# New Angular Momentum Conservation Laws for Gauge Fields in QED

Farhad Khosravi1,3, Li-Ping Yang2,3, Pronoy Das3 and Zubin Jacob3,∗ 11QBit,
Evergreen Building, Vancouver, British Columbia V6E 4B1, Canada  2Center for
Quantum Sciences and School of Physics, Northeast Normal University, Changchun
130024, China  3Elmore Family School of Electrical and Computer Engineering,
Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA
<EMAIL_ADDRESS>
###### Abstract

Quantum electrodynamics (QED) deals with the relativistic interaction of
bosonic gauge fields and fermionic charged particles. In QED, global
conservation laws of angular momentum for light-matter interactions are well-
known. However, local conservation laws, i.e. the conservation law of angular
momentum at every point in space, remain unexplored. Here, we use the QED
Lagrangian and Noether’s theorem to derive a new local conservation law of
angular momentum for Dirac-Maxwell fields in the form of the continuity
relation for linear momentum. We separate this local conservation law into
four coupled motion equations for spin and orbital angular momentum (OAM)
densities. We introduce a helicity current tensor, OAM current tensor, and
spin-orbit torque in the motion equations to shed light on on the local
dynamics of spin-OAM interaction and angular momentum exchange between
Maxwell-Dirac fields. We elucidate how our results translate to classical
electrodynamics using the example of plane wave interference as well as a
dual-mode optical fiber. Our results shine light on phenomena related to the
spin of gauge bosons.

  * March 2024

##  1 Introduction

Global conservation laws for angular momentum are well known in quantum
electrodynamics. However, the spatial distribution of angular momentum (AM)
i.e. AM density is relatively unexplored in QED. Here, the gauge fields take
center stage but no local conservation laws are known for the U(1) gauge field
in QED. Fig. 1a highlights the difference between global and local
conservation of total angular momentum. It also highlights unique physical
quantities related to angular momentum for both Dirac fields and Maxwell
fields. We emphasize that in the presence of sources, a comprehensive, gauge-
independent, locally applicable conservation equation becomes essential in
understanding the local dynamics of the angular momentum (Fig. 1b).

On the other hand, the developments in classical electrodynamics have focused
on both global and local angular momentum. The local approach towards field
quantities such as helicity, chirality, and angular momentum has proven useful
in the study of conservation laws as well as geometrical properties of EM
fields [1, 2, 3, 4, 5, 6, 7, 8, 9]. However, the procedure employed in these
derivations is based on the duality symmetry of the source-free EM Lagrangian.
Since the duality symmetry is only maintained in source-free regions, this
form of Lagrangian neglects any form of interaction with fermionic fields and
cannot be employed to present a local conservation law for the angular
momentum of gauge fields in QED. These properties are captured in the
manifestly covariant construction of the Dirac equation [10]. Connections
between the fermionic field of the Dirac equation and the bosonic fields of
Maxwell’s equations is the focus of Dirac-Maxwell correspondence where the
relativistic parallels between photons and electrons are studied [11, 12].
These studies show that, although different in nature, electronic and
electromagnetic fields can exhibit many analogous properties. This
correspondence is evident in phenomena such as spin-momentum locking which
emerges in the Dirac equation [13] as well as the Maxwell’s equations [14,
15].

(a)  (b)

Behavior | Scalar | Vector | Tensor
---|---|---|---
Physical quantity | Chirality / | Spin Angular | Orbital | Spin-orbit | Helicity | Orbital Angular
Helicity | Momentum | Angular Momentum | Torque | Current | Momentum Current
Dirac Field | $\frac{\hbar}{2}\psi^{\dagger}\gamma^{5}\psi$ | $\frac{\hbar}{2}\psi^{\dagger}\bm{\Sigma}\psi$ | $\mathcal{R}\\{\psi^{\dagger}(\bm{r}\times\bm{p}_{\parallel})\psi\\}$ | $-c\mathcal{R}\left\\{\bar{\psi}\left(\bm{\gamma}\times\bm{p}_{\parallel}\right)\psi\right\\}$ | $0$ | $\mathcal{R}\left\\{\bar{\psi}\gamma_{i}(\bm{r}\times\bm{p}_{\parallel})_{j}\psi\right\\}$
Maxwell Field |  | $\epsilon_{0}(\bm{E}^{\bot}\times\bm{A}^{\bot})$ | $\epsilon_{0}\left[E_{i}^{\bot}(\bm{r}\times\nabla)A_{i}^{\bot}\right]$ | $\frac{1}{\mu_{0}}(\bm{B}\cdot\nabla)\bm{A}^{\bot}$ |  | $-\frac{1}{\mu_{0}}\left[\varepsilon_{ikl}B_{k}(\bm{r}\times\bm{\nabla})_{j}A_{l}^{\bot}\right]$
$\frac{1}{\mu_{0}c}\left(\bm{A}^{\bot}\cdot\bm{B}\right)$ | $-\epsilon_{0}\left(\frac{\partial\bm{E}^{\parallel}}{\partial t}\times\bm{A}^{\bot}\right)$ | $-\frac{1}{\mu_{0}}(A^{\bot}_{i}B_{j})$ | $-\epsilon_{0}A_{i}^{\bot}\left(\bm{r}\times\frac{\partial\bm{E}^{\parallel}}{\partial t}\right)_{j}$

(c)

Figure 1: Conservation laws of angular momentum in light-matter interacting
systems (a) conventional conservation of global total photonic and electronic
angular momentum. This conservation law applies to closed systems and does not
include angular momentum exchange due to near-field interactions. (b) Local
conservation of angular momentum applicable to all regions of interaction. The
conventional conservation of sum of angular momenta is replaced by the
conservation Eq. (37). (c) Table of quantities defined in this paper,
representing the spatial and temporal densities of the scalar, vector, or
tensor observables pertinent to angular momentum. Highlighted quantities are
the new terms defined in this paper. The local conservation of angular
momentum equation (Eq. (37)) connects the well-known terms such as spin
density, OAM density, helicity and chirality to these newly defined
quantities.

In this paper, we study angular momentum of light using the Lorentz symmetry
of the QED Lagrangian and local $U(1)$ gauge invariance [16]. With the
application of Noether’s theorem [17], we find unique local conservation laws
pertinent to the angular momentum of the gauge field interacting with
fermionic fields. We show that, in the near-field, the electronic angular
momentum can be transferred not only to the optical angular momentum, but also
to other field quantities that represent angular momentum current [18]. These
extra terms include electromagnetic helicity density [19], fermionic chirality
density [10], electromagnetic helicity current tensor, as well as the
electromagnetic and electronic orbital angular momentum (OAM) current tensors.
We further study these conservation laws and angular-momentum-carrying terms
for the classical electromagnetic solutions of a dual-mode optical fiber. This
connection between QED and classical electrodynamics will be of interest for
future experiments related to OAM and SAM of light.

##  2 Noether’s theorem, QED Lagrangian, and Lorentz transformation

An important problem in QED relates to the angular momentum of the gauge
field. Particularly, the goal is to decompose the total angular momentum of
gauge bosons into spin and orbital contributions [20]. In early work, in both
Belinfante’s [21] and Ji’s [22] decompositions, only the total angular
momentum of photons has been explored in detail. Jaffe and Manohar split the
photonic angular momentum into spin and orbital parts [23]. However, their
decomposition does not satisfy the gauge-invariant requirement. By splitting
the vector potential into transverse and longitudinal parts, Chen et al. [24]
and Wakamatsu [25] propose gauge-invariant decompositions of QED angular
momentum. However, these existing decompositions have not resolved the
conflict between the gauge-invariant requirement and canonical angular
momentum commutation relations. Recently, this problem has been solved by
quantizing the U(1) gauge field in the Lorenz gauge and merging the spin and
OAM of virtural photons with the OAM of Dirac fermions [26, 27].

The application of Noether’s theorem to the Dirac Lagrangian has been used in
the Quantum Chromodynamics (QCD) and QED communities to derive the expressions
for the angular momentum of the nuclei and the gauge fields [22, 24]. In QCD
and QED interactions, conservation of the total integrated angular momentum
suffices to describe the angular momentum transfer in scattering processes. On
the other hand, for a fast moving electron, its spin and momentum degrees of
freedom are highly correlated [28]. Explorations of a generalized definition
for the spin operator of an electron as well as for gauge bosons is still a
problem of widespread interest [29, 30] in QED and QCD [20, 22, 24]. While the
focus has mainly been on global conservation laws for angular momentum, our
goal in this paper is to focus on local space dependent AM density effects. In
nanophotonic and condensed matter systems, atoms can interact locally with an
external EM field. Therefore, for such systems, a local conservation equation
for the angular momentum density is necessary in the realization of naonscale
applications.

In this paper, we start with the real (also called symmetrized) Lagrangian
density of a Dirac field coupled to EM field [31],

$\mathfrak{L}=\bar{\psi}\left[c\gamma^{\mu}\left(\frac{1}{2}i\hbar\overleftrightarrow{\partial}_{\mu}-eA_{\mu}\right)-mc^{2}\right]\psi-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu},$
(1)

where $\bar{\psi}=\psi^{\dagger}\gamma^{0}$ and $\gamma^{\mu}$ are the gamma
matrices, $F_{\mu\nu}$ the electromagnetic tensor, $A_{\mu}=(\phi/c,-\bm{A})$
the electromagnetic four-potential, $\psi$ Dirac fields, $\hbar$ Planck’s
constant, and $\mu_{0}$ the vacuum permeability (Appendix). Note that
summation over repeated indices is assumed throughout this paper. By applying
Noether’s theorem [17], we obtain the conserved current for the given
Lagrangian of Eq. (1) by finding,

$\mathcal{M}^{\lambda}=\frac{\partial\mathfrak{L}}{\partial(\partial_{\lambda}\psi)}\delta\psi+\delta\bar{\psi}\frac{\partial\mathfrak{L}}{\partial(\partial_{\lambda}\bar{\psi})}+\frac{\partial\mathfrak{L}}{\partial\left(\partial_{\lambda}A^{\mu}\right)}\delta
A^{\mu}+\mathfrak{L}\delta x^{\lambda}.$ (2)

Lorentz transformations are defined as the coordinate transformations such
that the coordinates transform as $x^{\mu}\to{\Lambda^{\mu}}_{\nu}x^{\mu}$,
with

${\Lambda^{\mu}}_{\nu}=e^{-\frac{i}{2}\omega_{\kappa\sigma}{(\hat{S}^{\kappa\sigma})^{\mu}}_{\nu}},\quad$
(3)

where
${(\hat{S}^{\kappa\sigma})^{\mu}}_{\nu}=i\left(\eta^{\kappa\mu}{\eta^{\sigma}}_{\nu}-\eta^{\sigma\mu}{\eta^{\kappa}}_{\nu}\right)$
are the generators of rotation and boost in the four-dimensional real space
[32] and $\omega_{\kappa\sigma}$ are the rotation and boost parameters. It can
be shown that, under the Lorentz transformation of Eq. (3), the Maxwell and
Dirac fields change as,

$\delta
A^{\mu}(x)=-\frac{i}{2}\omega_{\kappa\sigma}{(\hat{M}_{\text{em}}^{\kappa\sigma})^{\mu}}_{\nu},\quad\delta\psi(x)=-\frac{i}{2}\omega_{\mu\nu}\hat{M}_{\text{D}}^{\mu\nu}\psi(x)$
(4)

where $\hat{M}_{\text{em}}$ and $\hat{M}_{\text{D}}$ are the angular momentum
operators of the Dirac and Maxwell’s fields, respectively (Appendix).
Specializing to the rotations ($\mu,\nu=i,j=1,2,3$), we obtain the general
continuity equation associated with rotational symmetry of the QED Lagrangian
[32](Appendix),

$\begin{split}\partial_{\lambda}\mathcal{M}^{ij,\lambda}=&\partial_{\lambda}\left(\mathcal{S}^{ij,\lambda}_{\text{D}}+\mathcal{L}^{ij,\lambda}_{\text{D}}+\mathcal{S}^{ij,\lambda}_{\text{em}}+\mathcal{L}^{ij,\lambda}_{\text{em}}\right)=0,\end{split}$
(5)

where $\mathcal{M}^{ij,\lambda}$ is the total angular current tensor. This
tensor can be split into spin $\mathcal{S}^{ij,\lambda}_{\text{D(em)}}$
(related to the rotation of the internal degrees of freedom) and OAM
$\mathcal{L}^{ij,\lambda}_{\text{D(em)}}$ (related to the coordinate
dependence of the fields) parts of the Dirac and EM fields, respectively (Fig.
1). For non-interacting fields, we obtain separate continuity equations for
the Dirac and EM fields. Note that the roman indices take the values
$i,j=1,2,3$ while the Greek indices are $\lambda=0,1,2,3$.

