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# From quantum electrodynamics to a geometric gauge theory of classical
electromagnetism

Adam Marsh Electronic address<EMAIL_ADDRESS>
( )

###### Abstract

A relativistic version of the correspondence principle, a limit in which
classical electrodynamics may be derived from QED, has never been clear,
especially when including gravitational mass. Here we introduce a novel
classical field theory formulation of electromagnetism, and then show that it
approximates QED in the limit of a quantum state which corresponds to a
classical charged continua. Our formulation of electromagnetism features a
Lagrangian which is gauge invariant, includes a classical complex field from
which a divergenceless four-current may be derived, and reproduces all aspects
of the classical theory of charged massive continua without any quantum
effects. Taking a geometric approach, we identify the four-current as being in
the direction of extremal phase velocity of the classical field; the field
equations of motion determine this phase velocity as being equal to the mass,
which makes the rest density proportional to the squared modulus of the field.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Geometry
    3. 1.3 Overview
  2. 2 Geometric $U(1)$ gauge theory
    1. 2.1 A geometric view of gauge theory
    2. 2.2 The gauge potential
    3. 2.3 Inner products
    4. 2.4 The spacetime gradient
    5. 2.5 Matter field rotation
    6. 2.6 Quasi-gauge electromagnetism
    7. 2.7 Klein-Gordon theory
  3. 3 Matter field electromagnetism
    1. 3.1 The Lagrangian
    2. 3.2 Matter field equations of motion
    3. 3.3 Gauge potential equations of motion
    4. 3.4 Geometry of the four-current
    5. 3.5 Metric equations of motion
    6. 3.6 Noether’s theorem
  4. 4 From QED to classical electromagnetism
    1. 4.1 The QED Lagrangian
    2. 4.2 Plane wave solutions
    3. 4.3 Electron packets
    4. 4.4 Classical four-current configuration
    5. 4.5 Spinor component equations of motion
    6. 4.6 From the QED to the MFEM Lagrangian
  5. 5 Summary and discussion
    1. 5.1 Summary of results
    2. 5.2 The discrete four-current
    3. 5.3 Multiple matter fields

##  1 Introduction

###  1.1 Motivation

The (quantized, minimally coupled) Dirac equation describes (relativistic,
quantum mechanical) electrons (and positrons) interacting electromagnetically.
The Dirac spinor matter field comprises four complex components smoothly
defined at each point of spacetime; its values as operators provide a
description of multiple particles via quantum electrodynamics (QED). The Dirac
equation may be derived from the Dirac Lagrangian, which is invariant under
$U(1)$ gauge transformations.

Maxwell’s equations describe (relativistic, classical) charged massive
continua interacting electromagnetically. The four-current is a divergenceless
four-vector field on spacetime whose direction is that of the continuum of
particles at that point, and whose length is the number of particles per unit
space-like volume orthogonal to that direction. Maxwell’s equations are
usually derived from a Lagrangian which includes this four-current, does not
include a matter field, and is not invariant under $U(1)$ gauge
transformations.

While the classical electromagnetic gauge potential equation of motion may be
obtained from QED by the stationary phase approximation, there is no way to
obtain a classical matter field yielding a classical four-current using this
approximation, since the scalars obtained must anti-commute (i.e. they are
Grassmann numbers).

Our aim in this paper is to construct an alternative Lagrangian for classical
charged massive continua interacting electromagnetically which (1) is based
upon a classical matter field whose equations of motion yield a classical
four-current, (2) is invariant under gauge transformations, and (3) may be
obtained as a limit of quantum electrodynamics. An additional goal is a
detailed presentation of this alternative theory in terms of both real and
complex geometry.

###  1.2 Geometry

The evolution of gauge theory was from its beginning geometric and tied to
electromagnetism. Weyl [6] first introduced the concept of gauge invariance as
local spacetime scale invariance (thus the name) in an attempt to unify
general relativity with electromagnetism, later repurposing it more
successfully [7] as an extension of the global phase invariance of the wave
function in Dirac theory to local phase invariance, which yields Maxwell’s
equations. The resulting “gauge principle,” formulated as a procedure for
extending global symmetries to local symmetries, was then generalized to
higher dimensional complex “rotations” by Yang and Mills [8].

This set of ideas was eventually given an even more geometric formulation in
the language of fiber bundles (see e.g. [1]), which also accounts for global
considerations. A scalar matter field (a generic term to encompass both scalar
and operator field values) is defined as a section of a complex vector bundle
over a (possibly curved) spacetime manifold; in analogy with tangent vector
fields, the gauge potential is a connection defining parallel transport of
these vectors, the field strength is the curvature of this connection, and a
gauge transformation is a change of the frame defining the matter field
components.

In the viewpoint adopted here, this geometric formulation positions the matter
field as the primary quantity, with the gauge potential defining its parallel
transport. Classical electromagnetism however, despite being the prototypical
gauge theory, is typically described as a gauge theory with the four-current
inserted by hand as an external quantity, while the notions of a gauge
potential and field strength are preserved even in the absence of an
associated matter field. Moreover, the Lagrangian is not gauge invariant,
which from a geometric point of view is nonsensical. We will therefore refer
to this formulation absent a matter field as “quasi-gauge electromagnetism.”

###  1.3 Overview

We would like to formulate the classical theory of charged massive continua as
a gauge theory including a matter field. This matter field should determine a
divergenceless four-current; but unlike in quantum theory, its equations of
motion should not result in solutions with quantum characteristics. In
particular, we should have no need to interpret the “on-shell” matter fields
which satisfy these equations in terms of particles or probabilities; instead,
they should determine a general classical four-current.

In the following we provide a geometric description of such a theory, which is
based upon a gauge-invariant Lagrangian which includes a matter field, and
which, like quasi-gauge electromagnetism, reproduces the full classical theory
of charged massive continua, without any quantum characteristics. In addition,
QED is shown to simplify to this theory under certain conditions. We will
refer to this theory as “matter field electromagnetism.”

In Section 2 we summarize our geometric view of gauge theory and describe
various geometrical quantities associated with our $U(1)$ gauge theory from
both the real and complex viewpoints, taking pains to construct a consistent
picture in the real case. Section 3 defines our Lagrangian for matter field
electromagnetism, in terms of both geometry and complex algebra, and derives
the equations of motion along with the results of Noether’s theorem. Section 4
constructs a quantum state corresponding to a classical current and shows
that, as desired, QED in the limit of such a state results in the matter field
and equations of motion from matter field electromagnetism.

Throughout the paper we will use natural units, where the constants
$c=G=\hbar=1$, and the mostly pluses spacetime metric signature, where in an
orthonormal frame the metric is
$g_{\mu\nu}=\mathrm{diag}\left(-1,1,1,1\right)$.

##  2 Geometric $U(1)$ gauge theory

###  2.1 A geometric view of gauge theory

A common way to describe a scalar gauge theory is in terms of a multiplet of
(usually complex) matter fields $\Phi^{a}$ and a Lagrangian which is written
in terms of these fields along with their coordinate derivatives
$\partial_{\mu}\Phi^{a}$. The Lagrangian is then noted to be invariant under a
global gauge transformation, a complex matrix transformation of the $\Phi^{a}$
treated as vector components, where the matrix is an element of a matrix
group, usually $U(1)$ or $SU(n)$ (hereafter referred to as simply $SU(n)$).
The principle of minimal coupling then prescribes the introduction of a gauge
potential to replace coordinate derivatives with gauge covariant derivatives.
This promotes the global gauge invariance to a local gauge invariance, wherein
the Lagrangian is invariant under multiplication by the matter field at each
point by a different but smoothly varying matrix in $SU(n)$.

The geometric view of scalar gauge theory we take here (detailed in [4]) is
based upon a vector bundle $(E,M,X)$ over spacetime $M$, with the fiber
$X\cong\mathbb{C}^{n}$ called the internal space. A matter field is a section
of $E$, or equivalently an $X$-valued 0-form (function) on $M$. A choice of
gauge in a region of $M$ is a frame, a choice of orthonormal basis for the
fiber $X_{p}$ at each point $p$ in the region, which allows us to express the
matter field as a gauge-dependent $\mathbb{C}^{n}$-valued 0-form we denote
$\vec{\Phi}$, with components $\Phi^{a}$ smoothly defined at each point. A
gauge transformation is a change of frame, which in a choice of gauge
corresponds to a matrix element smoothly defined at each point which we denote
$\check{\gamma}^{-1}\in SU\left(n\right)$; the check decoration indicates a
matrix value, and the element is defined as an inverse matrix applied to the
basis so that the components $\Phi^{a}$ at each point in the region transform
as $\vec{\Phi}\rightarrow\check{\gamma}\vec{\Phi}$. A gauge transformation is
thus a “rotation” of the internal space basis at each point, or more precisely
a linear transformation which leaves the complex inner product on the fiber
$X$ invariant, along with the complex volume element
$\mathsf{e}_{1}\wedge\cdots\wedge\mathsf{e}_{n}$ for $n>1$; here
$\mathsf{e}_{a}$ is a complex orthonormal basis of $X$, and we use a sans
serif font for internal space basis vectors to distinguish them from spacetime
tangent space basis vectors.

In analogy with the tangent bundle, we introduce a connection defining
parallel transport, an $su(n)$-valued 1-form we denote

$\check{\Gamma}\equiv-iq\check{A}\equiv-iqA^{a}{}_{b\mu},$ (2.1)

where the gauge potential $A^{a}{}_{b\mu}$ is thus a hermitian matrix-valued
1-form with a Greek spacetime index. Geometrically, $-iqA^{a}{}_{b\mu}$ is the
$a^{\mathrm{th}}$ component of the difference between the frame
$\mathsf{e}_{b}$ and its parallel transport in the direction $e_{\mu}$.

The gauge covariant derivative, which is geometrically the infinitesimal
difference between $\vec{\Phi}$ and its parallel transport, can then be
written in terms of forms and components as

$\displaystyle\mathrm{D}\vec{\Phi}$
$\displaystyle=\mathrm{d}\vec{\Phi}+\check{\Gamma}\vec{\Phi},$ (2.2)
$\displaystyle\mathrm{D}_{\mu}\Phi^{a}$
$\displaystyle=\partial_{\mu}\Phi^{a}-iqA^{a}{}_{b\mu}\Phi^{b}.$

The connection defines a $su(n)$-valued curvature 2-form written in terms of
the field strength $\check{F}$ as $-iqF^{a}{}_{b\mu\nu}$, where

$\displaystyle\check{F}$
$\displaystyle\equiv\mathrm{d}\check{A}-iq\check{A}\wedge\check{A},$ (2.3)
$\displaystyle F^{a}{}_{b\mu\nu}$
$\displaystyle=\partial_{\mu}A^{a}{}_{b\nu}-\partial_{\nu}A^{a}{}_{b\mu}-iq\left(A^{a}{}_{c\mu}A^{c}{}_{b\nu}-A^{a}{}_{c\nu}A^{c}{}_{b\mu}\right).$

Under a gauge transformation $\check{\gamma}^{-1}$, these quantities transform
as follows:

$\displaystyle\vec{\Phi}$ $\displaystyle\rightarrow\check{\gamma}\vec{\Phi}$
(2.4) $\displaystyle\check{F}$
$\displaystyle\rightarrow\check{\gamma}\check{F}\check{\gamma}^{-1}$
$\displaystyle\check{A}$
$\displaystyle\rightarrow\check{\gamma}\check{A}\check{\gamma}^{-1}+\frac{i}{q}\check{\gamma}\mathrm{d}\check{\gamma}^{-1}$

In the case of $U(1)$ gauge theory, the matter field is complex-valued and
denoted $\Phi$, while the gauge potential $A_{\mu}$ is real-valued. A gauge
transformation is then usually written $\gamma^{-1}\equiv e^{-iq\Lambda}$,
under which the above relations simplify to:

$\displaystyle\mathrm{D}_{\mu}\Phi$ $\displaystyle=\partial_{\mu}\Phi-
iqA{}_{\mu}\Phi$ (2.5) $\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}A{}_{\nu}-\partial_{\nu}A{}_{\mu}$
$\displaystyle\Phi$ $\displaystyle\rightarrow e^{iq\Lambda}\Phi$
$\displaystyle F_{\mu\nu}$ $\displaystyle\rightarrow F_{\mu\nu}$
$\displaystyle A_{\mu}$ $\displaystyle\rightarrow
A_{\mu}+\partial_{\mu}\Lambda$

Figure 2.1: A gauge theory is a bundle over spacetime $M$ with fiber a vector
space $X$, and matter field $\vec{\Phi}$ a section of this bundle. If we
choose a gauge, an orthonormal basis for each $X_{p}$, the matter field can be
written in terms of components $\Phi^{a}$, with a gauge transformation being a
smoothly defined change of orthonormal basis for each $X_{p}$. The gauge
covariant derivative in a given gauge can then be written
$\mathrm{D}_{\mu}\Phi^{a}=\partial_{\mu}\Phi^{a}+\Gamma^{a}{}_{b\mu}\Phi^{b}=\partial_{\mu}\Phi^{a}-iqA^{a}{}_{b\mu}\Phi^{b}$.
Geometrically,
$\varepsilon\mathrm{D}_{\mu}\Phi^{a}=\left.\vec{\Phi}\right|_{p+\varepsilon
e_{\mu}}-\parallel_{\varepsilon e_{\mu}}\left.\vec{\Phi}\right|_{p}$ is the
difference between the matter field $\left.\vec{\Phi}\right|_{p+\varepsilon
e_{\mu}}$ at a point infinitesimally displaced in the direction $e_{\mu}$ from
$p$ and its parallel transport from $p$, which we denote
$\parallel_{\varepsilon e_{\mu}}\left.\vec{\Phi}\right|_{p}$.

