in a separate work
Partanen2023c . In Ref. Partanen2023c , we also extend the theory to cover,
not just QED, but all fundamental interactions of the standard model. The
present work provides a detailed discussion of many physical and mathematical
concepts that are extensively used in Ref. Partanen2023c .

We present a self-contained description of the theory and reserve plenty of
space for the detailed study of the foundations so that the reader can become
convinced that the mathematical and physical formulations are technically
sound in every detail. This work is organized as follows: Section II
introduces the spinorial photon wavefunction and formulates the spinorial
photon equation, which is the basis of the present work. Section III is
devoted to the description of Lorentz transformations and their generators,
which provide a natural way to introduce a relativistically consistent spin
structure to the theory of light. The quantum operators in the first
quantization of the field are presented in Sec. IV. The spinorial photon
eigenstates and their density expectation values and CPT symmetry are
described in Sec. V. The conventional QED Lagrangian density is reformulated
using eight-spinors, and Euler-Lagrange equations are investigated in Sec. VI.
Section VII shows how the eight-spinor formalism of QED enables the
description of the generating Lagrangian density of gravity that acts as the
basis for the Yang-Mills gauge theory of gravity Partanen2023c . Related
elegant special-unitary-symmetry-based derivations of the electromagnetic and
Dirac field SEM tensors are also presented. Section VIII briefly presents the
foundations of the second quantization using the spinorial photon states. This
section is provided as a technical tool for interested readers to outline how
the key properties of QED emerge from the conventional quantization procedure
using the spinorial photon states. Finally, conclusions are drawn in Sec. IX.

##  II Spinorial photon equation

###  II.1 Photon and charge-current spinors

We start the formulation of the theory by introducing a _photon spinor_ $\Psi$
and a _charge-current spinor_ $\Phi$, which both have eight components. As we
will show below, the advantage of using an eight-dimensional photon spinor is
that it is _gauge-independent_. The eight-component spinors describe the
physics of _all four Maxwell’s equations_ including the charge and current
densities: one component is related to Gauss’s law of magnetism, three
components to the Ampère-Maxwell law, one component to Gauss’s law of
electricity, and three components to Faraday’s law. In any inertial frame, the
components of $\Psi(t,\mathbf{r})$ are composed of the electric and magnetic
fields, $\mathbf{E}(t,\mathbf{r})$ and $\mathbf{B}(t,\mathbf{r})$, and the
components of $\Phi(t,\mathbf{r})$ are composed of the total electric charge
and current densities $\rho_{\mathrm{e}}(t,\mathbf{r})$ and
$\mathbf{J}_{\mathrm{e}}(t,\mathbf{r})$. Omitting the time and position
arguments, we write

$\Psi=\sqrt{\frac{\varepsilon_{0}}{2}}\left[\begin{array}[]{c}0\\\
\mathbf{E}\\\ 0\\\ ic\mathbf{B}\end{array}\right],\hskip
14.22636pt\Phi=\sqrt{\frac{\varepsilon_{0}}{2}}\left[\begin{array}[]{c}0\\\
\mu_{0}c\mathbf{J}_{\mathrm{e}}\\\ \rho_{\mathrm{e}}/\varepsilon_{0}\\\
\mathbf{0}\end{array}\right].$ (1)

Here $\varepsilon_{0}$, $\mu_{0}$, and $c=1/\sqrt{\varepsilon_{0}\mu_{0}}$ are
the permittivity, permeability, and the speed of light in vacuum,
respectively. The zero elements of the photon spinor $\Psi$ are related to the
property of all photon states to be eigenstates of the spin operator squared,
having a well-defined spin $S=1$. The zero elements of the charge-current
spinor $\Phi$ are related to the nonexistence of magnetic monopoles as
elementary particles Jackson1999 . The definition of the covariant spin
operator will be given in Secs. IV.2 and IV.3, and the eigenstates are
discussed in Sec. V.1.

The fields appearing in the photon spinor in Eq. (1) can be either real- or
complex-valued solutions of the complete set of Maxwell’s equations. They will
then automatically satisfy the eight-component spinorial photon equation,
given in Sec. II.2. The spinors made of real-valued fields $\mathbf{E}_{\Re}$
and $\mathbf{B}_{\Re}$, charge density $\rho_{\mathrm{e\Re}}$, and current
density $\mathbf{J}_{\mathrm{e\Re}}$ are denoted by the subscript $\Re$ as
$\Psi_{\Re}$ and $\Phi_{\Re}$. The spinors are normalized so that
$\Psi_{\Re}^{\dagger}\Psi_{\Re}=\frac{1}{2}(\varepsilon_{0}\mathbf{E}_{\Re}^{2}+\mathbf{B}_{\Re}^{2}/\mu_{0})$
corresponds the classical expression of the energy density of the
electromagnetic field. Using complex-valued amplitudes of time-harmonic
fields, the single-photon energy density expectation value is
$\frac{1}{2}\Psi^{\dagger}\Psi=\frac{1}{4}(\varepsilon_{0}|\mathbf{E}|^{2}+|\mathbf{B}|^{2}/\mu_{0})$,
which corresponds to the expression of the energy density of a classical time-
harmonic field averaged over the harmonic cycle. When we consider the photon
state in Eq. (1) as a quantum mechanical concept, it corresponds to a single-
photon Fock state. Thus, the quantum mechanical information content is not
equal to that of a classical time-harmonic field, but limited by complete lack
of the knowledge of the phase of single-photon states in accordance with the
_uncertainty principle_ of the photon number and the phase Smithey1993 . The
density expectation values and the classical and quantum-mechanical
information content of the spinorial photon state in Eq. (1) will be addressed
in more detail in Sec. V.

The eight-component photon spinor wavefunction in Eq. (1) can be compared to
previous approaches to construct a photon wavefunction. One conventional
approach is to use the electromagnetic vector potential or four-potential in
the role of the photon wavefunction Landau1982 . The Lorentz transformation of
this photon wavefunction predicts, for a radiation field in a general inertial
frame, unobserved longitudinal and scalar photons Gupta1950 ; Yang2022 . The
present eight-component photon spinor wavefunction $\Psi$ is seen to be a
generalization of the six-component photon wavefunction
$\sqrt{\frac{\varepsilon_{0}}{2}}[\mathbf{E},c\mathbf{B}]^{T}$ used in some
previous literature Sommerfeld1910a ; Sommerfeld1910b ; Darwin1932 ; Berry1990
; Berry2009 ; Mohr2010 ; Barnett2014 ; Bliokh2014a ; Bliokh2015a ; Bliokh2017a
; Bliokh2017b , where $T$ indicates the transpose. Depending on the work, the
imaginary unit can be present as a factor of the electric or magnetic field
component. In the case of real-valued fields, the sum of the electric and
magnetic components of $\Psi_{\Re}$ is also seen to correspond to the three-
component Weber vector
$\sqrt{\frac{\varepsilon_{0}}{2}}(\mathbf{E}_{\Re}+ic\mathbf{B}_{\Re})$
Weber1900 ; Kiessling2018 , also known as the Riemann-Silberstein vector
Silberstein1907a ; Silberstein1907b ; Bateman1915 ; Stratton1941 ;
BialynickiBirula1996 ; BialynickiBirula2003 ; BialynickiBirula2011 ; Good1957
. An expression that reminds the Weber vector, but generalizes to complex-
valued fields, is the representation of the electromagnetic field in the
space-time algebra Hestenes1966 ; Hestenes1987 ; Hestenes2003 ; Hestenes2005 ;
Dressel2015  through a quantity
$\sqrt{\frac{\varepsilon_{0}}{2}}(\mathbf{E}+Ic\mathbf{B})$, where the
algebraic element $I$ satisfing $I^{2}=-1$ is not the imaginary unit, but a
geometric object.

###  II.2 Spinorial photon equation and the associated gamma and sigma
matrices

Our formulation of electrodynamics equations below is closely analogous to the
approach of Dirac starting from the Klein-Gordon equation to derive a first-
order equation, the Dirac equation Dirac1928 ; Dirac1958 . Dirac’s clever idea
was to take a square root of the wave operator, which is equivalent to finding
matrices $\boldsymbol{\gamma}^{\mu}$, which satisfy
$\mathbf{I}_{n}\partial^{\mu}\partial_{\mu}=(\boldsymbol{\gamma}^{\mu}\partial_{\mu})^{2}$,
where the Einstein summation convention is used and $\mathbf{I}_{n}$ is the
$n\times n$ identity matrix. As conventional, the Greek indices range over the
four dimensions of the Minkowski space-time, i.e.,
$\mu,\nu,\rho,\sigma\in\\{ct,x,y,z\\}$. Correspondingly, the latin indices in
the equations of this work will range over the three spatial dimensions, i.e.,
$i,j,k\in\\{x,y,z\\}$. Dirac found that
$\mathbf{I}_{4}\partial^{\mu}\partial_{\mu}=(\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\partial_{\mu})^{2}$
is satisfied with $4\times 4$ matrices
$\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}$, which became known as the Dirac
gamma matrices. Here we use the subscript F to indicate that these gamma
matrices are associated with fermionic fields in distinction to the gamma
matrices of bosonic fields introduced below.

Our goal is to define gamma matrices, which enable writing the well-known
second-order wave equation of the four-potential, given by
$\partial^{\mu}\partial_{\mu}A^{\nu}=\mu_{0}J_{\mathrm{e}}^{\nu}$, as a first-
order equation for the photon spinor $\Psi$ in Eq. (1), which, unlike the
Dirac equation, also includes the source term expressed by the charge-current
spinor $\Phi$. We next show that it is possible to construct $8\times 8$ gamma
matrices, which operate on the photon spinor $\Psi$ in Eq. (1) leading to
Maxwell’s equations and to $\Psi$ having desired properties. In analogy to the
Dirac theory, our gamma matrices $\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}$
satisfy
$\mathbf{I}_{8}\partial^{\mu}\partial_{\mu}=(\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu})^{2}$.
Here we use the subscript B to indicate that these gamma matrices are
associated with bosonic fields in distinction to the gamma matrices of
fermionic fields in the Dirac theory.

We write the spinorial photon equation in a form that closely reminds the
Dirac equation describing spin-$\frac{1}{2}$ particles in the case when the
mass of the particle is set to zero and when an external source term is added.
The spinorial photon equation, which, as shown in Sec. II.3, describes all of
classical electromagnetism, is written as

$\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}\Psi=-\Phi.$ (2)

In analogy to the Dirac equation for electrons, the spinorial photon equation
is a first-order differential equation that is covariant in Lorentz
transformations between inertial frames. The $8\times 8$ bosonic gamma
matrices $\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}$ in Eq. (2) are defined as

$\boldsymbol{\gamma}_{\mathrm{B}}^{0}=\left[\begin{array}[]{cc}\mathbf{I}_{4}&\mathbf{0}\\\
\mathbf{0}&-\mathbf{I}_{4}\end{array}\right],\hskip
14.22636pt\boldsymbol{\gamma}_{\mathrm{B}}^{i}=\left[\begin{array}[]{cc}\mathbf{0}&\boldsymbol{\sigma}_{\mathrm{B}}^{i}\\\
-\boldsymbol{\sigma}_{\mathrm{B}}^{i}&\mathbf{0}\end{array}\right].$ (3)

The $4\times 4$ bosonic sigma matrices $\boldsymbol{\sigma}_{\mathrm{B}}^{x}$,
$\boldsymbol{\sigma}_{\mathrm{B}}^{y}$, and
$\boldsymbol{\sigma}_{\mathrm{B}}^{z}$ in Eq. (3) are given in the Cartesian
basis by

$\boldsymbol{\sigma}_{\mathrm{B}}^{i}=\mathbf{K}_{\mathrm{boost}}^{i}+i\mathbf{K}_{\mathrm{rot}}^{i}.$
(4)

Here $\mathbf{K}_{\mathrm{boost}}^{i}$ and $\mathbf{K}_{\mathrm{rot}}^{i}$ are
generators of the proper orthochronous Lorentz group SO+(1,3). The Lorentz
boost generators $\mathbf{K}_{\mathrm{boost}}^{i}$ are given by Baskal2015

$\displaystyle\mathbf{K}_{\mathrm{boost}}^{x}=\left[\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right]\\!\\!,\hskip
2.84544pt\mathbf{K}_{\mathrm{boost}}^{y}=\left[\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&0\\\ 1&0&0&0\\\ 0&0&0&0\end{array}\right]\\!\\!,$ (13)
$\displaystyle\hskip
42.67912pt\mathbf{K}_{\mathrm{boost}}^{z}=\left[\begin{array}[]{cccc}0&0&0&1\\\
0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\end{array}\right]\\!\\!,$ (18)

and the SO(3) rotation generators $\mathbf{K}_{\mathrm{rot}}^{i}$ are given by

$\displaystyle\mathbf{K}_{\mathrm{rot}}^{x}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right]\\!\\!,\hskip
2.84544pt\mathbf{K}_{\mathrm{rot}}^{y}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&1\\\ 0&0&0&0\\\ 0&-1&0&0\end{array}\right]\\!\\!,$ (27)
$\displaystyle\hskip
42.67912pt\mathbf{K}_{\mathrm{rot}}^{z}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&-1&0\\\ 0&1&0&0\\\ 0&0&0&0\end{array}\right]\\!\\!.$ (32)

Apart from not having the imaginary unit and the reduced Planck constant
$\hbar$ as a factor, the lower right $3\times 3$ space components of the
rotation generators in Eq. (32) are equal to the well-known spin matrices of
the three-dimensional space deShalit1963 ; Bliokh2015a ; Berry1998 ;
Akhiezer1965 ; BialynickiBirula1996 .

The form of the bosonic gamma matrices
$\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}$ in Eq. (3) reminds the conventional
definition of the $4\times 4$ fermionic gamma matrices
$\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}$ in the Dirac theory Zee2010 , but the
dimensions and the corresponding representations of the sigma matrices are
different. In the Dirac theory, the $2\times 2$ sigma matrices
$\boldsymbol{\sigma}_{\mathrm{F}}^{i}$ are known as Pauli matrices. The
$8\times 8$ gamma matrices of the present theory can also be seen as the
generalization of the $6\times 6$ gamma matrices that have been used together
with the six-component photon wavefunction in previous literature Mohr2010 ;
Barnett2014 . In comparison to the $6\times 6$ gamma matrices, one particular
advantage of our $8\times 8$ gamma matrices is that all Maxwell’s equations
can be presented as a single equation allowing us to rewrite the QED in the
eight-spinor notation as presented in Sec. VI. The bosonic gamma matrices in
Eq. (3) can be used to define the spinorial Lorentz transformations as will be
shown in Sec. III. This makes the photon spinor in Eq. (1) a true spinor with
spin 1 in the full physical meaning of the term Penrose1987 ; Cartan1981 ;
Corson1953 .

###  II.3 Dynamical equations in terms of the fields, charges, and currents

When the expressions of the photon spinor $\Psi$ and the charge-current spinor
$\Phi$ from Eq. (1) are substituted into the spinorial photon equation in Eq.
(2), we obtain the full set of four Maxwell’s equations as

$\nabla\cdot\mathbf{B}=0,$ (33)
$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}_{\mathrm{e}}+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\mathbf{E},$ (34)
$\nabla\cdot\mathbf{E}=\frac{\rho_{\mathrm{e}}}{\varepsilon_{0}},$ (35)
$\nabla\times\mathbf{E}=-\frac{\partial}{\partial t}\mathbf{B}.$ (36)

Thus, the very compact representation of the spinorial photon equation in Eq.
(2) together with the photon and charge-current spinors in Eq. (1) is
_equivalent_ to the full theory of classical electromagnetism Jackson1999 .
Again, the fields and charge and current densities in Eqs. (33)–(36) can be
either real or complex valued.

Operating on Eq. (2) side by side with an operator
$\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}$ and using
$(\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu})^{2}=\mathbf{I}_{8}\partial^{\mu}\partial_{\mu}$
on the left-hand side, we obtain
$\partial^{\mu}\partial_{\mu}\Psi=-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}\Phi$.
This equation is equivalent to the _continuity equation of the electric four-
current density_
$J_{\mathrm{e}}^{\mu}=(c\rho_{\mathrm{e}},\mathbf{J}_{\mathrm{e}})$ and the
_inhomogeneous wave equations of the electric and magnetic fields_ as

$\frac{\partial\rho_{\mathrm{e}}}{\partial
t}+\nabla\cdot\mathbf{J}_{\mathrm{e}}=0,$ (37)
$\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\mathbf{E}=\mu_{0}\frac{\partial}{\partial
t}\mathbf{J}_{\mathrm{e}}+\frac{1}{\varepsilon_{0}}\nabla\rho_{\mathrm{e}},$
(38) $\nabla^{2}\mathbf{B}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\mathbf{B}=-\mu_{0}\nabla\times\mathbf{J}_{\mathrm{e}}.$ (39)

Therefore, in addition to Maxwell’s equations, the spinorial photon equation
also describes the physics related to the _conservation of the electric four-
current density_ Jackson1999 . If the hypothetical magnetic charge and current
densities were added in the theory through substituting them in the zero
elements of the charge-current spinor in Eq. (1), the conservation law of the
magnetic four-current density would also follow.