The time-components ($\lambda=0$) of these four angular momentum tensors give
the common spin and OAM densites for the Dirac and EM fields, which are
generally gauge dependent. The electronic part of the spin and OAM
respectively are
$\mathcal{S}^{ij,0}_{\text{D}}=\varepsilon_{ijk}\hbar\psi^{\dagger}\Sigma_{k}\psi/2$
($\varepsilon_{ijk}$ is the Levi-Civita symbol) and
$\mathcal{L}^{ij,0}_{\text{D}}=-i\varepsilon_{ijk}\hbar\mathcal{R}\left\\{\psi^{\dagger}\left(\bm{r}\times\nabla\right)_{k}\psi\right\\}$,
where $\bm{\Sigma}$ is the spin operator in Dirac equation and
$\mathcal{R}\\{\cdots\\}$ takes the real part of its argument. The EM part of
the spin and OAM, on the other hand, are
$\mathcal{S}^{ij,0}_{\text{em}}=\varepsilon_{ijk}(\epsilon_{0}\bm{E}\times\bm{A})_{k}$
and
$\mathcal{L}^{ij,0}_{\text{em}}=\varepsilon_{ijk}\left[\epsilon_{0}E_{l}(\bm{r}\times\nabla)_{k}A_{l}\right]$,
respectively (Appendix), where $\epsilon_{0}$ is vacuum permittivity. An
important observation is that OAM densities of the Dirac fields as well as the
spin and OAM of the EM field are gauge dependent and thus do not represent
observable physical quantities in a local frame. In fact, one can see that
most of the terms in the angular momentum tensors
$\mathcal{S}^{ij,\lambda}_{\rm D(E)}$ and $\mathcal{L}^{ij,\lambda}_{\rm
D(E)}$ are gauge dependent. Due to this problem, we rearrange the expressions
in these four tensors to write the conservation law in Eq. (28) in terms of
gauge-independent and physically observable quantities.

##  3 Local conservation law of angular momentum

The essence of Noether’s theorem in field theory lies in the local
conservation law with a general form $\partial_{\mu}j^{\mu}=0$. The global
charge $Q\equiv\int d^{3}xj^{0}(x)$ conserves only if all fields vanish at
spatial infinity, i.e., the vanishing boundary condition. As we mentioned
above, the local conservation law of angular momentum has not received the
attention it deserves both in experiments and in theory. We note that the
density of the conserved charge in the original continuity equation (28) of
angular momentum is a rank-2 tensor and the corresponding current is a rank-3
tensor. This local conservation law is extremely difficult to apply to
practical systems and almost impossible to verify in experiments. In this
section, we re-express this relation as a continuity equation of a vector
density similar to the local conservation law of linear momentum. The
corresponding current reduces to a rank-2 tensor akin to the Maxwell stress
tensor. We then go one step further by splitting our local conservation law
into four coupled motion equations, which describe the spin-OAM interaction
and angular momentum exchanging between Maxwell-Dirac fields.

By splitting the electromagnetic vector potential into transverse and
longitudinal parts, $\bm{A}=\bm{A}^{\bot}+\bm{A}^{\parallel}$, defined as
$\nabla\cdot\bm{A}^{\bot}=0$ and $\nabla\times\bm{A}^{\parallel}=0$ [33],
X.-S. Chen et al defined the gauge-independent angular momentum densities for
both the Dirac field and EM field [24]. However, a gauge-independent form for
the continuity equation has not been addressed. Here, following the same
approach used in Ref. [24], we obtain a standard (vector) continuity equation
for the angular momentum density of the combined system (see more details in
the Appendix,

$\frac{\partial M_{j}}{\partial t}+\nabla_{i}{T}_{ij}=0$ (6)

where $M_{j}$ are the components of the angular momentum density vector,
written as

$\displaystyle\bm{M}=$
$\displaystyle\frac{\hbar}{2}(\psi^{\dagger}\bm{\Sigma}\psi)+\mathcal{R}\left\\{\psi^{\dagger}\left(\bm{r}\times\bm{p}_{\parallel}\right)\psi\right\\}$
(7)
$\displaystyle+\epsilon_{0}(\bm{E}^{\bot}\times\bm{A}^{\bot})+\epsilon_{0}\left[E_{i}^{\bot}(\bm{r}\times\nabla)A_{i}^{\bot}\right],$

with gauge-independent transverse vector potential $\bm{A}^{\bot}$ and gauge-
independent electron momentum operator
$\bm{p}_{\parallel}=\left(-i\hbar\nabla-e\bm{A}^{\parallel}\right)$. The first
two terms in Eq. (7) are the spin and OAM densities of the electron, while the
last two terms are the gauge-independent spin and OAM densities of the EM
field [20, 24]. This shows that only the transverse part of the vector
potential contributes to the physically observable spin and OAM of EM field.

The angular momentum current tensor, $T_{ij}$, is a second rank tensor,
similar to the Maxwell stress tensor (EM momentum current) [34, 13]. The
tensor $T_{ij}$ is composed of three parts
:$\nabla_{i}T_{ij}=\nabla_{i}(\chi_{ij}+J_{ij}+N_{ij})$ with

$\chi_{ij}\\!=\Big{[}\overbrace{\frac{\hbar}{2}(\psi^{\dagger}\gamma^{5}\psi)}^{\text{chirality}}+\overbrace{\frac{1}{\mu_{0}}(\bm{A}^{\bot}\\!\cdot\\!\bm{B})}^{\text{helicity}}\Big{]}\delta_{ij}\equiv\chi~{}\delta_{ij},$
(8a)
$\begin{split}J_{ij}=&\overbrace{\mathcal{R}\left\\{\bar{\psi}\gamma_{i}\left(\bm{r}\times\bm{p}_{\parallel}\right)_{j}\psi\right\\}}^{\text{OAM
current tensor
(Dirac)}}-\overbrace{\frac{1}{\mu_{0}}(A_{i}^{\bot}B_{j})}^{\text{helicity
current tensor (Maxwell)}}\\\
-&\underbrace{\frac{1}{\mu_{0}}\left[\varepsilon_{ikl}B_{k}(\bm{r}\times\bm{\nabla})_{j}A_{l}^{\bot}\right]-\epsilon_{0}A_{i}^{\bot}\left(\bm{r}\times\frac{\partial\bm{E}^{\parallel}}{\partial
t}\right)_{j}}_{\text{OAM current tensor (Maxwell)}},\end{split}$ (8b)
$\nabla_{i}N_{ij}=(\bm{r}\times\nabla)\mathfrak{N}_{\text{em}},\quad\mathfrak{N}_{\text{em}}=\underbrace{\frac{\epsilon_{0}}{2}\bm{E}^{\bot}\cdot\bm{E}^{\bot}-\frac{1}{2\mu_{0}}\bm{B}\cdot\bm{B}}_{\text{free
photon Lagrangian}}+\epsilon_{0}\frac{\partial\bm{E}^{\parallel}}{\partial
t}\cdot\bm{A}^{\bot}.$ (8c)

where $\gamma^{5}$ is the chirality operator in Dirac equation (Appendix).

The term $\chi$ in Eq. (8a) describes the chirality/helicity of the fields.
The first term, $\hbar(\psi^{\dagger}\gamma^{5}\psi)/2$, is the density of
expectation value (in the classical sense of quantum mechanics) of the
chirality operator of the Dirac field. The operator $\gamma^{5}$ is widely
used to find the projections of fermionic fields into left-handed and right-
handed chiral states [16]. Therefore, the expecation value
$\psi^{\dagger}\gamma^{5}\psi$ represents the chirality of the Dirac field; a
negative (postivie) value means a left-handed (right-handed) chiral state. The
second term in Eq. (8a), on the other hand, is widely known as the helicity of
EM field [35, 36, 37]. Interestingly, Eq. (8a) shows that the chirality of the
fermionic field is identified with the helicity of the electromagnetic field
and the gradient of these quantities enter the continuity equation.

Note that the chirality of EM field defined in Ref. [38] does not appear in
the continuity equation of angular momentum. This shows that the helicity is
more fundamental for interacting systems. It is also important to note that
helicity can be defined for the Dirac field. Being defined as the projection
of spin on the momentum of the field, it is fundamentally different from
chirality as it is momentum dependent, and unlike chirality, it is not Lorentz
covariant. Interestingly, however, for the massless Dirac field, the
definition of chirality and helicity become identical [39]. This points to a
similar property in the massless EM fields of the Maxwell’s equations.

In the next three sections we define the new terms encountered in the
conservation equation.

###  3.1 Helicity current density tensor

The second terms in Eq. (8b) is reminiscent of the EM helicity in Eq. (8a)
with the difference that this quantity is a tensor, while helicity is a
scalar. In fact, helicity density in Eq. (8a) is equal to the trace of
$\frac{1}{\mu_{0}}A_{i}^{\bot}B_{j}$. We thus name this tensor quantity
helicity current density tensor since it transforms as a helicity.
Furthermore, the divergence of the helicity current tensor enters the
conservation of angular momentum equation meaning that it serves as a current
between different forms of Dirac or Maxwell angular momentum.

One important fact regarding this term is that it has no analog in the Dirac
field representation, as one might expect. The equivalent expression for
helicity current tensor, in the Dirac field, is normally considered to be spin
current represented by a tensor involving $\gamma$ matrices and spin
operators. However, our derivation shows that this term vanishes due to the
spinor nature of the Dirac fields (Appendix). This, of course, does not put
the numerous works on spin currents in condensed matter physics into question
[40, 41, 42, 43], since spin currents are generated by the collective motion
of electrons through spatially separating electrons with opposite spins in a
quantum cavity [44] or using ferromagnets with strong electron-electron
interactions [45]. The relativistic treatment of the Dirac equation in this
paper, however, only incorporates the wave function of a single electron in
vacuum and thus cannot account for the observed electronic spin currents. This
shows that a single electron in vacuum does not exhibit spin current and that
many-body Dirac equation should be considered for a relativistic account of
electronic spin currents [46, 47].

###  3.2 Orbital angular momentum current density tensor

Equation (8b) specifies the angular momentum current tensors for the fields.
This current tensor is specified by two directions: direction of angular
momentum and direction of the current. For instance, in the first term of Eq.
(8b), the direction of propagation of the current is specified by the gamma
matrices $\gamma^{i}$, while the direction of the angular momentum is
specified by the operator $(\bm{r}\times\bm{p}_{\parallel})$. Thus we call
this term OAM current density tensor of Dirac field. Similarly, the third term
in Eq. (8b) is the OAM current tensor density of the EM field. The last term
is the angular momentum current tensor due to longitudinal component of
electric field and we incorporate that into the OAM current tensor of the EM
field. Equation (8c) shows that the electromagnetic Lagrangian also
contributes to the continuity Eq. (37).

S. Barnett introduced the angular momentum carrying terms in the source-free
electromagnetism [18] as quantities that behave as the currents in the
conservation equations written for angular momentum of light. The terms
helicity and OAM current tensors introduced here for the electromagnetic field
have a similar tensorial nature and extend to the QED interactions. These
quantities are gauge-invariant and connect to similar terms coming from the
Dirac fields.

Note that the expression for the angular momentum due to the electromagnetic
field in Eq. (7) originally has contribution from the longitudinal electric
field, $\bm{E}^{\parallel}$ . This is the contribution due to the free charges
($\nabla\cdot\bm{E}^{\parallel}=\rho/\epsilon_{0}$) which decays as a function
of $1/r^{2}$ outside of the region of charges [34](Appendix). We have shown
that the contribution from this component, namely the term
$\bm{E}^{\parallel}\times\bm{A}^{\bot}+E_{i}^{\parallel}(\bm{r}\times\nabla)A_{i}^{\bot}$,
can be written as the divergence of some quantities involving
$\bm{E}^{\parallel}$ (Appendix). Therefore, when integrated over the entire
space, these terms vanish and thus the total global angular momentum of the
interacting field does not involve longitudinal electric field components.
Therefore, we have shown that the angular momentum density of the EM field
only depends on the transverse component of the electric field $\bm{E}^{\bot}$
since it is only these terms that give nonzero contribution to the total
angular momentum.

It should be emphasized that, when working with global quantities (integrated
over entire space), the contribution from the longitudinal electric field as
well as the contribution from $T_{ij}$ in Eq. (8) become surface terms and
vanish assuming that the field quantities are zero on the surface of the
integration volume. In such cases, the global angular momentum can be written
in equivalent forms which are different only by the divergence of a function
of the fields. In the local case, however, all of the terms in Eq. (8) are
observable quantities and each have a different interpretation.

|  |  time-derivative $+$ divergence of $+$ gradient of $=$ spin-orbit
---|---|---
|  |  of spin helicity current tensor helicity torque
| $\hat{\rho}$ |  |  |  |
Dual-Mode Optical Fiber | $\hat{\phi}$ |  |  |  |
| $\hat{z}$ |  |  |  |
Figure 2: Local conservation law for the spin [Eq. (12)] in a dual-mode
optical fiber. The individual terms in Eq. (12) are plotted on the first three
columns from left: The first column is the time-derivative of spin
($\frac{\partial}{\partial t}(\bm{E}^{\bot}\times A^{\bot})$), the second
column is the divergence of helicity current tensor
($-\nabla_{i}(A_{i}^{\bot}\bm{B})/\mu_{0}$) , and the third column is the
gradient of the helicity density ($\nabla(\bm{A}^{\bot}\cdot\bm{B})/\mu_{0}$)
. The fourth column is the EM spin-orbit torque given by Eq. (11) (in this
case $\tau_{\text{em}}=(\bm{B}\cdot\nabla)\bm{A}^{\bot}/\mu_{0}$). The three
rows show the local value of each vector along the three axes of optical fiber
problem: $\hat{\rho}$ radial direction, $\hat{\phi}$ azimuthal direction, and
$\hat{z}$ axis of the fiber. Note that adding the first three column on each
row together gives the last column $\bm{\tau}_{\text{em}}$; thus confirming
Eq. (12). The results are for an optical fiber of radius $50~{}\mu$m with the
two modes at the wavelengths $4.3~{}\mu$m and $4.29~{}\mu$m.