###  2.2 The gauge potential

In $U(1)$ gauge theory, the matter field value at each point in spacetime is a
complex number $\Phi\in\mathbb{C}$, which from our geometric viewpoint is the
single complex component of an intrinsic complex vector in a given gauge
(choice of unit length complex vector). A gauge transformation (new choice of
unit length complex vector) $\Phi\rightarrow e^{iq\Lambda}\Phi$, or
infinitesimally $\Phi\rightarrow\Phi+iq\Lambda\Phi$, changes the complex
component, but does not change the intrinsic complex vector. Note that a
choice of unit length complex vector is also a choice of volume element, which
gauge transformations are only allowed to alter in one complex dimension; i.e.
in one dimension we consider $U(1)$ instead of $SU(1)$.

Another unique feature of $U(1)$ gauge theory is that $U(1)\cong SO(2)$, the
complete group of rotations in the decomplexified space, instead of a subgroup
$SU(n)\subset SO(2n)$ of these rotations. The matter field at each point may
therefore be viewed as a real vector $\vec{\Phi}\in\mathbb{R}^{2}$, with a
gauge transformation a new choice of real orthonormal basis for the internal
space which leaves $\vec{\Phi}$ unchanged but transforms its components
$\Phi^{a}$. We will adopt this view as our primary one, and will take it quite
literally; but we will utilize the complex notation in parallel with the
vector notation, since electromagnetism is almost universally expressed this
way, and the definition of the gauge potential $A_{\mu}$ is dependent upon the
complex viewpoint.

We associate the rotation of a real vector $\vec{\Phi}\in\mathbb{R}^{2}$ by
$\theta$ radians with the complex multiplication $e^{i\theta}\Phi$ in
$\mathbb{C}$. This rotation is usually depicted as being in the
counterclockwise direction, since the real axis is usually depicted as the
horizontal axis. Therefore, going forward we will use the word “rotation” to
mean “counterclockwise rotation in this depiction” or “rotation from the
positive real towards the positive imaginary axis.”

Now, $-iqA_{\mu}$ is the imaginary number which when multiplied by a unit
basis vector gives the infinitesimal difference between the basis vector at a
displaced point and its parallel transport. Since this displacement is
infinitesimal and rotates a unit vector, the rotation is by $-qA_{\mu}$
radians, so that we can view $q$ as a conversion factor between units, i.e. a
rotational multiplier specifying internal space radians per unit spacetime
length for unit $A_{\mu}$. We may then write the (counterclockwise
instantaneous) angular velocity (relative to parallel transport) of the
internal space frame (gauge) in the $e_{\mu}$ direction as

$\displaystyle A_{\mu}^{\varangle}$ $\displaystyle\equiv-qA_{\mu}$ (2.6)

in radians per coordinate length; a positive gauge potential value then
corresponds to a clockwise frame rotation if $q$ is positive.

Figure 2.2: In a given choice of gauge, the $so(2)$-valued connection
$\check{\Gamma}$ is a matrix which applied to any frame vector
$\vec{\mathsf{e}}_{a}$ yields the difference between the frame vector
$\left.\vec{\mathsf{e}}_{a}\right|_{p+\varepsilon e_{\mu}}$ at the
infinitesimally displaced point and the parallel transported frame vector
$\parallel_{\varepsilon e_{\mu}}\vec{\mathsf{e}}_{a}$. In the complex view,
the connection is $-iqA$, a complex number which when multiplied by any frame
vector yields the difference between the frame vector at the new point and the
parallel transported frame vector, again as a complex number. For example, the
internal space frame vector $\vec{\mathsf{e}}_{2}$ corresponds to $i$, so that
the difference is $-iqAi=qA$, which is a positive real number if $q$ and $A$
are. $q$ may therefore be viewed as a conversion factor which in the chosen
units makes $qA_{\mu}$ the clockwise angular velocity of the frame (relative
to parallel transport) in the $e_{\mu}$ direction. Note that if $\mu\neq 0$
this “velocity” will be in radians per unit distance instead of radians per
unit time.

###  2.3 Inner products

The frame rotations of the previous section imply that we define the real
inner product on the internal space $\mathbb{R}^{2}\cong\mathbb{C}$ as the
real part of the complex inner product
$\left\langle\vec{\Phi},\vec{\Psi}\right\rangle_{\mathbb{R}}=\mathrm{Re}\left\langle\Phi,\Psi\right\rangle_{\mathbb{C}}$;
in particular, the squared length of a matter field vector may then be written

$\displaystyle\left\langle\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}$
$\displaystyle=\left\|\vec{\Phi}\right\|^{2}=\Phi^{a}\Phi_{a}$ (2.7)
$\displaystyle=\left\langle\Phi,\Phi\right\rangle_{\mathbb{C}}$
$\displaystyle=\Phi^{*}\Phi=\left|\Phi\right|^{2},$

where $\left|\Phi\right|$ is the modulus and $\Phi^{*}$ is the complex
conjugate of the complex number $\Phi$. For the field strength, we have

$\displaystyle\left\langle F,F\right\rangle$
$\displaystyle=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}$ (2.8)
$\displaystyle=\frac{1}{2}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)$
$\displaystyle=\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu},$

where we recall the $k$-form inner product relation
$\left\langle\varphi,\psi\right\rangle_{\textrm{form}}=\frac{1}{k!}\varphi_{\mu_{1}\dots\mu_{k}}\psi^{\mu_{1}\dots\mu_{k}}$
and in the last line we swap dummy indices and combine terms.

We can write the “squared length” of the gauge covariant derivative as

$\displaystyle\left\langle\mathrm{D}_{\mu}\vec{\Phi},\mathrm{D}_{\mu}\vec{\Phi}\right\rangle$
$\displaystyle\equiv\left(\mathrm{D}^{\mu}\Phi\right)^{*}\mathrm{D}_{\mu}\Phi$
(2.9)
$\displaystyle=\left(\partial^{\mu}\Phi^{*}+iqA^{\mu}\Phi^{*}\right)\left(\partial_{\mu}\Phi-
iqA_{\mu}\Phi\right)$
$\displaystyle=\partial^{\mu}\Phi^{*}\partial_{\mu}\Phi+iqA^{\mu}\left(\Phi^{*}\partial_{\mu}\Phi-\Phi\partial_{\mu}\Phi^{*}\right)+q^{2}\left|\Phi\right|^{2}A_{\mu}A^{\mu}.$

If we define

$\displaystyle\Phi$ $\displaystyle\equiv\left|\Phi\right|e^{i\varphi}$ (2.10)
$\displaystyle\Rightarrow\mathrm{D}_{\mu}\Phi$
$\displaystyle=\frac{\Phi}{\left|\Phi\right|}\mathrm{D}_{\mu}\left|\Phi\right|+i\Phi\mathrm{D}_{\mu}\varphi$
$\displaystyle=\partial_{\mu}\Phi-iqA_{\mu}\Phi$
$\displaystyle=\frac{\Phi}{\left|\Phi\right|}\partial_{\mu}\left|\Phi\right|+i\Phi\partial_{\mu}\varphi-
iqA_{\mu}\Phi,$

we see that geometrically

$\displaystyle\mathrm{D}_{\mu}\left|\Phi\right|$
$\displaystyle=\partial_{\mu}\left|\Phi\right|,$ (2.11)
$\displaystyle\mathrm{D}_{\mu}\varphi$ $\displaystyle=\partial_{\mu}\varphi-
qA_{\mu}$

are the difference in length and angle between the matter field and its
parallel transport per unit distance in the direction $e_{\mu}$. In terms of
these quantities, we have

$\displaystyle\left\langle\mathrm{D}_{\mu}\Phi,\mathrm{D}_{\mu}\Phi\right\rangle$
$\displaystyle=\left(\frac{\Phi^{*}}{\left|\Phi\right|}\mathrm{D}^{\mu}\left|\Phi\right|-i\Phi^{*}\mathrm{D}^{\mu}\varphi\right)\left(\frac{\Phi}{\left|\Phi\right|}\mathrm{D}_{\mu}\left|\Phi\right|+i\Phi\mathrm{D}_{\mu}\varphi\right)$
(2.12)
$\displaystyle=\left\langle\mathrm{D}_{\mu}\left|\Phi\right|,\mathrm{D}_{\mu}\left|\Phi\right|\right\rangle+\left|\Phi\right|^{2}\left\langle\mathrm{D}_{\mu}\varphi,\mathrm{D}_{\mu}\varphi\right\rangle.$

###  2.4 The spacetime gradient

$\left\langle\mathrm{D}_{\mu}\varphi,\mathrm{D}_{\mu}\varphi\right\rangle$ is
the magnitude of the four-vector $\mathrm{D}^{\mu}\varphi$, which is a
“spacetime gradient.” Since we find no ready reference, we characterize the
direction of this four-vector here. For any non-null spacetime four-vector
$V^{\mu}$, we may choose an orthonormal basis for the tangent space for which
$V=Le_{\parallel}$, with $\left\langle
e_{\parallel},e_{\parallel}\right\rangle=\pm 1$ and any other basis four-
vector $e_{\perp}$ orthogonal to $V$. For any unit four-vector $\left\langle
B,B\right\rangle=\pm 1$ which is a boost of $e_{\parallel}$ in the same part
of the light cone, we have

$\displaystyle B$ $\displaystyle=e_{\parallel}+be_{\parallel}+ce_{\perp}$
(2.13) $\displaystyle\Rightarrow V_{\mu}B^{\mu}$ $\displaystyle=\pm
L\left(1+b\right)$

for positive numbers $b$ and $c$ (see Figure 2.3); $e_{\parallel}$ is
therefore the direction $U$ for which the absolute value
$\left|V_{\mu}U^{\mu}\right|$ is at a minimum under boosts. Similarly, for any
unit four-vector $\left\langle R,R\right\rangle=\pm 1$ which is a rotation of
$e_{\parallel}$ in the same part of the light cone (excluding
$-e_{\parallel}$), we have

$\displaystyle R$ $\displaystyle=e_{\parallel}-re_{\parallel}+se_{\perp}$
(2.14) $\displaystyle\Rightarrow V_{\mu}R^{\mu}$ $\displaystyle=\pm
L\left(1-r\right)$

for positive numbers $r<2$ and $s$; $e_{\parallel}$ is therefore the direction
$U$ for which the absolute value $\left|V_{\mu}U^{\mu}\right|$ is at a maximum
under rotations.

Figure 2.3: For a time-like vector $V_{\mathrm{t}}$ and a space-like vector
$V_{\mathrm{s}}$, the unit vector $U$ parallel to $V$ is the unit vector for
which $\left|V_{\mu}U^{\mu}\right|$ is a minimum compared to any boosted unit
vector $B$, and is a maximum compared to any rotated unit vector $R$.

With these results in hand, we may geometrically describe
$\mathrm{D}^{\mu}\varphi$ as pointing in the direction $U^{\mu}$ in which
$\left|\mathrm{D}_{U}\varphi\right|=\left|U^{\mu}\mathrm{D}_{\mu}\varphi\right|$
is smallest under boosts and largest under rotations (the latter corresponding
to the usual gradient description as the “direction of steepest ascent”). In
the interest of brevity we may say that $\mathrm{D}^{\mu}\varphi$ points in
the direction $U$ in which $\mathrm{D}_{U}\varphi$ is extremal; in particular,
$\mathrm{D}_{U^{\perp}}\varphi=0$ for any $U^{\perp}$ orthogonal to $U$. We
may then say that
$\left|\left\langle\mathrm{D}_{\mu}\varphi,\mathrm{D}_{\mu}\varphi\right\rangle\right|$
is the squared angular difference between the matter field and its parallel
transport per unit distance in the direction in which this difference is
extremal.

Note that $\mathrm{D}_{\mu}\Phi$ has values which are complex and therefore
not ordered, disallowing the above interpretation, and there will in general
be different directions in which the change in the modulus and phase are
extremal.

###  2.5 Matter field rotation

The imaginary part of the complex inner product is just the real inner product
with one of the arguments rotated by $\pi/2$, i.e.

$\displaystyle\mathrm{Im}\left\langle\Psi,\Phi\right\rangle_{\mathbb{C}}$
$\displaystyle=\mathrm{Re}\left(-i\left\langle\Psi,\Phi\right\rangle_{\mathbb{C}}\right)$
$\displaystyle=\left\langle\vec{\Psi},-\overrightarrow{\left(i\Phi\right)}\right\rangle_{\mathbb{R}}$
$\displaystyle=\left\langle\overrightarrow{\left(i\Psi\right)},\vec{\Phi}\right\rangle_{\mathbb{R}}.$

Since the real inner product is geometrically a projection, we may express the
lengths of the components of $\vec{\Psi}$ parallel and perpendicular to
$\vec{\Phi}$ as

$\displaystyle\vec{\Psi}^{\parallel\vec{\Phi}}$
$\displaystyle=\frac{\left\langle\vec{\Psi},\vec{\Phi}\right\rangle_{\mathbb{R}}}{\left\|\vec{\Phi}\right\|}=\frac{\mathrm{Re}\left\langle\Psi,\Phi\right\rangle_{\mathbb{C}}}{\left|\Phi\right|}$
(2.15)
$\displaystyle=\frac{\left(\Phi^{*}\Psi+\Phi\Psi^{*}\right)}{2\left|\Phi\right|},$
$\displaystyle\vec{\Psi}^{\perp\vec{\Phi}}$
$\displaystyle=\frac{\left\langle\vec{\Psi},\overrightarrow{\left(i\Phi\right)}\right\rangle_{\mathbb{R}}}{\left\|\vec{\Phi}\right\|}=\frac{-\mathrm{Im}\left\langle\Psi,\Phi\right\rangle_{\mathbb{C}}}{\left|\Phi\right|}$
$\displaystyle=\frac{-i\left(\Phi^{*}\Psi-\Phi\Psi^{*}\right)}{2\left|\Phi\right|}.$