###  II.4 Potential spinor

For the Lagrangian formulation of QED using the present spinorial notation, to
be presented in Sec. VI, we next provide the spinorial representation of the
four-potential. The electromagnetic four-potential is a four-vector
$A^{\mu}=(\phi_{\mathrm{e}}/c,\mathbf{A})$, where $\phi_{\mathrm{e}}$ is the
electric scalar potential and $\mathbf{A}$ is the vector potential. Apart from
constant prefactors, the _potential spinor_ $\Theta$ is constructed from the
components of $A^{\mu}=(\phi_{\mathrm{e}}/c,\mathbf{A})$ in the same way as
the charge-current spinor is made of the components of the electric four-
current density
$J_{\mathrm{e}}^{\mu}=(c\rho_{\mathrm{e}},\mathbf{J}_{\mathrm{e}})$ as
presented in Eq. (1). Consequently, we write $\Theta$ as

$\Theta=\sqrt{\frac{\varepsilon_{0}}{2}}\left[\begin{array}[]{c}0\\\
c\mathbf{A}\\\ \phi_{\mathrm{e}}\\\ \mathbf{0}\end{array}\right].$ (40)

Then, operating on this spinor by the operator
$-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}$ we obtain

$\displaystyle-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}\Theta$
$\displaystyle=\sqrt{\frac{\varepsilon_{0}}{2}}\left[\begin{array}[]{c}0\\\
-\nabla\phi_{\mathrm{e}}-\frac{\partial}{\partial t}\mathbf{A}\\\
c\partial_{\mu}A^{\mu}\\\ ic\nabla\times\mathbf{A}\end{array}\right]$ (45)
$\displaystyle\hskip 11.38092pt\xrightarrow{\partial_{\mu}A^{\mu}\rightarrow
0}\sqrt{\frac{\varepsilon_{0}}{2}}\left[\begin{array}[]{c}0\\\ \mathbf{E}\\\
0\\\ ic\mathbf{B}\end{array}\right]=\Psi.$ (50)

Here we have used the conventional expressions of the fields in terms of the
vector and scalar potentials, given by
$\mathbf{E}=-\nabla\phi_{\mathrm{e}}-\frac{\partial}{\partial t}\mathbf{A}$
and $\mathbf{B}=\nabla\times\mathbf{A}$ Jackson1999 . Equation (50) shows that
the four-potential can be used to construct the photon spinor $\Psi$ by the
operator $-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}$ when the
Lorenz gauge condition $\partial_{\mu}A^{\mu}=0$ is satisfied. Otherwise, the
fifth component of the photon spinor would be nonzero as seen from Eq. (50).
This would violate the property of all photon states to be eigenstates of the
spin operator squared, having a well-defined spin of $S=1$, as presented in
Sec. V.1. Note the fact that the Lorenz gauge condition does not determine the
four-potential uniquely since there are remaining degrees of freedom
corresponding to gauge functions which satisfy the wave equation. This is
called residual gauge symmetry Swanson2022 . If the Coulomb gauge
$\nabla\cdot\mathbf{A}=0$ is used, then the Lorenz gauge condition must be
simultaneously satisfied. This means that we must separately have the time
derivative of the scalar potential equal to zero. If the electric and magnetic
fields are transverse, we can set the scalar potential to zero, and with this
further condition, the Coulomb gauge is called the radiation gauge Jackson1999
.

From the perspective of the four-potential, the formulation of the spinorial
photon equation can be seen as follows: The wave equation of the four-
potential in the Lorenz gauge is given by
$\partial^{\mu}\partial_{\mu}A^{\nu}=\mu_{0}J_{\mathrm{e}}^{\nu}$. To
transform this equation to an equation for the potential spinor, we use
$A^{\nu}$ and $J_{\mathrm{e}}^{\nu}$ to form spinors $\Theta$ and $\Phi$
according to Eqs. (1) and (40). Thus, the wave equation of the four-potential
is rewritten as

$\partial^{\mu}\partial_{\mu}\Theta=\Phi.$ (51)

To transform this equation into a first-order partial differential equation
for the photon spinor $\Psi$, we write the wave operator as
$\mathbf{I}_{8}\partial^{\mu}\partial_{\mu}=(-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu})^{2}$.
Thus, the left-hand-side of the wave equation of the potential spinor in Eq.
(51) is rewritten as
$\partial^{\mu}\partial_{\mu}\Theta=(-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu})^{2}\Theta=-\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}\Psi$,
where we have used Eq. (50). Thus, from Eq. (51), we obtain

$-\partial^{\mu}\partial_{\mu}\Theta=\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\partial_{\mu}\Psi=-\Phi.$
(52)

This equation is equivalent to the spinorial photon equation in Eq. (2).

###  II.5 Left-handed and right-handed spinor components and the fifth gamma
matrix

In analogy to the Dirac theory Peskin2018 , the photon spinor can be projected
onto its left-handed and right-handed chiral components $\Psi_{\mathrm{L}}$
and $\Psi_{\mathrm{R}}$ as

$\Psi_{\mathrm{L}}=\frac{\mathbf{I}_{8}-\boldsymbol{\gamma}_{\mathrm{B}}^{5}}{2}\Psi,\hskip
14.22636pt\Psi_{\mathrm{R}}=\frac{\mathbf{I}_{8}+\boldsymbol{\gamma}_{\mathrm{B}}^{5}}{2}\Psi.$
(53)

The components $\Psi_{\mathrm{L}}$ and $\Psi_{\mathrm{R}}$ are eigenstates of
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}$ with eigenvalues $\pm 1$ as
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}\Psi_{\mathrm{L}}=-\Psi_{\mathrm{L}}$ and
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}\Psi_{\mathrm{R}}=\Psi_{\mathrm{R}}$. The
fifth gamma matrix $\boldsymbol{\gamma}_{\mathrm{B}}^{5}$ is the chirality
operator, and it is defined in terms of the other gamma matrices by the
relation
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}=\frac{i}{4!}\varepsilon_{\mu\nu\rho\sigma}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\boldsymbol{\gamma}_{\mathrm{B}}^{\rho}\boldsymbol{\gamma}_{\mathrm{B}}^{\sigma}=i\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{x}\boldsymbol{\gamma}_{\mathrm{B}}^{y}\boldsymbol{\gamma}_{\mathrm{B}}^{z}$,
where $\varepsilon_{\mu\nu\rho\sigma}$ is the four-dimensional Levi-Civita
symbol. This relation is of the same form as the corresponding relation for
conventional Dirac gamma matrices Peskin2018 . The matrix
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}$ anticommutes with the other gamma
matrices as
$\\{\boldsymbol{\gamma}_{\mathrm{B}}^{5},\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\\}=\mathbf{0}$
and commutes with the Lorentz transformation matrices that will be presented
in Sec. III. For massless particles, such as for photons in the present case,
the chirality is equivalent to the helicity divided by $\hbar$ Peskin2018 .
For the definition of the helicity operator, see Sec. IV.2. For further
discussion of the helicity-chirality equivalence, see Appendix A. As will be
shown in Sec. VII, $\boldsymbol{\gamma}_{\mathrm{B}}^{5}$ plays a fundamental
role in the definition of the generating Lagrangian density of gravity and its
special unitary symmetry.

###  II.6 Local and non-local scalar products

Next, we define the scalar product of eight-component photon and charge-
current spinors. In analogy to the Dirac theory, we define the local scalar
product for two arbitrary eight-component spinors. Each of the spinors can be
a photon spinor, charge-current spinor, or any other eight-spinor. As an
example, using the photon spinor symbols $\Psi_{1}$ and $\Psi_{2}$, the local
scalar product is written as

$\bar{\Psi}_{1}\Psi_{2}=\Psi_{1}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}\Psi_{2}.$
(54)

Here $\bar{\Psi}_{1}$ is the bosonic Dirac adjoint spinor defined by
$\bar{\Psi}_{1}=\Psi_{1}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$, where
$\Psi_{1}^{\dagger}$ denotes the Hermitian adjoint of the spinor $\Psi_{1}$.
The Hermiticity of all quantum operators in this work is defined with respect
to the conventional non-local integral-form scalar product, written as

$\langle\Psi_{1}|\Psi_{2}\rangle=\int\Psi_{1}^{\dagger}\Psi_{2}d^{3}r.$ (55)

For a general Hermitian operator $\hat{A}$, we then have
$\langle\hat{A}\Psi_{1}|\Psi_{2}\rangle=\langle\Psi_{1}|\hat{A}\Psi_{2}\rangle$.
A fundamental property of the local scalar product in Eq. (54) is that, for
two photon spinors, $\bar{\Psi}_{1}\Psi_{2}$ transforms as a Lorentz scalar
between inertial frames, which is not the case for
$\Psi_{1}^{\dagger}\Psi_{2}$. Thus, the local scalar product of eight-
component spinors in Eq. (54) highlights the fundamental analogy between the
present eight-spinor theory of the electromagnetic field and the Dirac theory
of the electron-positron field. In previous constructions of the photon
wavefunction or the representation of the electromagnetic field in general
BialynickiBirula1996 ; Landau1982 ; Kiessling2018 ; Dressel2015 , such a
transparent analogy is not typically observable. However, the previous six-
component representation of the photon wavefunction Darwin1932 ; Mohr2010 ;
Barnett2014  can be seen to be analogous to the Dirac theory, but it neither
possess all properties of the present theory, such as the presentation of all
Maxwell’s equations by a single equation.

###  II.7 Algebraic properties of the Lorentz generators and sigma and gamma
matrices

The Lorentz boost and rotation generators $\mathbf{K}_{\mathrm{boost}}^{i}$
and $\mathbf{K}_{\mathrm{rot}}^{i}$ in Eqs. (18) and (32) satisfy the
commutation relations
$[\mathbf{K}_{\mathrm{boost}}^{i},\mathbf{K}_{\mathrm{boost}}^{j}]=-\varepsilon_{ijk}\mathbf{K}_{\mathrm{rot}}^{k}$,
$[\mathbf{K}_{\mathrm{rot}}^{i},\mathbf{K}_{\mathrm{rot}}^{j}]=\varepsilon_{ijk}\mathbf{K}_{\mathrm{rot}}^{k}$,
and
$[\mathbf{K}_{\mathrm{rot}}^{i},\mathbf{K}_{\mathrm{boost}}^{j}]=\varepsilon_{ijk}\mathbf{K}_{\mathrm{boost}}^{k}$,
where $\varepsilon_{ijk}$ is the three-dimensional Levi-Civita symbol. The
Lorentz Lie algebra relations and the presentation of the Lorentz
transformations in terms of the Lorentz generators are discussed in Sec. III.

The sigma matrices $\boldsymbol{\sigma}_{\mathrm{B}}^{i}$ in Eq. (4) are
Hermitian, involutory, and unitary satisfying
$\boldsymbol{\sigma}_{\mathrm{B}}^{x}\boldsymbol{\sigma}_{\mathrm{B}}^{x}=\boldsymbol{\sigma}_{\mathrm{B}}^{y}\boldsymbol{\sigma}_{\mathrm{B}}^{y}=\boldsymbol{\sigma}_{\mathrm{B}}^{z}\boldsymbol{\sigma}_{\mathrm{B}}^{z}=-i\boldsymbol{\sigma}_{\mathrm{B}}^{x}\boldsymbol{\sigma}_{\mathrm{B}}^{y}\boldsymbol{\sigma}_{\mathrm{B}}^{z}=\mathbf{I}_{4}$.
Their determinants are equal to unity as
$\mathrm{Det}(\boldsymbol{\sigma}_{\mathrm{B}}^{i})=1$ and their traces are
zero as $\mathrm{Tr}(\boldsymbol{\sigma}_{\mathrm{B}}^{i})=0$. The eigenvalues
of each $\boldsymbol{\sigma}_{\mathrm{B}}^{i}$ are $\pm 1$. The commutation
and anticommutation relations of the sigma matrices are given by
$[\boldsymbol{\sigma}_{\mathrm{B}}^{i},\boldsymbol{\sigma}_{\mathrm{B}}^{j}]=2i\varepsilon_{ijk}\boldsymbol{\sigma}_{\mathrm{B}}^{k}$
and
$\\{\boldsymbol{\sigma}_{\mathrm{B}}^{i},\boldsymbol{\sigma}_{\mathrm{B}}^{j}\\}=2\delta^{ij}\mathbf{I}_{4}$,
where $\delta^{ij}$ is the Kronecker delta. Apart from the sign of the
determinants of $\boldsymbol{\sigma}_{\mathrm{B}}^{i}$, these relations
correspond to the relations of the Pauli matrices in the conventional $2\times
2$-dimensional case.

In analogy to the Dirac theory, the time-like gamma matrix
$\boldsymbol{\gamma}_{\mathrm{B}}^{0}$ is Hermitian and the space-like gamma
matrices $\boldsymbol{\gamma}_{\mathrm{B}}^{x}$,
$\boldsymbol{\gamma}_{\mathrm{B}}^{y}$, and
$\boldsymbol{\gamma}_{\mathrm{B}}^{z}$ are anti-Hermitian. The gamma matrices
are generators of the gamma group $G_{1,3}$ Petitjean2020 . Furthermore, the
gamma matrices satisfy the anticommutation relation
$\\{\boldsymbol{\gamma}_{\mathrm{B}}^{\mu},\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\\}=2\eta^{\mu\nu}\mathbf{I}_{8}$,
where $\eta^{\mu\nu}$ is the Minkowski metric tensor with signature
$(+,-,-,-)$. The conventional Dirac gamma matrices are well-known to satisfy
the corresponding relation
$\\{\boldsymbol{\gamma}_{\mathrm{F}}^{\mu},\boldsymbol{\gamma}_{\mathrm{F}}^{\nu}\\}=2\eta^{\mu\nu}\mathbf{I}_{4}$
Peskin2018 . This is the defining relation of a Dirac algebra, which is a
Clifford algebra $\mathcal{C}\ell_{1,3}(\mathbb{C})$ over the four-dimensional
Minkowski space-time. Therefore, in spite of being $8\times 8$ matrices, the
present bosonic gamma matrices share the same group-theoretical and algebraic
properties with the conventional $4\times 4$ gamma matrices of the Dirac
theory. One particularly useful relation to note is
$\boldsymbol{\gamma}_{\mathrm{B}}^{\mu{\dagger}}=\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$.
To preserve the forms of the photon and charge-current spinors in Eq. (1) in
terms of their component fields in all inertial frames, the present theory
follows the convention that the spinors transform under Lorentz
transformations and the gamma matrices are constant having the same form in
all inertial frames Feynman1961 .

##  III Lorentz transformations

In the present theory, there are two types of Lorentz transformations for
eight-component spinors: the Lorentz transformation of spinors generated by
four-vectors, called four-vector spinors, such as the charge-current spinor
and the potential spinor; and the Lorentz transformation of spin-1 field
spinors, such as the photon spinor. These transformations preserve the forms
of the photon and charge-current spinors in Eq. (1) and the potential spinor
in Eq. (40) in all inertial frames. The transformations are analogous to the
Lorentz transformation of Dirac spinors in their construction through the
Lorentz generators in a way that they differ from the Lorentz transformation
of four-vectors. For completeness, the Lorentz transformation of the
conventional spin-$\frac{1}{2}$ Dirac field spinors is also presented. In
addition, we construct the Lorentz transformation of spin-2 fields, which can
be presented using $8\times 8$ matrices in the present theory Partanen2023c .
The Lorentz transformations are described in detail in the subsections below.

###  III.1 Lorentz transformation of four-vectors

We start with the conventional Lorentz transformation of four-vectors. In
terms of the generators of the proper orthochronous Lorentz group SO+(1,3) in
Eqs. (18) and (32), the Lorentz boost of four-vectors is given by
$\boldsymbol{\Lambda}_{\mathrm{boost}}(\boldsymbol{\zeta})=\exp(-\boldsymbol{\zeta}\cdot\mathbf{K}_{\mathrm{boost}})$
and the spatial rotation of four-vectors is given by
$\boldsymbol{\Lambda}_{\mathrm{rot}}(\boldsymbol{\theta})=\exp(\boldsymbol{\theta}\cdot\mathbf{K}_{\mathrm{rot}})$,
where
$\boldsymbol{\zeta}=(\zeta_{x},\zeta_{y},\zeta_{z})=\arctan(v/c)\mathbf{v}/|\mathbf{v}|$
is the rapidity vector, in which $\mathbf{v}$ is the boost velocity, and
$\boldsymbol{\theta}=(\theta_{x},\theta_{y},\theta_{z})$ is the axis-angle
vector. The total combined Lorentz transformation of four-vectors is given by
$\boldsymbol{\Lambda}(\boldsymbol{\boldsymbol{\zeta},\theta})=\exp(-\boldsymbol{\zeta}\cdot\mathbf{K}_{\mathrm{boost}}+\boldsymbol{\theta}\cdot\mathbf{K}_{\mathrm{rot}})$,
where
$\mathbf{K}_{\mathrm{boost}}=(\mathbf{K}_{\mathrm{boost}}^{x},\mathbf{K}_{\mathrm{boost}}^{y},\mathbf{K}_{\mathrm{boost}}^{z})$
and
$\mathbf{K}_{\mathrm{rot}}=(\mathbf{K}_{\mathrm{rot}}^{x},\mathbf{K}_{\mathrm{rot}}^{y},\mathbf{K}_{\mathrm{rot}}^{z})$.