###  3.3 Spin-orbit torque

We now study the spin-orbital angular momentum exchange and the angular
momentum transfer between the Dirac and EM fields. The continuity equation in
Eqs. (28) and (37) only give the dynamics of the total angular momentum
density. To obtain insight into the detailed interaction between spin and OAM
of the Dirac and EM fields, we separately write the four-divergence of the
gauge-independent spin and OAM tensors of the Dirac
($\text{S}^{ij,\lambda}_{\text{D}}$ and $\text{L}^{ij,\lambda}_{\text{D}}$) as
well as the EM fields ($\text{S}^{ij,\lambda}_{\text{em}}$ and
$\text{L}^{ij,\lambda}_{\text{em}}$). By incorporating the Dirac and Maxwell’s
equations into the gauge-invariant angular momentum densities in Eqs. (7) and
(8), we obtain their detailed conservation equations (see Appendix),

$\partial_{\lambda}\text{S}_{\text{D}}^{ij,\lambda}=\varepsilon_{ijk}\left[\bm{\tau}_{\text{D}}+\bm{j}_{c}\times\bm{A}^{\bot}\right]_{k},$
(9a)
$\\!\\!\\!\partial_{\lambda}\text{L}_{\text{D}}^{ij,\lambda}\\!=\\!\varepsilon_{ijk}\\!\left[-\bm{\tau}_{\text{D}}\\!+\\!\rho(\bm{r}\times\bm{E}^{\parallel})\\!+\\!j_{c,l}(\bm{r}\times\bm{\nabla})A_{l}^{\bot}\right]_{k}\\!\\!,$
(9b)
$\partial_{\lambda}\text{S}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\left[\bm{\tau}_{\text{em}}-\bm{j}_{c}\times\bm{A}^{\bot}\right]_{k},$
(9c)
$\\!\\!\\!\partial_{\lambda}\text{L}_{\text{em}}^{ij,\lambda}\\!=\\!\varepsilon_{ijk}\\!\left[-\bm{\tau}_{\text{em}}\\!-\\!\rho(\bm{r}\times\bm{E}^{\parallel})\\!-\\!j_{c,l}(\bm{r}\times\bm{\nabla})A^{\bot}_{l}\right]_{k}\\!\\!,$
(9d)

where we have defined

$\bm{\tau}_{\text{D}}=-c\mathcal{R}\left\\{\bar{\psi}\left(\bm{\gamma}\times\bm{p}_{\parallel}\right)\psi\right\\}$
(10)

and

$\bm{\tau}_{\text{em}}=\frac{1}{\mu_{0}}(\bm{B}\cdot\nabla)\bm{A}^{\bot}-\epsilon_{0}\left(\frac{\partial\bm{E}^{\parallel}}{\partial
t}\times\bm{A}^{\bot}\right)$ (11)

as the Dirac and Maxwell spin-orbit torque, respectively, since it gives the
amount of torque exerted on the spin from the OAM of the fields and vice
versa. This nomenclature is further motivated by the resemblance of the Dirac
spin-orbit torque (Eq. (10)) of the Rashba spin-orbit coupling Hamiltonian
[48]. The direct connection between these terms, however, is out of scope of
this article and is the focus of a future work. Note that
$\bm{\gamma}=\gamma^{1}\hat{x}+\gamma^{2}\hat{y}+\gamma^{3}\hat{z}$ and
$\bm{j}_{c}=ec\bar{\psi}\bm{\gamma}\psi$ is the electric charge current
density in Eqs. (9) and (10).

Equation (9) clearly shows that the spin-orbit torques contribute to the spin-
OAM exchange in both Dirac and EM fields. Moreover, it is evident from Eqs.
(29a), (29b), (33a), and (33b) that the charge-field coupling terms,
$\bm{j}_{c}\times\bm{A}^{\bot}$, $j_{c,l}(\bm{r}\times\nabla)A_{l}^{\bot}$,
and $\rho(\bm{r}\times\bm{E}^{\parallel})$, are responsible for the angular
momentum transfer between the Dirac and EM fields [49]. In fact, the first
terms gives rise to an optical torque exerted on dipoles due to a circularly
polarized optical field [50]. Our results are significantly different from
scalar continuity equations of EM helicity in Refs. [3] and [4], which derive
a scalar conservation law for the dual-symmetric expressions of spin and
helicity of EM field. The helicity continuity equation obtained from the free-
space Maxwell equations can not characterize the spin-OAM exchange and
specifically the angular momentum transfer between Dirac and EM fields.

##  4 Source-free problems

We now show the importance of these continuity equations by demonstrating the
spin-OAM exchange via the spin-orbit torque for propagating EM fields. We
evaluate the terms in $\partial_{\lambda}\text{S}^{ij,\lambda}_{\text{em}}$
and the EM spin-orbit torque $\bm{\tau}_{\text{em}}$ (Eq. (11)) for two simple
EM problems in source-free regions. Similar spin-orbit signature can also be
observed in a cylindrical geometry for the Dirac fields [13]. However, a
thorough study of each individual term in Eq. (8) for Dirac and EM fields is
not the purpose of this paper.

The general form of $\partial_{\lambda}\text{S}^{ij,\lambda}_{\text{em}}$, in
a source-free regions, is

$\epsilon\frac{\partial}{\partial
t}(\bm{E}^{\bot}\times\bm{A}^{\bot}\\!)-\frac{1}{\mu_{0}}\nabla_{i}(A_{i}^{\bot}\\!\bm{B})\\!+\\!\frac{1}{\mu_{0}}\\!\nabla(\\!\bm{A}^{\bot}\\!\cdot\\!\bm{B})\\!=\bm{\tau}_{em}.\\!$
(12)

Here, the extra term on the right hand side results from the coupling to OAM.
The spin-orbit torque in the source-free case reduces to
$\bm{\tau}_{em}\\!=\\!(\bm{B}\cdot\nabla)\bm{A}^{\bot}/\mu_{0}$. We emphasize
that different from free-space case [3, 4], the near-field spin-OAM exchange
can still exist in the presence of sources.

###  4.1 dual-mode optical fiber

We first consider a dual-mode optical fiber, which is placed along the $z$
axis with radius $a$. The two modes have the same propagation constant,
$\beta$, and different frequencies of $\omega_{1}$ and $\omega_{2}$. Figure 2
shows the four terms in Eq. (12). The three rows show the components of these
quantities along the radius $\hat{\rho}$, azimuthal $\hat{\phi}$, and $z$ axis
of the fiber. Note that for all three rows, the sum of the first three terms
is equal to $\bm{\tau}_{\text{em}}$; thus satisfying Eq. (12). These figures
show that, for a dual-mode optical fiber, the helicity current current tensor,
helicity density, spin density, as well as the spin-orbit torque are all non-
zero and can play a role in a local interaction between optical modes and
atomic sources. Note that all three components become zero as the fiber radius
$a\to\infty$, which confirms that the spin-orbit torque
$\bm{\tau}_{\text{em}}$ becomes zero for plane wave solutions (Appendix).

(a)  (b)

Figure 3: Dynamics of the terms in Eq. (12) versus (a) $z$ and (b) time. The
plots only show the $\hat{z}$ component of each term. These plots show that
the spin-orbit torque ($\bm{\tau}_{\text{em}}$) is equal to the sum of the
other terms. The inset in panel (a) shows the location where the terms are
evaluated for both of the figures.

Figure 3 shows the dynamics of the time-derivative of spin, divergence of
helicity current density, and gradient of helicity (the terms in Eq. (12)) as
a function of $z$ (Fig. (3a)) and time (Fig. (3b)). The curves show the value
of these terms at a point $0.05~{}a$ from the fiber as shown in the insets.
The black curve with the circle markers show the EM spin-orbit torque,
$\bm{\tau}_{\text{em}}$, which is essentially the torque that is transferred
to the OAM of the fields. Note that the sum of other curves equals the black
curve $\bm{\tau}_{\text{em}}$; satisfying the continuity Eq. (12).

###  4.2 double plane wave interference

The second example is the interference of two circularly polarized plane waves
at two different frequencies. For the plane wave solutions, the current term
$\nabla_{i}(A_{i}^{\bot}\bm{B})$ and the spin-orbit torque $\bm{\tau}_{em}$
vanish (Appendix). Therefore, we arrive at a simpler conservation law for the
second problem,

$\frac{\partial}{\partial
t}\left(\epsilon_{0}\bm{E}^{\bot}\times\bm{A}^{\bot}\right)+c\bm{\nabla}\left(\frac{1}{\mu_{0}c}\bm{A}^{\bot}\cdot\bm{B}\right)=0,$
(13)

which shows the spin density propagating in space with light speed $c$. For
circularly polarized plane waves propagating along $z$ direction, the electric
field is written as
$\bm{E}=\mathcal{R}\\{\bm{\mathcal{E}}_{1}e^{-i(\omega_{1}t-k_{1}z)}+\bm{\mathcal{E}}_{2}e^{-i(\omega_{2}t-k_{2}z)}\\}$,
where $\bm{\mathcal{E}}_{i}=\mathcal{E}_{i}(\hat{x}+i\hat{y})/\sqrt{2}$ are
the complex electric field amplitudes of the two modes with frequencies
$\omega_{i}/c=k_{i}$. For these fields, we find (Appendix)

$\begin{split}\frac{1}{\mu_{0}}&\nabla(\bm{A}^{\bot}\cdot\bm{B})=-\epsilon_{0}\frac{\partial}{\partial
t}(\bm{E}\times\bm{A}^{\bot})\\\
=&\epsilon_{0}\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2\omega_{1}\omega_{2}}\mathcal{I}\left\\{\mathcal{E}_{1}\mathcal{E}_{2}^{*}e^{-i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\},\end{split}$
(14)

hence satisfying the conservation Eq. (13). This clearly shows that the change
in time of the spin is compensated by the gradient of the helicity of the EM
field.

##  5 Experimental perspective

The interaction between the electronic spin and optical angular momentum has
been widely used in devices such as optical circulators, insulators, electro-
optic modulators, and Faraday rotators. In these devices, conservation of
angular momentum combined with the field properties of the electromagnetic
(EM) radiation gives rise to non-reciprocal effects that can be manipulated
for a wide variety of applications. These devices incorporate the interaction
between EM radiation and the magnetic materials in the far-field regions where
a conservation law based on the total electronic and photonic angular momenta
suffices to explain the underlying physical phenomena [51].

Due to the possibility of optical control of the quantum spin states, the
dynamics of angular momentum between the electronic and optical fields has
recently gained attention for the systems interacting in the near-field
region. Coupling to the local spin of EM field has been observed for cold
atoms in the vicinity of an optical fiber [52], magnons interacting with
spherical whispering gallery modes (WGMs) [53], quantum dots in photonic
crystals [54], as well as quantum sources coupling to other waveguide systems
[55, 56, 8]. Non-classical spin texture and non-local photonic spin noise has
been shown in quantum structured light [57, 58], which can be measured in
experiment via nano-scale quantum sensors for photonic spin density [59]. In
such systems, since the interaction between the source and the EM field occurs
in the near-field rather than the far-field regions, a local approach to the
governing dynamical equations of angular momentum becomes important. These
studies reveal the new insight that the local interaction with classical
electromagnetic field has to offer.

With recent advances in the cold atom and quantum dot communities, local
interactions between atomic sources and optical fields are gaining more
attention due to the emergent new phenomena. These experiments emphasize the
need for a local conservation law describing the dynamics of angular momentum
in light-matter interactions.

We applied the Noether’s theorem to the Dirac Lagrangian interacting with the
EM field to find the local conservation laws of angular momentum for the most
general electrodynamics problem. The results developed here can be applied to
near-field as well as far-field to study the transformation of angular
momentum between different fields in these regions. Our results show that, in
consideration of local conservation laws, other quantities including helicity
and OAM current tensors, EM helicity, and electronic chirality should also be
considered in addition to the spin and OAM of the EM and Dirac fields.

Equation (37) holds everywhere in space and time. This shows that the total
angular momentum density $\bm{M}$ is not locally conserved. In other words,
$\frac{\partial\bm{M}}{\partial t}\neq 0$ and the conservation law can only be
written after including all the other terms in Eqs. (37) and (8). Integrating
Eq. (37) over some volume $V$, on the surface of which both Dirac and EM
fields become zero, $\psi\to 0$, $\bm{E}\to 0$, gives the usual global
conservation law, $\frac{\partial}{\partial t}\int_{V}\bm{M}d^{3}x=0$. This
states that the integrated values of the spin and OAM densities of electron
and EM field over the entire space is a conserved quantity (Fig. 1a).

We can also get a simpler continuity equation than Eq. (37) if we limit
ourselves to regions outside the Dirac fields. To do so, we take the integral
of Eq. (37) in the volume $V^{\prime}$ on which only the Dirac fields become
zero $\psi=0$. Doing so we get the semi-local conservation law (Appendix),

$\frac{\partial\tilde{\bm{M}}_{\text{D}}}{\partial
t}=-\left[\frac{\partial\tilde{\bm{M}}_{\text{em}}}{\partial
t}+\tilde{\bm{J}}+\tilde{\bm{h}}+\int_{S^{\prime}}\hat{\bm{n}}\times(\bm{r}{\mathfrak{N}}_{\text{em}})da\right]$
(15)

which expresses the time-evolution of the angular momentum of electron
$\bm{\tilde{}}{\bm{M}}_{\text{D}}$ in terms of EM dependent quantities.
$\tilde{\bm{M}}_{\text{em}}$ is the EM angular momentum in the region of the
source, while $\tilde{\bm{J}}$ and $\tilde{\bm{h}}$ are the EM angular
momentum current tensor (projected onto the normal of the surface) and
helicity (multiplied by the normal of the surface), integrated on the surface
of the volume surrounding the current charges (Appendix). Also,
$\mathfrak{N}_{\text{em}}$ is given by Eq. (8c). Equation (15) can be used to
find the conservation of angular momentum in the near-field of current sources
and how is it transferred to the electromagnetic fields.