In particular, defining

$\displaystyle\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}$
$\displaystyle\equiv\left(\mathrm{D}_{\mu}\vec{\Phi}\right)^{\parallel\vec{\Phi}}$
$\displaystyle\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$
$\displaystyle\equiv\left(\mathrm{D}_{\mu}\vec{\Phi}\right)^{\perp\vec{\Phi}},$

we have

$\displaystyle
2\left\|\vec{\Phi}\right\|\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}$
$\displaystyle=2\left\langle\mathrm{D}_{\mu}\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}$
(2.16)
$\displaystyle=2\mathrm{Re}\left\langle\mathrm{D}_{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}$
$\displaystyle=\Phi^{*}\mathrm{D}_{\mu}\Phi+\Phi\left(\mathrm{D}_{\mu}\Phi\right)^{*}$
$\displaystyle=\Phi^{*}\partial_{\mu}\Phi+\Phi\partial_{\mu}\Phi^{*}$
$\displaystyle=\partial_{\mu}\left|\Phi\right|^{2},$ $\displaystyle
2\left\|\vec{\Phi}\right\|\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$
$\displaystyle=2\left\langle\mathrm{D}_{\mu}\vec{\Phi},\overrightarrow{\left(i\Phi\right)}\right\rangle_{\mathbb{R}}$
$\displaystyle=2\mathrm{Im}\left\langle\Phi,\mathrm{D}_{\mu}\Phi\right\rangle_{\mathbb{C}}$
$\displaystyle=-i\left(\Phi^{*}\mathrm{D}_{\mu}\Phi-\Phi\left(\mathrm{D}_{\mu}\Phi\right)^{*}\right)$
$\displaystyle=-i\left(\Phi^{*}\partial_{\mu}\Phi-\Phi\partial_{\mu}\Phi^{*}-2iq\left|\Phi\right|^{2}A_{\mu}\right).$

As expected, since it measures the change in the length of $\vec{\Phi}$, we
can see that $\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}$ is independent of
$A_{\mu}$, which affects only rotations; it may also be written

$\displaystyle\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}$
$\displaystyle=\partial_{\mu}\left|\Phi\right|$ (2.17)
$\displaystyle=\mathrm{D}_{\mu}\left|\Phi\right|.$

Figure 2.4: The internal space vector $\mathrm{D}_{\mu}\vec{\Phi}$ may be
split into components parallel and perpendicular to $\vec{\Phi}$. The length
of the parallel component is the orthogonal projection of
$\mathrm{D}_{\mu}\vec{\Phi}$ on to $\vec{\Phi}$, which is the real inner
product over the length of $\vec{\Phi}$, i.e.
$\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}=\left\langle\mathrm{D}_{\mu}\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}/\left\|\vec{\Phi}\right\|$.
The length of the perpendicular component $\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$
is the orthogonal projection of $\mathrm{D}_{\mu}\vec{\Phi}$ on to the
$\pi/2$-rotated $\vec{\Phi}$, which is
$\left\langle\mathrm{D}_{\mu}\vec{\Phi},\overrightarrow{\left(i\Phi\right)}\right\rangle_{\mathbb{R}}=\mathrm{Im}\left\langle\Phi,\mathrm{D}_{\mu}\Phi\right\rangle_{\mathbb{C}}$
over the length of $\vec{\Phi}$. In the figure we see that since
$\overrightarrow{\left(i\Phi\right)}$ is rotated in the counterclockwise
direction, a positive value of $\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$ represents
a counterclockwise rotation of $\vec{\Phi}$ relative to parallel transport.

Now, $\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$ is the length of the component
perpendicular to $\vec{\Phi}$ of the difference between $\vec{\Phi}$ and its
parallel transport in the $e_{\mu}$ direction, i.e. it is the distance by
which $\vec{\Phi}$ moves due to rotation, ignoring the change in its length.
Since the rotation is infinitesimal, the sine is equal to the angle in
radians; hence the angular velocity of $\vec{\Phi}$ relative to parallel
transport per unit distance in the $e_{\mu}$ direction is

$\displaystyle\begin{aligned}
\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}&\equiv\frac{\mathrm{D}_{\mu}^{\perp}\vec{\Phi}}{\left\|\vec{\Phi}\right\|}\\\
&=\frac{\mathrm{Im}\left\langle\Phi,\mathrm{D}_{\mu}\Phi\right\rangle_{\mathbb{C}}}{\left|\Phi\right|^{2}}\\\
&=\frac{\mathrm{Im}\left\langle\Phi,\partial_{\mu}\Phi\right\rangle_{\mathbb{C}}}{\left|\Phi\right|^{2}}-qA_{\mu}\\\
&=\frac{\partial_{\mu}^{\perp}\vec{\Phi}}{\left\|\vec{\Phi}\right\|}+A_{\mu}^{\varangle}\\\
&=\partial_{\mu}^{\varangle}\vec{\Phi}+A_{\mu}^{\varangle}\end{aligned}$
(2.18)

in radians per coordinate length. Note that as expected, this is the
counterclockwise angular velocity of the components in the chosen gauge plus
the counterclockwise angular velocity of the coordinate axes relative to
parallel transport. We can also see this by using (2.10), whereby the third
line above yields

$\displaystyle\begin{aligned}
\mathrm{D}_{\mu}^{\varangle}\Phi&=\frac{-i\left(\Phi^{*}\partial_{\mu}\Phi-\Phi\partial_{\mu}\Phi^{*}\right)}{2\left|\Phi\right|^{2}}-qA_{\mu}\\\
&=\partial_{\mu}\varphi-qA_{\mu}\\\ &=\mathrm{D}_{\mu}\varphi,\end{aligned}$
(2.19)

which again splits into the angular velocity of the coordinate plus the
angular velocity of the coordinate axes.

The above results allow us to use (2.12) to express the squared length of the
perpendicular component in terms of $\vec{\Phi}\in\mathbb{R}^{2}$ , i.e.

$\displaystyle\begin{aligned}
\left\langle\mathrm{D}_{\mu}^{\perp}\vec{\Phi},\mathrm{D}_{\mu}^{\perp}\vec{\Phi}\right\rangle&=\left\|\vec{\Phi}\right\|^{2}\left\langle\mathrm{D}_{\mu}^{\varangle}\vec{\Phi},\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\right\rangle=\left|\Phi\right|^{2}\left\langle\mathrm{D}_{\mu}\varphi,\mathrm{D}_{\mu}\varphi\right\rangle\\\
&=\left\langle\mathrm{D}_{\mu}\Phi,\mathrm{D}_{\mu}\Phi\right\rangle-\left\langle\mathrm{D}_{\mu}\left|\Phi\right|,\mathrm{D}_{\mu}\left|\Phi\right|\right\rangle\\\
&=\left\langle\mathrm{D}_{\mu}\vec{\Phi},\mathrm{D}_{\mu}\vec{\Phi}\right\rangle-\left\langle\mathrm{D}_{\mu}^{\parallel}\vec{\Phi},\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}\right\rangle\\\
&=\mathrm{D}_{\mu}\Phi^{a}\mathrm{D}^{\mu}\Phi_{a}-\frac{1}{\left\|\vec{\Phi}\right\|^{2}}\left\langle\mathrm{D}_{\mu}\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}\left\langle\mathrm{D}^{\mu}\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}\\\
&=\mathrm{D}_{\mu}\Phi^{a}\mathrm{D}^{\mu}\Phi_{a}-\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}_{\mu}\Phi_{a}\right)\Phi^{b}\left(\mathrm{D}^{\mu}\Phi_{b}\right).\end{aligned}$
(2.20)

Finally, multiplying by the area per radian yields

$\displaystyle\begin{aligned}
\mathrm{D}_{\mu}^{\leftslice}\vec{\Phi}&\equiv\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\frac{\pi\left\|\vec{\Phi}\right\|^{2}}{2\pi}\\\
&=\frac{1}{2}\left\|\vec{\Phi}\right\|^{2}\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\\\
&=\frac{1}{2}\left\|\vec{\Phi}\right\|\mathrm{D}_{\mu}^{\perp}\vec{\Phi},\end{aligned}$
(2.21)

the (counterclockwise) area swept out by the matter field (relative to
parallel transport) per unit distance in the $e_{\mu}$ direction.

Figure 2.5: The infinitesimal counterclockwise rotation of the matter field
vector relative to parallel transport per unit distance in the $e_{\mu}$
direction is
$\mathrm{D}_{\mu}^{\varangle}\Phi=\partial_{\mu}\varphi+A_{\mu}^{\varangle}$,
the rotation of the angle in $\Phi\equiv\left|\Phi\right|e^{i\varphi}$ plus
the rotation of the coordinate axes. The infinitesimal area swept out by the
rotation of the matter field is
$\mathrm{D}_{\mu}^{\leftslice}\vec{\Phi}=\frac{1}{2}\left\|\vec{\Phi}\right\|\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$.

###  2.6 Quasi-gauge electromagnetism

Classical electromagnetism, excluding mass and gravity, is usually defined
(using natural units and the mostly pluses metric signature in flat spacetime)
as a $U(1)$ gauge theory with no matter field and Lagrangian

$\displaystyle L_{\widetilde{\mathrm{EM}}}$ $\displaystyle\equiv
A_{\mu}J_{q}^{\mu}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$ (2.22)

where

$J_{q}^{\mu}\equiv qJ^{\mu}\equiv q\rho_{0}U^{\mu}$ (2.23)

is the electromagnetic four-current, which may be defined in terms of the
charge $q$ and the matter four-current $J$, which in turn may be defined in
terms of the matter rest density $\rho_{0}$ and the unit time-like four-vector
$U$ in the direction of the four-current. The charge $q$ may be viewed as
being per particle number (theoretically fractional for a continuum), which
substitutes “particle number” for “matter” above.

The equations of motion (EOM) which result from varying the gauge potential
are (see Section 3.3, or for a treatment in terms of forms see Section 4.1 in
[3]) Maxwell’s equations

$\displaystyle J_{q}^{\nu}$ $\displaystyle=\nabla_{\mu}F^{\nu\mu}$ (2.24)
$\displaystyle\Rightarrow\nabla_{\nu}J_{q}^{\nu}$
$\displaystyle=\nabla_{\nu}\nabla_{\mu}F^{\nu\mu}=0,$ (2.25)

where $\nabla$ is the Levi-Civita covariant derivative, so that (2.25) implies
$J_{q}$ is divergenceless. These EOM are used to excuse the fact that the
Lagrangian is not gauge invariant (and is therefore ill-defined from the
geometric point of view), since under a gauge transformation $e^{-iq\Lambda}$
we have

$\displaystyle A_{\mu}$ $\displaystyle\rightarrow
A_{\mu}+\partial_{\mu}\Lambda$ (2.26) $\displaystyle\Rightarrow
L_{\widetilde{\mathrm{EM}}}$ $\displaystyle\rightarrow
L_{\widetilde{\mathrm{EM}}}-J_{q}^{\mu}\partial_{\mu}\Lambda$
$\displaystyle=L_{\widetilde{\mathrm{EM}}}-\nabla_{\mu}\left(J_{q}^{\mu}\Lambda\right)+\Lambda\nabla_{\mu}J_{q}^{\mu},$

where the first extra term is a divergence and therefore does not change the
equations of motion, while the second vanishes via the EOM. Varying the metric
(see Section 3.5) identifies the Hilbert stress energy momentum (SEM) tensor
as

$\displaystyle T_{\mathrm{EM}}^{\mu\nu}$ $\displaystyle\equiv
F^{\mu\lambda}F^{\nu}{}_{\lambda}-\frac{1}{4}F^{\lambda\sigma}F_{\lambda\sigma}g^{\mu\nu}.$
(2.27)

Following e.g. [2], we may include gravity and associate a mass $m$ as well as
a charge with the four-current by defining

$\displaystyle L_{\mathrm{G}\widetilde{\mathrm{EM}}}$
$\displaystyle\equiv-m\sqrt{-J^{\mu}J^{\nu}g_{\mu\nu}}+qA_{\mu}J^{\mu}-\frac{1}{4}g^{\mu\nu}g^{\lambda\sigma}F_{\mu\lambda}F_{\nu\sigma}+\frac{1}{16\pi}R,$
(2.28)

where $R$ is the spacetime scalar curvature. The new terms leave the gauge
potential EOM unchanged, but upon variation of the metric yield a Hilbert SEM
tensor with a time-like dust term

$\displaystyle T_{\mathrm{GEM}}^{\mu\nu}$ $\displaystyle\equiv
m\rho_{0}U^{\mu}U^{\nu}+T_{\mathrm{EM}}^{\mu\nu}.$ (2.29)

This tensor is proportional to the Einstein tensor and hence must be
divergenceless, which (see Section 3.5) yields the equations of geodesic
deviation

$m\rho_{0}\left(\nabla_{U}U\right)^{\mu}=F^{\mu}{}_{\nu}J_{q}^{\nu},$ (2.30)

better known as the Lorentz force law.

###  2.7 Klein-Gordon theory

For reference, we here summarize Klein-Gordon theory (complex, minimally
coupled, and in flat spacetime; sometimes then called scalar electrodynamics
or scalar QED), which is usually defined (again using natural units and the
mostly pluses metric signature) as a $U(1)$ gauge theory with a complex scalar
matter field and Lagrangian

$\displaystyle L_{\mathrm{KG}}$
$\displaystyle\equiv-\left(\mathrm{D}_{\mu}\Phi\right)^{*}\mathrm{D}^{\mu}\Phi-m^{2}\Phi^{*}\Phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$
(2.31)

This is of course usually considered as a quantum theory, but here we want to
explore whether it can provide at least inspiration for a model of classical
electromagnetism.