A more compact presentation of the full Lorentz transformation involving both
a boost and rotation is obtained by substituting the components of the
rapidity and axis-angle vectors in an antisymmetric Lorentz parameter matrix
$\Omega_{\rho\sigma}$, given by

$\Omega_{\rho\sigma}=\left[\begin{array}[]{cccc}0&-\zeta_{x}&-\zeta_{y}&-\zeta_{z}\\\
\zeta_{x}&0&\theta_{z}&-\theta_{y}\\\ \zeta_{y}&-\theta_{z}&0&\theta_{x}\\\
\zeta_{z}&\theta_{y}&-\theta_{x}&0\end{array}\right],$ (56)

and combining the representation of the Lorentz generators in Eqs. (18) and
(32) by defining Maggiore2005

$(K^{\rho\sigma})^{\mu}_{\;\,\nu}=\eta^{\rho\mu}\delta^{\sigma}_{\;\,\nu}-\eta^{\sigma\mu}\delta^{\rho}_{\;\,\nu}.$
(57)

For different values of indices $\rho$ and $\sigma$, this expression gives
matrices corresponding to the Lorentz generators in Eqs. (18) and (32). Using
Eq. (57), one can show that the generators $K^{\rho\sigma}$ obey the Lorentz
Lie algebra relations,
$[K^{\mu\nu},K^{\rho\sigma}]=\eta^{\nu\rho}K^{\mu\sigma}-\eta^{\mu\rho}K^{\nu\sigma}+\eta^{\mu\sigma}K^{\nu\rho}-\eta^{\nu\sigma}K^{\mu\rho}$,
and the Lorentz transformation of four-vectors is given by Maggiore2005

$\textstyle\boldsymbol{\Lambda}=\exp(\frac{1}{2}\Omega_{\rho\sigma}K^{\rho\sigma}).$
(58)

Using this Lorentz transformation matrix, a four-vector $X$ transforms between
inertial frames as $X^{\prime}=\boldsymbol{\Lambda}X$. For boosts, the real-
valued Lorentz transformation matrix $\boldsymbol{\Lambda}$ in Eq. (58) is
symmetric, i.e., $\boldsymbol{\Lambda}=\boldsymbol{\Lambda}^{T}$. For SO(3)
rotations, it is orthogonal, i.e.,
$\boldsymbol{\Lambda}\boldsymbol{\Lambda}^{T}=\mathbf{I}_{4}$.

###  III.2 Lorentz transformation of four-vector spinors

The commutator of the boost parts of the bosonic gamma matrices determines the
Lorentz transformation of eight-component four-vector spinors, such as the
charge-current spinor in Eq. (1) and the potential spinor in Eq. (40). With
this commutator, we define $8\times 8$ antisymmetric matrices
$\Sigma^{\rho\sigma}_{\mathrm{L}}$, the generators of Lorentz transformations
on four-vector spinors, as

$\Sigma_{\mathrm{L}}^{\rho\sigma}=[\boldsymbol{\gamma}_{\mathrm{B,boost}}^{\rho},\boldsymbol{\gamma}_{\mathrm{B,boost}}^{\sigma}].$
(59)

The boost parts $\boldsymbol{\gamma}_{\mathrm{B,boost}}^{\mu}$ of the gamma
matrices are determined through Eq. (3) with
$\boldsymbol{\sigma}_{\mathrm{B}}^{i}$ replaced by
$\mathbf{K}_{\mathrm{boost}}^{i}$ and $\mathbf{I}_{4}$ replaced by
$\frac{1}{2}\mathbf{I}_{4}$. The generators $\Sigma^{\rho\sigma}_{\mathrm{L}}$
obey the Lorentz Lie algebra relations,
$[\Sigma^{\mu\nu}_{\mathrm{L}},\Sigma^{\rho\sigma}_{\mathrm{L}}]=\eta^{\nu\rho}\Sigma^{\mu\sigma}_{\mathrm{L}}-\eta^{\mu\rho}\Sigma^{\nu\sigma}_{\mathrm{L}}+\eta^{\mu\sigma}\Sigma^{\nu\rho}_{\mathrm{L}}-\eta^{\nu\sigma}\Sigma^{\mu\rho}_{\mathrm{L}}$.

Under the Lorentz transformation, four-vector spinors transform as
$\Phi^{\prime}=\boldsymbol{\Lambda}_{\mathrm{L}}\Phi$, where the $8\times 8$
transformation matrix $\boldsymbol{\Lambda}_{\mathrm{L}}$ is in analogy to Eq.
(58) given by

$\textstyle\boldsymbol{\Lambda}_{\mathrm{L}}=\exp(\frac{1}{2}\Omega_{\rho\sigma}\Sigma_{\mathrm{L}}^{\rho\sigma}).$
(60)

The appearance of the Lorentz parameter matrix $\Omega_{\rho\sigma}$ in Eq.
(60) ensures that this transformation is a manifestation of the same Lorentz
transformation as presented for four-vectors in Eq. (58). For boosts, the
Lorentz transformation matrix $\boldsymbol{\Lambda}_{\mathrm{L}}$ in Eq. (60)
is Hermitian, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{L}}=\boldsymbol{\Lambda}_{\mathrm{L}}^{\dagger}$.
For SO(3) rotations, it is unitary, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{L}}\boldsymbol{\Lambda}_{\mathrm{L}}^{\dagger}=\mathbf{I}_{8}$.
In analogy to the Dirac theory Peskin2018 , using the identity
$\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\Lambda}_{\mathrm{L}}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}=\boldsymbol{\Lambda}_{\mathrm{L}}^{-1}$,
for the Lorentz transformation of
$\bar{\Phi}=\Phi^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$, we obtain
$\bar{\Phi}^{\prime}=\bar{\Phi}\boldsymbol{\Lambda}_{\mathrm{L}}^{-1}$. Thus,
$\boldsymbol{\Lambda}_{\mathrm{L}}$ preserves the scalar product in Eq. (54)
for arbitrary four-vector spinors $\Phi_{1}$ and $\Phi_{2}$ as
$\overline{\boldsymbol{\Lambda}_{\mathrm{L}}\Phi_{1}}\boldsymbol{\Lambda}_{\mathrm{L}}\Phi_{2}=\bar{\Phi}_{1}\Phi_{2}$.

The consistency of the Lorentz transformation of our eight-component four-
vector spinors in Eq. (60) can be straightforwardly verified by observing that
it reproduces the conventional Lorentz transformation of the electric four-
current density, when it is used to transform a generic charge-current spinor
of the form in Eq. (1).

###  III.3 Lorentz transformation of spin-$\frac{1}{2}$ field spinors

The commutator of the conventional fermionic gamma matrices determines the
Lorentz transformation of spin-$\frac{1}{2}$ field spinors, i.e., the Dirac
spinors. With this commutator, one defines antisymmetric spin matrices
$\Sigma^{\rho\sigma}_{\mathrm{F}}$, the generators of Lorentz transformations
on spin-$\frac{1}{2}$ field spinors as Peskin2018

$\Sigma^{\rho\sigma}_{\mathrm{F}}=\frac{1}{4}[\boldsymbol{\gamma}_{\mathrm{F}}^{\rho},\boldsymbol{\gamma}_{\mathrm{F}}^{\sigma}].$
(61)

The generators $\Sigma^{\rho\sigma}_{\mathrm{F}}$ obey the Lorentz Lie algebra
relations,
$[\Sigma^{\mu\nu}_{\mathrm{F}},\Sigma^{\rho\sigma}_{\mathrm{F}}]=\eta^{\nu\rho}\Sigma^{\mu\sigma}_{\mathrm{F}}-\eta^{\mu\rho}\Sigma^{\nu\sigma}_{\mathrm{F}}+\eta^{\mu\sigma}\Sigma^{\nu\rho}_{\mathrm{F}}-\eta^{\nu\sigma}\Sigma^{\mu\rho}_{\mathrm{F}}$.

Under the Lorentz transformation, spin-$\frac{1}{2}$ field spinors $\psi$
transform as $\psi^{\prime}=\boldsymbol{\Lambda}_{\mathrm{F}}\psi$, where the
$4\times 4$ transformation matrix $\boldsymbol{\Lambda}_{\mathrm{F}}$ is in
analogy to Eq. (58) given by Peskin2018

$\textstyle\boldsymbol{\Lambda}_{\mathrm{F}}=\exp(\frac{1}{2}\Omega_{\rho\sigma}\Sigma_{\mathrm{F}}^{\rho\sigma}).$
(62)

Again, the appearance of the Lorentz parameter matrix $\Omega_{\rho\sigma}$ in
the transformation in Eq. (62) ensures that this transformation manifests the
same Lorentz transformation for spin-$\frac{1}{2}$ field spinors as was
defined for four-vectors in Eq. (58). For boosts, the Lorentz transformation
matrix $\boldsymbol{\Lambda}_{\mathrm{F}}$ in Eq. (62) is Hermitian, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{F}}=\boldsymbol{\Lambda}_{\mathrm{F}}^{\dagger}$.
For SO(3) rotations, it is unitary, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{F}}\boldsymbol{\Lambda}_{\mathrm{F}}^{\dagger}=\mathbf{I}_{4}$.
Using the identity
$\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\Lambda}_{\mathrm{F}}^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}=\boldsymbol{\Lambda}_{\mathrm{F}}^{-1}$,
for the Lorentz transformation of
$\bar{\psi}=\psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}$, we obtain
$\bar{\psi}^{\prime}=\bar{\psi}\boldsymbol{\Lambda}_{\mathrm{F}}^{-1}$
Peskin2018 . Thus, $\boldsymbol{\Lambda}_{\mathrm{F}}$ preserves the scalar
product $\bar{\psi}_{1}\psi_{2}$ for arbitrary Dirac spinors $\psi_{1}$ and
$\psi_{2}$ as
$\overline{\boldsymbol{\Lambda}_{\mathrm{F}}\psi_{1}}\boldsymbol{\Lambda}_{\mathrm{F}}\psi_{2}=\bar{\psi}_{1}\psi_{2}$.

###  III.4 Lorentz transformation of spin-1 field spinors

The commutator of the rotational parts of the bosonic gamma matrices
determines the Lorentz transformation of photon spinors or any other spinors
corresponding to a spin-1 field. With this commutator, we define $8\times 8$
antisymmetric spin matrices $\Sigma^{\rho\sigma}_{\mathrm{S}}$, the generators
of Lorentz transformations on spin-1 field spinors as

$\Sigma^{\rho\sigma}_{\mathrm{S}}=[\boldsymbol{\gamma}_{\mathrm{B,rot}}^{\rho},\boldsymbol{\gamma}_{\mathrm{B,rot}}^{\sigma}].$
(63)

The rotational parts $\boldsymbol{\gamma}_{\mathrm{B,rot}}^{\mu}$ of the gamma
matrices are determined through Eq. (3) with
$\boldsymbol{\sigma}_{\mathrm{B}}^{i}$ replaced by
$i\mathbf{K}_{\mathrm{rot}}^{i}$ and $\mathbf{I}_{4}$ replaced by
$\frac{1}{2}\mathbf{I}_{4}$. The generators $\Sigma^{\rho\sigma}_{\mathrm{S}}$
obey the Lorentz Lie algebra relations,
$[\Sigma^{\mu\nu}_{\mathrm{S}},\Sigma^{\rho\sigma}_{\mathrm{S}}]=\eta^{\nu\rho}\Sigma^{\mu\sigma}_{\mathrm{S}}-\eta^{\mu\rho}\Sigma^{\nu\sigma}_{\mathrm{S}}+\eta^{\mu\sigma}\Sigma^{\nu\rho}_{\mathrm{S}}-\eta^{\nu\sigma}\Sigma^{\mu\rho}_{\mathrm{S}}$.

Under the Lorentz transformation, spin-1 field spinors transform as
$\Psi^{\prime}=\boldsymbol{\Lambda}_{\mathrm{S}}\Psi$, where the $8\times 8$
transformation matrix $\boldsymbol{\Lambda}_{\mathrm{S}}$ is in analogy to
Eqs. (58), (60), and (62) given by

$\textstyle\boldsymbol{\Lambda}_{\mathrm{S}}=\exp(\frac{1}{2}\Omega_{\rho\sigma}\Sigma_{\mathrm{S}}^{\rho\sigma}).$
(64)

Again, the appearance of the Lorentz parameter matrix $\Omega_{\rho\sigma}$ in
the transformation in Eq. (64) ensures that this transformation is a
manifestation of the same Lorentz transformation for spin-1 field spinors as
was defined for four-vectors in Eq. (58). For boosts, the Lorentz
transformation matrix $\boldsymbol{\Lambda}_{\mathrm{S}}$ in Eq. (64) is
Hermitian, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{S}}=\boldsymbol{\Lambda}_{\mathrm{S}}^{\dagger}$.
For SO(3) rotations, it is unitary, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{S}}\boldsymbol{\Lambda}_{\mathrm{S}}^{\dagger}=\mathbf{I}_{8}$.
In analogy to the Dirac theory Peskin2018 , using the identity
$\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\Lambda}_{\mathrm{S}}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}=\boldsymbol{\Lambda}_{\mathrm{S}}^{-1}$,
for the Lorentz transformation of
$\bar{\Psi}=\Psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$, we obtain
$\bar{\Psi}^{\prime}=\bar{\Psi}\boldsymbol{\Lambda}_{\mathrm{S}}^{-1}$. Thus,
$\boldsymbol{\Lambda}_{\mathrm{S}}$ preserves the scalar product in Eq. (54)
for arbitrary photon spinors $\Psi_{1}$ and $\Psi_{2}$ as
$\overline{\boldsymbol{\Lambda}_{\mathrm{S}}\Psi_{1}}\boldsymbol{\Lambda}_{\mathrm{S}}\Psi_{2}=\bar{\Psi}_{1}\Psi_{2}$.

The consistency of the Lorentz transformation in Eq. (64) for our eight-
component spin-1 field spinors can be straightforwardly verified by observing
that it reproduces the conventional Lorentz transformation of the electric and
magnetic fields, when it is used to transform a generic photon spinor of the
form in Eq. (1).

###  III.5 Lorentz transformation of spin-2 fields

The commutator of the full bosonic gamma matrices determines the Lorentz
transformation of spin-2 fields, which can be presented using $8\times 8$
matrices in the present theory Partanen2023c . With this commutator, we define
$8\times 8$ antisymmetric matrices $\Sigma^{\rho\sigma}_{\mathrm{J}}$, the
generators of Lorentz transformations on spin-2 fields as

$\Sigma^{\rho\sigma}_{\mathrm{J}}=\frac{1}{4}[\boldsymbol{\gamma}_{\mathrm{B}}^{\rho},\boldsymbol{\gamma}_{\mathrm{B}}^{\sigma}].$
(65)

The generators $\Sigma^{\rho\sigma}_{\mathrm{J}}$ obey the Lorentz Lie algebra
relations,
$[\Sigma^{\mu\nu}_{\mathrm{J}},\Sigma^{\rho\sigma}_{\mathrm{J}}]=\eta^{\nu\rho}\Sigma^{\mu\sigma}_{\mathrm{J}}-\eta^{\mu\rho}\Sigma^{\nu\sigma}_{\mathrm{J}}+\eta^{\mu\sigma}\Sigma^{\nu\rho}_{\mathrm{J}}-\eta^{\nu\sigma}\Sigma^{\mu\rho}_{\mathrm{J}}$.