It is also important to note that the conservation of angular momentum derived
here is based on the spatial components of Lorentz transformations (i.e.
rotations). Time-space components of the Lorentz transformation can also give
conservation equations related to the boost operators in the relativistic
equations of motion. These equations provide a new conservation equation for
the Dirac-Maxwell fields which can be of interest to applications
investigating relativistic behaviour of the particles and their pertinent
conserved quantities when interacting. A brief discussion about these
conservation equations is given in the Appendix.

The method presented in this paper can be further extended to find the
dynamics of magnetization in different materials. The simplest system can be
regarded as the interaction between an externally applied EM field and the
electrons in a non-magnetic metal. A weak probe signal can then be used to
study spin dynamics of the metal. Such a system can be closely modeled as a
non-interacting electron gas whose dynamics can be described by the Dirac
equation. Although the solutions of the Dirac equation interacting with an
externally applied plane wave can be rigorously found [60, 61], these
solutions are extremely complicated and only apply to the particular case of
free electrons in a plane wave. Our method, circumvents the problem of solving
the Dirac Hamiltonian to find the spin dynamics of the electrons, by using
Noether’s theorem and finding the conservation equation governing angular
momentum dynamics of the electrons. By knowing the properties of the
externally applied EM field, a simpler equation such as Eq. (15) can be used
to find the dynamics of angular momentum without the need of electronic
wavefunctions. Since in our derivations we make no simplifying assumption on
the EM field, these equations can be easily used for interactions that take
place in the near-field region and where EM fields cannot be regarded as plane
waves.

The Dirac equation can be further extended to model ferromagnetic materials.
Dirac-Kohn-Sham (DKS) equation is an extension to the Dirac equation which
accounts for the Kohn-Sham potential as well as the spin-polarized part of the
exchange correlation potential inside a magnetic material [62, 63].
Corrections from DKS equation can be added to the usual Pauli Hamiltonian to
account for the terms in the Landau-Lifshitz-Gilbert (LLG) equation and to add
corrections accounting for higher order terms dependent on external and
internal parameters [64]. The method presented in this paper can also be
applied to the DKS Hamiltonian to derive the conservation equation for the
angular momentum of electrons in magnetic materials.

##  6 Acknowledgements

This work is partially supported by the funding from Army Research Office
(W911NF-21-1-0287). L.P.Y is funded by National Key R&D Program of China (No.
2021YFE0193500). L.P.Y was a post-doctoral scholar and Farhad Khosravi was a
visiting scholar at Purdue University when this collaborative research work
was performed.

##  7 Data Availability Statement

All data that support the findings of this study are included within the
article (and any supplementary files).

##  Appendix A  Conserved Angular Momentum Tensor

Symmetrized Dirac Lagrangian, with the minimal coupling term [31], is written
as

$\mathfrak{L}=\bar{\psi}\left[c\gamma^{\mu}(\frac{i\hbar}{2}\overleftrightarrow{\partial}_{\mu}-eA_{\mu})-mc^{2}\right]\psi-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}$
(16)

where
$\overleftrightarrow{\partial}=\overleftarrow{\partial}+\overrightarrow{\partial}$
with $\overleftarrow{\partial}$ and $\overrightarrow{\partial}$ acting only on
$\bar{\psi}$ and $\psi$, respectively,

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=\begin{pmatrix}0&{E_{x}}/{c}&{E_{y}}/{c}&{E_{z}}/{c}\\\
-{E_{x}}/{c}&0&-B_{z}&B_{y}\\\ -{E_{y}}/{c}&B_{z}&0&-B_{x}\\\
-{E_{z}}/{c}&-B_{y}&B_{x}&0\end{pmatrix}$ (17)

is the electromagnetic tensor, and $\gamma^{\mu}$ are the Dirac gamma matrices
with the property $\\{\gamma^{\mu},\gamma^{\nu}\\}=\eta^{\mu\nu}$, where

$\eta^{\mu\nu}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}$ (18)

is the Minkowski metric tensor with the signature $(+---)$. We can get the
conserved currents related to the rotational symmetry of the Lagrangian, using
the Noether’s theorem, as [17, 31]:

$\mathcal{M}^{\mu\nu,\lambda}=\left(-\frac{i}{2}\frac{\partial\mathfrak{L}}{\partial(\partial_{\lambda}\psi)}\hat{M}_{\text{D}}^{\mu\nu}\psi\right)+\left(-\frac{i}{2}\bar{\psi}\hat{M}_{\text{D}}^{\mu\nu}\frac{\partial\mathfrak{L}}{\partial(\partial_{\lambda}\bar{\psi})}\right)+\left(-\frac{i}{2}\frac{\partial\mathfrak{L}}{\partial(\partial_{\lambda}A^{\kappa})}\right){(\hat{M}_{\text{em}}^{\mu\nu})^{\kappa}}_{\sigma}A^{\sigma}+\frac{1}{2}\left(\eta^{\lambda\mu}\mathfrak{L}x^{\nu}-\eta^{\lambda\nu}\mathfrak{L}x^{\mu}\right)$
(19)

where

$\hat{M}_{\text{D}}^{\mu\nu}=\hat{L}^{\mu\nu}+\hat{\Sigma}^{\mu\nu}$ (20)

is the angular momentum operator for the Dirac fields with

$\hat{\Sigma}^{\mu\nu}=\frac{1}{2}\sigma^{\mu\nu},\quad\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$
(21) $\hat{L}^{\mu\nu}=x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu},$ (22)

and

${(\hat{M}_{\text{em}}^{\mu\nu})^{\kappa}}_{\sigma}=\hat{L}^{\mu\nu}\delta^{\kappa}_{\sigma}+{(\hat{S}^{\mu\nu})^{\kappa}}_{\sigma}$
(23)

is the angular momentum operator for the electromagnetic fields with
$\hat{L}^{\mu\nu}$ given by Eq. (22), $\delta^{\kappa}_{\sigma}$ being the
Kronecker delta function, and

${(\hat{S}^{\mu\nu})^{\kappa}}_{\sigma}=i\left(\eta^{\mu\kappa}{\eta^{\nu}}_{\sigma}-{\eta^{\mu}}_{\sigma}\eta^{\nu\kappa}\right).$
(24)

Plugging these equations into Eq. (19), we get for the angular momentum
currents

$\mathcal{M}^{\mu\nu,\lambda}=\mathcal{M}_{\text{D}}^{\mu\nu,\lambda}+\mathcal{M}_{\text{em}}^{\mu\nu,\lambda}$
(25)

where

$\mathcal{M}_{\text{D}}^{\mu\nu,\lambda}=\mathcal{S}_{\text{D}}^{\mu\nu,\lambda}+\mathcal{L}_{\text{D}}^{\mu\nu,\lambda}$
(26a) $\mathcal{S}_{\text{D}}^{\mu\nu,\lambda}=\frac{\hbar
c}{4}\bar{\psi}\left(\gamma^{\lambda}\sigma^{\mu\nu}+\sigma^{\mu\nu}\gamma^{\lambda}\right)\psi$
(26b) $\mathcal{L}_{\text{D}}^{\mu\nu,\lambda}=\hbar
c\mathcal{R}\left\\{\bar{\psi}\gamma^{\lambda}(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})\psi\right\\}+(\eta^{\lambda\mu}x^{\nu}-\eta^{\lambda\nu}x^{\nu})\mathfrak{L}_{\text{D}}$
(26c) $\mathfrak{L}_{\text{D}}=\bar{\psi}\left[i\hbar
c\frac{1}{2}\gamma^{\mu}\overleftrightarrow{\partial}_{\mu}-mc^{2}\right]\psi-
ce\bar{\psi}\gamma^{\mu}\psi A_{\mu}$ (26d)

is the contribution due to the Dirac field and

$\mathcal{M}_{\text{em}}^{\mu\nu,\lambda}=\mathcal{S}_{\text{em}}^{\mu\nu,\lambda}+\mathcal{L}_{\text{em}}^{\mu\nu,\lambda}$
(27a)
$\mathcal{S}_{\text{em}}^{\mu\nu,\lambda}=-\frac{1}{\mu_{0}}\left(F^{\lambda\mu}A^{\nu}-F^{\lambda\nu}A^{\mu}\right)$
(27b)
$\mathcal{L}_{\text{em}}^{\mu\nu,\lambda}=-\frac{1}{\mu_{0}}\left[{F^{\lambda}}_{\kappa}(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})A^{\kappa}\right]+(\eta^{\lambda\mu}x^{\nu}-\eta^{\lambda\nu}x^{\mu})\mathfrak{L}_{\text{em}}$
(27c)
$\mathfrak{L}_{\text{em}}=-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2\mu_{0}}\left(\frac{\bm{E}\cdot\bm{E}}{c^{2}}-\bm{B}\cdot\bm{B}\right)$
(27d)

is the contribution due to the electromagnetic field. As a consequence of
Noether theorem, the angular momentum tensor $\mathcal{M}^{\mu\nu,\lambda}$ is
conserved. In other words,

$\partial_{\lambda}\mathcal{M}^{\mu\nu,\lambda}=0.$ (28)

For the tensors given in Eqs. (26) and (27), and using Maxwell and Dirac
equations, one can show that Eq. (28) holds for the total angular momentum
tensor.

For the case that $\mu\nu=ij$, where $i,j=1,2,3$, we find the angular momentum
currents due to rotations. We find for the spin and OAM currents of the Dirac
field

$\partial_{\lambda}\mathcal{S}_{\text{D}}^{ij,\lambda}=\varepsilon_{ijk}\left[\hbar\frac{\partial}{\partial
t}(\psi^{\dagger}\bm{\Sigma}\psi)+\frac{\hbar
c}{2}\nabla(\psi^{\dagger}\gamma^{5}\psi)\right]_{k},$ (29a)
$\partial_{\lambda}\mathcal{L}_{\text{D}}^{ij,\lambda}=\varepsilon_{ijk}\left[-\hbar\frac{\partial}{\partial
t}\mathcal{R}\left\\{i\psi^{\dagger}(\bm{r}\times\nabla)\psi\right\\}-\hbar
c\nabla\cdot\mathcal{R}\left\\{i\bar{\psi}\bm{\gamma}(\bm{r}\times\nabla)\psi\right\\}\right]_{k}$
(29b)

where

$\Sigma_{i}=\frac{1}{2}\varepsilon_{ijk}\sigma^{jk}=\frac{i}{4}\varepsilon_{ijk}[\gamma^{j},\gamma^{k}]=\frac{1}{2}\begin{pmatrix}\sigma_{i}&0\\\
0&\sigma_{i}\end{pmatrix},$ (30)

with $\sigma_{i}$ being the Pauli matrices, $\mathcal{R}\\{\cdots\\}$ takes
the real part of its argument,
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ is the chirality
operator in the Dirac equation [31], and

$\bm{\gamma}=\gamma^{1}\hat{x}+\gamma^{2}\hat{y}+\gamma^{3}\hat{z}.$ (31)

Note that we have used the fact that, using the Dirac equation,

$\left(i\hbar\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}A_{\mu}-mc\right)\psi=0,$
(32)

we get $\mathfrak{L}_{\text{D}}=0$ for the fields that follow Dirac equation
of motion. This is straightforward to show by multiplying Eq. (32) from left
by $c\bar{\psi}$. For the spin and OAM currents of the electromagnetic field
we find

$\partial_{\lambda}\mathcal{S}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\left[\epsilon\frac{\partial}{\partial
t}(\bm{E}\times\bm{A})-\frac{1}{\mu_{0}}\nabla\cdot(\bm{A}\bm{B})+\frac{1}{\mu_{0}}\nabla(\bm{A}\cdot\bm{B})\right]_{k}$
(33a)
$\partial_{\lambda}\mathcal{L}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\left\\{\epsilon\frac{\partial}{\partial
t}\left[\bm{E}\cdot(\bm{r}\times\nabla)\bm{A}\right]-\nabla\cdot\left[\frac{1}{\mu_{0}}\bm{B}\times(\bm{r}\times\nabla)\bm{A}-\epsilon\bm{E}(\bm{r}\times\nabla\phi)\right]+(\bm{r}\times\nabla)\mathfrak{L}_{\text{em}}\right\\}_{k}$
(33b)

The problem with Eqs. (29b) and (33) is that they are gauge-dependent, which
means that under the transformations $\psi\rightarrow\psi
e^{\frac{i}{e}\zeta}$ and $A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\zeta$
these expressions change. Therefore, the individual terms do not represent any
physically meaningful quantity. The fundamental equation to hold is Eq. (28).
Therefore, as long as this relation is satisfied, we can cast Eqs. (29b) and
(33) into gauge-independent forms. To do so, we break $\bm{A}$ into two
longitudinal and transverse parts as
$\bm{A}=\bm{A}^{\parallel}+\bm{A}^{\bot}$, where $\nabla\cdot\bm{A}^{\bot}=0$
and $\nabla\times\bm{A}^{\parallel}=0$ by definition. After some algebra, we
find for the new spin and OAM tensors of Dirac and electromagnetic field