Expanding the first and last terms (see Section 3.1), we find the EOM from
varying the gauge potential are

$\displaystyle-
iq\left(\Phi^{*}\mathrm{D}^{\nu}\Phi-\Phi\left(\mathrm{D}^{\nu}\Phi\right)^{*}\right)$
$\displaystyle=\nabla_{\mu}F^{\nu\mu},$ (2.32)

allowing us to identify the divergenceless matter four-current

$\displaystyle J^{\nu}$
$\displaystyle\equiv-i\left(\Phi^{*}\mathrm{D}^{\nu}\Phi-\Phi\left(\mathrm{D}^{\nu}\Phi\right)^{*}\right),$
(2.33)

in terms of which (2.32) is Maxwell’s equations. Varying the matter field,
however, results in the Klein-Gordon equation

$\displaystyle\mathrm{D}_{\mu}\mathrm{D}^{\mu}\Phi$ $\displaystyle=m^{2}\Phi,$
(2.34)

from which point it is difficult to proceed classically. Firstly, these EOM do
not constrain the particle density associated with $J$ to be positive, and
secondly, we have no way to eliminate $\Phi$ from the Lagrangian in order to
obtain the classical Hilbert SEM tensor $T_{\mathrm{GEM}}^{\mu\nu}$ of (2.29).

We nevertheless may obtain some results which might act as inspiration for a
classical theory by taking $A=0$. Then (free complex) Klein-Gordon theory no
longer includes electromagnetism, but the associated Klein-Gordon equation

$\displaystyle\begin{aligned}
\partial_{\mu}\partial^{\mu}\Phi&=m^{2}\Phi\end{aligned}$ (2.35)

has a general solution which is a linear combination of plane wave solutions

$\displaystyle\Phi_{P}$
$\displaystyle\equiv\left|\Phi\right|e^{iP_{\mu}x^{\mu}},$ (2.36)

where the complex modulus $\left|\Phi\right|$ and the four-vector $P$ are
constant. Taking derivatives, we have

$\displaystyle\begin{aligned} \partial_{\mu}\Phi_{P}&=iP_{\mu}\Phi_{P}\\\
\Rightarrow\partial_{\mu}\partial^{\mu}\Phi_{P}&=-P_{\mu}P^{\mu}\Phi_{P}=m^{2}\Phi_{P}\\\
\Rightarrow P^{\mu}&=mU^{\mu},\end{aligned}$ (2.37)

where $U$ is again a unit time-like four-vector, while

$\displaystyle\begin{aligned} J_{P}^{\mu}&=2\left|\Phi\right|^{2}P^{\mu}\\\
\Rightarrow\rho_{0}&=2m\left|\Phi\right|^{2}.\end{aligned}$ (2.38)

Geometrically, these are matter fields which rotate in the time-like $P^{\mu}$
direction with an angular velocity of $m$ radians per unit proper time, and
whose constant length squared is proportional to the rest density. Moreover,
on-shell for a plane wave the Lagrangian may be written as

$\displaystyle L_{\mathrm{KG}-P}$
$\displaystyle=-\frac{1}{2}m\rho_{0}U_{\mu}U_{\nu}g^{\mu\nu}-\frac{1}{2}m\rho_{0},$
(2.39)

which upon varying the metric (see Section 3.5) yields a time-like dust
Hilbert SEM tensor of $m\rho_{0}U^{\mu}U^{\nu}$.

This is all quite suggestive, but faces two issues as inspiration for
constructing a classical gauge theory of electromagnetism: (1) we eliminated
the electromagnetic interaction to get solutions, and (2) we cannot construct
a general composite classical four-current by combining plane wave solutions,
since their phases interfere with each other, i.e. they are quantum in nature.
In Section 3 which follows, we arrive at an alternative Lagrangian which
avoids these issues while taking advantage of the above observations.

##  3 Matter field electromagnetism

###  3.1 The Lagrangian

With the geometric quantities of Section 2 in hand, we now define matter field
electromagnetism (MFEM) (again using natural units and the mostly pluses
metric signature) as a geometric $U(1)$ gauge theory with Lagrangian

$\displaystyle\begin{aligned}
L_{\mathrm{EM}}&\equiv-\left\langle\mathrm{D}_{\mu}^{\perp}\vec{\Phi},\mathrm{D}_{\mu}^{\perp}\vec{\Phi}\right\rangle-m^{2}\left\langle\vec{\Phi},\vec{\Phi}\right\rangle-\frac{1}{2}\left\langle
F,F\right\rangle\\\
&=\frac{1}{4\left|\Phi\right|^{2}}\left(\Phi^{*}\mathrm{D}_{\mu}\Phi-\Phi\left(\mathrm{D}_{\mu}\Phi\right)^{*}\right)\left(\Phi^{*}\mathrm{D}^{\mu}\Phi-\Phi\left(\mathrm{D}^{\mu}\Phi\right)^{*}\right)-m^{2}\left|\Phi\right|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.\end{aligned}$
(3.1)

Geometrically, the absolute value of the dynamical term is the squared
perpendicular distance between the matter field and its parallel transport per
unit distance in the direction in which this distance is extremal. Its
negative can be written in terms of various quantities as

$\displaystyle\begin{aligned}
-L_{\mathrm{EM-D}}&=\left\|\mathrm{D}_{\mu}^{\perp}\vec{\Phi}\right\|^{2}\\\
&=\left\|\vec{\Phi}\right\|^{2}\left\langle\mathrm{D}_{\mu}^{\varangle}\vec{\Phi},\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\right\rangle\\\
&=\left\|\mathrm{D}_{\mu}\vec{\Phi}\right\|^{2}-\frac{1}{\left\|\vec{\Phi}\right\|^{2}}\left\langle\mathrm{D}_{\mu}\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}\left\langle\mathrm{D}^{\mu}\vec{\Phi},\vec{\Phi}\right\rangle_{\mathbb{R}}\\\
&=\mathrm{D}_{\mu}\Phi^{a}\mathrm{D}^{\mu}\Phi_{a}-\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}_{\mu}\Phi_{a}\right)\Phi^{b}\left(\mathrm{D}^{\mu}\Phi_{b}\right)\\\
&=\frac{1}{\left|\Phi\right|^{2}}\mathrm{Im}\left\langle\mathrm{D}_{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}\mathrm{Im}\left\langle\mathrm{D}^{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}\\\
&=\left|\Phi\right|^{2}\left(\partial_{\mu}\varphi-
qA_{\mu}\varphi\right)\left(\partial^{\mu}\varphi-
qA^{\mu}\varphi\right).\end{aligned}$ (3.2)

Note that if $\mathrm{D}_{\mu}^{\parallel}\vec{\Phi}=0$, i.e. if the matter
field has constant length, then
$\mathrm{D}_{\mu}^{\perp}\vec{\Phi}=\mathrm{D}_{\mu}\vec{\Phi}$ and our
Lagrangian is identical to the Klein-Gordon Lagrangian.

In order to vary the gauge potential, we must express the Lagrangian
explicitly in terms of $A_{\mu}$. Expanding out the gauge covariant
derivatives yields a dynamical term for gauge potential variations of

$\displaystyle\begin{aligned}
L_{\mathrm{EM-D}}&=\frac{1}{4\left|\Phi\right|^{2}}\left\|\Phi^{*}\partial_{\mu}\Phi-\Phi\partial_{\mu}\Phi^{*}-2iq\left|\Phi\right|^{2}A_{\mu}\right\|^{2}\\\
\Rightarrow
L_{\mathrm{EM-D}}\left(A_{\mu}\right)&=-iqA_{\mu}\left(\Phi^{*}\partial^{\mu}\Phi-\Phi\partial^{\mu}\Phi^{*}\right)-q^{2}\left|\Phi\right|^{2}A_{\mu}A^{\mu}.\end{aligned}$
(3.3)

Note that this is identical to that of the Klein-Gordon dynamical term

$\displaystyle\begin{aligned} L_{\mathrm{KG-D}}&=-\left(\partial_{\mu}\Phi-
iqA_{\mu}\Phi\right)\left(\partial^{\mu}\Phi^{*}+iqA^{\mu}\Phi^{*}\right)\\\
\Rightarrow
L_{\mathrm{KG-D}}\left(A_{\mu}\right)&=-iqA_{\mu}\left(\Phi^{*}\partial^{\mu}\Phi-\Phi\partial^{\mu}\Phi^{*}\right)-q^{2}\left|\Phi\right|^{2}A_{\mu}A^{\mu},\end{aligned}$
(3.4)

which means we will end up with the same gauge potential EOM.

###  3.2 Matter field equations of motion

The Euler-Lagrange equation may be written

$\displaystyle\frac{\partial L}{\partial\Phi}$
$\displaystyle=\mathrm{D}_{\mu}p_{\Phi}^{\mu}$ (3.5)

in terms of the canonical momentum

$\displaystyle p_{\Phi}^{\mu}$ $\displaystyle\equiv\frac{\partial
L}{\partial\left(\mathrm{D}_{\mu}\Phi\right)}.$ (3.6)

In keeping with our geometrical viewpoint, we opt to work here in terms of the
real vector components $\Phi^{a}$, but in Section 3.6 we derive the same
results in terms of $\Phi$ and $\Phi^{*}$ as independent quantities, as is
more common. Recalling (2.20), we have

$\displaystyle\begin{aligned}
L_{\mathrm{EM}}\left(\Phi^{a},\mathrm{D}_{\mu}\Phi^{a}\right)&=-\mathrm{D}_{\mu}\Phi^{a}\mathrm{D}^{\mu}\Phi_{a}+\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}_{\mu}\Phi_{a}\right)\Phi^{b}\left(\mathrm{D}^{\mu}\Phi_{b}\right)-m^{2}\Phi_{a}\Phi^{a},\\\
\frac{\partial
L_{\mathrm{EM}}}{\partial\Phi^{d}}&=-2\left(\Phi_{c}\Phi^{c}\right)^{-2}\Phi_{d}\Phi^{a}\left(\mathrm{D}_{\mu}\Phi_{a}\right)\Phi^{b}\left(\mathrm{D}^{\mu}\Phi_{b}\right)\\\
&\phantom{{}=}+2\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}_{\mu}\Phi_{a}\right)\left(\mathrm{D}^{\mu}\Phi_{d}\right)-2m^{2}\Phi_{d},\\\
p_{\Phi^{d}}^{\mu}=\frac{\partial
L_{\mathrm{EM}}}{\partial\left(\mathrm{D}_{\mu}\Phi^{d}\right)}&=-2\mathrm{D}^{\mu}\Phi_{d}+2\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}^{\mu}\Phi_{a}\right)\Phi_{d}\\\
\Rightarrow\mathrm{D}_{\mu}p_{\Phi^{d}}^{\mu}&=-2\mathrm{D}_{\mu}\mathrm{D}^{\mu}\Phi_{d}-4\left(\Phi_{c}\Phi^{c}\right)^{-2}\Phi^{b}\left(\mathrm{D}_{\mu}\Phi_{b}\right)\Phi^{a}\left(\mathrm{D}^{\mu}\Phi_{a}\right)\Phi_{d}\\\
&\phantom{{}=}+2\left(\Phi_{c}\Phi^{c}\right)^{-1}\mathrm{D}_{\mu}\Phi^{a}\left(\mathrm{D}^{\mu}\Phi_{a}\right)\Phi_{d}\\\
&\phantom{{}=}+2\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}_{\mu}\mathrm{D}^{\mu}\Phi_{a}\right)\Phi_{d}+2\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}^{\mu}\Phi_{a}\right)\mathrm{D}_{\mu}\Phi_{d}.\end{aligned}$
(3.7)

But

$2\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}_{\mu}\mathrm{D}^{\mu}\Phi_{a}\right)\Phi_{d}=2\left(\mathrm{D}_{\mu}\mathrm{D}^{\mu}\Phi_{d}\right)$
(3.8)

since applying each side to $\Phi^{d}$ yields the same result. Therefore the
Euler-Lagrange equation is

$\displaystyle\begin{aligned}
-2m^{2}\Phi_{d}&=-2\left(\Phi_{c}\Phi^{c}\right)^{-2}\Phi^{b}\left(\mathrm{D}_{\mu}\Phi_{b}\right)\Phi^{a}\left(\mathrm{D}^{\mu}\Phi_{a}\right)\Phi_{d}+2\left(\Phi_{c}\Phi^{c}\right)^{-1}\mathrm{D}_{\mu}\Phi^{a}\mathrm{D}^{\mu}\Phi_{a}\Phi_{d}\\\
&=-2\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi_{d}\left(-\mathrm{D}_{\mu}\Phi^{a}\mathrm{D}^{\mu}\Phi_{a}+\left(\Phi_{c}\Phi^{c}\right)^{-1}\Phi^{a}\left(\mathrm{D}^{\mu}\Phi_{a}\right)\Phi^{b}\left(\mathrm{D}_{\mu}\Phi_{b}\right)\right)\\\
\Rightarrow-m^{2}&=\left\langle\mathrm{D}_{\mu}^{\varangle}\vec{\Phi},\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\right\rangle.\end{aligned}$
(3.9)

We obtain this same result if, as is more common, we use the equivalent Euler-
Lagrange expression $\partial L/\partial\Phi=\partial_{\mu}\left(\partial
L/\partial\left(\partial_{\mu}\Phi\right)\right)$ in flat spacetime.

Using the results of Section 2.3, we may geometrically describe the solutions
of this equation as those for which at every point in spacetime, the unit
length four-vector $U^{\mu}$ in the direction of extremal angular velocity is
time-like, and in this direction the angular velocity is constant and equal to
$m$:

$\mathrm{D}_{U}^{\varangle}\vec{\Phi}=\pm m$ (3.10)

So when the matter field satisfies the EOM (is on-shell), $m$ is identified
with the (constant) angular velocity of the matter field (relative to parallel
transport) per unit proper time in the direction for which this angular
velocity is extremal. In Section 3.5, $m$ will be also identified as the mass
associated with the matter four-current, which in Section 3.3 we will see
points in this same direction.

We may also note that the matter field part of $L_{\mathrm{EM}}$ vanishes on-
shell (as expected from Euler’s homogeneous function theorem, since it is
homogeneous of degree 2 and
$\nabla_{\mu}\left(\Phi^{d}p_{\Phi^{d}}^{\mu}\right)=0$); and we now have yet
another way to express the negative dynamical term, this time as the square of
the canonical momentum

$\displaystyle\begin{aligned}
-4L_{\mathrm{EM-D}}&=\left\|p_{\Phi}\right\|^{2},\end{aligned}$ (3.11)

reminiscent of Lagrangians for relativistic particles (the canonical momentum
obtained in Section 3.6 by differentiating by $\Phi$ independently of
$\Phi^{*}$ even removes the factor of four).