Under the Lorentz transformation, a tensor field $\mathbf{H}$ transforms as
$\mathbf{H}^{\prime}=\boldsymbol{\Lambda}_{\mathrm{J}}\mathbf{H}\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}$,
where the $8\times 8$ transformation matrix
$\boldsymbol{\Lambda}_{\mathrm{J}}$ is in analogy to Eqs. (58), (60), (62),
and (64) given by

$\textstyle\boldsymbol{\Lambda}_{\mathrm{J}}=\exp(\frac{1}{2}\Omega_{\rho\sigma}\Sigma_{\mathrm{J}}^{\rho\sigma}).$
(66)

The appearance of the Lorentz parameter matrix $\Omega_{\rho\sigma}$ in the
transformation in Eq. (66) ensures that this transformation is a manifestation
of the same Lorentz transformation for gamma matrices as was defined for four-
vectors and other spinors above. For boosts, the Lorentz transformation matrix
$\boldsymbol{\Lambda}_{\mathrm{J}}$ in Eq. (66) is Hermitian, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{J}}=\boldsymbol{\Lambda}_{\mathrm{J}}^{\dagger}$.
For SO(3) rotations, it is unitary, i.e.,
$\boldsymbol{\Lambda}_{\mathrm{J}}\boldsymbol{\Lambda}_{\mathrm{J}}^{\dagger}=\mathbf{I}_{8}$.
Using the identity
$\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\Lambda}_{\mathrm{J}}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}=\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}$,
for the Lorentz transformation of
$\bar{\mathbf{H}}=\boldsymbol{\gamma}_{\mathrm{B}}^{0}\mathbf{H}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$,
we obtain
$\bar{\mathbf{H}}^{\prime}=\boldsymbol{\Lambda}_{\mathrm{J}}\bar{\mathbf{H}}\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}$.

For a general Lorentz transformation, using the gamma matrix algebra, it can
be verified that the operation of $\boldsymbol{\Lambda}_{\mathrm{J}}$ on the
matrix product of two arbitrary complex-conjugated gamma matrices is invariant
as
$\boldsymbol{\Lambda}_{\mathrm{J}}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu*}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu*}\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}=\boldsymbol{\gamma}_{\mathrm{B}}^{\mu*}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu*}$.
Similarly, we have
$\boldsymbol{\Lambda}_{\mathrm{J}}\boldsymbol{\gamma}_{\mathrm{B}}^{5*}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu*}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu*}\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}=\boldsymbol{\gamma}_{\mathrm{B}}^{5*}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu*}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu*}$.
From this relation, it follows that the representation of a spin-2 tensor
field as a linear combination of the symmetry transform generators in the
special unitary symmetry studied in Sec. VII.2 is invariant. This observation
is utilized in the Yang-Mills gauge theory of gravity that we elaborate in a
separate work in Ref. Partanen2023c . It is known that a massless spin-2 field
can be associated with gravity Misner1973 . Therefore, due to the connection
to gravity, we call the Lorentz transformation
$\boldsymbol{\Lambda}_{\mathrm{J}}$ the Lorentz transformation of spin-2
fields. However, since spin-2 fields are invariant with respect to
$\boldsymbol{\Lambda}_{\mathrm{J}}$ as briefly mentioned above,
$\boldsymbol{\Lambda}_{\mathrm{J}}$ does not represent an actual
transformation of such fields.

##  IV Quantum operators in the first quantization

In this section, we determine the key quantum mechanical operators for single-
photon spinorial states given in Eq. (1). These operators are in the
literature called the operators in the first quantization. For the quantum
field of many photons, one will need the corresponding second quantization
operators described in Sec. VIII.

###  IV.1 Hamiltonian operator

By defining the $8\times 8$ matrices
$\boldsymbol{\alpha}_{\mathrm{B}}^{i}=\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}$
and $\boldsymbol{\beta}_{\mathrm{B}}=\boldsymbol{\gamma}_{\mathrm{B}}^{0}$,
the associated vector
$\boldsymbol{\alpha}_{\mathrm{B}}=(\boldsymbol{\alpha}_{\mathrm{B}}^{x},\boldsymbol{\alpha}_{\mathrm{B}}^{y},\boldsymbol{\alpha}_{\mathrm{B}}^{z})$,
and the momentum operator $\hat{\mathbf{p}}=-i\hbar\nabla$, the spinorial
photon equation in Eq. (2), corresponding to Maxwell’s equations in Eqs.
(33)–(36), can be written as

$c\boldsymbol{\alpha}_{\mathrm{B}}\cdot\hat{\mathbf{p}}\Psi-i\hbar
c\boldsymbol{\beta}_{\mathrm{B}}\Phi=i\hbar\frac{\partial\Psi}{\partial t}.$
(67)

Here the time derivative has been separated to the right-hand side. For
nonzero charge-current spinors $\Phi$, the spinorial photon equation in Eq.
(67) is not of the conventional form of a time-dependent wave equation of a
particle. However, for external field components or for the total field in the
absence of charges and currents, we can write the spinorial photon equation in
Eq. (67) as an explicit time-dependent wave equation in the conventional form,
given by

$\hat{H}\Psi=i\hbar\frac{\partial\Psi}{\partial t}.$ (68)

Here the photon field Hamiltonian operator $\hat{H}$ is simply obtained from
the first term of Eq. (67) as

$\hat{H}=c\boldsymbol{\alpha}_{\mathrm{B}}\cdot\hat{\mathbf{p}}.$ (69)

The form of the spinorial photon equation in Eq. (68) corresponds to a photon
propagating in vacuum, and it is analogous to the corresponding equation for
_free_ particles in the Dirac theory Dirac1958 . However, note that, in the
present theory, the square of the photon spinor describes the electromagnetic
energy density expectation value, while the square of the Dirac spinor
describes the probability density of the Dirac fermions. For narrow-frequency-
band spinorial photon wavepacket states, the probability density can be
defined as discussed in Sec. V.3.

###  IV.2 Scalar and three-dimensional quantum operators

Next, we consider other relevant quantum mechanical operators and their
commutation relations. We start with the most trivial operator, which is the
photon number operator that is, by definition, an identity operator for
single-photon states as

$\hat{n}=1.$ (70)

The single-photon number operator naturally commutes with all other operators
defined for single-photon states.

Then, we write the conventional energy and momentum operators corresponding to
the pertinent classical quantities. The energy operator $\hat{E}$ and the
momentum operator $\hat{\mathbf{p}}$, also used in Eqs. (67)–(69) above, are
given by Messiah1961 ; Sakurai1967

$\hat{E}=i\hbar\frac{\partial}{\partial t},$ (71)
$\hat{\mathbf{p}}=-i\hbar\nabla.$ (72)

With space and time coordinates, the energy and momentum operators satisfy the
well-known commutation relations $[r_{i},\hat{p}_{j}]=i\hbar\delta_{ij}$,
$[t,\hat{E}]=-i\hbar$, $[r_{i},\hat{E}]=0$, and $[t,\hat{p}_{i}]=0$. The
energy operator $\hat{E}$ and the momentum operator components $\hat{p}_{i}$
commute with each other and with the Hamiltonian operator as
$[\hat{E},\hat{p}_{i}]=0$, $[\hat{p}_{i},\hat{p}_{j}]=0$,
$[\hat{E},\hat{H}]=\mathbf{0}$ and $[\hat{p}_{i},\hat{H}]=\mathbf{0}$. The
commutation relation between the position and the Hamiltonian operator is
given by $[\mathbf{r},\hat{H}]=i\hbar c\boldsymbol{\alpha}_{\mathrm{B}}$.

The angular momentum is the rotational analog of linear momentum. As
conventional in quantum mechanics, the total angular momentum operator
$\hat{\mathbf{J}}$ is defined as a sum of the orbital angular momentum
operator $\hat{\mathbf{L}}$ and the spin angular momentum operator
$\hat{\mathbf{S}}$ as Messiah1961 ; Sakurai1967

$\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}.$ (73)

The orbital angular momentum operator is the quantum mechanical counterpart of
the classical angular momentum. It follows from the classical definition of
the angular momentum as a cross product between the position and momentum
vectors of a particle by replacing the classical momentum with the momentum
operator in Eq. (72) Messiah1961 ; Sakurai1967 . With an additional
multiplication by an identity matrix, the components of the orbital angular
momentum operator are matrices, given by

$\hat{\mathbf{L}}_{i}=-i\hbar\mathbf{I}_{8}(\mathbf{r}\times\nabla)_{i},$ (74)

For light beams, the orbital angular momentum is known to be associated with
helical wavefronts following from a phase proportional to the azimuthal angle
Allen1992b ; Piccirillo2013 ; Andrews2013 ; Barnett2016 ; Ji2020 ; Devlin2017
.

The spin angular momentum operator has in general no classical counterpart. It
is a purely quantum-mechanical property related to elementary particles. The
spin angular momentum operator depends on the type of the elementary particle
as it operates in the internal vector space associated with the elementary
particle. In the present formulation of spinorial photon states, the
components of the spin angular momentum operator are matrices, given by

$\hat{\mathbf{S}}_{i}=\left[\begin{array}[]{cc}i\hbar\mathbf{K}_{\mathrm{rot}}^{i}&\mathbf{0}\\\
\mathbf{0}&i\hbar\mathbf{K}_{\mathrm{rot}}^{i}\end{array}\right].$ (75)

Even though the spin has in general no classical counterpart, for the
electromagnetic field, the spin is known to be related to the polarization of
light Allen1992b ; Andrews2013 ; Mechelen2018 .

The orbital and spin angular momentum operators satisfy the conventional
commutation relations, given by
$[\hat{\mathbf{L}}_{i},\hat{\mathbf{L}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\mathbf{L}}_{k}$,
$[\hat{\mathbf{S}}_{i},\hat{\mathbf{S}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\mathbf{S}}_{k}$,
and $[\hat{\mathbf{L}}_{i},\hat{\mathbf{S}}_{j}]=\mathbf{0}$. For the energy
and momentum operators, the commutation relations are given by
$[\hat{\mathbf{L}}_{i},\hat{E}]=\mathbf{0}$,
$[\hat{\mathbf{S}}_{i},\hat{E}]=\mathbf{0}$,
$[\hat{\mathbf{L}}_{i},\hat{p}_{j}]=i\hbar\mathbf{I}_{8}\varepsilon_{ijk}\hat{p}_{k}$,
and $[\hat{\mathbf{S}}_{i},\hat{p}_{j}]=\mathbf{0}$. In the same way as in the
Dirac theory Bransden2000 , the spin and orbital angular momentum operators do
not commute with the Hamiltonian operator. Their commutation relations with
the Hamiltonian operator in Eq. (69) are given by
$[\hat{\mathbf{L}},\hat{H}]=i\hbar
c\boldsymbol{\alpha}_{\mathrm{B}}\times\hat{\mathbf{p}}$ and
$[\hat{\mathbf{S}},\hat{H}]=-i\hbar
c\boldsymbol{\alpha}_{\mathrm{B}}\times\hat{\mathbf{p}}$. Thus, the total
angular momentum operator $\hat{\mathbf{J}}$ in Eq. (73) commutes with the
Hamiltonian operator as $[\hat{\mathbf{J}},\hat{H}]=\mathbf{0}$. The total
angular momentum operator satisfies the same commutation relations as the
orbital and spin angular momentum operators, given by
$[\hat{\mathbf{J}}_{i},\hat{\mathbf{J}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\mathbf{J}}_{k}$.

By their definition, it is seen that the spin and orbital angular momentum
operators commute with each other, but neither of them commutes with the
Hamiltonian operator. This implies that they cannot be independently constants
of motion simultaneously with energy. Unlike in the case of the Dirac
electron-positron field in an external electromagnetic field Landau1982 , it
is not possible to define a separate interaction energy for the coupling of
the spin and orbital angular momenta of light. The interaction of magnetic
momenta related to orbital and spin angular momenta of an electron is best
understood in the nonrelativistic limit, which cannot be taken for the
electromagnetic field.

In this section, we rely on the definition of the spin and orbital angular
momentum operators in the conventional first-quantized quantum mechanics of
particles. Thus, our approach is analogous to the Dirac theory before the
quantization of the field resulting in the many-particle picture. In previous
literature, there are several classical and quantum field formulations of the
spin and orbital angular momenta of light in terms of the electric, magnetic
and vector-potential fields and the corresponding operators vanEnk1994a ;
vanEnk1994b ; Yang2022 ; Andrews2013 ; Barnett2016 ; Piccirillo2013 . These
approaches are not directly comparable to the first-quantized operators of
this section since they are based on the many-photon picture obtained in the
second quantization. Such a picture is obtained also in the present theory
after the quantization of the field as presented in Sec. VIII. Detailed
comparison of the present theory with the extensive previous literature on the
spin and orbital angular momenta of light is a topic of a separate work.

Related to the energy and momentum operators, it is convenient to define the
orbital-boost-momentum operator $\hat{\mathbf{N}}$ Bliokh2013b ; Bliokh2018a ;
Smirnova2018 ; Partanen2019a ; Partanen2021b . This operator is needed in the
determination of the four-dimensional orbital angular momentum operator as
presented in the next section. The components of $\hat{\mathbf{N}}$ are
defined as matrix operators, given by

$\hat{\mathbf{N}}_{i}=\mathbf{I}_{8}\Big{(}\frac{\hat{E}}{c^{2}}r_{i}+\hat{p}_{i}t\Big{)}.$
(76)

From the commutation relations of the energy and momentum operators, it
follows that the orbital-boost-momentum operator satisfies the commutation
relations $[\hat{\mathbf{N}}_{i},\hat{E}]=-i\hbar\mathbf{I}_{8}\hat{p}_{i}$,
$[\hat{\mathbf{N}}_{i},\hat{p}_{j}]=i\hbar\mathbf{I}_{8}\delta_{ij}\hat{E}/c^{2}$,
$[\hat{\mathbf{N}}_{i},\hat{\mathbf{N}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\mathbf{L}}_{k}/c^{2}$,
$[\hat{\mathbf{N}}_{i},\hat{\mathbf{L}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\mathbf{N}}_{k}$,
and
$[\hat{\mathbf{N}},\hat{H}]=i\hbar\boldsymbol{\alpha}_{\mathrm{B}}\hat{E}/c$.

For use in the determination of the four-dimensional spin angular momentum
operator in the next section, we define the spin-boost-momentum operator
$\hat{\boldsymbol{\mathcal{N}}}$. The components of
$\hat{\boldsymbol{\mathcal{N}}}$ are defined as matrix operators, given by

$\hat{\boldsymbol{\mathcal{N}}}_{i}=\boldsymbol{\gamma}_{\mathrm{B}}^{5}\hat{\mathbf{S}}_{i}/c.$
(77)

It follows that the spin-boost-momentum operator satisfies the commutation
relations
$[\hat{\boldsymbol{\mathcal{N}}}_{i},\hat{\boldsymbol{\mathcal{N}}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\mathbf{S}}_{k}/c^{2}$,
$[\hat{\boldsymbol{\mathcal{N}}}_{i},\hat{\mathbf{S}}_{j}]=i\hbar\varepsilon_{ijk}\hat{\boldsymbol{\mathcal{N}}}_{k}$,
and
$[\hat{\boldsymbol{\mathcal{N}}},\hat{H}]=-i\hbar\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\alpha}_{\mathrm{B}}\times\hat{\mathbf{p}}$.
These relations are analogous to the commutation relations of the orbital-
boost-momentum operator above.

The helicity operator is defined as the projection of the spin angular
momentum operator along the direction of the momentum, and it is written as
Landau1982 ; Sakurai1967 ; Messiah1961 ; Alpeggiani2018 ; Peskin2018

$\hat{\mathfrak{h}}=\frac{\hat{\mathbf{S}}\cdot\hat{\mathbf{p}}}{|\mathbf{p}|}.$
(78)

By studying the action of the helicity operator in Eq. (78) on the spinorial
photon states, it is found that the helicity operator divided by $\hbar$ swaps
the upper and lower components of the photon states. This corresponds to the
action of the chirality operator $\boldsymbol{\gamma}_{\mathrm{B}}^{5}$
discussed in Sec. II.5. The correspondence
$\hat{\mathfrak{h}}/\hbar\leftrightarrow\boldsymbol{\gamma}_{\mathrm{B}}^{5}$
for spinorial photon states is discussed further in Appendix A. The helicity
operator commutes with the Hamiltonian operator as
$[\hat{\mathfrak{h}},\hat{H}]=\mathbf{0}$.

###  IV.3 Four-dimensional quantum operators

The energy and momentum operators in Eqs. (71) and (72) are components of the
four-momentum operator $\hat{P}^{\mu}$ whose contravariant form is given by

$\hat{P}^{\mu}=i\hbar\partial^{\mu}=i\hbar\eta^{\mu\nu}\partial_{\nu}=(\hat{E}/c,\hat{\mathbf{p}}).$
(79)

The four-vector corresponding to the four-momentum operator in Eq. (79) can be
transformed between inertial frames using the conventional Lorentz
transformation of four-vectors discussed in Sec. III.1

The components of the angular momentum do not form a four-vector Fayngold2008
; Misner1973 . Therefore, it is useful to form a tensor operator, through
which the components of the angular momentum can be transformed between
inertial frames by using the Lorentz transformation of this tensor.
Consequently, the relativistic four-dimensional orbital angular momentum is
_defined_ as a tensor operator $\hat{L}^{\mu\nu}$ that is different from a
three-vector matrix operator $\hat{\mathbf{L}}$, whose components are given in
Eq. (74). This second-order antisymmetric tensor operator, called the orbital
angular momentum tensor operator, also includes components of the boost-
momentum operator $\hat{\mathbf{N}}$ in Eq. (76), and, in analogy to the
relativistic structure of the corresponding classical quantity Fayngold2008 ,
it is given by

$\hat{L}^{\mu\nu}=\left[\begin{array}[]{cccc}\mathbf{0}&-c\hat{\mathbf{N}}_{x}&-c\hat{\mathbf{N}}_{y}&-c\hat{\mathbf{N}}_{z}\\\
c\hat{\mathbf{N}}_{x}&\mathbf{0}&\hat{\mathbf{L}}_{z}&-\hat{\mathbf{L}}_{y}\\\
c\hat{\mathbf{N}}_{y}&-\hat{\mathbf{L}}_{z}&\mathbf{0}&\hat{\mathbf{L}}_{x}\\\
c\hat{\mathbf{N}}_{z}&\hat{\mathbf{L}}_{y}&-\hat{\mathbf{L}}_{x}&\mathbf{0}\end{array}\right].$
(80)

The corresponding expressions for the components of $\hat{L}^{\mu\nu}$ are
given by $\hat{L}^{i0}=c\hat{\mathbf{N}}_{i}$,
$\hat{L}^{0i}=-c\hat{\mathbf{N}}_{i}$, and
$\hat{L}^{ij}=\varepsilon_{ijk}\hat{\mathbf{L}}_{k}$.