$\partial_{\lambda}\text{S}_{\text{D}}^{ij,\lambda}=\varepsilon_{ijk}\left[\hbar\frac{\partial}{\partial
t}(\psi^{\dagger}\bm{\Sigma}\psi)+\frac{\hbar
c}{2}\nabla(\psi^{\dagger}\gamma^{5}\psi)\right]_{k},$ (34a)
$\partial_{\lambda}\text{L}_{\text{D}}^{ij,\lambda}=\varepsilon_{ijk}\left[\frac{\partial}{\partial
t}\mathcal{R}\left\\{\psi^{\dagger}(\bm{r}\times\bm{p}_{\parallel})\psi\right\\}+c\nabla\cdot\mathcal{R}\left\\{\bar{\psi}\bm{\gamma}(\bm{r}\times\bm{p}_{\parallel})\psi\right\\}\right]_{k},$
(34b)
$\partial_{\lambda}\text{S}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\left[\epsilon\frac{\partial}{\partial
t}(\bm{E}\times\bm{A}^{\bot})-\frac{1}{\mu_{0}}\nabla\cdot(\bm{A}^{\bot}\bm{B})+\frac{1}{\mu_{0}}\nabla(\bm{A}^{\bot}\cdot\bm{B})\right]_{k},$
(34c)
$\partial_{\lambda}\text{L}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\left\\{\epsilon\frac{\partial}{\partial
t}\left[\bm{E}\cdot(\bm{r}\times\nabla)\bm{A}^{\bot}\right]-\nabla\cdot\left[\frac{1}{\mu_{0}}\bm{B}\times(\bm{r}\times\nabla)\bm{A}^{\bot}+\epsilon\bm{E}(\bm{r}\times\bm{E}^{\parallel})\right]+(\bm{r}\times\nabla)\mathfrak{L}_{\text{em}}\right\\}_{k}.$
(34d)

where $\bm{p}_{\parallel}=-i\hbar\nabla-e\bm{A}^{\parallel}$ is the gauge-
independent covariant momentum operator of electron. We can further separate
the contribution of longitudinal electric field $\bm{E}^{\parallel}$ to the
angular momentum. This contribution can be written as

$\bm{E}^{\parallel}\times\bm{A}^{\bot}+E_{i}^{\parallel}(\bm{r}\times\nabla)A_{i}^{\bot}=-\nabla\times\left[\bm{r}(\bm{E}^{\parallel}\cdot\bm{A}^{\bot})\right]-\nabla\cdot\left[\bm{A}^{\bot}(\bm{r}\times\bm{E}^{\parallel})\right]$
(35)

When integrated over the entire space, both of these terms on r.h.s of this
expression become zero due to the Stokes theorem and thus the longitudinal
electric field does not contribute to the global angular momentum of the
electromagnetic field. For this reason, we move this term into the angular
momentum current terms so that the global angular momentum represents the
integrated angular momentum density. Making this change, we get for the new
components of the electromagnetic angular momentum currents

$\partial_{\lambda}\text{S}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\left[\epsilon\frac{\partial}{\partial
t}(\bm{E}^{\bot}\times\bm{A}^{\bot})-\frac{1}{\mu_{0}}\nabla\cdot(\bm{A}^{\bot}\bm{B})+\frac{1}{\mu_{0}}\nabla(\bm{A}^{\bot}\cdot\bm{B})\right]_{k},$
(36a)
$\begin{split}\partial_{\lambda}\text{L}_{\text{em}}^{ij,\lambda}=\varepsilon_{ijk}\Bigg{\\{}&\epsilon\frac{\partial}{\partial
t}\left[\bm{E}^{\bot}\cdot(\bm{r}\times\nabla)\bm{A}^{\bot}\right]-\nabla\cdot\left[\frac{1}{\mu_{0}}\bm{B}\times(\bm{r}\times\nabla)\bm{A}^{\bot}+\epsilon_{0}\bm{A}^{\bot}\left(\bm{r}\times\frac{\partial\bm{E}^{\parallel}}{\partial
t}\right)\right]\\\
+&(\bm{r}\times\nabla)\left[-\frac{1}{2\mu_{0}}\bm{B}\cdot\bm{B}+\frac{\epsilon_{0}}{2}\bm{E}^{\bot}\cdot\bm{E}^{\bot}+\epsilon_{0}\frac{\partial\bm{E}^{\parallel}}{\partial
t}\cdot\bm{A}^{\bot}\right]\Bigg{\\}}_{k}.\end{split}$ (36b)

Adding these four equations together we get Eq. (6) in the main manuscript,

$\frac{\partial\bm{M}}{\partial
t}+\nabla_{i}{J}_{ij}+\nabla\chi+\nabla_{i}N_{ij}=0$ (37)

where the terms are given in Eqs. (7) and (8) of the manuscript. Eq. (37) can
be written as

$\frac{\partial\bm{M}}{\partial t}+\nabla\cdot\overleftrightarrow{\bm{T}}=0$
(38)

where

$\overleftrightarrow{\bm{T}}=\chi\overleftrightarrow{I}+\overleftrightarrow{\bm{J}}+\overleftrightarrow{N}$
(39)

with
$\overleftrightarrow{I}=\hat{x}\hat{x}+\hat{y}\hat{y}+\hat{z}\hat{z}=\delta_{ij}\hat{x}_{i}\hat{x}_{j}$
, and

$\overleftrightarrow{N}=\varepsilon_{ijk}\hat{x_{i}}\hat{x_{j}}x_{k}\mathfrak{N}_{\text{em}}=\begin{pmatrix}0&z&-y\\\
-z&0&x\\\
y&-x&0\end{pmatrix}\mathfrak{N}_{\text{em}},\quad\mathfrak{N}_{em}=\frac{\epsilon_{0}}{2}\bm{E}^{\bot}\cdot\bm{E}^{\bot}-\frac{1}{2\mu_{0}}\bm{B}\cdot\bm{B}+\epsilon_{0}\frac{\partial\bm{E}^{\parallel}}{\partial
t}\cdot\bm{A}^{\bot}.$ (40)

Note that the first two terms of $\mathfrak{N}_{em}$ describe the Lagrangian
due to the transverse electric field, which can be interpreted as the
Lagrangian of the free photon [26], while the last term shows the interaction
between the transverse vector potential and the currents due to the
longitudinal electric fields. Note that the term
$\frac{\epsilon_{0}}{2}\bm{E}^{\parallel}\cdot\bm{E}^{\parallel}$ in the
Lagrangian of the electromagnetic field cancels out with the same term coming
from $\nabla_{i}J_{ij}$. The expressions for $\overleftrightarrow{\bm{J}}$ and
$\chi$ are given in the main manuscript.

Also note that the EM Lagrangian does not include the interaction term
$j_{c}^{\mu}A_{\mu}$ because it appears in the Dirac part of the Lagrangian
$\mathfrak{L}_{\text{D}}$. Dirac Lagrangian, as mentioned earlier, vanishes
from the expression for the OAM of the Dirac field since it satisfies the
Dirac equation. For this reason, the contribution from the Dirac Lagrangian
$\mathfrak{L}_{\text{D}}$ disappears from the conservation equations.

Equation (38) describes the local conservation law for the angular momentum
currents $\overleftrightarrow{\bm{T}}$ and the angular momentum density
(charge) $\bm{M}$. When integrated over the entire space, and assuming that
the fields vanish on the boundary of the this surface, the second term becomes
an integral over this surface and thus vanishes. In this case, we arrive at
the usual global conservation of angular momentum equation which states that
the total angular momentum of the Dirac and Maxwell fields is a constant.
However, in situations where the problem under consideration is an open
dissipative system, this simplification cannot be made and surface terms of
the angular momentum current can carry angular momentum out of the system.

##  Appendix B Spin-Orbit Torque

In this section we show that Eq. (34) leads to Eq. (9) in the main manuscript.
Starting with the equation for the spin of the Dirac fields, we get for the
$z$ component for instance,

$\partial_{\lambda}\text{S}_{\text{D}}^{12,\lambda}=\frac{\hbar}{2}\partial_{t}(\psi^{\dagger}\Sigma_{z}\psi)+\frac{\hbar
c}{2}\partial_{z}(\psi^{\dagger}\gamma^{5}\psi)$ (41)

where $\partial_{t}\equiv\frac{\partial}{\partial t}$ and
$\partial_{i}\equiv\frac{\partial}{\partial x_{i}}$. Using Dirac equation we
get

$\hbar\partial_{t}\psi=-\hbar
c\gamma^{0}(\bm{\gamma}\cdot\bm{\nabla}\psi)-iceA_{\mu}\gamma^{0}\gamma^{\mu}\psi-
imc^{2}\gamma^{0}\psi$ (42a) $\hbar\partial_{t}\psi^{\dagger}=\hbar
c(\bm{\nabla}\psi^{\dagger}\cdot\bm{\gamma})\gamma^{0}+iceA_{\mu}\psi^{\dagger}\gamma^{0}\gamma^{\mu}+imc^{2}\psi^{\dagger}\gamma^{0}$
(42b)

Using these equations we find

$\begin{split}\partial_{\lambda}\text{S}_{\text{D}}^{12,\lambda}=&\frac{\hbar}{2}\left\\{(\partial_{t}\psi^{\dagger})\Sigma_{z}\psi+\psi^{\dagger}\Sigma_{z}(\partial_{t}\psi)+c\partial_{z}(\psi^{\dagger}\gamma^{5}\psi)\right\\}\\\
+&\frac{1}{2}\Big{\\{}\hbar
c(\bm{\nabla}\psi^{\dagger}\cdot\bm{\gamma})i\gamma^{0}\gamma^{1}\gamma^{2}\psi-
ceA_{\mu}\psi^{\dagger}\gamma^{0}\gamma^{\mu}\gamma^{1}\gamma^{2}\psi-
mc^{2}\psi^{\dagger}\gamma^{0}\gamma^{1}\gamma^{2}\psi\\\ -&i\hbar
c\psi^{\dagger}\gamma^{1}\gamma^{2}\gamma^{0}(\bm{\gamma}\cdot\bm{\nabla}\psi)+ceA_{\mu}\psi^{\dagger}\gamma^{1}\gamma^{2}\gamma^{0}\gamma^{\mu}\psi+mc^{2}\psi^{\dagger}\gamma^{1}\gamma^{2}\gamma^{0}\psi+\hbar
c\partial_{z}(\psi^{\dagger}\gamma^{5}\psi)\Big{\\}}\\\
=&\frac{1}{2}\Big{\\{}\\!\\!-\\!\hbar
c\partial_{z}(\psi^{\dagger}\gamma^{5}\psi)\\!+\\!i\hbar
c(\partial_{x}\psi^{\dagger})\gamma^{0}\gamma^{2}\psi-i\hbar
c\psi^{\dagger}\gamma^{0}\gamma^{2}(\partial_{x}\psi)-i\hbar
c(\partial_{y}\psi^{\dagger})\gamma^{0}\gamma^{1}\psi+i\hbar
c\psi^{\dagger}\gamma^{0}\gamma^{1}(\partial_{y}\psi)\\\
+&2ceA_{1}\psi^{\dagger}\gamma^{0}\gamma^{2}\psi-2ceA_{2}\psi^{\dagger}\gamma^{0}\gamma^{1}\psi+\hbar
c\partial_{z}(\psi^{\dagger}\gamma^{5}\psi)\Big{\\}}\\\ =&\hbar
c\mathcal{R}\left\\{i\bar{\psi}(\gamma^{1}\partial_{y}-\gamma^{2}\partial_{x})\psi\right\\}-ce(A_{x}^{\parallel}\bar{\psi}\gamma^{2}\psi-
A_{y}^{\parallel}\bar{\psi}\gamma^{1}\psi)-(A_{x}^{\bot}j_{c,y}-A_{y}^{\bot}j_{c,x})\\\
=&-c\mathcal{R}\left\\{\bar{\psi}\left[\gamma^{1}(-i\hbar\partial_{y}-eA_{y}^{\parallel})-\gamma^{2}(-i\hbar\partial_{x}-eA_{x}^{\parallel})\right]\psi\right\\}-(A_{x}^{\bot}j_{c,y}-A_{y}^{\bot}j_{c,x})\\\
=&-c\mathcal{R}\left\\{\bar{\psi}(\bm{\gamma}\times\bm{p}_{\parallel})_{z}\psi\right\\}+(\bm{j}_{c}\times\bm{A}^{\bot})_{z}.\end{split}$
(43)

In this derivation we have used the facts that
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}$,
$(\gamma^{i})^{2}=-1$, $A_{1}=-A_{x}$, $A_{2}=-A_{y}$, and
$\bm{j}_{c}=ce\bar{\psi}\bm{\gamma}\psi$. We can do a similar derivation for
other components of $\partial_{\lambda}\text{S}_{\text{D}}^{ij,\lambda}$.
Doing so we find Eq. (9a) in the main manuscript. Note that the first term on
the r.h.s. of Eq. (43) is nothing but the spin-orbit torque of the Dirac field
$\bm{\tau}_{\text{D}}$ given in Eq. (10) of the paper.