###  3.3 Gauge potential equations of motion

From (3.3) and (2.8), the Lagrangian for gauge potential variations is

$\displaystyle\begin{aligned}
L_{\mathrm{EM}}\left(A_{\mu},\partial_{\nu}A_{\mu}\right)&=-iqA_{\mu}\left(\Phi^{*}\partial^{\mu}\Phi-\Phi\partial^{\mu}\Phi^{*}\right)-q^{2}\left|\Phi\right|^{2}A_{\mu}A^{\mu}\\\
&\phantom{{}=}-\frac{1}{2}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}\right).\end{aligned}$
(3.12)

Our dynamical term does not depend on the derivative of the gauge potential,
so the electromagnetic four-current will be $\partial L/\partial A_{\nu}$ as
is usual in other EM gauge theories. Explicitly,

$\displaystyle\frac{\partial L}{\partial A_{\nu}}$
$\displaystyle=-iq\left(\Phi^{*}\partial^{\nu}\Phi-\Phi\partial^{\nu}\Phi^{*}\right)-2q^{2}\left|\Phi\right|^{2}A^{\nu}$
(3.13)
$\displaystyle=-iq\left(\Phi^{*}\mathrm{D}^{\nu}\Phi-\Phi\left(\mathrm{D}^{\nu}\Phi\right)^{*}\right),$
$\displaystyle p_{A_{\nu}}^{\mu}\equiv\frac{\partial
L}{\partial\left(\partial_{\mu}A_{\nu}\right)}$
$\displaystyle=-\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)$
$\displaystyle=-F^{\mu\nu},$

yielding an Euler-Lagrange equation of

$\displaystyle
iq\left(\Phi^{*}\mathrm{D}^{\nu}\Phi-\Phi\left(\mathrm{D}^{\nu}\Phi\right)^{*}\right)$
$\displaystyle=\nabla_{\mu}F^{\mu\nu},$ (3.14)

which identifies the electromagnetic four-current as

$\displaystyle J_{q}^{\nu}$ $\displaystyle\equiv-
iq\left(\Phi^{*}\mathrm{D}^{\nu}\Phi-\Phi\left(\mathrm{D}^{\nu}\Phi\right)^{*}\right).$
(3.15)

As in (2.25) $J_{q}$ is therefore divergenceless when the gauge potential EOM
are satisfied; reversing the indices of $F$ yields the usual form of Maxwell’s
equations

$\displaystyle J_{q}^{\nu}$ $\displaystyle=\nabla_{\mu}F^{\nu\mu}.$ (3.16)

###  3.4 Geometry of the four-current

The matter four-current may be written in terms of the various quantities we
have defined as

$\displaystyle J_{\mu}$
$\displaystyle=-i\left(\Phi^{*}\mathrm{D}_{\nu}\Phi-\Phi\left(\mathrm{D}_{\nu}\Phi\right)^{*}\right)$
(3.17)
$\displaystyle=-i\left(\Phi^{*}\partial_{\mu}\Phi-\Phi\partial_{\mu}\Phi^{*}-2iq\left|\Phi\right|^{2}A_{\mu}\right)$
$\displaystyle=2\left\|\vec{\Phi}\right\|\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$
$\displaystyle=2\left\|\vec{\Phi}\right\|^{2}\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}$
$\displaystyle=4\mathrm{D}_{\mu}^{\leftslice}\vec{\Phi}$
$\displaystyle=2\left|\Phi\right|^{2}\left(\partial_{\mu}\varphi+A_{\mu}^{\varangle}\right)$
$\displaystyle=2\mathrm{Im}\left\langle\mathrm{D}_{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}.$

The fourth expression in (3.17) explicitly identifies the direction of the
four-current at a given point as being the direction in which the matter field
angular velocity (relative to parallel transport) is extremal. Recall from
Section 3.2 that the matter field EOM require the unit length four-vector
$U^{\mu}$ in this direction to be time-like. We may then write

$\displaystyle J_{\mu}$ $\displaystyle=\rho_{0}U_{\mu}$ (3.18)
$\displaystyle\Rightarrow\rho_{0}$ $\displaystyle=-U^{\mu}J_{\mu}$
$\displaystyle=-2\left\|\vec{\Phi}\right\|^{2}\mathrm{D}_{U}^{\varangle}\vec{\Phi}$
$\displaystyle=-4\mathrm{D}_{U}^{\leftslice}\vec{\Phi},$

so that geometrically the matter per volume orthogonal to the four-current is
proportional to the internal space area swept out clockwise by the matter
field per unit proper time along the four-current.

Since the matter field EOM are $\mathrm{D}_{U}^{\varangle}\vec{\Phi}=\pm m$,
and particle density must be positive, we henceforth only consider the
negative value

$\mathrm{D}_{U}^{\varangle}\vec{\Phi}=-m$ (3.19)

to be physical, so that we have

$\displaystyle\begin{aligned}
\rho_{0}&=2\left\|\vec{\Phi}_{\mathrm{EL}}\right\|^{2}m\\\ \Rightarrow
J^{\mu}&=2\left\|\vec{\Phi}_{\mathrm{EL}}\right\|^{2}mU^{\mu}\end{aligned}$
(3.20)

in terms of the on-shell matter field $\vec{\Phi}_{\mathrm{EL}}$, whose
rotation relative to parallel transport in the direction $U$ is always
clockwise. The reversed sign matter field EOM may be associated with a
positive particle density by defining the electromagnetic four-current with
the opposite sign, which is equivalent to simply flipping the sign of $q$.
Thus positively charged four-currents rotate clockwise, while negatively
charged four-currents rotate counterclockwise.

We presently see an immediate distinction with the classical theory of
continua: the metric dependence of $\rho_{0}$ and $J^{\mu}$. In the classical
theory, $\rho_{0}$ and $J^{\mu}$ both depend upon the metric such that the
four-current density $\mathfrak{J}\equiv J\sqrt{-\det\left(g_{\mu\nu}\right)}$
is metric-independent. In MFEM, $\rho_{0}$ has a different metric dependency
and $J$ is a metric-independent 1-form, except on-shell where $\rho_{0}$ is
metric-independent and $J$ is proportional to a unit vector. This will be
explained in Section 5.2 when viewing MFEM as an approximation of QED.

It is important to note that despite the appearance of $m$ in the on-shell
expression for $J$, it is the expression for the matter (particle number)
four-current; the electromagnetic four-current is $J_{q}\equiv qJ$, just as
the mass four-current is $J_{m}\equiv mJ$, which will therefore have a factor
$m^{2}$. We also may note that we can now write the negative dynamical term of
the Lagrangian on-shell as

$\displaystyle\begin{aligned} -L_{\mathrm{EM-D-
EL}}&=\frac{1}{4\left\|\vec{\Phi}_{\mathrm{EL}}\right\|^{2}}\left\langle
J,J\right\rangle\\\ &=\frac{1}{2}\frac{m}{\rho_{0}}\left\langle
J,J\right\rangle\\\ &=\frac{1}{2}m\rho_{0}\left\langle
U,U\right\rangle,\end{aligned}$ (3.21)

which is the form of a Lagrangian for relativistic dust.

###  3.5 Metric equations of motion

As noted in the previous section, both the four-current 1-form
$J_{\nu}=-i\left(\Phi^{*}\mathrm{D}_{\nu}\Phi-\Phi\left(\mathrm{D}_{\nu}\Phi\right)^{*}\right)$
and the rest density $\rho_{0}=2\left\|\vec{\Phi}_{\mathrm{EL}}\right\|^{2}m$
are metric-independent, so that for an on-shell matter field we use (3.21) and
(3.20) to write the action as

$\displaystyle S_{\mathrm{EM}}\left(g_{\mu\nu}\right)$
$\displaystyle=\int\left(-\frac{1}{2}\frac{m}{\rho_{0}}J_{\mu}J_{\nu}g^{\mu\nu}-\frac{1}{2}m\rho_{0}-\frac{1}{4}g^{\mu\nu}g^{\lambda\sigma}F_{\mu\lambda}F_{\nu\sigma}\right)\sqrt{g}\mathrm{d}^{4}x.$
(3.22)

Recalling that $\delta g^{\mu\nu}=-g^{\mu\lambda}g^{\nu\sigma}\delta
g_{\lambda\sigma}$ and
$\delta\left(\sqrt{g}\right)=\frac{1}{2}\sqrt{g}g^{\mu\nu}\delta g_{\mu\nu}$,
the variation of the action yields

$\displaystyle\delta S_{\mathrm{EM}}\left(g_{\mu\nu}\right)$
$\displaystyle=\frac{1}{2}\int\left(\frac{m}{\rho_{0}}J^{\mu}J^{\nu}\sqrt{g}-\frac{m}{\rho_{0}}J_{\lambda}J_{\sigma}g^{\lambda\sigma}\frac{1}{2}\sqrt{g}g^{\mu\nu}-m\rho_{0}\frac{1}{2}\sqrt{g}g^{\mu\nu}\right)\delta
g_{\mu\nu}\mathrm{d}^{4}x$ (3.23)
$\displaystyle\phantom{{}=}-\frac{1}{4}\int\left(2g^{\lambda\sigma}F_{\mu\lambda}F_{\nu\sigma}\sqrt{g}\delta
g^{\mu\nu}+F^{\lambda\sigma}F_{\lambda\sigma}\frac{1}{2}\sqrt{g}g^{\mu\nu}\delta
g_{\mu\nu}\right)\mathrm{d}^{4}x$
$\displaystyle=\frac{1}{2}\int\left(m\rho_{0}U^{\mu}U^{\nu}+T_{\mathrm{EM}}^{\mu\nu}\right)\delta
g_{\mu\nu}\sqrt{g}\mathrm{d}^{4}x,$

where $T_{\mathrm{EM}}^{\mu\nu}$ is defined per (2.27) and the quantity in
parentheses in the last line defines the Hilbert SEM tensor, which is equal to
$T_{\mathrm{GEM}}^{\mu\nu}$ from (2.29), matching that of quasi-gauge EM as
desired.

This implies the Lorentz force law, completing our equivalence with classical
electromagnetism, as we quickly review (see e.g. [2]).
$T_{\mathrm{GEM}}^{\mu\nu}$ is proportional to the Einstein tensor and hence
must be divergenceless, which yields the equations of geodesic deviation

$\displaystyle\nabla_{\nu}T_{\mathrm{GEM}}^{\mu\nu}$
$\displaystyle=\nabla_{\nu}\left(m\rho_{0}U^{\mu}U^{\nu}+g_{\lambda\sigma}F^{\mu\lambda}F^{\nu\sigma}-\frac{1}{4}F^{\lambda\sigma}F_{\lambda\sigma}g^{\mu\nu}\right)$
(3.24)
$\displaystyle=mU^{\mu}\nabla_{\nu}J^{\nu}+m\rho_{0}U^{\nu}\nabla_{\nu}U^{\mu}$
$\displaystyle\phantom{{}=}+g_{\lambda\sigma}F^{\mu\lambda}\nabla_{\nu}F^{\nu\sigma}+g_{\lambda\sigma}F^{\nu\sigma}\nabla_{\nu}F^{\mu\lambda}-\frac{1}{2}F^{\lambda\sigma}\nabla_{\nu}F_{\lambda\sigma}g^{\mu\nu}$
$\displaystyle=m\rho_{0}\left(\nabla_{U}U\right)^{\mu}+F^{\mu}{}_{\sigma}\nabla_{\nu}F^{\nu\sigma}$
$\displaystyle\phantom{{}=}+\frac{1}{2}F^{\lambda\sigma}g^{\mu\nu}\left(\nabla_{\lambda}F_{\nu\sigma}-\nabla_{\sigma}F_{\nu\lambda}-\nabla_{\nu}F_{\lambda\sigma}\right)$
$\displaystyle\Rightarrow m\rho_{0}\left(\nabla_{U}U\right)^{\mu}$
$\displaystyle=F^{\mu}{}_{\sigma}J_{q}^{\sigma},$

where in the penultimate equality we use $\nabla_{\nu}J^{\nu}=0$ and the anti-
symmetry of $F$, yielding the three terms in parentheses which vanish due to
the second Bianchi identity, and in the last line we use the gauge potential
EOM. $U$ is the unit four-vector in the direction of the proper time $\tau$ of
the four-current, so that at a point in flat spacetime this equation becomes

$\displaystyle\partial_{\tau}P^{\mu}$
$\displaystyle=qF^{\mu}{}_{\sigma}U^{\sigma},$ (3.25)

where $P=mU$. In an inertial frame we then have components

$\displaystyle\partial_{\tau}\begin{pmatrix}E&p^{x}&p^{y}&p^{z}\end{pmatrix}$
$\displaystyle=q\begin{pmatrix}0&E^{x}&E^{y}&E^{z}\\\ E^{x}&0&B^{z}&-B^{y}\\\
E^{y}&-B^{z}&0&B^{x}\\\
E^{z}&B^{y}&-B^{x}&0\end{pmatrix}\begin{pmatrix}\gamma\\\ \gamma v^{x}\\\
\gamma v^{y}\\\ \gamma v^{z}\end{pmatrix}$ (3.26)
$\displaystyle\Rightarrow\partial_{t}\left(\mathbf{p}\right)$
$\displaystyle=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right),$

where $\gamma$ is the Lorentz factor, we use
$\partial_{\tau}=\gamma\partial_{t}$, and recall that $\mathbf{p}\equiv\gamma
m\mathbf{v}$ is the relativistic momentum, which in the non-relativistic limit
is just the momentum.