Similarly, the relativistic four-dimensional spin angular momentum is
_defined_ as a tensor operator $\hat{S}^{\mu\nu}$, which is different from a
three-vector matrix operator $\hat{\mathbf{S}}$, whose components are given in
Eq. (75). This second-order antisymmetric tensor operator is made of
generators $\Sigma_{\mathrm{S}}^{\mu\nu}$ of the Lorentz transformation on
photon spinors, given in Eq. (63). This is analogous to the case of the spin
angular momentum in the Dirac theory Peskin2018 . Consequently, in the
covariant formulation, $\hat{\mathbf{S}}$ is replaced by a four-dimensional
spin operator, given by

$\displaystyle\hat{S}^{\mu\nu}$
$\displaystyle=i\hbar\left[\begin{array}[]{cccc}\Sigma_{\mathrm{S}}^{00}&i\Sigma_{\mathrm{S}}^{0x}&i\Sigma_{\mathrm{S}}^{0y}&i\Sigma_{\mathrm{S}}^{0z}\\\
i\Sigma_{\mathrm{S}}^{x0}&\Sigma_{\mathrm{S}}^{xx}&\Sigma_{\mathrm{S}}^{xy}&\Sigma_{\mathrm{S}}^{xz}\\\
i\Sigma_{\mathrm{S}}^{y0}&\Sigma_{\mathrm{S}}^{yx}&\Sigma_{\mathrm{S}}^{yy}&\Sigma_{\mathrm{S}}^{yz}\\\
i\Sigma_{\mathrm{S}}^{z0}&\Sigma_{\mathrm{S}}^{zx}&\Sigma_{\mathrm{S}}^{zy}&\Sigma_{\mathrm{S}}^{zz}\end{array}\right]$
(85)
$\displaystyle=\left[\begin{array}[]{cccc}\mathbf{0}&-c\hat{\boldsymbol{\mathcal{N}}}_{x}&-c\hat{\boldsymbol{\mathcal{N}}}_{y}&-c\hat{\boldsymbol{\mathcal{N}}}_{z}\\\
c\hat{\boldsymbol{\mathcal{N}}}_{x}&\mathbf{0}&\hat{\mathbf{S}}_{z}&-\hat{\mathbf{S}}_{y}\\\
c\hat{\boldsymbol{\mathcal{N}}}_{y}&-\hat{\mathbf{S}}_{z}&\mathbf{0}&\hat{\mathbf{S}}_{x}\\\
c\hat{\boldsymbol{\mathcal{N}}}_{z}&\hat{\mathbf{S}}_{y}&-\hat{\mathbf{S}}_{x}&\mathbf{0}\end{array}\right].$
(90)

The operator $\hat{S}^{\mu\nu}$ is antisymmetric. All the component matrices
can be expressed in terms of the components $\hat{\mathbf{S}}_{i}$ of the
three-vector spin operator as
$\hat{S}^{i0}=c\hat{\boldsymbol{\mathcal{N}}}_{i}$,
$\hat{S}^{0i}=-c\hat{\boldsymbol{\mathcal{N}}}_{i}$, and
$\hat{S}^{ij}=\varepsilon_{ijk}\hat{\mathbf{S}}_{k}$. These relations are of
the same form as the corresponding relations in the Dirac theory Peskin2018 .
The appearance of the spin-boost-momentum operators in the definition of the
four-tensor spin operator in Eq. (90), in analogy to the Dirac theory,
strongly indicates that the present spinorial formulation of the theory is
necessary for the consistent determination of the relativistic spin structure
of light.

The four-dimensional total angular momentum operator is a sum of the four-
tensor orbital angular momentum operator in Eq. (80) and the four-tensor spin
operator in Eq. (90) as

$\hat{J}^{\mu\nu}=\hat{L}^{\mu\nu}+\hat{S}^{\mu\nu}.$ (91)

This relation is again analogous to the corresponding relation in the Dirac
theory Peskin2018 .

###  IV.4 Density operators

For systematic calculation of density expectation values of quantum operators,
we present an arbitrary first-quantized density operator
$\hat{\rho}_{\hat{O}}$ corresponding to the pertinent first-quantized operator
$\hat{O}$ as

$\hat{\rho}_{\hat{O}}(\mathbf{r},\mathbf{r}^{\prime})=\frac{1}{2}\\{\hat{O}(\mathbf{r}^{\prime}),\delta(\mathbf{r}-\mathbf{r}^{\prime})\\}.$
(92)

The argument $\mathbf{r}$ indicates the position at which the density is
calculated and $\mathbf{r}^{\prime}$ is the wavefunction coordinate. The
anticommutator is necessary in Eq. (92) for obtaining real-valued density
expectation values in the case of operators that do not commute with the
position operator. Equation (92) is justified by the physically meaningful
results obtained for the density expectation values in Sec. V.2 and for the
second-quantized density operators in Sec. VIII.2.

As an example of the use of the first-quantized density operator in Eq. (92),
we study the number density. Substituting the trivial single-photon number
operator from Eq. (70) into Eq. (92) gives the single-photon number density
operator as
$\hat{\rho}_{\hat{n}}(\mathbf{r},\mathbf{r}^{\prime})=\delta(\mathbf{r}-\mathbf{r}^{\prime})$
Fetter1971 . Thus, for the first-quantized number-density expectation value,
we obtain

$\displaystyle\rho_{\hat{n}}^{\mathrm{(M)}}(\mathbf{r})$
$\displaystyle=\frac{\int\Psi^{\dagger}(\mathbf{r}^{\prime})\hat{\rho}_{\hat{n}}(\mathbf{r},\mathbf{r}^{\prime})\Psi(\mathbf{r}^{\prime})d^{3}r^{\prime}}{\int\Psi^{\dagger}(\mathbf{r}^{\prime})\Psi(\mathbf{r}^{\prime})d^{3}r^{\prime}}$
$\displaystyle=\frac{\Psi^{\dagger}(\mathbf{r})\Psi(\mathbf{r})}{\int\Psi^{\dagger}(\mathbf{r}^{\prime})\Psi(\mathbf{r}^{\prime})d^{3}r^{\prime}}.$
(93)

The superscript $\mathrm{(M)}$ is used to indicate that the expectation value
density is calculated for the Maxwell photon spinor field. Equation (93) shows
that the number density of first-quantized photon states is simply equal to
the squared norm of the spinorial photon wavefunction normalized by the volume
integral of the same quantity. The state must be normalizable for Eq. (93) to
be well-defined. Equation (93) is in accordance with Born’s probability
interpretation of the wavefunction Liboff1980 . Since the photon number of
single-photon states is fixed to one, according to the _uncertainty principle_
of the photon number and the phase Smithey1993 , there is complete lack of the
knowledge of the _phase of the photon state_. For the photon number density
and other density expectation values in the second-quantization picture, see
Sec. VIII.

##  V Spinorial photon eigenstates

In this section, we will present selected well-known special cases of photon
eigenstates in the photon spinor formulation. To be able to define the second-
quantized electromagnetic field, we need a complete set of spinorial photon
wavefunctions. Eigenvalue equations of quantum operators, presented below,
cannot be satisfied for a spinor in Eq. (1) if it is made of classical real-
valued fields. Therefore, the eigenvalue equations of quantum operators
require the determination of complex-valued spinorial wavefunctions of
photons. The real-valued physical quantities are recovered in this formalism
as expectation values.

Complete sets of complex-valued spinorial photon wavefunctions are
conveniently given in the plane-wave and spherical state bases. Their
mathematical forms are presented in Appendix A. These sets describe a
transverse electromagnetic radiation field. In contrast to the conventional
vector-potential eigenstates, the spinorial photon states remain transverse in
the Lorentz transformation defined in Sec. III. In other words, these sets are
closed under the action of the Lorentz group. Therefore, the spinorial photon
states avoid the gauge dependence problem of the conventional vector-potential
eigenstates. The radiation gauge is not Lorentz invariant, so each inertial
coordinate system requires its own gauge condition.

###  V.1 Eigenvalues of physical observables in the plane-wave and spherical
state bases

When the field is an eigenstate of the energy, momentum, and helicity
operators, we obtain the following set of eigenvalue equations of commuting
observables, written as

$\left\\{\begin{array}[]{l}\hat{\mathbf{S}}^{2}\Psi_{\mathbf{k},q}=\hbar^{2}S(S+1)\Psi_{\mathbf{k},q},\\\
\hat{H}\Psi_{\mathbf{k},q}=\hat{E}\Psi_{\mathbf{k},q}=\hbar\omega_{\mathbf{k}}\Psi_{\mathbf{k},q},\\\
\hat{\mathbf{p}}\Psi_{\mathbf{k},q}=\hbar\mathbf{k}\Psi_{\mathbf{k},q},\\\
\hat{\mathfrak{h}}\Psi_{\mathbf{k},q}=\hbar
q\Psi_{\mathbf{k},q}.\end{array}\right.$ (94)

Here $S=1$ is the spin quantum number, $\mathbf{k}$ is the wave vector,
$\omega_{\mathbf{k}}=c|\mathbf{k}|$, and $q=\pm 1$ is the helicity quantum
number. We call the spinorial photon states $\Psi_{\mathbf{k},q}$ plane-wave
states. One can produce identical photons corresponding to plane-wave states
given in Eq. (94) and measure any of the quantum numbers
$\omega_{\mathbf{k}}$, $\mathbf{k}$, and $q$. In repeated measurements, one
then always obtains the same results. Thus, these quantum numbers refer to
physical quantities, which are constants of motion.

As presented in Appendix A, the spinor $\Psi_{\mathbf{k},q}$ is formed from
the complex-valued normalized plane-wave electric and magnetic field
amplitudes $\boldsymbol{\mathcal{E}}_{\mathbf{k},q}$ and
$\boldsymbol{\mathcal{B}}_{\mathbf{k},q}$ Loudon2000  corresponding to Eq. (1)
as
$\Psi_{\mathbf{k},q}=[0,\boldsymbol{\mathcal{E}}_{\mathbf{k},q},0,i\boldsymbol{\mathcal{B}}_{\mathbf{k},q}]^{T}$.
The operator
$\hat{\mathbf{S}}^{2}=\hat{\mathbf{S}}_{x}^{2}+\hat{\mathbf{S}}_{y}^{2}+\hat{\mathbf{S}}_{z}^{2}$,
formed using the operators for the components of the spin in Eq. (75), is a
diagonal matrix whose first and fifth diagonal elements are zero and its
eigenvalue corresponds to $S=1$. As briefly mentioned in the context of the
definition of the photon spinor in Eq. (1), this corresponds to the property
of all photon states to be eigenstates of $\hat{\mathbf{S}}^{2}$ with the
first and fifth spinor components being zero. Together with this condition,
from the spinorial photon equation in Eq. (68), it follows that the field must
be transverse and the helicity eigenvalues correspond to the helicity quantum
number $q=1$ for right-handed circular polarization and $q=-1$ for the left-
handed circular polarization.

Correspondingly, when the field is an eigenstate of the energy operator, and
the square and $z$ component of the total angular momentum operator, we obtain

$\left\\{\begin{array}[]{l}\hat{\mathbf{S}}^{2}\Psi_{\omega,J,M}=\hbar^{2}S(S+1)\Psi_{\omega,J,M},\\\
\hat{H}\Psi_{\omega,J,M}=\hat{E}\Psi_{\omega,J,M}=\hbar\omega\Psi_{\omega,J,M},\\\
\hat{\mathbf{J}}^{2}\Psi_{\omega,J,M}=\hbar^{2}J(J+1)\Psi_{\omega,J,M},\\\
\hat{\mathbf{J}}_{z}\Psi_{\omega,J,M}=\hbar
M\Psi_{\omega,J,M}.\end{array}\right.$ (95)

We call these photon states spherical states. As presented in Appendix A,
using the definition of the photon spinor $\Psi_{\omega,J,M}$ in Eq. (1), we
can write $\Psi_{\omega,J,M}$ using the well-known complex-valued spherical
photon electric and magnetic field amplitudes
$\boldsymbol{\mathcal{E}}_{\omega,J,M}$ and
$\boldsymbol{\mathcal{B}}_{\omega,J,M}$ Landau1982  as
$\Psi_{\omega,J,M}=[0,\boldsymbol{\mathcal{E}}_{\omega,J,M},0,i\boldsymbol{\mathcal{B}}_{\omega,J,M}]^{T}$.

Both sets of quantum numbers, $(\mathbf{k},q)$ and $(\omega,J,M)$, considered
above, unambiguously specify the quantum state of the photon. Accordingly,
both
$(\hat{\mathbf{S}}^{2},\hat{H},\hat{E},\hat{\mathbf{p}},\hat{\mathfrak{h}})$
and
$(\hat{\mathbf{S}}^{2},\hat{H},\hat{E},\hat{\mathbf{J}}^{2},\hat{\mathbf{J}}_{z})$
are complete sets of mutually commuting operators Sakurai1994 . Furthermore,
the wavefunctions corresponding to the latter set can be split into two
linearly independent parts associated with the electric and magnetic
transitions of atoms. This will be discussed in Appendix A.

###  V.2 Density expectation values

Next, we consider the density expectation values for eigenstates of the
Hamiltonian operator, which are complex-valued spinors as discussed above. We
assume that the state is normalizable as $\frac{1}{2}\int\Psi^{\dagger}\Psi
d^{3}r=\hbar\omega$, where $\omega$ is the angular frequency of the field.
This normalization condition effectively assumes a narrow-frequency-band
wavepacket state since the single-photon eigenstates of the Hamiltonian
operator do not vanish sufficiently rapidly at infinity and are, thus, only
normalizable with respect to a delta function as presented in Appendix A.
Without this effective narrow-frequency-band wavepacket approximation, the
probability density cannot be defined for photons using a wavefunction
picture. This is discussed in previous literature Sipe1995 ; Landau1982 ;
Newton1949 ; BialynickiBirula1996 ; Hawton2019 ; Hawton1999 ; SmithB2007 .