We now turn into the equation for the OAM of the Dirac field Eq. (34b). We get
for the $z$ component, for instance

$\begin{split}\partial_{\lambda}\text{L}_{\text{D}}^{12,\lambda}=&\partial_{t}\mathcal{R}\left\\{\psi^{\dagger}(xp_{\parallel,y}-yp_{\parallel,x})\psi\right\\}+c\bm{\nabla}\cdot\mathcal{R}\left\\{\bar{\psi}\bm{\gamma}(xp_{\parallel,y}-yp_{\parallel,x})\psi\right\\}\\\
=&\mathcal{R}\Big{\\{}(\partial_{t}\psi^{\dagger})(xp_{\parallel,y}-yp_{\parallel,x})\psi+\psi^{\dagger}(xp_{\parallel,y}-yp_{\parallel,x})(\partial_{t}\psi)+c\bm{\nabla}\cdot\bar{\psi}\bm{\gamma}(xp_{\parallel,y}-yp_{\parallel,x})\psi\\\
-&e\psi^{\dagger}\psi(x\partial_{t}A_{y}^{\parallel}-y\partial_{t}A_{x}^{\parallel})\Big{\\}}\\\
=&\mathcal{R}\Big{\\{}c(\bm{\nabla}\psi^{\dagger}\cdot\bm{\gamma})\gamma^{0}(xp_{\parallel,y}\\!-\\!yp_{\parallel,x})\psi+\\!\frac{ice}{\hbar}A_{\mu}\psi^{\dagger}\gamma^{0}\gamma^{\mu}(xp_{\parallel,y}\\!\\!-\\!yp_{\parallel,x})\psi+\\!\frac{imc^{2}}{\hbar}\psi^{\dagger}\gamma^{0}(xp_{\parallel,y}\\!-\\!yp_{\parallel,x})\psi\\\
-&c\psi^{\dagger}(xp_{\parallel,y}\\!-\\!yp_{\parallel,x})\gamma^{0}(\bm{\gamma}\cdot\bm{\nabla}\psi)-\frac{ice}{\hbar}\psi^{\dagger}(xp_{\parallel,y}\\!-\\!yp_{\parallel,x})\gamma^{0}\gamma^{\mu}(A_{\mu}\psi)-\frac{imc^{2}}{\hbar}\psi^{\dagger}(xp_{\parallel,y}\\!-\\!yp_{\parallel,x})\gamma^{0}\psi\\\
+&c\bm{\nabla}\cdot\bar{\psi}\bm{\gamma}(xp_{\parallel,y}-yp_{\parallel,x})\psi-e\psi^{\dagger}\psi(x\partial_{t}A_{y}^{\parallel}-y\partial_{t}A_{x}^{\parallel})\Big{\\}}\\\
=&\mathcal{R}\Big{\\{}\\!\\!-\\!c\bm{\nabla}\cdot\bar{\psi}\bm{\gamma}(xp_{\parallel,y}-yp_{\parallel,x})\psi+c\bar{\psi}\left[\bm{\gamma}\cdot\bm{\nabla}(xp_{\parallel,y}\\!-\\!yp_{\parallel,x})\right]\psi-
ce\bar{\psi}\gamma^{\mu}\psi(x\partial_{y}-y\partial_{x})A_{\mu}\\\
+&c\bm{\nabla}\cdot\bar{\psi}\bm{\gamma}(xp_{\parallel,y}-yp_{\parallel,x})\psi-e\psi^{\dagger}\psi(x\partial_{t}A_{y}^{\parallel}-y\partial_{t}A_{x}^{\parallel})\Big{\\}}\\\
=&\mathcal{R}\left\\{c\bar{\psi}(\gamma^{1}p_{\parallel,y}-\gamma^{2}p_{\parallel,x})\psi\right\\}-ce\bar{\psi}\bm{\gamma}\psi(x\bm{\nabla}A_{y}^{\parallel}-y\bm{\nabla}A_{x}^{\parallel})-ce\bar{\psi}\gamma^{\mu}\psi(x\partial_{y}-y\partial_{x})A_{\mu}\\\
-&e\psi^{\dagger}\psi(x\partial_{t}A_{y}^{\parallel}-y\partial_{t}A_{x}^{\parallel})\\\
=&\mathcal{R}\left\\{c\bar{\psi}(\gamma^{1}p_{\parallel,y}-\gamma^{2}p_{\parallel,x})\psi\right\\}-ce\bar{\psi}\Big{[}\gamma^{1}(x\partial_{x}A_{y}^{\parallel}-y\partial_{x}A_{x}^{\parallel})+\gamma^{2}(x\partial_{y}A_{y}^{\parallel}-y\partial_{y}A_{x}^{\parallel})\\\
+&\gamma^{3}(x\partial_{z}A_{y}^{\parallel}-y\partial_{z}A_{x}^{\parallel})-\gamma^{1}(x\partial_{y}A_{x}-y\partial_{x}A_{x})-\gamma^{2}(x\partial_{y}A_{y}-y\partial_{x}A_{y})-\gamma^{3}(x\partial_{y}A_{z}-y\partial_{x}A_{z})\\\
+&\frac{1}{c}\gamma^{0}(x\partial_{y}\phi-y\partial_{x}\phi)+\frac{1}{c}\psi^{\dagger}(x\partial_{t}A_{y}^{\parallel}-y\partial_{t}A_{x}^{\parallel})\Big{]}\psi\\\
=&\mathcal{R}\left\\{c\bar{\psi}(\gamma^{1}p_{\parallel,y}-\gamma^{2}p_{\parallel,x})\psi\right\\}-ce\bar{\psi}\gamma^{1}\psi
x(\partial_{x}A_{y}^{\parallel}-\partial_{y}A_{x}^{\parallel})+ce\bar{\psi}\gamma^{1}\psi(x\partial_{y}-y\partial_{x})A_{x}^{\bot}\\\
-&ce\bar{\psi}\gamma^{2}\psi
y(\partial_{x}A_{y}^{\parallel}-\partial_{y}A_{x}^{\parallel})+ce\bar{\psi}\gamma^{2}\psi(x\partial_{y}-y\partial_{x})A_{y}^{\bot}-ce\bar{\psi}\gamma^{3}\psi
x(\partial_{z}A_{y}^{\parallel}-\partial_{y}A_{z}^{\parallel})\\\
+&ce\bar{\psi}\gamma^{3}\psi
y(\partial_{z}A_{x}^{\parallel}\\!-\\!\partial_{x}A_{z}^{\parallel})+\\!ce\bar{\psi}\gamma^{3}\psi(x\partial_{y}\\!-\\!\\!\\!y\partial_{x})A_{z}^{\bot}+e\psi^{\dagger}\psi\left[x(-\partial_{y}\phi\\!-\\!\partial_{t}A_{y}^{\parallel})\\!-\\!y(-\partial_{x}\phi-\partial_{t}A_{x}^{\parallel})\right]\\\
=&~{}c\mathcal{R}\left\\{\bar{\psi}(\bm{\gamma}\times\bm{p}_{\parallel})_{z}\psi\right\\}+j_{c,i}(\bm{r}\times\bm{\nabla})_{z}A_{i}^{\bot}+\rho(\bm{r}\times\bm{E}^{\parallel})_{z}\end{split}$
(44)

where we have used the facts that $\bm{\nabla}\times\bm{A}^{\parallel}=0$ and
$\bm{E}^{\parallel}=-\bm{\nabla}\phi-\partial_{t}\bm{A}^{\parallel}$. Writing
similar equations for the other components we get Eq. (9b) in the main
manuscript.

We can repeat this derivation for the electromagnetic spin and OAM currents as
well. Using Eq. (36a) we find, again for the $z$ component for instance,

$\begin{split}\partial_{\lambda}\text{S}_{\text{em}}^{12,\lambda}=&\epsilon_{0}\partial_{t}(\bm{E}^{\bot}\times\bm{A}^{\bot})_{z}-\frac{1}{\mu_{0}}\bm{\nabla}\cdot(\bm{A}^{\bot}B_{z})+\frac{1}{\mu_{0}}\partial_{z}(\bm{A}^{\bot}\cdot\bm{B})\\\
=&\epsilon_{0}\partial_{t}(\bm{E}^{\bot}\times\bm{A}^{\bot})_{z}-\frac{1}{\mu_{0}}B_{z}(\bm{\nabla}\cdot\bm{A}^{\bot})-\frac{1}{\mu_{0}}(\bm{A}^{\bot}\cdot\bm{\nabla})B_{z}\\\
+&\frac{1}{\mu_{0}}\left\\{(\bm{A}^{\bot}\cdot\bm{\nabla})B_{z}+(\bm{B}\cdot\bm{\nabla})A_{z}^{\bot}+[\bm{A}^{\bot}\times(\bm{\nabla}\times\bm{B})]_{z}+[\bm{B}\times(\bm{\nabla}\times\bm{A}^{\bot})]_{z}\right\\}\\\
=&\epsilon_{0}\partial_{t}(\bm{E}^{\bot}\times\bm{A}^{\bot})+\frac{1}{\mu_{0}}(\bm{B}\cdot\bm{\nabla})A_{z}^{\bot}-\epsilon_{0}[(\partial_{t}\bm{E})\times\bm{A}^{\bot}]_{z}-(\bm{j}_{c}\times\bm{A}^{\bot})_{z}\\\
=&\epsilon_{0}[\bm{E}^{\bot}\times(\partial_{t}\bm{A}^{\bot})]_{z}+\frac{1}{\mu_{0}}(\bm{B}\cdot\bm{\nabla})A_{z}^{\bot}-\epsilon_{0}\left[\frac{\partial\bm{E}^{\parallel}}{\partial
t}\times\bm{A}^{\bot}\right]-(\bm{j}_{c}\times\bm{A}^{\bot})_{z}\\\
=&\frac{1}{\mu_{0}}(\bm{B}\cdot\bm{\nabla})A_{z}^{\bot}-\epsilon_{0}\left(\frac{\partial\bm{E}^{\parallel}}{\partial
t}\times\bm{A}^{\bot}\right)_{z}-(\bm{j}_{c}\times\bm{A}^{\bot})_{z},\end{split}$
(45)

where we have used the facts that $\bm{\nabla}\cdot\bm{A}^{\bot}=0$,
$\bm{B}=\bm{\nabla}\times\bm{A}^{\bot}$,
$\bm{E}^{\bot}=-\partial_{t}\bm{A}^{\bot}$, and
$\bm{\nabla}\times\bm{B}=\frac{1}{c^{2}}\partial_{t}\bm{E}+\mu_{0}\bm{j}_{c}$
. By repeating this derivation for the other components we get Eq. (9c) in the
paper. Note that the first two terms are the spin-orbit torque of the
electromagnetic field, $\bm{\tau}_{\text{em}}$, given in Eq. (11) of the main
manuscript.

We can repeat this derivation to find the equation for
$\partial_{\lambda}\text{L}^{ij,\lambda}_{\text{em}}$. However, using the
continuity condition,

$\partial_{\lambda}(\text{S}_{\text{D}}^{ij,\lambda}+\text{L}_{\text{D}}^{ij,\lambda}+\text{S}_{\text{em}}^{ij,\lambda}+\text{L}_{\text{em}}^{ij,\lambda})=0,$
(46)

it is straightforward to show that Eq. (9d) of the main manuscript holds.

##  Appendix C Semi-Local Conservation Law

We can get the semi-local conservation laws by integrating the terms in Eqs.
(34a), (34b), and (36) over the volume $V^{\prime}$, on which surface the
Dirac eigenfunctions $\psi$ become zero. Using Gauss’s theorem, the integral
of the terms $\nabla(\psi^{\dagger}\gamma^{5}\psi)$ and
$\nabla\cdot\mathcal{R}\\{\bar{\psi}\bm{\gamma}(\bm{r}\times\bm{p})\psi\\}$
vanish because they become surface integrals of functions of $\psi$. We
therefore arrive at the semi-local conservation law

$\frac{\partial\tilde{\bm{M}}_{\text{D}}}{\partial
t}+\frac{\partial\tilde{\bm{M}}_{\text{em}}}{\partial
t}+\tilde{{\bm{J}}}_{\text{A}}+\tilde{\bm{h}}+\int_{V^{\prime}}\nabla\times(\bm{r}\mathfrak{N}_{\text{em}})dV^{\prime}=0$
(47)

where $\tilde{\bm{M}}_{\text{D}}$ is the total angular momentum of the
electron given by

$\tilde{\bm{M}}_{\text{D}}=\hbar\int_{V^{\prime}}\left[\frac{1}{2}(\psi^{\dagger}\bm{\Sigma}\psi)+\mathcal{R}\\{\psi^{\dagger}(\bm{r}\times\bm{p})\psi\\}\right]dV^{\prime},$
(48)

$\tilde{\bm{M}}_{\text{em}}$ is the EM total angular momentum in the volume
$V^{\prime}$,

$\tilde{\bm{M}}_{\text{em}}=\int_{V^{\prime}}\epsilon(\bm{E}^{\bot}\times\bm{A}^{\bot})d^{3}x+\int_{V^{\prime}}\epsilon\left[\bm{E}^{\bot}\cdot(\bm{r}\times\nabla)\bm{A}^{\bot}\right]d^{3}x,$
(49)

and

$\tilde{{\bm{J}}}_{\text{A}}=-\int_{V^{\prime}}\nabla\cdot\left[\frac{1}{\mu_{0}}\bm{A}^{\bot}\bm{B}+\frac{1}{\mu_{0}}\bm{B}\times(\bm{r}\times\nabla)\bm{A}^{\bot}+\epsilon\bm{A}^{\bot}\left(\bm{r}\times\frac{\bm{\partial}E^{\parallel}}{\partial
t}\right)\right]dV^{\prime},$ (50a)
$\tilde{\bm{h}}=\int_{V^{\prime}}\nabla(\bm{A}^{\bot}\cdot\bm{B})dV^{\prime}.$
(50b)

Using Gauss’s theorem the volume integrals in Eqs. (47), (50a) and (50b) can
be converted into surface integrals. We find

$\tilde{\bm{J}}_{\text{A}}=-\int_{S^{\prime}}\hat{n}_{i}\Big{[}A_{i}^{\bot}\bm{B}+\epsilon_{ijk}B_{j}(\bm{r}\times\nabla)A_{k}^{\bot}+\epsilon
A^{\bot}_{i}\left(\bm{r}\times\frac{\bm{\partial}E^{\parallel}}{\partial
t}\right)\Big{]}da$ (51a)

$\tilde{\bm{h}}=\int_{S^{\prime}}\hat{\bm{n}}(\bm{A}^{\bot}\cdot\bm{B})da$
(51b)

where $\hat{\bm{n}}$ is the unit vector normal to the surface of the volume
$V^{\prime}$, and $da$ its surface element. Equation (47) presents an equation
for the time evolution of angular momentum of the electron in terms of the EM
fields.