###  3.6 Noether’s theorem

In order to take advantage of the simplicity of complex gauge transformations,
we here re-derive the matter field EOM in complex notation. As with similar
theories, we may vary $\Phi$ and $\Phi^{*}$ as independent quantities. Using
the complex second expression of (3.1), we write the Lagrangian for matter
field variations in terms of the four-current as

$\displaystyle\begin{aligned}
L_{\mathrm{EM}}\left(\Phi,\mathrm{D}_{\mu}\Phi\right)&=\frac{1}{4\Phi^{*}\Phi}\left(iJ_{\mu}\right)\left(iJ^{\mu}\right)-m^{2}\Phi^{*}\Phi,\\\
iJ_{\mu}&=\Phi^{*}\mathrm{D}_{\mu}\Phi-\Phi\left(\mathrm{D}_{\mu}\Phi\right)^{*}.\end{aligned}$
(3.27)

Calculating derivatives then yields

$\displaystyle\begin{aligned} \frac{\partial
L_{\mathrm{EM}}}{\partial\Phi}&=\frac{J_{\mu}J^{\mu}\Phi^{*}}{4\left(\Phi^{*}\Phi\right)^{2}}-\frac{iJ^{\mu}\left(\mathrm{D}_{\mu}\Phi\right)^{*}}{2\Phi^{*}\Phi}-m^{2}\Phi^{*},\\\
p_{\Phi}^{\mu}=\frac{\partial
L_{\mathrm{EM}}}{\partial\left(\mathrm{D}_{\mu}\Phi\right)}&=\frac{\Phi^{*}iJ^{\mu}}{2\Phi^{*}\Phi}\\\
\Rightarrow\mathrm{D}_{\mu}\left(p_{\Phi}^{\mu}\right)&=-\frac{\Phi^{*}iJ^{\mu}}{2\left(\Phi^{*}\Phi\right)^{2}}\partial_{\mu}\left|\Phi\right|^{2}+\frac{\left(\mathrm{D}_{\mu}\Phi\right)^{*}iJ^{\mu}}{2\Phi^{*}\Phi}+\frac{\Phi^{*}i\nabla_{\mu}J^{\mu}}{2\Phi^{*}\Phi},\end{aligned}$
(3.28)

where we use the facts that $\mathrm{D}$ is $\partial$ if applied to a scalar
and $\nabla$ if applied to a vector, and
$\mathrm{D}_{\mu}\left(\Phi^{*}\right)=\left(\mathrm{D}_{\mu}\Phi\right)^{*}$.
If we multiply the Euler-Lagrange equation by $\Phi$, we arrive at

$\displaystyle\begin{aligned}
-m^{2}\Phi^{*}\Phi&=-\frac{J_{\mu}J^{\mu}}{4\Phi^{*}\Phi}-\frac{iJ^{\mu}}{2\Phi^{*}\Phi}\partial_{\mu}\left|\Phi\right|^{2}+\frac{\left(\mathrm{D}_{\mu}\Phi\right)^{*}\Phi
iJ^{\mu}}{\Phi^{*}\Phi}+\frac{i}{2}\nabla_{\mu}J^{\mu}.\end{aligned}$ (3.29)

But using (2.16) we have

$\displaystyle\begin{aligned}
\left\langle\mathrm{D}_{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}iJ^{\mu}&=\left(i\mathrm{Re}\left\langle\mathrm{D}_{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}-\mathrm{Im}\left\langle\mathrm{D}_{\mu}\Phi,\Phi\right\rangle_{\mathbb{C}}\right)J^{\mu}\\\
&=\left(\frac{i}{2}\partial_{\mu}\left|\Phi\right|^{2}+\frac{1}{2}J_{\mu}\right)J^{\mu},\end{aligned}$
(3.30)

so that the Euler-Lagrange equation multiplied by $\Phi$ is

$\displaystyle\begin{aligned}
-m^{2}\Phi^{*}\Phi&=\frac{J_{\mu}J^{\mu}}{4\Phi^{*}\Phi}+\frac{i}{2}\nabla_{\mu}J^{\mu},\end{aligned}$
(3.31)

whose real and imaginary parts yield

$\displaystyle\begin{aligned}
-m^{2}&=\left\langle\mathrm{D}_{\mu}^{\varangle}\vec{\Phi},\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\right\rangle,\\\
0&=\nabla_{\mu}J^{\mu}.\end{aligned}$ (3.32)

As we see below, the second equation is redundant. Differentiating instead by
$\Phi^{*}$ yields a canonical momentum of

$\displaystyle\begin{aligned} p_{\Phi^{*}}^{\mu}&=-\frac{\Phi
iJ^{\mu}}{2\Phi^{*}\Phi},\end{aligned}$ (3.33)

with an Euler-Lagrange equation multiplied by $\Phi^{*}$ of

$\displaystyle\begin{aligned}
-m^{2}\Phi^{*}\Phi&=\frac{J_{\mu}J^{\mu}}{4\Phi^{*}\Phi}-\frac{i}{2}\nabla_{\mu}J^{\mu},\end{aligned}$
(3.34)

whose second term we note has a reversed sign from (3.31). Again, we obtain
the same results if we use the equivalent Euler-Lagrange expression $\partial
L/\partial\Phi=\partial_{\mu}\left(\partial
L/\partial\left(\partial_{\mu}\Phi\right)\right)$ in flat spacetime.

With the matter field EOM confirmed and a complex canonical momentum in hand,
we may apply Noether’s theorem in complex form. The MFEM Lagrangian is
invariant under global infinitesimal gauge transformations, which in complex
notation transform the matter field according to

$\displaystyle\begin{aligned} \Phi&\rightarrow e^{iq\varepsilon}\Phi\\\
&=\Phi+iq\varepsilon\Phi,\end{aligned}$ (3.35)

yielding a corresponding Noether current of

$\displaystyle iq\Phi p_{\Phi}^{\mu}$ $\displaystyle=-\frac{1}{2}J_{q}^{\mu}$
(3.36) $\displaystyle\Rightarrow\nabla_{\mu}J_{q}^{\mu}$ $\displaystyle=0.$

Thus the four-current is divergenceless if the matter field EOM are satisfied,
even if the gauge potential EOM are not; this is commonly described in flat
spacetime as “global gauge invariance results in conservation of charge.” This
result implies that the second EOM in (3.32) is redundant, since it will be
true for any matter field for which the action vanishes upon its variation.

##  4 From QED to classical electromagnetism

In this section we arrive at MFEM as a limit of QED by setting up a quantum
state configuration that corresponds to a classical four-current, i.e. a
configuration similar to the “in” state when defining the scattering matrix,
except assumed to hold locally for all spacetime. Explicitly, this
configuration will consist of spatially separated electron wave packets
interacting only via an electromagnetic field which is too weak for pair
production or bound states. Our treatment is for spin up electrons, but spin
down electrons and positrons of both spins may be treated similarly.

We begin with a summary of the relevant aspects of QED to fix notation and
conventions. Our presentation is a bit idiosyncratic but allows for
calculations without the usual clutter of integrals and sums.

###  4.1 The QED Lagrangian

Quantum electrodynamics (in flat spacetime) is defined by quantizing the
classical field theory of the Dirac Lagrangian (minimally coupled, using
natural units and the mostly pluses metric signature)

$\displaystyle L_{\mathrm{DIRAC}}$
$\displaystyle\equiv-\mathrm{Re}\left(\overline{\psi}\gamma^{\mu}\mathrm{D}_{\mu}\psi\right)-m\overline{\psi}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$
(4.1)

where in the first term we only take the negative real part, $\psi$ is a
complex four-component spinor matter field, $\gamma^{\mu}$ are the Dirac
matrices, and we follow Weinberg[5] in defining
$\overline{\psi}\equiv\psi^{\dagger}i\gamma^{0}$ as the Dirac adjoint of the
matter field, where $\psi^{\dagger}$ is the Hermitian conjugate. The Dirac
matrices are a complex matrix representation of an arbitrary constant
orthonormal spacetime dual frame, with matrix multiplication the action of
Clifford multiplication.

Expanding the first and last terms of the Lagrangian, we find the EOM from
varying the gauge potential are

$\displaystyle q\overline{\psi}i\gamma^{\nu}\psi$
$\displaystyle=\nabla_{\mu}F^{\nu\mu},$ (4.2)

allowing us to identify the real divergenceless matter four-current

$\displaystyle J^{\nu}$ $\displaystyle\equiv\overline{\psi}i\gamma^{\nu}\psi,$
(4.3)

in terms of which (4.2) is Maxwell’s equations.

We note that this expression determines the components of $J$, so that for
example if in our inertial frame the only non-zero component is $J^{0}$, then
we may write $J^{\nu}=\overline{\psi}i\gamma^{0}\psi U^{\nu}$, where $U$ is
the unit vector in the $x^{0}$ direction; this expression then holds in any
inertial frame, since under a Lorentz transformation the spinor $\psi$ is
multiplied by the matrix representation of this transformation to make it so.
We also note that the particle density associated with $J$ may be shown to be
$\psi^{\dagger}\psi$, which is positive, and this definition allows us to
write the Lagrangian as

$\displaystyle L_{\mathrm{DIRAC}}$
$\displaystyle=-\mathrm{Re}\left(\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi\right)+qJ^{\mu}A_{\mu}-m\overline{\psi}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$
(4.4)

which omitting the terms in $\psi$ is identical to the quasi-gauge EM
Lagrangian.

Varying the matter field results in the Dirac equation

$\displaystyle\gamma^{\mu}\mathrm{D}_{\mu}\psi$ $\displaystyle=-m\psi.$ (4.5)

The Dirac operator $\gamma^{\mu}\mathrm{D}_{\mu}$ is the “square root” of the
Laplacian, i.e. generalized to the gauge covariant derivative we have

$\displaystyle\left(\gamma^{\mu}\mathrm{D}_{\mu}\right)\left(\gamma^{\nu}\mathrm{D}_{\nu}\right)$
$\displaystyle=\gamma^{\mu}\gamma^{\nu}\mathrm{D}_{\mu}\mathrm{D}_{\nu}=\eta^{\mu\nu}\mathrm{D}_{\mu}\mathrm{D}_{\nu}=\mathrm{D}^{\mu}\mathrm{D}_{\mu},$
(4.6)

using the properties of the Dirac matrices $\gamma^{\mu}$ (see [4]). Thus the
components of $\psi$ satisfy

$\displaystyle 0$
$\displaystyle=\left(\gamma^{\mu}\mathrm{D}_{\mu}+m\right)\psi$ (4.7)
$\displaystyle\Rightarrow 0$
$\displaystyle=\left(\gamma^{\mu}\mathrm{D}_{\mu}-m\right)\left(\gamma^{\mu}\mathrm{D}_{\mu}+m\right)\psi$
$\displaystyle=\left(\mathrm{D}^{\mu}\mathrm{D}_{\mu}-m^{2}\right)\psi,$

the Klein-Gordon equation.

Varying the frame results in the on-shell Hilbert SEM tensor

$\displaystyle T_{\mathrm{DIRAC}}^{\mu\nu}$
$\displaystyle=\frac{1}{2}\mathrm{Re}\left(\overline{\psi}\gamma^{\mu}\mathrm{D}^{\nu}\psi+\overline{\psi}\gamma^{\nu}\mathbf{\mathrm{D}}^{\mu}\psi\right)+T_{\mathrm{EM}}^{\mu\nu}.$
(4.8)

###  4.2 Plane wave solutions

The canonically quantized Dirac field is based on the free quantum Dirac
equation

$\gamma^{\mu}\partial_{\mu}\hat{\psi}=-m\hat{\psi},$ (4.9)

where $\hat{\psi}$ is now the time-dependent Dirac spinor operator in the
Heisenberg picture.

This equation has plane wave solutions, which we express using the mostly
pluses chiral basis from [5], in which the Dirac matrices are

$\gamma^{0}\equiv-i\begin{pmatrix}0&I\\\
I&0\end{pmatrix},\;\gamma^{i}\equiv-i\begin{pmatrix}0&\sigma_{i}\\\
-\sigma_{i}&0\end{pmatrix},$ (4.10)

where $\sigma_{i}$ are the Pauli matrices. With this choice we may define an
(operator-valued) spin up electron plane wave solution of four-momentum $P=mU$
by aligning $\gamma^{0}$ and the coordinate $x^{0}$ with $U$, so that
$e^{iP_{\mu}x^{\mu}}=e^{-imx^{0}}$ and we write

$\displaystyle\begin{aligned}
\hat{\psi}_{P}&\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ 0\\\ 1\\\
0\end{pmatrix}\hat{a}_{P}e^{iP_{\mu}x^{\mu}}\\\ &\equiv
u\hat{a}_{P}e^{-imx^{0}},\end{aligned}$ (4.11)

where $\hat{a}_{P}$ is postulated to be the annihilation operator for a spin
up single electron state $\rvert P\rangle$ of four-momentum $P$. The spinor
$u$ satisfies

$\displaystyle u^{\dagger}u$ $\displaystyle=1,$ (4.12)
$\displaystyle\gamma^{0}u$ $\displaystyle=-iu,$ $\displaystyle
u^{\dagger}\gamma^{0}\gamma^{\nu\neq 0}u$ $\displaystyle=0,$

so that we may verify that the free Dirac equation is satisfied:

$\displaystyle\begin{aligned}
\gamma^{\mu}\partial_{\mu}\hat{\psi}_{P}&=\gamma^{0}u\partial_{0}\hat{a}_{P}e^{-imx^{0}}\\\
&=-um\hat{a}_{P}e^{-imx^{0}}\\\ &=-m\hat{\psi}_{P}\end{aligned}$ (4.13)

Keeping $\gamma^{0}$ constant, a spin up electron plane wave solution of
arbitrary four-momentum $K$ may be written

$\hat{\psi}_{K}=\Lambda_{K}u\hat{a}_{K}e^{iK_{\mu}x^{\mu}},$ (4.14)

where $\Lambda_{K}$ is the matrix representation of the Lorentz transformation
which aligns $\gamma^{0}$ with $K$. The general free solution $\hat{\psi}$ is
then a (complex) linear combination (integral) of such plane wave solutions,
along with similar solutions for spin down electrons and positrons of both
spins, and if $\rvert 0\rangle$ is the ground state of the free theory, then
we have