Within the narrow-frequency-band wavepacket approximation, the photon spinor
eigenstates, expressed in terms of the _general_ complex-valued electric and
magnetic field amplitudes using Eq. (1), can be applied to obtain the
following density expectation values for selected key quantum operators.
Omitting the function arguments for brevity, we obtain

$\displaystyle\rho_{\hat{n}}^{\mathrm{(M)}}$
$\displaystyle=\frac{\int\Psi^{\dagger}\hat{\rho}_{\hat{n}}\Psi
d^{3}r^{\prime}}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{\Psi^{\dagger}\Psi}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{\Psi^{\dagger}\Psi}{2\hbar\omega}=\frac{\rho_{\hat{H}}^{\mathrm{(M)}}}{\hbar\omega}$
$\displaystyle=\frac{1}{4\hbar\omega}\Big{(}\varepsilon_{0}|\mathbf{E}|^{2}+\frac{1}{\mu_{0}}|\mathbf{B}|^{2}\Big{)},$
(96) $\displaystyle\rho_{\hat{H}}^{\mathrm{(M)}}$
$\displaystyle=\frac{\int\Psi^{\dagger}\hat{\rho}_{\hat{H}}\Psi
d^{3}r^{\prime}}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{\Psi^{\dagger}\hat{H}\Psi}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{\Psi^{\dagger}\Psi}{2}=\hbar\omega\rho_{\hat{n}}^{\mathrm{(M)}}$
$\displaystyle=\frac{1}{4}\Big{(}\varepsilon_{0}|\mathbf{E}|^{2}+\frac{1}{\mu_{0}}|\mathbf{B}|^{2}\Big{)},$
(97) $\displaystyle\boldsymbol{\rho}_{\hat{\mathbf{p}}}^{\mathrm{(M)}}$
$\displaystyle=\frac{\int\Psi^{\dagger}\hat{\boldsymbol{\rho}}_{\hat{\mathbf{p}}}\Psi
d^{3}r^{\prime}}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{1}{2}\frac{\Psi^{\dagger}\hat{\mathbf{p}}\Psi-(\hat{\mathbf{p}}\Psi^{\dagger})\Psi}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}$
$\displaystyle=\frac{1}{4\omega}\mathrm{Im}\Big{[}\varepsilon_{0}\mathbf{E}^{*}\cdot(\nabla)\mathbf{E}+\frac{1}{\mu_{0}}\mathbf{B}^{*}\cdot(\nabla)\mathbf{B}\Big{]},$
(98) $\displaystyle\boldsymbol{\rho}_{\hat{\mathbf{L}}}^{\mathrm{(M)}}$
$\displaystyle=\frac{\int\Psi^{\dagger}\hat{\boldsymbol{\rho}}_{\hat{\mathbf{L}}}\Psi
d^{3}r^{\prime}}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{1}{2}\frac{\Psi^{\dagger}\hat{\mathbf{L}}\Psi-(\hat{\mathbf{L}}\Psi^{\dagger})\Psi}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\mathbf{r}\times\boldsymbol{\rho}_{\hat{\mathbf{p}}}^{\mathrm{(M)}}$
$\displaystyle=\frac{1}{4\omega}\mathbf{r}\times\mathrm{Im}\Big{[}\varepsilon_{0}\mathbf{E}^{*}\cdot(\nabla)\mathbf{E}+\frac{1}{\mu_{0}}\mathbf{B}^{*}\cdot(\nabla)\mathbf{B}\Big{]},$
(99) $\displaystyle\boldsymbol{\rho}_{\hat{\mathbf{S}}}^{\mathrm{(M)}}$
$\displaystyle=\frac{\int\Psi^{\dagger}\hat{\boldsymbol{\rho}}_{\hat{\mathbf{S}}}\Psi
d^{3}r^{\prime}}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{\Psi^{\dagger}\hat{\mathbf{S}}\Psi}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}$
$\displaystyle=\frac{1}{4\omega}\mathrm{Im}\Big{[}\varepsilon_{0}\mathbf{E}^{*}\times\mathbf{E}+\frac{1}{\mu_{0}}\mathbf{B}^{*}\times\mathbf{B}\Big{]},$
(100) $\displaystyle\rho_{\hat{\mathfrak{h}}}^{\mathrm{(M)}}$
$\displaystyle=\frac{\int\Psi^{\dagger}\hat{\rho}_{\hat{\mathfrak{h}}}\Psi
d^{3}r^{\prime}}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}=\frac{\Psi^{\dagger}\hat{\mathfrak{h}}\Psi}{\int\Psi^{\dagger}\Psi
d^{3}r^{\prime}}$
$\displaystyle=\frac{c}{4\omega^{2}}\mathrm{Re}\Big{[}\varepsilon_{0}\mathbf{E}^{*}\cdot(\nabla\times\mathbf{E})+\frac{1}{\mu_{0}}\mathbf{B}^{*}\cdot(\nabla\times\mathbf{B})\Big{]}$
$\displaystyle=\frac{\varepsilon_{0}c}{2\omega}\mathrm{Im}(\mathbf{B}^{*}\cdot\mathbf{E}).$
(101)

Boldface is used for the density operators and density expectation values of
vector quantities. All density expectation values in Eqs. (96)–(101) are
independent of time since they are given for an eigenstate of the energy
operator. If the energy eigenstate is also an eigenstate of the momentum
operator, the density expectation values are additionally independent of the
position. The results in Eqs. (96)–(101) are seen to be consistent with
previous literature Bliokh2015a ; Bliokh2017b , as expected.

When comparing the energy density expectation value in Eq. (97) to the well-
known energy density of classical field, one must account for the complete
lack of knowledge of the phase of single-photon states. In Eq. (97), the phase
information is lost in the squares of the absolute values of the complex-
valued fields. Thus, the density expectation value of energy in Eq. (97)
corresponds to the expression of the energy density of a classical time-
harmonic field averaged over the harmonic cycle as discussed in Sec. II.1. The
lack of phase information applies similarly to the other density expectation
values in Eqs. (96)–(101).

###  V.3 Photon probability current four-vector

Next, we consider the photon probability current four-vector
$\mathcal{I}_{\mathrm{M}}^{\nu}=(\rho_{\hat{n}}^{\mathrm{(M)}},\mathbf{j}_{\mathrm{M}}/c)$
for eigenstates of the Hamiltonian operator. Again, we effectively assume the
normalizability of a narrow-frequency-band wavepacket state, given by
$\frac{1}{2}\int\Psi^{\dagger}\Psi d^{3}r=\hbar\omega$. The quantity
$\mathbf{j}_{\mathrm{M}}$ is the conventional probability current three-
vector. The time component $\rho_{\hat{n}}^{\mathrm{(M)}}$ of the photon
probability current four-vector is equal to the photon probability density in
Eq. (96) as

$\rho_{\hat{n}}^{\mathrm{(M)}}=\frac{\Psi^{\dagger}\Psi}{2\hbar\omega}=\frac{1}{4\hbar\omega}\Big{(}\varepsilon_{0}|\mathbf{E}|^{2}+\frac{|\mathbf{B}|^{2}}{\mu_{0}}\Big{)}.$
(102)

The fundamental relation between the photon probability density
$\rho_{\hat{n}}^{\mathrm{(M)}}$ and the probability current
$\mathbf{j}_{\mathrm{M}}$ is given by

$\frac{d}{dt}\int_{V}\rho_{\hat{n}}^{\mathrm{(M)}}d^{3}r=-\int_{V}\nabla\cdot\mathbf{j}_{\mathrm{M}}d^{3}r=-\int_{\partial
V}\mathbf{j}_{\mathrm{M}}\cdot d\mathbf{S}.$ (103)

This equation is required to apply to an arbitrary volume $V$ and its boundary
$\partial V$.

Next, we elaborate an expression for the photon probability current three-
vector $\mathbf{j}_{\mathrm{M}}$ in terms of the photon spinor. First, we
write the Hamiltonian operator in Eq. (69) as

$\hat{H}=c\boldsymbol{\alpha}_{\mathrm{B}}\cdot\hat{\mathbf{p}}=-i\hbar
c\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\partial_{i}.$
(104)

The dynamical equations for the photon spinor and its conjugate transpose are
then given by

$\hat{H}\Psi=i\hbar\frac{\partial\Psi}{\partial t}=-i\hbar
c\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\partial_{i}\Psi,$
(105) $\Psi^{\dagger}\hat{H}=i\hbar\frac{\partial\Psi^{\dagger}}{\partial
t}=-i\hbar
c\partial_{i}\Psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}.$
(106)

Using these equations, the partial time derivative of $\Psi^{\dagger}\Psi$ is
written as

$\displaystyle\frac{\partial}{\partial t}(\Psi^{\dagger}\Psi)$
$\displaystyle=\frac{\partial\Psi^{\dagger}}{\partial
t}\Psi+\Psi^{\dagger}\frac{\partial\Psi}{\partial t}$
$\displaystyle=-c(\partial_{i}\Psi^{\dagger})\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\Psi-c\Psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}(\partial_{i}\Psi)$
$\displaystyle=-\partial_{i}(c\Psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\Psi)$
$\displaystyle=-\partial_{i}(c\bar{\Psi}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\Psi).$
(107)

For eigenstates of the Hamiltonian operator, the photon energy $\hbar\omega$
is constant. Therefore, we can place $2\hbar\omega$ as the numerator inside
the derivatives of Eq. (107) so that the left-hand side corresponds to the
partial time derivative of the photon probability density in Eq. (102). Thus,
we obtain

$\frac{\partial\rho_{\hat{n}}^{\mathrm{(M)}}}{\partial
t}=\frac{\partial}{\partial
t}\Big{(}\frac{\Psi^{\dagger}\Psi}{2\hbar\omega}\Big{)}=-\partial_{i}\Big{(}\frac{c\bar{\Psi}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\Psi}{2\hbar\omega}\Big{)}=-\nabla\cdot\mathbf{j}_{\mathrm{M}},$
(108)

where the components of the probability current three-vector
$\mathbf{j}_{\mathrm{M}}$ are, thus, given by

$j_{\mathrm{M}}^{i}=\frac{c\bar{\Psi}\boldsymbol{\gamma}_{\mathrm{B}}^{i}\Psi}{2\hbar\omega}.$
(109)

Furthermore, in terms of the electric and magnetic field amplitudes, we can
write the photon probability current three-vector as

$\mathbf{j}_{\mathrm{M}}=\frac{\epsilon_{0}c^{2}}{2\hbar\omega}\mathrm{Re}(\mathbf{E}^{*}\times\mathbf{B}).$
(110)

Thus, the spatial components of the probability current four-vector are equal
to the components of the Poynting vector of a classical time-harmonic field
averaged over the harmonic cycle and normalized by the photon energy.

Using Eqs. (102) and (109), the photon probability current four-vector for
single-photon states is then given by

$\mathcal{I}_{\mathrm{M}}^{\mu}=(\rho_{\hat{n}}^{\mathrm{(M)}},\mathbf{j}_{\mathrm{M}}/c)=\frac{\bar{\Psi}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\Psi}{2\hbar\omega}.$
(111)

The covariance of this representation can be verified for plane-wave states,
discussed in Sec. V, by checking that the Lorentz transformation of spinors
and the angular frequency gives the same result as the direct Lorentz
transformation of the probability current four-vector as
$\Lambda^{\mu}_{\;\nu}\mathcal{I}_{\mathrm{M}}^{\nu}=\frac{1}{2\hbar\omega^{\prime}}\overline{\boldsymbol{\Lambda}_{\mathrm{S}}\Psi}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\Lambda}_{\mathrm{S}}\Psi$,
where $\omega^{\prime}$ is the Doppler-shifted angular frequency. The Lorentz
transformations are discussed in more detail in Sec. III.

In the case of spherical photon states, discussed in Sec. V, it is important
to note that these states when transformed to another inertial frame are no
longer eigenstates of the energy operator in the transformed inertial frame.
This is because the field components propagating in different directions
experience different Doppler shifts. Thus, the angular frequency of a
spherical state is not a global state parameter in the transformed inertial
frame. Therefore, Eq. (111) cannot be applied to spherical states in
transformed inertial frames.

We find that the probability current three-vector of a single-photon state in
Eq. (110) is related to the momentum and spin density expectation values in
Eqs. (98) and (100) as

$\displaystyle\mathbf{j}_{\mathrm{M}}$
$\displaystyle=\frac{c^{2}}{\hbar\omega}\Big{(}\boldsymbol{\rho}_{\hat{\mathbf{p}}}^{\mathrm{(M)}}+\frac{1}{2}\nabla\times\boldsymbol{\rho}_{\hat{\mathbf{S}}}^{\mathrm{(M)}}\Big{)}$
$\displaystyle=\frac{c^{2}}{2\hbar\omega}\frac{\Psi^{\dagger}\hat{\mathbf{p}}\Psi-(\hat{\mathbf{p}}\Psi^{\dagger})\Psi+\nabla\times(\Psi^{\dagger}\hat{\mathbf{S}}\Psi)}{\int\Psi^{\dagger}\Psi
d^{3}r}.$ (112)

In nonrelativistic quantum mechanics of spinless particles with a rest mass
$m_{0}$, a formula equivalent to Eq. (112) is well known to give the
probability current of the particle. In this case, the photon energy
$\hbar\omega$ is replaced by the rest mass energy $m_{0}c^{2}$ and the spin
term is dropped out Shankar1994 ; Landau1977 . In the case of Dirac fermions,
the momentum operator is replaced by the corresponding components of the
gauge-covariant momentum operator containing a dependence on the
electromagnetic four-potential, and the coefficient of the spin term is
different by the electron spin g-factor Peleg1998 . The four-dimensional
generalization of the relation in Eq. (112) is given by

$\mathcal{I}_{\mathrm{M}}^{\mu}=\frac{c}{2\hbar\omega}\frac{\Psi^{\dagger}\hat{P}^{\mu}\Psi-(\hat{P}^{\mu}\Psi^{\dagger})\Psi+\partial_{\nu}(\Psi^{\dagger}\hat{S}^{\mu\nu}\Psi)}{\int\Psi^{\dagger}\Psi
d^{3}r}.$ (113)

Here $\hat{P}^{\mu}$ is the four-momentum operator in Eq. (79) and
$\hat{S}^{\mu\nu}$ is the four-tensor spin operator in Eq. (90). Equations
above strongly indicate that, assuming the narrow-frequency-band wavepacket
approximation, the spinorial photon wavefunction of the present work is
_equally consistent_ as the electron wavefunction of the Dirac theory. The
obvious difference comes from changing the number of particles in the system
since any single-photon state overlaps in energy scale with an infinite number
of many-photon states. No such overlapping takes place in the case of Dirac
fermions at low energies. For Dirac fermions, one must create a particle-
antiparticle pair, which has a large threshold energy of
$2m_{\mathrm{e}}c^{2}$, where $m_{\mathrm{e}}$ is the electron rest mass. The
eight-component spinor representation of the photon wavefunction and Eqs.
(111) and (113) enable discovering photon-particle form similarities that are
not equally obvious in the conventional four-potential-based formulation of
QED.

###  V.4 CPT symmetry

The CPT symmetry is the combined charge conjugation (C), parity (P), and time
reversal (T) symmetry that is observed to be an exact symmetry of the laws of
nature. It is formed by discrete Lorentz transformations Tully2011 ;
Schwartz2014 . Electromagnetic interaction is known to satisfy each of these
symmetries separately. Here we limit our study to the transformation
properties of the spinorial photon states. Another part of the CPT symmetry is
the transformation of the derivatives in the equations of motion, which we do
not consider here. We define the time reversal $\hat{T}$, parity $\hat{P}$,
and charge conjugation $\hat{C}$ transformations of spinorial photon states as

$\hat{T}\Psi(t,\mathbf{r})=\Psi(-t,\mathbf{r}),$ (114)
$\hat{P}\Psi(t,\mathbf{r})=\Psi(t,-\mathbf{r}),$ (115)
$\hat{C}\Psi(t,\mathbf{r})=\eta_{c}\mathbf{C}\bar{\Psi}^{T}(t,\mathbf{r}).$
(116)

Here $\mathbf{C}$ is the charge conjugation matrix and $\eta_{c}$ is a phase
factor. Since photons are their own antiparticles, the charge conjugation
matrix does not change the particle type when it operates on spinorial photon
states. Instead, for spinorial photon states, it is related to the inversion
of the angular momentum. Accordingly, for plane-wave photon states, the charge
conjugation matrix operates on the spinor components so that the helicity is
inverted as

$\hat{C}\Psi_{\mathbf{k},q}(t,\mathbf{r})=-\bar{\Psi}_{\mathbf{k},-q}^{T}(t,\mathbf{r}).$
(117)

For spherical photon states, the charge conjugation matrix inverts the angular
momentum $z$ component as

$\hat{C}\Psi_{\omega,J,M}^{(\eta)}(t,\mathbf{r})=(-1)^{J+M+\delta_{\eta,m}}\bar{\Psi}_{\omega,J,-M}^{(\eta)T}(t,\mathbf{r}).$
(118)

General spinorial photon states do not remain unchanged in each of the three
symmetry transformations in Eqs. (114)–(116) separately, but the combination
of the three symmetry transformations preserves spinorial photon states
unchanged as

$\hat{C}\hat{P}\hat{T}\Psi(t,\mathbf{r})=\Psi(t,\mathbf{r}).$ (119)

This means that the spinorial photon states satisfy the CPT symmetry. Equation
(119) is straightforward to verify by using the explicit expressions of the
plane-wave and spherical photon state spinors, given in Appendix A.

##  VI QED Lagrangian density and Euler-Lagrange equations

In this section, we investigate the derivation of the present spinorial photon
theory from the Lagrangian density. Thus, we follow the conventional approach
of deriving field theories. There are formal differences in our spinorial
representation compared to the conventional four-potential-based QED, but the
resulting theory is equivalent to the conventional QED as discussed in more
detail below.

###  VI.1 Lagrangian density of QED

Here we present the conventional Lagrangian density of the coupled system of
the electromagnetic field and the Dirac field. It will be given in terms of
the potential spinor $\Theta_{\Re}$ in Eq. (40) and the electron-positron
field $\psi$ since the interaction between the electromagnetic field and the
Dirac field is known in terms of the four-potential and the direct
representation in terms of the electric and magnetic fields is unknown.