##  Appendix D Symmetrized Angular Momentum Tensor

In our derivation, we have used the canonical form of the angular momentum
tensor which is derived directly from the application of Noether theorem to
the QED Lagrangian in Eq. (16). The angular momentum tensor can be written in
terms of the canonical energy-momentum tensor, $T^{\mu\nu}$, as

$\mathcal{M}^{\mu\nu,\lambda}=x^{\mu}T^{\lambda\nu}-x^{\nu}T^{\lambda\mu}+S^{\mu\nu,\lambda}.$
(52)

Conservation of angular momentum implies that, when
$\partial_{\lambda}S^{\mu\nu,\lambda}\neq 0$,

$T^{\mu\nu}\neq T^{\nu\mu}$ (53)

This in unpleasant because in general relativity, the energy-momentum tensor
is directly proportional to the metric tensor which is symmetric in $\mu$ and
$\nu$. To overcome this problem, the energy momentum tensor can be modified to
the so-called Bellifante-Resenfeld energy-momentum tensor as [21, 65]

${T}^{{}^{\prime}\mu\nu}=T^{\mu\nu}+\frac{1}{2}\partial_{\lambda}(S^{\nu\lambda,\mu}+S^{\mu\lambda,\nu}-S^{\nu\mu,\lambda})$
(54)

It can be shown that this new energy-momentum tensor is symmetric and does not
change the conservation law of the energy-momentum tensor,
$\partial_{\mu}{T}^{{}^{\prime}\mu\nu}=0$. We can therefore write a new
symmetrized angular momentum tensor, ${M}^{{}^{\prime}\mu\nu,\lambda}$ as

$\mathcal{M}^{{}^{\prime}\mu\nu,\lambda}=x^{\mu}{T}^{{}^{\prime}\lambda\nu}-x^{\nu}{T}^{{}^{\prime}\lambda\mu}$
(55)

where we do not need to include the additional spin tensor because it is
already present in the symmetric energy-momentum tensor. This symmetrized
angular momentum tensor is of course different from the canonical one we
derived in Eqs. (26) and (27). However, it is still not gauge invariant and
the conservation law of angular momentum still holds. In other words

$\partial_{\lambda}\mathcal{M}^{{}^{\prime}\mu\nu,\lambda}=\partial_{\lambda}\mathcal{M}^{\mu\nu,\lambda}=0$
(56)

We can follow a similar procedure as we did in the previous section to derive
the expression for the gauge-independent forms of the symmetrized angular
momentum tensor for the electromagnetic and Dirac fields. We find, setting
$\mu,\nu=i,j$,

$\frac{\partial\bm{M}^{{}^{\prime}}}{\partial
t}+\nabla\cdot\overleftrightarrow{\bm{J}^{{}^{\prime}}}+\nabla{\chi^{\prime}}-\nabla\times(\bm{r}\mathcal{U}_{\text{em}})=0$
(57)

where

${\bm{M}^{{}^{\prime}}}=\frac{\hbar}{2}\left(\psi^{\dagger}\bm{\Sigma}\psi\right)+\mathcal{R}\left\\{\psi^{\dagger}(\bm{r}\times\bm{p}_{\parallel})\psi\right\\}+\epsilon_{0}~{}\bm{r}\times(\bm{E}\times\bm{B})-\rho(\bm{r}\times\bm{A}^{\bot})$
(58a)
$\nabla\cdot{\overleftrightarrow{\bm{J}^{{}^{\prime}}}}=c\nabla_{i}\mathcal{R}\\{\bar{\psi}\gamma^{i}(\bm{r}\times\bm{p}_{\parallel})\psi\\}+\epsilon_{0}~{}\nabla_{i}E_{i}(\bm{r}\times\bm{E})+\frac{1}{\mu_{0}}\nabla_{i}\left[B_{i}(\bm{r}\times\bm{B})\right]-\nabla_{i}\left[J_{i}(\bm{r}\times\bm{A}^{\bot})\right]$
(58b) $\tilde{\chi}=\frac{\hbar c}{2}(\psi^{\dagger}\gamma^{5}\psi)$ (58c)
$\mathcal{U}_{\text{em}}=\frac{1}{2}\left(\epsilon_{0}\bm{E}\cdot\bm{E}+\frac{1}{\mu_{0}}\bm{B}\cdot\bm{B}\right)$
(58d)

We emphasize that Eq. (57) is identical to the conservation Eq. (6) in the
main text. In fact, the terms related to the Dirac field are exactly the same
as the one with the four-divergence of the canonical angular momentum tensor.
The main difference is that we lose separate physically observable expressions
for the spin and OAM densities and currents of the electromagnetic field and
instead we get expressions for the total angular momentum,

$\epsilon_{0}\bm{r}\times(\bm{E}\times\bm{B})-\rho(\bm{r}\times\bm{A}^{\bot}),$
(59)

and total angular momentum currents,

$\epsilon_{0}~{}\bm{E}(\bm{r}\times\bm{E})+\frac{1}{\mu_{0}}\bm{B}(\bm{r}\times\bm{B})-\bm{J}(\bm{r}\times\bm{A}^{\bot}),$
(60)

of the electromagnetic field. Assuming that the symmetric energy-momentum
tensor is more fundamental than the canonical one, however, does not prevent
us from writing conservation laws that involve canonical angular momentum
tensor and taking its gauge-independent terms as physically meaningful
quantities.

##  Appendix E Boost Conservation Relations

We started our derivation by applying the Noether’s theorem to the Dirac-
Maxwell fields under the Lorentz transformation. To derive the angular
momentum conservation equations, however, we only focused on the spatial
rotations of the coordinates i.e. the space-space components of the angular
momentum tensor current $\mathcal{M}^{\mu\nu,\lambda}$. In this section, we
look at the conservation equations for the time-space components of the
Lorentz transformations, namely the boosts, of the Dirac-Maxwell fields.

Using a similar approach to the one used in the first section, the
conservation equation resulting from the boost components of the Lorentz
transformation is given by,

$\begin{split}\partial_{\lambda}\mathcal{M}^{i0,\lambda}=&\partial_{\lambda}(\mathcal{M}_{\text{D}})^{i0,\lambda}+\partial_{\lambda}(\mathcal{M}_{\text{E}})^{i0,\lambda}\\\
=&\frac{\partial}{\partial
t}\mathcal{R}\left\\{i\psi^{\dagger}\left[\frac{\bm{r}}{c}p_{0}-ct\bm{p}_{\parallel}\right]\psi\right\\}-c\nabla\times(\frac{\hbar}{2}\psi^{\dagger}\bm{\Sigma}\psi)+c\nabla.\mathcal{R}\left\\{i\bar{\psi}\bm{\gamma}\left[\frac{\bm{r}}{c}p_{0}-ct\bm{p}_{\parallel}\right]\psi\right\\}\\\
-&\epsilon\frac{\partial}{\partial
t}\left[\bm{E}.(\frac{\bm{r}}{c}\partial_{t}+ct\nabla)\bm{A}^{\bot}\right]+\frac{1}{\mu_{0}}\nabla.\left[\bm{B}\times(\frac{\bm{r}}{c}\partial_{t}+ct\nabla)\bm{A}^{\bot}\right]+\frac{1}{\mu_{0}c}\bm{B}\times\bm{E}^{\bot}+\frac{1}{\mu_{0}c}\bm{E}\cdot(\nabla)\bm{A}^{\bot}\\\
+&c\rho\bm{A}^{\bot}-\left(\frac{\bm{r}}{c}\right)\bm{J}.\bm{E}^{\parallel}+ct\rho\bm{E}^{\parallel}+\epsilon\bm{E}.\left(\frac{\bm{r}}{c}\partial_{t}+ct\nabla\right)\bm{E}^{\parallel}+c\left(\frac{\bm{r}}{c^{2}}\partial_{t}+t\nabla\right)\mathcal{L}_{\text{E}}=0\end{split}$
(61)

where $p_{0}=i\hbar\partial_{t}-e\phi$ and
$\bm{p}_{\parallel}=-i\hbar\nabla-e\bm{A}^{\parallel}$ are the gauge-
independent time-derivative and momentum operators of the Dirac field,
respectively. It is a matter of straightforward algebra to show that
$\bm{r}\times\partial_{\lambda}\bm{M}^{0,\lambda}=\epsilon_{ijk}x_{j}\partial_{\lambda}\mathcal{M}^{k0,\lambda}$
gives the conservation equation of the angular momentum in Eq. (38).

##  Appendix F Two Plane Wave Interference

We now evaluate the terms in Eq. (33a) for the interference of two plane waves
at different frequencies. For a plane wave propagating along
$\bm{k}/|\bm{k}|$, with the wavevector $k$ we have
$\bm{k}\cdot\bm{E}=\bm{k}\cdot\bm{A}^{\bot}=\bm{k}\cdot\bm{B}=0$. Therefore we
get

$\nabla\cdot(\bm{A}^{\bot}\bm{B})=(\bm{k}\cdot\bm{A}^{\bot})\bm{B}=0.$ (62)

Note also that the EM spin-orbit torque $\bm{\tau}_{\text{em}}$ also vanishes
for planes waves because

$(\bm{B}\cdot\nabla)\bm{A}^{\bot}=(\bm{B}\cdot\bm{k})\bm{A}^{\bot}=0.$ (63)

Thus the only relevant terms in finding the conservation law for the spin
current of the two plane wave interference are the time-derivative of spin and
gradient of helicity.

The electric field for two plane waves propagating along $z$ direction can be
written as

$\bm{E}=\mathcal{R}\\{\mathbfcal{\bm{E}}_{1}e^{-i(\omega_{1}t-k_{1}z)}+\mathbfcal{\bm{E}}_{2}e^{-i(\omega_{2}t-k_{2}z)}\\}$
(64)

where $\mathbfcal{E}_{i}=\frac{\mathcal{E}_{i}}{\sqrt{2}}(\hat{x}+i\hat{y})$
are the complex electric field amplitudes of the two modes with frequencies
$\omega_{i}/c=k_{i}$. Using Maxwell equation
$\nabla\times\bm{E}=-\frac{\partial\bm{B}}{\partial t}$ and
$\bm{E}=-\frac{\partial\bm{A}}{\partial t}$, we get

$\bm{B}=\mathcal{R}\left\\{-\frac{k_{1}}{i\omega_{1}}\mathbfcal{\bm{E}}_{1}e^{-i(\omega_{1}t-k_{1}z)}-\frac{k_{2}}{i\omega_{2}}\mathbfcal{\bm{E}}_{2}e^{-i(\omega_{2}t-k_{2}z)}\right\\}$
(65a)
$\bm{A}^{\bot}=\mathcal{R}\left\\{\frac{1}{i\omega_{1}}\mathbfcal{\bm{E}}_{1}e^{-i(\omega_{1}t-k_{1}z)}+\frac{1}{i\omega_{2}}\mathbfcal{\bm{E}}_{2}e^{-i(\omega_{2}t-k_{2}z)}\right\\}$
(65b)

Using these equations we get for the spin of the two plane waves

$\epsilon(\bm{E}^{\bot}\times\bm{A}^{\bot})=\frac{\epsilon}{2}\left[\frac{1}{\omega_{1}}\mathcal{I}\\{\mathbfcal{E}_{1}^{*}\times\mathbfcal{E}_{1}\\}+\frac{1}{\omega_{2}}\mathcal{I}\\{\mathbfcal{E}^{*}_{2}\times\mathbfcal{E}_{2}\\}+\left(\frac{1}{\omega_{1}}+\frac{1}{\omega_{2}}\right)\mathcal{I}\left\\{\mathbfcal{E}^{*}_{1}\times\mathbfcal{E}_{2}e^{+i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\}\right]$
(66)

and thus we get

$\begin{split}\epsilon\frac{\partial}{\partial
t}(\bm{E}^{\bot}\times\bm{A}^{\bot})=&-\epsilon\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2\omega_{1}\omega_{2}}\mathcal{R}\left\\{\mathbfcal{E}_{1}^{*}\times\mathbfcal{E}_{2}e^{+i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\}\\\
=&-\epsilon\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2\omega_{1}\omega_{2}}\mathcal{I}\left\\{\mathcal{E}_{1}\mathcal{E}_{2}^{*}e^{-i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\}\hat{z}\end{split}$
(67)

For the helicity density we find

$\bm{A}^{\bot}\cdot\bm{B}=-\frac{k_{1}}{\omega_{1}^{2}}|\mathcal{E}_{1}|^{2}-\frac{k_{2}}{\omega_{2}^{2}}|\mathcal{E}_{2}|^{2}-\frac{k_{1}+k_{2}}{2\omega_{1}\omega_{2}}\mathcal{R}\left\\{\mathcal{E}_{1}\mathcal{E}_{2}^{*}e^{-i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\}$
(68)

and thus

$\begin{split}\frac{1}{\mu_{0}}\nabla(\bm{A}^{\bot}\cdot\bm{B})=&\frac{1}{\mu_{0}}\frac{k_{1}^{2}-k_{2}^{2}}{2\omega_{1}\omega_{2}}\mathcal{I}\left\\{\mathcal{E}_{1}\mathcal{E}_{2}^{*}e^{-i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\}\hat{z}\\\
=&\epsilon\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2\omega_{1}\omega_{2}}\mathcal{I}\left\\{\mathcal{E}_{1}\mathcal{E}_{2}^{*}e^{-i[(\omega_{1}-\omega_{2})t-(k_{1}-k_{2})z]}\right\\}\end{split}$
(69)

which confirms the conservation Eq. (14) in the main manuscript.