$\displaystyle\left\langle 0\middle\rvert\hat{\psi}\middle\rvert
P\right\rangle$ $\displaystyle=ue^{-imx^{0}}.$ (4.15)

In the interacting theory, we choose inertial coordinates and define an
(operator-valued) plane wave solution for a spin up electron of four-momentum
$P=mU=\left(E_{\mathbf{p}},\mathbf{p}\right)$ at time $t_{0}$ to be

$\hat{\psi}_{\mathbf{p}}=\Lambda_{\mathbf{p}}u\hat{a}_{\mathbf{p}}\left(t_{0}\right)e^{i\mathbf{p}\cdot\mathbf{x}},$
(4.16)

where $\hat{a}_{\mathbf{p}}\left(t_{0}\right)$ is now postulated to be the
annihilation operator for a spin up single electron state
$\rvert\mathbf{p}\rangle_{t_{0}}$ of momentum $\mathbf{p}$ at time $t_{0}$.
The general solution at time $t_{0}$ is again a linear combination of such
plane wave solutions, so that if $\rvert\Omega\rangle$ is the ground state of
the interacting theory and $\gamma^{0}$ and $t=x^{0}$ are aligned with $U$, we
have

$\displaystyle\left\langle\Omega\middle\rvert\hat{\psi}\middle\rvert\mathbf{p}\right\rangle_{t_{0}}$
$\displaystyle=ue^{-imt_{0}}.$ (4.17)

###  4.3 Electron packets

We may now construct a spin up electron packet. We define the quantum state
$\rvert\phi_{\mathbf{p}}\rangle_{t_{0}}$ to be an integral of electron states
$\rvert\mathbf{k}\rangle_{t_{0}}$ with momenta clustered around $\mathbf{p}$
such that the spinor-valued wave packet

$\displaystyle\begin{aligned}
\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)&\equiv\left\langle\Omega\middle\rvert\hat{\psi}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}\\\
&\sim\left|\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\right|ue^{-imt_{0}}\end{aligned}$
(4.18)

is smooth and normalized, i.e. its modulus is close to a smooth envelope whose
square integrated over space is unity. Propagation in time is defined by

$\displaystyle\begin{aligned}
\gamma^{\mu}\mathrm{D}_{\mu}\phi_{\mathbf{p}}\left(\mathbf{x},t\right)&=\left\langle\Omega\middle\rvert\gamma^{\mu}\mathrm{D}_{\mu}\hat{\psi}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}\\\
&=-\left\langle\Omega\middle\rvert
m\hat{\psi}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}\\\
&=-m\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right),\end{aligned}$ (4.19)

i.e. $\phi_{\mathbf{p}}\left(\mathbf{x},t\right)$ satisfies the Dirac
equation.

It is important to note that a Gaussian wave packet at $t_{0}$ will not remain
so as it propagates in time, even in the free theory due to the Lorentz
factors in the $\rvert\mathbf{k}\rangle_{t_{0}}$. For fermionic ladder
operators we are also constrained to only have one plane wave for each four-
momentum value; however, if the packet is in an unbound state, this has no
practical effect since $\mathbf{k}$ can take a continuum of values.

Recalling (4.12), the matter four-current operator for
$\hat{\psi}_{\mathbf{p}}$ is

$\displaystyle\hat{J_{P}}^{\nu}$
$\displaystyle=\overline{\hat{\psi}}_{P}i\gamma^{\nu}\hat{\psi}_{P}$ (4.20)
$\displaystyle=-u^{\dagger}\hat{a}_{P}^{\dagger}e^{-iP_{\mu}x^{\mu}}\gamma^{0}\gamma^{\nu}u\hat{a}_{P}e^{iP_{\mu}x^{\mu}}$
$\displaystyle\Rightarrow\hat{J}_{\mathbf{p}}^{0}\left(t_{0}\right)$
$\displaystyle=-\hat{a}_{\mathbf{p}}^{\dagger}\left(t_{0}\right)e^{-imt_{0}}\gamma^{0}\gamma^{0}\hat{a}_{\mathbf{p}}\left(t_{0}\right)e^{imt_{0}}$
$\displaystyle=\hat{a}_{\mathbf{p}}^{\dagger}\left(t_{0}\right)\hat{a}_{\mathbf{p}}\left(t_{0}\right),$

the number operator, and the other components vanish due to our alignment of
$\gamma^{0}$. When computing the matter four-current, the term for positrons
is reversed to also give the number operator, so that at time $t_{0}$ we have

$\displaystyle\left\langle\phi_{\mathbf{p}}\middle\rvert\hat{J}^{0}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}$
$\displaystyle\sim\left|\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\right|^{2}\left\langle\phi_{\mathbf{p}}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}$
(4.21)
$\displaystyle\sim\overline{\phi}_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)i\gamma^{0}\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\left\langle\phi_{\mathbf{p}}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}.$

For arbitrary $\gamma^{0}$ we then have

$\displaystyle\left\langle\phi_{\mathbf{p}}\middle\rvert\hat{J}^{\nu}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}$
$\displaystyle\sim\overline{\phi}_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)i\gamma^{\nu}\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\left\langle\phi_{\mathbf{p}}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}.$
(4.22)

###  4.4 Classical four-current configuration

We now define a quantum state configuration that corresponds to a classical
continuous four-current. We make the following assumptions:

  1. 1.

Durable localized packets: The quantum state at any time corresponds to some
number of non-overlapping packets $\sum_{n}\phi_{\mathbf{p}_{n}}$, each of
whose four-momenta changes slowly from packet to packet. The state evolves in
time such that the packets remain smooth, localized, and separated (a
nontrivial assumption per the previous section), and without pair production.

  2. 2.

Smooth packet distribution: Spacetime can be split up into cells, each of
which contains an integral number of packets with approximately equal four-
momenta $P=mU$, such that the change in the number of packets and the change
in momentum per unit of space are both much smaller than the change in the
phase of the packets per unit time.

  3. 3.

Classical gauge potential: The electromagnetic state is dominated at larger
length scales by the classical solution, i.e. the stationary phase
approximation holds, so that the classical gauge potential equation of motion
also holds for real-valued $A_{\mu}$.

Within any cell we may align $t=x^{0}$ and $\gamma^{0}$ with $U$, and choose a
time $t_{0}$ near the center of a cell. The first two assumptions mean that
for each packet in the cell we may make the approximation

$\displaystyle\begin{aligned}
\phi_{\mathbf{p}}\left(\mathbf{x},t\right)&\sim\left|\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\right|ue^{-im\left(t-t_{0}\right)}.\end{aligned}$
(4.23)

By choosing the Weyl gauge ($A_{0}=0$) in the cell, we therefore can neglect
any change in $\phi_{\mathbf{p}}$ except in the $U$ direction, since

$\displaystyle\begin{aligned}
\gamma^{\mu}\mathrm{D}_{\mu}\phi_{\mathbf{p}}&\sim-
im\gamma^{0}\left|\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\right|ue^{-im\left(t-t_{0}\right)}+\gamma^{j}\mathrm{D}_{j}\phi_{\mathbf{p}}\\\
&=-m\phi_{\mathbf{p}}+\gamma^{j}\mathrm{D}_{j}\phi_{\mathbf{p}}\\\
\Rightarrow\gamma^{j}\mathrm{D}_{j}\phi_{\mathbf{p}}&\sim 0.\end{aligned}$
(4.24)

The Dirac equation approximately satisfied by each $\phi_{\mathbf{p}}$ is then

$\displaystyle\gamma^{0}\mathrm{D}_{U}\phi_{\mathbf{p}}$
$\displaystyle=-m\phi_{\mathbf{p}},$ (4.25)

which we will call the Dirac packet equation. Note that $U$ will be different
for each cell.

Again using the first two assumptions, we may then define a spinor-valued
field $\phi$ on all of spacetime by “smearing” the $\phi_{\mathbf{p}_{n}}$
across cells; more precisely, we again choose inertial coordinates and Dirac
matrices which align $x^{0}$ and $\gamma^{0}$ with $U$ and in a space-like
hyperplane across each cell $s\equiv\left(\mathbf{x},t_{0}\right)$ define

$\displaystyle\phi_{s}$
$\displaystyle\equiv\frac{1}{\sqrt{2V_{s}}}\int_{V_{s}}\sum_{n}\phi_{\mathbf{p}_{n}}\left(\mathbf{x},t_{0}\right)\mathrm{d}^{3}x$
(4.26)
$\displaystyle\sim\frac{1}{\sqrt{2V_{s}}}ue^{-imt_{0}}\sum_{n}\sqrt{2}\int_{V_{s}}\left|\phi_{\mathbf{p}_{n}}\left(\mathbf{x},t_{0}\right)\right|^{2}\mathrm{d}^{3}x$
$\displaystyle=\sqrt{\rho_{0}}ue^{-imt_{0}},$

where we assume a Gaussian envelope and at a given time

$\rho_{0}\equiv\frac{N_{s}}{V_{s}}$ (4.27)

is the number of wave packets in the space-like hyperplane of the cell divided
by the volume of the hyperplane, i.e. it is the matter rest density. We then
smoothly interpolate $\mathbf{p}$ and $\rho_{0}$ between cells to arrive at a
globally defined $\phi$. At any point we therefore have

$\displaystyle\overline{\phi}i\gamma^{0}\phi$
$\displaystyle=\overline{\phi}\phi$ $\displaystyle=\rho_{0},$

and the Dirac packet equation, which is only dependent upon the phase, remains
valid at any point as

$\displaystyle\begin{aligned}
\gamma^{0}\mathrm{D}_{U}\phi&=-m\phi.\end{aligned}$ (4.28)

Figure 4.1: The quantum state configuration that corresponds to a classical
continuous four-current is a smooth distribution of durable localized packets.
By dividing spacetime into cells and aligning $t=x^{0}$ and $\gamma^{0}$ with
$P=mU$, we may express each packet at time $t_{0}$ as
$\left\langle\Omega\middle\rvert\hat{\psi}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t_{0}}\sim\left|\phi_{\mathbf{p}}\left(\mathbf{x},t_{0}\right)\right|ue^{-imt_{0}}$
. In the figure the value of the modulus is represented by the shading of each
packet, while the packet phase is depicted as rotating counterclockwise as $t$
increases. Assuming the packet momenta are approximately equal within a cell
lets us define a spinor-valued $\phi_{s}$ per cell, which may be interpolated
to yield a global spinor-valued $\phi$. Since the change in each
$\phi_{\mathbf{p}}$ and therefore $\phi$ is overwhelmingly due to the change
in phase, the Dirac packet equation $\gamma^{0}\mathrm{D}_{U}\phi=-m\phi$ is
approximately satisfied at any point, where $U=\partial/\partial t$.

Using the last assumption and (4.22), each packet approximately satisfies the
equation of motion

$\displaystyle
q\left\langle\phi_{\mathbf{p}}\middle\rvert\hat{J}^{\nu}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t}$
$\displaystyle=\nabla_{\mu}F^{\nu\mu}\left\langle\phi_{\mathbf{p}}\middle\rvert\phi_{\mathbf{p}}\right\rangle_{t}$
(4.29) $\displaystyle\Rightarrow
q\overline{\phi}_{\mathbf{p}}i\gamma^{\nu}\phi_{\mathbf{p}}$
$\displaystyle\sim\nabla_{\mu}F^{\nu\mu}.$

The construction of $\phi$ makes the field homogeneous within each cell, and
therefore so is $F$. Aligning $\gamma^{0}$ with $P$ in a cell, we can then
integrate over space to arrive at

$\displaystyle\int_{V_{s}}\sum_{n}q\overline{\phi}_{\mathbf{p}}i\gamma^{\nu}\phi_{\mathbf{p}}\mathrm{d}^{3}x$
$\displaystyle\sim\int_{V_{s}}\nabla_{\mu}F^{\nu\mu}\mathrm{d}^{3}x$ (4.30)
$\displaystyle\Rightarrow
q\int_{V_{s}}\sum_{n}\left|\phi_{\mathbf{p}_{n}}\left(\mathbf{x},t_{0}\right)\right|^{2}\mathrm{d}^{3}x$
$\displaystyle\sim\nabla_{\mu}F^{\nu\mu}\int_{V_{s}}\mathrm{d}^{3}x$
$\displaystyle\Rightarrow qN_{s}$ $\displaystyle=V_{s}\nabla_{\mu}F^{\nu\mu}$
$\displaystyle\Rightarrow q\rho_{0}$ $\displaystyle=\nabla_{\mu}F^{\nu\mu}$
$\displaystyle=q\overline{\phi}i\gamma^{\nu}\phi,$

i.e. $\phi$ satisfies the classical Dirac gauge potential equations of motion,
so that the continua of packets propagates according to Maxwell’s equations.
Thus we see that the Dirac packet equation determines $\phi$, with $U$
determined by Maxwell’s equations.