Using our spinor notation, the derivation of the theory below differs from the
conventional four-potential-based derivation, even though the resulting
dynamical equations are equivalent. The conventional four-potential is a real-
valued gauge potential. Therefore, here we use the real-valued form of the
potential spinor in Eq. (40). The conventional QED Lagrangian density
$\mathcal{L}_{\mathrm{QED}}$ is, thus, written in the eight-spinor notation as

$\displaystyle\mathcal{L}_{\mathrm{QED}}$
$\displaystyle=\mathcal{L}_{\mathrm{QED,D}}+\mathcal{L}_{\mathrm{QED,M}}+\mathcal{L}_{\mathrm{QED,DM}}$
$\displaystyle=\frac{i\hbar
c}{2}\bar{\psi}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\vec{D}-\reflectbox{$\vec{\reflectbox{$D$}}$}\boldsymbol{\gamma}_{\mathrm{F}})\psi-
m_{\mathrm{e}}c^{2}\bar{\psi}\psi+\bar{\Psi}_{\Re}\Psi_{\Re},$
$\displaystyle\mathcal{L}_{\mathrm{QED,D}}$ $\displaystyle=\frac{i\hbar
c}{2}\bar{\psi}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\vec{\partial}-\reflectbox{$\vec{\reflectbox{$\partial$}}$}\boldsymbol{\gamma}_{\mathrm{F}})\psi-
m_{\mathrm{e}}c^{2}\bar{\psi}\psi$ $\displaystyle\mathcal{L}_{\mathrm{QED,M}}$
$\displaystyle=\bar{\Psi}_{\Re}\Psi_{\Re}=\bar{\Theta}_{\Re}\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\Theta_{\Re}=-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu},$
$\displaystyle\mathcal{L}_{\mathrm{QED,DM}}$
$\displaystyle=\bar{\Phi}_{\Re}\Theta_{\Re}+\bar{\Theta}_{\Re}\Phi_{\Re}=\frac{g_{\mathrm{e}}}{2}\bar{\psi}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})\psi$
$\displaystyle=-J_{\mathrm{e\Re}}^{\mu}A_{\Re\mu}.$ (120)

Here $\mathcal{L}_{\mathrm{QED,D}}$ is the Lagrangian density of the free
Dirac field and $\mathcal{L}_{\mathrm{QED,M}}$ is the Lagrangian density of
the free Maxwell electromagnetic field. The Lagrangian density
$\mathcal{L}_{\mathrm{QED,DM}}$ describes the minimal coupling interaction
between the Dirac and electromagnetic fields. The quantity
$g_{\mathrm{e}}=q_{\mathrm{e}}\sqrt{2/\varepsilon_{0}}$ in Eq. (120) is the
electric coupling constant of the present theory, where $q_{\mathrm{e}}=\pm e$
is the electric charge of the particle with $e$ being the elementary charge.
The Dirac field is normalized so that $\psi^{\dagger}\psi$ corresponds to the
number density. The quantity
$\bar{\psi}=\psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}$ in Eq. (120)
is the conventional Dirac adjoint. The vector arrows above symbols in Eq.
(120) indicate the direction in which the differential operators operate. We
have defined the eight-spinor electromagnetic-gauge-covariant derivative
operator $\vec{D}$ and its adjoint  $\vec{\reflectbox{$D$}}$ as

$\displaystyle\vec{D}$
$\displaystyle=\vec{\partial}-\frac{ig_{\mathrm{e}}}{\hbar
c}\Theta_{\Re}=[0,\vec{D}_{x},\vec{D}_{y},\vec{D}_{z},-\vec{D}_{0},0,0,0]^{T},$

$\vec{\reflectbox{$D$}}$

$\displaystyle=\reflectbox{$\vec{\reflectbox{$\partial$}}$}+\frac{ig_{\mathrm{e}}}{\hbar
c}\bar{\Theta}_{\Re}=[0,\reflectbox{$\vec{\reflectbox{$D$}}$}_{x},\reflectbox{$\vec{\reflectbox{$D$}}$}_{y},\reflectbox{$\vec{\reflectbox{$D$}}$}_{z},\reflectbox{$\vec{\reflectbox{$D$}}$}_{0},0,0,0].$
(121)

Here the well-known electromagnetic-gauge-covariant derivative of the Dirac
field is given by Peskin2018

$\vec{D}_{\mu}=\vec{\partial}_{\mu}+i\frac{q_{\mathrm{e}}}{\hbar}A_{\Re\mu},\hskip
14.22636pt\reflectbox{$\vec{\reflectbox{$D$}}$}_{\mu}=\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\mu}-i\frac{q_{\mathrm{e}}}{\hbar}A_{\Re\mu}.$
(122)

The eight-spinor partial differential operator $\vec{\partial}$, its adjoint
$\vec{\reflectbox{$\partial$}}$ , the four-vector-type eight-spinor
$\boldsymbol{\gamma}_{\mathrm{F}}$, made of the Dirac gamma matrices, and its
adjoint $\bar{\boldsymbol{\gamma}}_{\mathrm{F}}$ are given by

$\displaystyle\vec{\partial}$
$\displaystyle=[0,\vec{\partial}_{x},\vec{\partial}_{y},\vec{\partial}_{z},-\vec{\partial}_{0},0,0,0]^{T},$

$\vec{\reflectbox{$\partial$}}$

$\displaystyle=[0,\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{x},\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{y},\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{z},\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{0},0,0,0],$
$\displaystyle\boldsymbol{\gamma}_{\mathrm{F}}$
$\displaystyle=[\mathbf{0},\boldsymbol{\gamma}_{\mathrm{F}}^{x},\boldsymbol{\gamma}_{\mathrm{F}}^{y},\boldsymbol{\gamma}_{\mathrm{F}}^{z},\boldsymbol{\gamma}_{\mathrm{F}}^{0},\mathbf{0},\mathbf{0},\mathbf{0}]^{T},$
$\displaystyle\bar{\boldsymbol{\gamma}}_{\mathrm{F}}$
$\displaystyle=[\mathbf{0},\bar{\boldsymbol{\gamma}}_{\mathrm{F}}^{x},\bar{\boldsymbol{\gamma}}_{\mathrm{F}}^{y},\bar{\boldsymbol{\gamma}}_{\mathrm{F}}^{z},-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}^{0},\mathbf{0},\mathbf{0},\mathbf{0}].$
(123)

Here
$\bar{\boldsymbol{\gamma}}_{\mathrm{F}}^{\mu}=\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu{\dagger}}\boldsymbol{\gamma}_{\mathrm{F}}^{0}=\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}$.
In the equality between the first two forms of $\mathcal{L}_{\mathrm{DM}}$ in
Eq. (120), we have used

$\Phi_{\Re}=\frac{1}{2}g_{\mathrm{e}}\bar{\psi}(\boldsymbol{\gamma}_{\mathrm{F}})\psi,\hskip
14.22636pt\bar{\Phi}_{\Re}=\Phi_{\Re}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}=\frac{1}{2}g_{\mathrm{e}}\bar{\psi}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}})\psi.$
(124)

These forms are obtained from the definition of the charge-current spinor in
Eq. (1) by using the well-known expression of the four-current density in
terms of the Dirac spinors, given by
$J_{\mathrm{e}\Re}^{\mu}=(c\rho_{\mathrm{e}},\mathbf{J}_{\mathrm{e}\Re})=q_{\mathrm{e}}c\bar{\psi}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\psi$
Landau1982 . In the last form of $\textstyle\mathcal{L}_{\mathrm{DM}}$ in Eq.
(120), we have used the covariant form of the electromagnetic four-potential,
given by
$A_{\Re\mu}=\eta_{\nu\mu}A_{\Re}^{\nu}=(\phi_{\mathrm{e\Re}}/c,-\mathbf{A}_{\Re})$.
In the last form of $\textstyle\mathcal{L}_{\mathrm{M}}$ in Eq. (120),
$F^{\mu\nu}$ is the electromagnetic tensor, which is given in terms of the
four-potential and the corresponding electric and magnetic fields as
Jackson1999 ; Landau1989

$\displaystyle F^{\mu\nu}$
$\displaystyle=\partial^{\mu}A_{\Re}^{\nu}-\partial^{\nu}A_{\Re}^{\mu}$
$\displaystyle=\left[\begin{array}[]{cccc}0&-E_{\Re}^{x}/c&-E_{\Re}^{y}/c&-E_{\Re}^{z}/c\\\
E_{\Re}^{x}/c&0&-B_{\Re}^{z}&B_{\Re}^{y}\\\
E_{\Re}^{y}/c&B_{\Re}^{z}&0&-B_{\Re}^{x}\\\
E_{\Re}^{z}/c&-B_{\Re}^{y}&B_{\Re}^{x}&0\end{array}\right].$ (129)

###  VI.2 Euler-Lagrange equations of the potential spinor field

When writing the Euler-Lagrange equations, the spinors $\Theta_{\Re}$ and
$\bar{\Theta}_{\Re}$ are treated as independent dynamical variables. The
Euler-Lagrange equation of $\bar{\Theta}_{\Re}$ is given by

$\frac{\partial\mathcal{L}_{\mathrm{QED}}}{\partial\bar{\Theta}_{\Re}}-\partial_{\mu}\Big{[}\frac{\partial\mathcal{L}_{\mathrm{QED}}}{\partial(\partial_{\mu}\bar{\Theta}_{\Re})}\Big{]}=0.$
(130)

Using the Lagrangian density in Eq. (120), we obtain
$\partial\mathcal{L}_{\mathrm{QED}}/\partial\bar{\Theta}_{\Re}=\Phi_{\Re}$ and
$\partial\mathcal{L}_{\mathrm{QED}}/\partial(\partial_{\mu}\bar{\Theta}_{\Re})=\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\partial_{\nu}\Theta_{\Re}$.
Substituting these derivatives into Eq. (130), using
$\partial_{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\partial_{\nu}=\mathbf{I}_{8}\partial^{\mu}\partial_{\mu}$,
and rearranging the terms, the Euler-Lagrange equations of motion become

$\partial^{\mu}\partial_{\mu}\Theta_{\Re}=\Phi_{\Re}.$ (131)

Equation (131) is equal to the real part of the equation of the potential
spinor in Eq. (51). Using Eq. (52), we can see that it is also equal to the
spinorial photon equation in Eq. (2). Thus, the Lagrangian density in Eq.
(120) enables a compact derivation of the full set of Maxwell’s equations. In
contrast, in the conventional Lagrangian theory derivation of electrodynamics
using the four-potential, one obtains only Gauss’s law for electricity and the
Ampère-Maxwell law directly from the Euler-Lagrange equations Landau1989 ;
Bliokh2013b ; Partanen2019b . Faraday’s law of induction and Gauss’s law for
magnetism are obtained from the definition of the electromagnetic field tensor
in terms of the four-potential by using the Bianchi identity.

As a further consistency check, we investigate the Euler-Lagrange equations
for $\Theta_{\Re}$, which are identical to Eq. (130) with $\bar{\Theta}_{\Re}$
replaced by $\Theta_{\Re}$. In this case, we have
$\partial\mathcal{L}_{\mathrm{QED}}/\partial\Theta_{\Re}=\bar{\Phi}_{\Re}$ and
$\partial\mathcal{L}_{\mathrm{QED}}/\partial(\partial_{\mu}\Theta_{\Re})=\partial_{\nu}\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}$.
Substituting these derivatives into the Euler-Lagrange equation, using
$\partial_{\mu}\partial_{\nu}\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}=\partial^{\mu}\partial_{\mu}\bar{\Theta}_{\Re}$,
and rearranging the terms, we obtain
$-\partial^{\mu}\partial_{\mu}\bar{\Theta}_{\Re}=-\bar{\Phi}_{\Re}$. Using
$\bar{\Theta}_{\Re}=\Theta_{\Re}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$
and
$\bar{\Phi}_{\Re}=\Phi_{\Re}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$,
we obtain
$-\partial^{\mu}\partial_{\mu}\Theta_{\Re}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}=-\Phi_{\Re}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$.
Taking the conjugate transpose of this equation, multiplying the resulting
equation by $\boldsymbol{\gamma}_{\mathrm{B}}^{0}$ from the left, and
rearranging the terms leads to
$\partial^{\mu}\partial_{\mu}\Theta_{\Re}=\Phi_{\Re}$. This is equal to Eq.
(131).

###  VI.3 Euler-Lagrange equations of the Dirac field

Next, we consider dynamical equations of the Dirac field. As conventional, we
treat $\psi$ and
$\bar{\psi}=\psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}$ as independent
dynamical variables and write the Euler-Lagrange equations for $\bar{\psi}$ as
Peskin2018

$\frac{\partial\mathcal{L}_{\mathrm{QED}}}{\partial\bar{\psi}}-\partial_{\mu}\Big{[}\frac{\partial\mathcal{L}_{\mathrm{QED}}}{\partial(\partial_{\mu}\bar{\psi})}\Big{]}=0.$
(132)

Using the Lagrangian density in Eq. (120) with the identities
$\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\vec{\partial}=\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\vec{\partial}_{\mu}$
and
$\reflectbox{$\vec{\reflectbox{$\partial$}}$}\boldsymbol{\gamma}_{\mathrm{F}}=\reflectbox{$\vec{\reflectbox{$\partial$}}$}_{\mu}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}$,
we obtain
$\partial\mathcal{L}_{\mathrm{QED}}/\partial\bar{\psi}=\frac{i}{2}\hbar
c\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\partial_{\mu}\psi-
m_{\mathrm{e}}c^{2}\psi+\frac{1}{2}g_{\mathrm{e}}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})\psi$
and
$\partial\mathcal{L}_{\mathrm{QED}}/\partial(\partial_{\mu}\bar{\psi})=-\frac{i}{2}\hbar
c\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\psi$. Substituting these derivatives
into Eq. (132), the Euler-Lagrange equations of motion become

$i\hbar c\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\partial_{\mu}\psi-
m_{\mathrm{e}}c^{2}\psi+\textstyle\frac{1}{2}g_{\mathrm{e}}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})\psi=0.$
(133)

Using the relation
$\textstyle\frac{1}{2}g_{\mathrm{e}}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})=-q_{\mathrm{e}}c\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}A_{\Re\mu}$,
and the definition of the electromagnetic-gauge-covariant derivative of the
Dirac field in Eq. (122), we can write Eq. (133) compactly as

$i\hbar c\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}\vec{D}_{\mu}\psi-
m_{\mathrm{e}}c^{2}\psi=0.$ (134)

In terms of the components of the four-potential
$A_{\Re\mu}=(\phi_{\mathrm{e\Re}}/c,-\mathbf{A}_{\Re})$, momentum operator
$\hat{\mathbf{p}}$, Dirac alpha and beta matrices
$\boldsymbol{\alpha}_{\mathrm{F}}^{i}=\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\gamma}_{\mathrm{F}}^{i}$
and $\boldsymbol{\beta}_{\mathrm{F}}=\boldsymbol{\gamma}_{\mathrm{F}}^{0}$,
and the associated vector
$\boldsymbol{\alpha}_{\mathrm{F}}=(\boldsymbol{\alpha}_{\mathrm{F}}^{x},\boldsymbol{\alpha}_{\mathrm{F}}^{y},\boldsymbol{\alpha}_{\mathrm{F}}^{z})$,
Eq. (134) can be further rewritten as

$\hat{H}\psi=i\hbar\frac{\partial}{\partial t}\psi.$ (135)

Here the Hamiltonian operator $\hat{H}$ takes for the Dirac field the form

$\hat{H}=c\boldsymbol{\alpha}_{\mathrm{F}}\cdot(\hat{\mathbf{p}}-q_{\mathrm{e}}\mathbf{A}_{\Re})+\boldsymbol{\beta}_{\mathrm{F}}m_{\mathrm{e}}c^{2}+q_{\mathrm{e}}\phi_{\mathrm{e\Re}}\mathbf{I}_{4}.$
(136)

Equations (134)–(136) show that the dynamical equation of the Dirac field is
equivalent to the conventional minimally coupled Dirac equation in the
presence of interaction with the electromagnetic field Landau1982 , as
expected.

For checking the consistency, we derive the Dirac equation from the Euler-
Lagrange equations for $\psi$, which is identical to Eq. (132) with
$\bar{\psi}$ replaced by $\psi$. In this case, we have
$\partial\mathcal{L}_{\mathrm{QED}}/\partial\psi=-\frac{i}{2}\hbar
c\partial_{\mu}\bar{\psi}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}-m_{\mathrm{e}}c^{2}\bar{\psi}+\frac{1}{2}g_{\mathrm{e}}\bar{\psi}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})$
and
$\partial\mathcal{L}_{\mathrm{QED}}/\partial(\partial_{\mu}\psi)=\frac{i}{2}\hbar
c\bar{\psi}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}$. Substituting these
derivatives into the Euler-Lagrange equation, we obtain $-i\hbar
c\partial_{\mu}\bar{\psi}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}-m_{\mathrm{e}}c^{2}\bar{\psi}+\frac{1}{2}g_{\mathrm{e}}\bar{\psi}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})$.
Taking the conjugate transpose of this equation, using
$\bar{\psi}=\psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}$, and
multiplying the resulting equation by $\boldsymbol{\gamma}_{\mathrm{F}}^{0}$
from the left leads to $i\hbar
c\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu{\dagger}}\boldsymbol{\gamma}_{\mathrm{F}}^{0}\partial_{\mu}\psi-\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\gamma}_{\mathrm{F}}^{0}m_{\mathrm{e}}c^{2}\psi+\frac{1}{2}g_{\mathrm{e}}\boldsymbol{\gamma}_{\mathrm{F}}^{0}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}\psi$.
Using
$\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\gamma}_{\mathrm{F}}^{\mu{\dagger}}\boldsymbol{\gamma}_{\mathrm{F}}^{0}=\boldsymbol{\gamma}_{\mathrm{F}}^{\mu}$,
$\boldsymbol{\gamma}_{\mathrm{F}}^{0}\boldsymbol{\gamma}_{\mathrm{F}}^{0}=\mathbf{I}_{8}$,
and
$\boldsymbol{\gamma}_{\mathrm{F}}^{0}(\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}})^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}=\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\Theta_{\Re}+\bar{\Theta}_{\Re}\boldsymbol{\gamma}_{\mathrm{F}}$,
we finally obtain the Dirac equation in the form given in Eq. (133).