##  Appendix G Meaning of the Notations

Throughout this paper, the usual expressions for dot and cross product are
assumed. Terms like $\bm{A}^{\bot}\bm{B}$ are tensorial expressions which can
be expanded as

$\begin{split}\bm{A}^{\bot}\bm{B}=&A_{x}^{\bot}B_{x}\hat{x}\hat{x}+A_{x}^{\bot}B_{y}\hat{x}\hat{y}+A_{x}^{\bot}B_{z}\hat{x}\hat{z}\\\
+&A_{y}^{\bot}B_{x}\hat{y}\hat{x}+A_{y}^{\bot}B_{y}\hat{y}\hat{y}+A_{y}^{\bot}B_{z}\hat{y}\hat{z}\\\
+&A_{z}^{\bot}B_{x}\hat{z}\hat{x}+A_{z}^{\bot}B_{y}\hat{z}\hat{y}+A_{z}^{\bot}B_{z}\hat{z}\hat{z}\end{split}$
(70)

Similar expressions can be written the terms like
$\bm{E}(\bm{r}\times\bm{E})$, $\bm{E}(\bm{r}\times\bm{E})$,
$\bm{J}(\bm{r}\times\bm{A}^{\bot})$, and so on. Therefore, the expression
$\nabla\cdot(\bm{A}^{\bot}\bm{B})$ means

$\begin{split}\nabla\cdot(\bm{A}^{\bot}\bm{B})=&\left[\partial_{x}(A_{x}^{\bot}B_{x})+\partial_{y}(A_{y}^{\bot}B_{x})+\partial_{z}(A_{z}^{\bot}B_{x})\right]\hat{x}\\\
+&\left[\partial_{x}(A_{x}^{\bot}B_{y})+\partial_{y}(A_{y}^{\bot}B_{y})+\partial_{z}(A_{z}^{\bot}B_{y})\right]\hat{y}\\\
+&\left[\partial_{x}(A_{x}^{\bot}B_{z})+\partial_{y}(A_{y}^{\bot}B_{z})+\partial_{z}(A_{z}^{\bot}B_{z})\right]\hat{x}\end{split}$
(71)

which is a vector. We can similar expand

$\begin{split}\bm{B}\times(\bm{r}\times\nabla)\bm{A}^{\bot}=&\left[B_{y}(y\partial_{z}-z\partial_{y})A^{\bot}_{z}-B_{z}(y\partial_{z}-z\partial_{y})A^{\bot}_{y}\right]\hat{x}\hat{x}\\\
+&\left[B_{z}(y\partial_{z}-z\partial_{y})A^{\bot}_{x}-B_{x}(y\partial_{z}-z\partial_{y})A^{\bot}_{z}\right]\hat{y}\hat{x}\\\
+&\left[B_{x}(y\partial_{z}-z\partial_{y})A^{\bot}_{y}-B_{y}(y\partial_{z}-z\partial_{y})A^{\bot}_{x}\right]\hat{z}\hat{x}\\\
+&\left[B_{y}(z\partial_{x}-x\partial_{z})A^{\bot}_{z}-B_{z}(z\partial_{x}-x\partial_{z})A^{\bot}_{y}\right]\hat{x}\hat{y}\\\
+&\left[B_{z}(z\partial_{x}-x\partial_{z})A^{\bot}_{x}-B_{x}(z\partial_{x}-x\partial_{z})A^{\bot}_{z}\right]\hat{y}\hat{y}\\\
+&\left[B_{x}(z\partial_{x}-x\partial_{z})A^{\bot}_{y}-B_{y}(z\partial_{x}-x\partial_{z})A^{\bot}_{x}\right]\hat{z}\hat{y}\\\
+&\left[B_{y}(x\partial_{y}-y\partial_{x})A^{\bot}_{z}-B_{z}(x\partial_{y}-y\partial_{x})A^{\bot}_{y}\right]\hat{x}\hat{z}\\\
+&\left[B_{z}(x\partial_{y}-y\partial_{x})A^{\bot}_{x}-B_{x}(x\partial_{y}-y\partial_{x})A^{\bot}_{z}\right]\hat{y}\hat{z}\\\
+&\left[B_{x}(x\partial_{y}-y\partial_{x})A^{\bot}_{y}-B_{y}(x\partial_{y}-y\partial_{x})A^{\bot}_{x}\right]\hat{z}\hat{z}\end{split}$
(72)

and we can take its divergence similar to Eq. (71).

## References

  * [1] Willner A E, Pang K, Song H, Zou K and Zhou H 2021 Applied Physics Reviews 8
  * [2] Bustamante C J, Chemla Y R, Liu S and Wang M D 2021 Rev. Dis. Primers 1(1) 25
  * [3] Crimin F, Mackinnon N, Götte J B and Barnett S M 2019 Applied Sciences 9 828
  * [4] Bliokh K Y, Bekshaev A Y and Nori F 2013 New Journal of Physics 15 033026
  * [5] Berry M V 2009 J. Opt. A: Pure Appl. Opt. 11 094001
  * [6] Nienhuis G 2016 Phys. Rev. A 93(2) 023840
  * [7] Fernandez-Corbaton I, Zambrana-Puyalto X, Tischler N, Vidal X, Juan M L and Molina-Terriza G 2013 Phys. Rev. Lett. 111(6) 060401
  * [8] Tang Y and Cohen A E 2010 Phys. Rev. Lett. 104(16) 163901
  * [9] Philbin T G 2013 Phys. Rev. A 87(4) 043843
  * [10] Greiner W et al. 1990 Relativistic quantum mechanics vol 3 (Springer)
  * [11] Barnett S M 2014 New Journal of Physics 16 093008
  * [12] Bialynicki-Birula I 1996 36 245 – 294
  * [13] Khosravi F, Van Mechelen T and Jacob Z 2019 Phys. Rev. B 100(15) 155105
  * [14] Van Mechelen T and Jacob Z 2019 Phys. Rev. B 99(20) 205146
  * [15] Van Mechelen T and Jacob Z 2019 Nanophotonics 8 1399–1416
  * [16] Greiner W, Müller B and Bromley D A 1996 Gauge theory of weak interactions vol 5 (Springer, Berlin)
  * [17] Noether E 1971 Transport Theory and Statistical Physics 1 186–207
  * [18] Barnett S M 2002 Journal of Optics B: Quantum and Semiclassical Optics 4 S7–S16
  * [19] Calkin M G 1965 American Journal of Physics 33 958–960 (Preprint https://doi.org/10.1119/1.1971089)
  * [20] Leader E and Lorcé C 2014 Physics Reports 541 163 – 248
  * [21] Belinfante F 1940 Physica 7 449–474
  * [22] Ji X 1997 Phys. Rev. Lett. 78(4) 610–613
  * [23] Jaffe R L and Manohar A 1990 Nuclear Physics B 337 509–546
  * [24] Chen X S, Lü X F, Sun W M, Wang F and Goldman T 2008 Phys. Rev. Lett. 100(23) 232002
  * [25] Wakamatsu M 2010 Phys. Rev. D 81(11) 114010
  * [26] Yang L P, Khosravi F and Jacob Z 2020 arXiv preprint arXiv:2004.03771 URL https://arxiv.org/abs/2004.03771
  * [27] Das P, Yang L P and Jacob Z 2024 arXiv:2403.01054
  * [28] Foldy L L and Wouthuysen S A 1950 Phys. Rev. 78(1) 29–36
  * [29] Terno D R 2003 Phys. Rev. A 67(1) 014102
  * [30] Fujikawa K, Oh C H and Zhang C 2014 Phys. Rev. D 90(2) 025028
  * [31] Greiner W and Reinhardt J 2013 Field quantization (Springer Science & Business Media)
  * [32] Kleinert H 2016 Particles and quantum fields (World Scientific Singapore)
  * [33] Cohen-Tannoudji C, Dupont-Roc J and Grynberg G 1997 Photons and Atoms-Introduction to Quantum Electrodynamics
  * [34] Jackson J D 1999 Classical electrodynamics (Wiley, New York) 3rd Edition, Chap. 6.7
  * [35] Woltjer L 1958 Proc. Natl. Acad. Sci. USA 44 489
  * [36] Coles M M and Andrews D L 2012 Phys. Rev. A 85(6) 063810
  * [37] Leeder J M, Haniewicz H T and Andrews D L 2015 J. Opt. Soc. Am. B 32 2308–2313
  * [38] Lipkin D M 1964 Journal of Mathematical Physics 5 696–700 (Preprint https://doi.org/10.1063/1.1704165)
  * [39] Meredith L 2018
  * [40] McCreary K M, Swartz A G, Han W, Fabian J and Kawakami R K 2012 Phys. Rev. Lett. 109(18) 186604
  * [41] Zhao H, Loren E J, van Driel H M and Smirl A L 2006 Phys. Rev. Lett. 96(24) 246601
  * [42] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Molnár S, Roukes M L, Chtchelkanova A Y and Treger D M 2001 Science 294 1488–1495 (Preprint https://science.sciencemag.org/content/294/5546/1488.full.pdf)
  * [43] Sharma P 2005 Science 307 531–533 (Preprint https://science.sciencemag.org/content/307/5709/531.full.pdf)
  * [44] Watson S K, Potok R M, Marcus C M and Umansky V 2003 Phys. Rev. Lett. 91(25) 258301
  * [45] Kashiwaya S, Tanaka Y, Yoshida N and Beasley M R 1999 Phys. Rev. B 60(5) 3572–3580
  * [46] Crater H W and Alstine] P V 1983 Annals of Physics 148 57 – 94
  * [47] Salpeter E E and Bethe H A 1951 Phys. Rev. 84(6) 1232–1242
  * [48] Manchon A, Koo H C, Nitta J, Frolov S and Duine R 2015 Nature materials 14 871–882
  * [49] Enk S V and Nienhuis G 1994 Journal of Modern Optics 41 963–977 (Preprint https://doi.org/10.1080/09500349414550911)
  * [50] Nieto-Vesperinas M 2015 Opt. Lett. 40 3021–3024
  * [51] Malinowski G, Dalla Longa F, Rietjens J, Paluskar P, Huijink R, Swagten H and Koopmans B 2008 Nature Physics 4 855
  * [52] Mitsch R, Sayrin C, Albrecht B, Schneeweiss P and Rauschenbeutel A 2014 Nature communications 5 5713
  * [53] Schmiegelow C T, Schulz J, Kaufmann H, Ruster T, Poschinger U G and Schmidt-Kaler F 2016 Nature communications 7 12998
  * [54] Young A B, Thijssen A C T, Beggs D M, Androvitsaneas P, Kuipers L, Rarity J G, Hughes S and Oulton R 2015 Phys. Rev. Lett. 115(15) 153901
  * [55] Söllner I, Mahmoodian S, Hansen S L, Midolo L, Javadi A, Kiršanskė G, Pregnolato T, El-Ella H, Lee E H, Song J D et al. 2015 Nature nanotechnology 10 775
  * [56] Barik S, Karasahin A, Mittal S, Waks E and Hafezi M 2019 Chiral quantum optics using a topological resonator (Preprint 1906.11263)
  * [57] Das P, Bharadwaj S and Jacob Z 2024 arXiv:2403.01054
  * [58] Yang L P and Jacob Z 2021 Comm. Phys. 4(1) 1–9
  * [59] Kalhor F, Yang L P, Bauer L and Jacob Z 2021 Phys. Rev. Research 3(4) 043007
  * [60] Wolkow D M 1935 Zeitschrift für Physik 94 250–260
  * [61] Redmond P J 1965 Journal of Mathematical Physics 6 1163–1169 (Preprint https://doi.org/10.1063/1.1704385)
  * [62] MacDonald A H and Vosko S H 1979 Journal of Physics C: Solid State Physics 12 2977–2990
  * [63] Eschrig H and Servedio V D P 1999 Journal of Computational Chemistry 20 23–30
  * [64] Mondal R, Berritta M and Oppeneer P M 2018 Phys. Rev. B 98(21) 214429
  * [65] Rosenfeld L J H C 1940 Sur le tenseur d’impulsion-énergie (Palais des Académies)