Putting this all together, the equations of motion for the spinor-valued field
$\phi$ may be derived from the Lagrangian

$\displaystyle L_{\mathrm{DIRAC-\phi}}$
$\displaystyle\equiv-\mathrm{Re}\left(\overline{\phi}\gamma^{\mu}U_{\mu}\mathrm{D}_{U}\phi\right)-m\overline{\phi}\phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$
(4.31)

###  4.5 Spinor component equations of motion

We now determine that the MFEM equations of motion are satisfied by the rest
frame components of the spinor-valued $\phi$. At any point in spacetime we can
align $\gamma^{0}$ with $P$ and write

$\displaystyle\begin{aligned} \Phi&\equiv\sqrt{\rho_{0}}e^{-imt_{0}}\\\
\Rightarrow\phi&=\frac{1}{\sqrt{2}}\begin{pmatrix}\Phi\\\ 0\\\ \Phi\\\
0\end{pmatrix}\\\ \Rightarrow J^{0}&=\overline{\phi}i\gamma^{0}\phi,\\\
&=\overline{\phi}\phi\\\ &=\left|\Phi\right|^{2}\\\ &=\rho_{0}.\end{aligned}$
(4.32)

This may be contrasted with the MFEM result
$\rho_{0}=2m\left|\Phi\right|^{2}$, and means that the gauge potential
equations of motion (4.30) may be written

$\displaystyle q\left|\Phi\right|^{2}U^{\nu}$
$\displaystyle=\nabla_{\mu}F^{\nu\mu}.$ (4.33)

If we now left multiply the Dirac packet equation by $\overline{\phi}$ and
note that the right side is real, we have

$\displaystyle\begin{aligned}
\mathrm{Re}\left(\overline{\phi}\gamma^{0}\mathrm{D}_{U}\phi\right)&=-m\overline{\phi}\phi\\\
\Rightarrow\mathrm{Re}\left(\phi^{\dagger}i\gamma^{0}\gamma^{0}\mathrm{D}_{U}\phi\right)&=\mathrm{Re}\left(-i\Phi^{*}\mathrm{D}_{U}\Phi\right)\\\
&=\mathrm{Im}\left\langle\Phi,\mathrm{D}_{U}\Phi\right\rangle_{\mathbb{C}}\\\
&=-\left|\Phi\right|^{2}m\\\
\Rightarrow-m&=\frac{\mathrm{Im}\left\langle\Phi,\mathrm{D}_{U}\Phi\right\rangle_{\mathbb{C}}}{\left|\Phi\right|^{2}}\\\
&=\mathrm{D}_{U}^{\varangle}\vec{\Phi}.\end{aligned}$ (4.34)

Thus we see that for the spinor-valued field $\phi$ which represents the
“smeared” distribution of electron packets, the Dirac equation is equivalent
to the MFEM EOM for the rest frame components $\Phi$. Also note that there
only one sign of solution, but that building a positron packet would have
resulted in an opposite sign in the EOM; thus both theories include anti-
particles.

Lastly, at any point in spacetime we can again align $x^{0}$ and $\gamma^{0}$
with $U$ and choose the Weyl gauge so that counting only non-zero components
we have

$\displaystyle\begin{aligned}
\overline{\phi}\gamma^{\mu}\mathrm{D}^{\nu}\phi&=\phi^{\dagger}i\gamma^{0}\gamma^{\mu}\eta^{00}\partial_{0}\phi\\\
&=im\phi^{\dagger}i\gamma^{0}\gamma^{\mu}\phi\\\
&=-m\phi^{\dagger}\gamma^{0}\gamma^{0}\phi\\\ &=m\left|\Phi\right|^{2}\\\
\Rightarrow\overline{\phi}\gamma^{\mu}\mathrm{D}^{\nu}\phi&=m\rho_{0}U^{\mu}U^{\nu},\end{aligned}$
(4.35)

where in the third line we have used 4.12. Thus for an on-shell matter field
the Hilbert SEM tensor using (4.8) is

$\displaystyle\begin{aligned}
T_{\mathrm{DIRAC-\phi}}^{\mu\nu}&=\frac{1}{2}\mathrm{Re}\left(\overline{\phi}\gamma^{\mu}\mathrm{D}^{\nu}\phi+\overline{\phi}\gamma^{\nu}\mathrm{D}^{\mu}\phi\right)+T_{\mathrm{EM}}^{\mu\nu}\\\
&=m\rho_{0}U^{\mu}U^{\nu}+T_{\mathrm{EM}}^{\mu\nu},\end{aligned}$ (4.36)

which includes the desired time-like dust term.

###  4.6 From the QED to the MFEM Lagrangian

The gauge potential and matter field EOM we have detailed for the spinor
component in the previous section may be extracted from the Lagrangian

$\displaystyle\begin{aligned}
L_{\mathrm{DIRAC-\Phi}}&=-\mathrm{Im}\left\langle\Phi,U^{\mu}\mathrm{D}_{\mu}\Phi\right\rangle_{\mathbb{C}}-m\Phi^{*}\Phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\\\
&=-\left\|\vec{\Phi}\right\|^{2}\mathrm{D}_{U}^{\varangle}\vec{\Phi}-m\left\|\vec{\Phi}\right\|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\end{aligned}$
(4.37)

where $\Phi$ is now an arbitrary complex-valued field to be determined by the
equations of motion, and we have also expressed the Lagrangian in terms of the
real vector valued field $\vec{\Phi}$. This Lagrangian, however, has an extra
free variable: the time-like unit vector $U$ at each point, which is parallel
to the four-current. It is determined by the gauge potential EOM, but we would
like to eliminate it to obtain a standard gauge theory Lagrangian.

We may eliminate $U$ by taking advantage of the geometric observations of
Section 2.4 and squaring the distinct parts of the dynamical and mass terms in
the Lagrangian. This results in the MFEM Lagrangian:

$\displaystyle\begin{aligned}
L_{\mathrm{DIRAC-\Phi}}&\overset{\mathrm{"squared"}}{\rightarrow}-\left\|\vec{\Phi}\right\|^{2}\left(\mathrm{D}_{U}^{\varangle}\vec{\Phi}\right)^{2}-m^{2}\left\|\vec{\Phi}\right\|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\\\
&=-\left\|\vec{\Phi}\right\|^{2}\left\langle\mathrm{D}_{\mu}^{\varangle}\vec{\Phi},\mathrm{D}_{\mu}^{\varangle}\vec{\Phi}\right\rangle-m^{2}\left\|\vec{\Phi}\right\|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\\\
&=L_{\mathrm{MFEM}}\end{aligned}$ (4.38)

Note that the resulting matter field EOM will include both the solutions
before squaring and the opposite sign solutions, thereby incorporating the
classical positron four-current quantum states we omitted. As we saw in
Section 3, the expression of the four-current in terms of the field changes,
but this change combined with the altered Lagrangian yields the correct on-
shell Hilbert SEM tensor.

##  5 Summary and discussion

###  5.1 Summary of results

With the Lagrangian for MFEM defined as

$\displaystyle\begin{aligned}
L_{\mathrm{EM}}&\equiv-\left\langle\mathrm{D}_{\mu}^{\perp}\vec{\Phi},\mathrm{D}_{\mu}^{\perp}\vec{\Phi}\right\rangle-m^{2}\left\langle\vec{\Phi},\vec{\Phi}\right\rangle-\frac{1}{2}\left\langle
F,F\right\rangle\\\
&=\frac{1}{4\left|\Phi\right|^{2}}\left(\Phi^{*}\mathrm{D}_{\mu}\Phi-\Phi\left(\mathrm{D}_{\mu}\Phi\right)^{*}\right)\left(\Phi^{*}\mathrm{D}^{\mu}\Phi-\Phi\left(\mathrm{D}^{\mu}\Phi\right)^{*}\right)-m^{2}\left|\Phi\right|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},\end{aligned}$
(5.1)

and the definition

$\displaystyle J_{\mu}$ $\displaystyle\equiv
2\left\|\vec{\Phi}\right\|\mathrm{D}_{\mu}^{\perp}\vec{\Phi}$ (5.2)
$\displaystyle=-i\left(\Phi^{*}\mathrm{D}_{\mu}\Phi-\Phi\left(\mathrm{D}_{\mu}\Phi\right)^{*}\right),$

the EOM corresponding to the variation of the matter field, gauge potential
and metric yield

$\displaystyle\mathrm{D}_{U}^{\varangle}\vec{\Phi}$ $\displaystyle=-m$ (5.3)
$\displaystyle qJ^{\nu}$ $\displaystyle=\nabla_{\mu}F^{\nu\mu}$ $\displaystyle
T^{\mu\nu}$
$\displaystyle=\frac{1}{4\left\|\vec{\Phi}\right\|^{2}}J^{\mu}J^{\nu}+F^{\mu\lambda}F^{\nu}{}_{\lambda}-\frac{1}{4}F^{\sigma\lambda}F_{\sigma\lambda}g^{\mu\nu},$

while the Noether current corresponding to the global gauge symmetry is

$\nabla_{\mu}J^{\mu}=0,$ (5.4)

which is also implied by the gauge potential EOM.

If we write $J=\rho_{0}U$, these EOM imply that

$\displaystyle\left\|\vec{\Phi}\right\|^{2}$ $\displaystyle\begin{aligned}
=\frac{\rho_{0}}{2m}\end{aligned},$ (5.5)

which allows us to write the on-shell Hilbert SEM tensor as

$\displaystyle T^{\mu\nu}$
$\displaystyle=\frac{1}{2}m\rho_{0}U^{\mu}U^{\nu}+F^{\mu\lambda}F^{\nu}{}_{\lambda}-\frac{1}{4}F^{\sigma\lambda}F_{\sigma\lambda}g^{\mu\nu},$
(5.6)

whereupon taking the divergence of both sides yields the equations of geodesic
deviation

$m\rho_{0}\left(\nabla_{U}U\right)^{\mu}=F^{\mu}{}_{\sigma}J_{q}^{\sigma},$
(5.7)

which is equivalent to the Lorentz force law.

One may define a QED quantum state configuration that corresponds to a
classical continuous four-current: a smooth distribution of durable localized
packets which follow the classical gauge potential EOM. A spinor-valued field
whose EOM may be derived from

$\displaystyle L_{\mathrm{DIRAC-\phi}}$
$\displaystyle\equiv-\mathrm{Re}\left(\overline{\phi}\gamma^{\mu}U_{\mu}\mathrm{D}_{U}\phi\right)-m\overline{\phi}\phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
(5.8)

is defined by smearing this state across spacetime; the complex-valued rest
frame components of this spinor then satisfy the MFEM EOM, which may be
derived from

$\displaystyle\begin{aligned}
L_{\mathrm{DIRAC-\Phi}}&=-\left(\mathrm{D}_{U}^{\varangle}\vec{\Phi}-m\right)\left\|\vec{\Phi}\right\|^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.\end{aligned}$
(5.9)

This Lagrangian includes an extra free variable $U$ which is determined by the
gauge potential EOM; squaring each of the terms in brackets eliminates $U$ and
yields the MFEM Lagrangian.

###  5.2 The discrete four-current

The definition of the four-current $J=\rho_{0}U$ corresponding to classical
matter continua is a four-vector whose direction $U$ at each point is tangent
to worldline of the continua and whose “length” $\sqrt{-\left\langle
J,J\right\rangle}=\rho_{0}$ is equal to the particle number rest density
$\rho_{0}$. The rest density is the worldline rest frame particle number per
unit volume (i.e. the infinitesimal particle number per unit space-like
hypersurface orthogonal to $U$), which like $U$ is metric-dependent, so that
$J$ is as well, and these dependencies are such that the four-current density
$\mathfrak{J}\equiv J\sqrt{-\det\left(g_{\mu\nu}\right)}$ is independent of
the metric.

The curious fact that the $\rho_{0}$ defined in MFEM (as in Dirac and Klein-
Gordon theory) is not metric dependent is here explained by the QED
approximation; $\rho_{0}$ is describing a unit volume of matter that is
discrete, not continuous. This means that while $\rho_{0}$ depends upon the
chosen units, it does not change with an infinitesimal change of the metric,
since an infinitesimal change in the unit space-like hypersurface orthogonal
to $U$ does not alter the integral number of packets enclosed.

Note that the various expressions for the four-current

$\displaystyle\overline{\psi}i\gamma^{\nu}\psi$
$\displaystyle\rightarrow\overline{\phi}i\gamma^{\nu}\phi$
$\displaystyle\rightarrow\Phi^{*}U^{\nu}\Phi$

are all metric-dependent, since they involve the frame; moreover, their
dependency remains consistent with discrete matter, since they compensate for
altered time-like lengths under metric variations to keep
$\sqrt{-J^{\mu}J^{\nu}g_{\mu\nu}}=\rho_{0}$ constant. The transition to the
MFEM Lagrangian results in a four-current which is a metric-independent
1-form, with $J^{\mu}=g^{\mu\nu}J_{\nu}$ no longer transforming like a
discrete four-current under metric variations. However, unlike the various
expressions for the rest density

$\displaystyle\psi^{\dagger}\psi$ $\displaystyle\rightarrow\phi^{\dagger}\phi$
$\displaystyle\rightarrow\Phi^{*}\Phi$

which are metric-independent, the rest density
$-2\left\|\vec{\Phi}\right\|^{2}\mathrm{D}_{U}^{\varangle}\vec{\Phi}$ in MFEM
from (3.18) is metric-dependent via the unit vector $U$; when applying the
matter field EOM in (3.20), we instead arrive at the on-shell four-current
$J^{\mu}=2\left|\Phi_{\mathrm{EL}}\right|^{2}mU^{\mu}$, which regains its
consistency with discrete matter.

###  5.3 Multiple matter fields

Our matter field configuration is associated with a single four-current
defined on all of spacetime. Unlike in quantum theory, where there is a single
field and interacting particles are identified as Fourier components of this
field which may interfere, here we have a single continuum four-current with
no explicit reference to constituent particles.

We may also consider multiple matter fields, each of which has its own terms
in the Lagrangian and is associated with a separate four-current; each matter
field is a separate section of the vector bundle, but all these sections are
associated with the same principal bundle with a single parallel transport and
curvature, and with the same spacetime base space with a single metric. These
four-currents will gravitationally interact as relativistic dust via the
spacetime metric, and also electromagnetically interact via the
electromagnetic gauge potential according to the EOM
$\sum_{i}qJ_{i}^{\nu}=\nabla_{\mu}F^{\nu\mu}$.

Note that all matter fields are associated with the same charge per particle
number since they share a single field strength term, but each may be
associated with its own mass per particle number; thus the mass per charge of
each matter field is arbitrary.

One may also define a QED quantum state configuration that corresponds to
multiple classical continuous four-currents, as long as the packets remain
discrete, localized, and smoothly distributed. This results in multiple
spinor-valued fields whose multiple rest frame components correspond to
multiple MFEM matter fields.

## References

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