##  VII Generating Lagrangian density of gravity, special unitary symmetry,
and the SEM tensor

In the sections above, we have presented how the conventional QED is expressed
in the eight-spinor formalism. Next, we introduce _the generating Lagrangian
density of gravity_ that plays, in the definition of the gauge theory of
gravity Partanen2023c , a similar role as the conventional Lagrangian density
of the free Dirac field plays in the definition of the gauge theory of QED
Peskin2018 ; Schwartz2014 ; Weinberg1996 . The generating Lagrangian density
of gravity is associated with a special unitary symmetry of the quantum fields
in the standard model, and it enables an elegant derivation of the symmetric
SEM tensors as described in detail in the subsections below. The Yang-Mills
gauge theory of gravity that follows is presented in a separate work
Partanen2023c .

###  VII.1 Generating Lagrangian density of gravity

As the starting point for defining the generating Lagrangian density of
gravity, we use the analogy with standard gauge theory as it is applied in
deriving the full QED from the Lagrangian density of the free Dirac field,
$\mathcal{L}_{\mathrm{QED,0}}=\mathcal{L}_{\mathrm{QED,D}}$, in Eq. (120). We
have presented a concise summary of the conventional gauge theory of QED in
Appendix B. As shown in Appendix B, the unitary transformation
$\psi\rightarrow e^{i\theta}\psi$ with parameter $\theta$ and the
electromagnetic-gauge-covariant derivative
$\vec{D}_{\mu}=\vec{\partial}_{\mu}+i\frac{q_{\mathrm{e}}}{\hbar}A_{\Re\mu}$
in Eq. (122) _generate_ from the Lagrangian density
$\mathcal{L}_{\mathrm{QED,0}}$ the full QED. In Appendix B, we also show that
the variation of the generating Lagrangian density of QED,
$\delta\mathcal{L}_{\mathrm{QED,0}}=-\frac{\hbar}{q_{\mathrm{e}}}J_{\mathrm{e\Re}}^{\mu}\partial_{\mu}\theta$,
is proportional to the electric four-current density, which is the source term
of the electromagnetic field.

Based on the known properties of gravitational interaction, we must set
certain conditions that we require the generating Lagrangian density of
gravity and the resulting full gauge theory of gravity to satisfy:

  1. 1.

The SEM tensor must act as the source term of the gravitational field. It
follows that the gravitational field is a tensor gauge field in contrast to
the vector gauge fields of the standard model.

  2. 2.

Instead of the Abelian U(1) gauge theory of QED, we must use the non-Abelian
Yang-Mills gauge theory analogous to the theories of weak and strong
interactions Peskin2018 ; Schwartz2014 ; Weinberg1996 . This is because four
symmetry generators are needed for the description of the tensor gauge field.

  3. 3.

To enable unification of gravity with the fundamental interactions of the
standard model, the gauge theory of gravity must be based on an internal
special unitary symmetry or subsymmetry of the quantum fields in the standard
model.

  4. 4.

The theory must contain a new coupling constant $g_{\mathrm{g}}$. The
variation of the generating Lagrangian density of gravity with respect to the
symmetry transformation parameters must be directly proportional to the SEM
tensor divided by $g_{\mathrm{g}}$.

  5. 5.

The gauge theory of gravity must enable writing the dynamical equations for
the gravitational field through the Euler-Lagrange equations. In the limit of
weak gravitational field, the gravitational field equations must reproduce the
experimentally verified predictions of general relativity.

  6. 6.

Through the Euler-Lagrange equations, we must also obtain the generalized
equations of motion containing gravitational coupling for all the fundamental
interactions of the standard model Peskin2018 ; Schwartz2014 ; Weinberg1996 .
In the Minkowski metric limit of the gravitational gauge field, these
equations must reproduce the dynamical equations of the standard model.

In the formulation of the generating Lagrangian density of gravity, we must
aim for the simplest form possible that satisfies the requirements above. The
generating Lagrangian density of gravity can be seen as the _fundamental
hypothesis_ of the theory that unifies the standard model and gravity. All
quantum fields of the standard model can be included on equal footing, but,
for simplicity, here we present the theory using the Dirac electron-positron
field and the electromagnetic field only. In a process of trial and error, we
have heuristically ended up to the generating Lagrangian density of gravity,
given by

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\mathcal{L}_{\mathrm{D,0}}+\mathcal{L}_{\mathrm{M,0}}+\mathcal{L}_{\mathrm{DM,0}}$
$\displaystyle=\frac{\hbar
c}{4g_{\mathrm{g}}}\bar{\psi}(\reflectbox{$\vec{\reflectbox{$D$}}$}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\boldsymbol{\gamma}_{\mathrm{F}}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\vec{D})\psi$
$\displaystyle\hskip
14.22636pt+\frac{i}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}\Psi_{\Re}-m_{\mathrm{e}}c^{2}\bar{\psi}\psi+\bar{\Psi}_{\Re}\Psi_{\Re}$
$\displaystyle\mathcal{L}_{\mathrm{D,0}}$ $\displaystyle=\frac{\hbar
c}{4g_{\mathrm{g}}}\bar{\psi}(\reflectbox{$\vec{\reflectbox{$\partial$}}$}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\boldsymbol{\gamma}_{\mathrm{F}}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\vec{\partial})\psi$
$\displaystyle\hskip 14.22636pt-m_{\mathrm{e}}c^{2}\bar{\psi}\psi$
$\displaystyle\mathcal{L}_{\mathrm{M,0}}$
$\displaystyle=\frac{i}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}\Psi_{\Re}+\bar{\Psi}_{\Re}\Psi_{\Re}$
$\displaystyle\mathcal{L}_{\mathrm{DM,0}}$
$\displaystyle=\frac{ig_{\mathrm{e}}}{4g_{\mathrm{g}}}\bar{\psi}(\bar{\Theta}_{\Re}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\boldsymbol{\gamma}_{\mathrm{F}}$
$\displaystyle\hskip
14.22636pt+\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\Theta_{\Re})\psi.$
(137)

The term $\mathcal{L}_{\mathrm{D,0}}$ of the generating Lagrangian density of
gravity in Eq. (137) describes the Dirac field, the term
$\mathcal{L}_{\mathrm{M,0}}$ describes the electromagnetic field, and the term
$\mathcal{L}_{\mathrm{DM,0}}$ describes the interaction between the Dirac and
electromagnetic fields. The partial derivatives $\vec{\partial}_{\nu}$ in
front of $\mathbf{I}_{8}$ and $\bar{\mathbf{I}}_{8}^{\dagger}$ in Eq. (137)
act on $\mathbf{I}_{8}$ and do not extend to the spinors $\Psi_{\Re}$ and
$\psi$. Since $\mathbf{I}_{8}$ is a constant identity matrix, the terms
$\vec{\partial}_{\nu}\mathbf{I}_{8}$ and
$\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}$ are actually identically
zero. These terms become nonzero after the partial derivatives
$\vec{\partial}_{\nu}$ are replaced by gravitational-gauge-covariant
derivatives as shown in the Yang-Mills gauge theory of gravity in Ref.
Partanen2023c . The terms $\vec{\partial}_{\nu}\mathbf{I}_{8}$ and
$\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}$ become nonzero also when
the related space-time-dependent symmetry transformation is carried out for
$\mathbf{I}_{8}$. This symmetry transformation will be elaborated below.

###  VII.2 SU(8)4D symmetry

While the theories of the electromagnetic, weak, and strong interactions
utilize U(1), SU(2), and SU(3) symmetries Peskin2018 ; Schwartz2014 ;
Weinberg1996 , here we utilize a four-dimensional subsymmetry of SU(8),
defined below and denoted by SU(8)4D. In analogy to the description of gauge
symmeties in the conventional quantum field theory Peskin2018 ; Schwartz2014 ;
Zee2010 ; Weinberg1996 , we consider the SU(8)4D symmetry transformation under
which the generating Lagrangian density in Eq. (137) is globally invariant and
whose space-time-dependent variation without introducing the gauge field is
related to the source term of the gauge field. In our case, this source term
is the SEM tensor, while in the case of the U(1) symmetry of QED it is the
electric four-current density as briefly presented in Appendix B. The
generating Lagrangian density of gravity is invariant with respect to any
global, i.e., space-time-independent, symmetry transformation of
$\mathbf{I}_{8}$, since such transformations do not make
$\vec{\partial}_{\nu}\mathbf{I}_{8}$ and
$\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}$ nonzero. Thus, the
generating Lagrangian density of gravity is trivially globally invariant with
respect to the SU(8)4D symmetry.

The SU(8)4D symmetry differs from the U(1) symmetry of the conventional QED:
While the U(1) symmetry operates primarily on the Dirac field and generates
the electromagnetic field as a gauge field, the SU(8)4D symmetry operates
primarily on both the Dirac and electromagnetic fields and generates the
gravitational field as a gauge field. Such an operation is necessary since
gravity is known to affect all fields and matter. Here we limit our study of
the SU(8)4D symmetry to its relation to the SEM tensors of the Dirac and
electromagnetic fields without introducing the gravitational gauge field. The
Yang-Mills gauge theory of gravity arising from the SU(8)4D symmetry is
studied in Ref. Partanen2023c . We define the SU(8)4D symmetry transformation
as

$\mathbf{I}_{8}\rightarrow\mathbf{U}\mathbf{I}_{8},\hskip
14.22636pt\text{where }\mathbf{U}=\exp(i\phi_{\mu}\mathbf{t}^{\mu}).$ (138)

The symmetry transformation matrix $\mathbf{U}$ has determinant 1. Thus, it is
an element of the special unitary group SU(8). The real-valued four-vector
$\phi_{\mu}$ in Eq. (138) defines the symmetry transformation parameters. The
four transformation generators $\mathbf{t}^{\mu}$ are constant traceless
Hermitian matrices, given in terms of the complex-conjugated bosonic gamma
matrices as
$\mathbf{t}^{\mu}=(\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\mu})^{*}$.
This set of matrices generates the Clifford algebra
$\mathcal{C}\ell_{4,0}(\mathbb{C})$ with the anticommutation relation
$\\{\mathbf{t}^{\mu},\mathbf{t}^{\nu}\\}=2\delta^{\mu\nu}\mathbf{I}_{8}$. The
commutation relation of the generators is given by
$[\mathbf{t}^{\mu},\mathbf{t}^{\nu}]=if_{\rho}^{\;\,\mu\nu}\mathbf{t}^{\rho}$,
where $f_{\rho}^{\;\,\mu\nu}=2\varepsilon^{0\rho\mu\nu}$. Thus,
$\mathbf{t}^{\mu}$ also generate a Lie algebra, which is a prerequisite for
developing the Yang-Mills gauge theory in Ref. Partanen2023c . The generators
are Lorentz invariant satisfying
$\boldsymbol{\Lambda}_{\mathrm{J}}\mathbf{t}^{\mu}\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}=\mathbf{t}^{\mu}$.
The matrix $\mathbf{U}$ satisfies the commutation relations
$[\mathbf{U},\boldsymbol{\gamma}_{\mathrm{B}}^{5}]=\mathbf{0}$,
$[\mathbf{U},\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}]=\mathbf{0}$,
and $[\mathbf{U},\boldsymbol{\Lambda}_{\mathrm{J}}]=\mathbf{0}$. These
relations are necessary for the gravitational gauge invariance of the
Lagrangian density in the Yang-Mills gauge theory of gravity Partanen2023c .
The group of transformations defined by $\mathbf{U}$ corresponds to a four-
dimensional subgroup of SU(8), which we denote SU(8)4D. For comparison with
the present SU(8)4D generators $\mathbf{t}^{\mu}$, in the case of the SU(2)
symmetry of the electroweak interaction, the generators are the three Pauli
matrices, and in the case of the SU(3) symmetry of quantum chromodynamics
(QCD), the generators are the eight Gell-Mann matrices Schwartz2014 .

###  VII.3 Symmetric SEM tensor

In this section, we generalize the global SU(8)4D symmetry transformation of
Eq. (138) into a local symmetry transformation by making the symmetry
transformation parameters $\phi_{\mu}$ space-time dependent as
$\phi_{\mu}=\phi(t,\mathbf{r})$. Would we follow the Yang-Mills gauge theory,
we should simultaneously introduce the gauge-covariant derivative that makes
the derivative terms of the generating Lagrangian density of gravity in Eq.
(137) invariant. The consequences of adding the gauge-covariant derivative are
elaborated in the separate work on the Yang-Mills gauge theory of gravity in
Ref. Partanen2023c . Here we consider the variation of the generating
Lagrangian density of gravity without adding the gauge field. We recover the
profound relationship between the SU(8)4D symmetry and the symmetric SEM
tensor source term of gravity. This relationship is completely analogous to
the relation between the U(1) symmetry of QED and the electric four-current
density source term of electromagnetism as briefly presented in Appendix B.

The infinitesimal variation of $\mathbf{I}_{8}$ in the SU(8)4D symmetry
transformation of Eq. (138) with respect to the transformation parameters
$\phi_{\mu}$ is given by

$\delta\mathbf{I}_{8}=i\mathbf{t}^{\mu}\delta\phi_{\mu}.$ (139)

Using this inifinitesimal variation, it is straightforward to calculate the
variation of the generating Lagrangian density of gravity in Eq. (137) with
respect to $\phi_{\mu}$ as

$\displaystyle\delta\mathcal{L}_{0}$ $\displaystyle=\frac{\hbar
c}{4g_{\mathrm{g}}}\bar{\psi}[\reflectbox{$\vec{\reflectbox{$D$}}$}\overline{(\delta\mathbf{I}_{8})}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\boldsymbol{\gamma}_{\mathrm{F}}+\reflectbox{$\vec{\reflectbox{$D$}}$}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}(\delta\mathbf{I}_{8})\boldsymbol{\gamma}_{\mathrm{F}}$
$\displaystyle\hskip
14.22636pt-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\overline{(\delta\mathbf{I}_{8})}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\vec{D}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}(\delta\mathbf{I}_{8})\vec{D}]\psi$
$\displaystyle\hskip
14.22636pt+\frac{i}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}(\delta\mathbf{I}_{8})^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}\Psi_{\Re}$
$\displaystyle\hskip
14.22636pt+\frac{i}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\overline{(\delta\mathbf{I}_{8})}^{\dagger}\Psi_{\Re}$
$\displaystyle=\frac{i\hbar
c}{4g_{\mathrm{g}}}\bar{\psi}[\reflectbox{$\vec{\reflectbox{$D$}}$}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\mathbf{t}^{\mu}\boldsymbol{\gamma}_{\mathrm{F}}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\mathbf{t}^{\mu}\vec{D}]\psi\vec{\partial}_{\nu}\delta\phi_{\mu}$
$\displaystyle\hskip
14.22636pt+\frac{1}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}\mathbf{t}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\Psi_{\Re}\vec{\partial}_{\nu}\delta\phi_{\mu}$
$\displaystyle=\frac{1}{g_{\mathrm{g}}}\bar{T}^{\mu\nu}\vec{\partial}_{\nu}\delta\phi_{\mu}.$
(140)

In the second equality of Eq. (140), we have used an identity
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\bar{\mathbf{t}}^{\mu}=-\mathbf{t}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\boldsymbol{\gamma}_{\mathrm{B}}^{5}$
for the electromagnetic field term. The quantity $\bar{T}^{\mu\nu}$, defined
by the last equality of Eq. (140), is found to be equal to the so-called
trace-reverse of the total SEM tensor $T^{\mu\nu}$ of the Dirac and
electromagnetic fields. In terms of the SEM tensor, it is written as
$\bar{T}^{\mu\nu}=T^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}T$ Carroll2004 , where
$T=\eta_{\rho\sigma}T^{\rho\sigma}$. The trace-reversed SEM tensor is, thus,
given by

$\displaystyle\bar{T}^{\mu\nu}$ $\displaystyle=\frac{i\hbar