# Double roton-minima in bosonic fractional quantum Hall states

Moumita Indra 1 111Orcid ID: 0000-0002-7900-7947<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>Deepak Jain 2 Sandip Mondal1<EMAIL_ADDRESS>1
Indian Institute of Technology Bombay, Mumbai 400076, India  2 Indian
Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

###### Abstract

We have studied the collective spin-conserving collective excitation spectra
in rotating diluted ultra-cold Bose atoms. Double roton-minima have been
observed in the fractional quantum Hall (FQH) states for the two filling
fractions ($\nu$) of the first series of Jain’s composite fermion sequences.
The obtained roton-minima for $\nu=1/4$ are at the wave-vectors $1.26$ and
$2.38$ and the roton-minima for $\nu=1/6$ have been shifted to $1.08$ and
$2.06$. Such shift of roton-minima is attributed due to strong correlation
between the particles in bosonic FQH-system. Moreover, the number of roton
minima observed depends upon number of attached fluxes as well as the ranges
of interaction between the particles.

Collective excitation of strongly correlated fractional quantum Hall (FQH)
fluids is an insightful investigation in a two-dimensional electron system
(2DES). Strongly correlated many-body quantum phenomenon, such as the
fractional quantum Hall effect (FQHE) fqhe ; fqhe1 ; fqhe2 ; QHE  can be
explained with the concept of quasi-particles. Composite fermions (CFs) CFs
are the hypothetical quasi-particles introduced by J. K. Jain to explain the
fractional plateus plateau , obtained in Hall resistivity. It is assumed that
electrons attach even number of flux quanta with themselves from the external
magnetic field and form new quasi-particles, named CFs. The majority of the
FQH-states observed in experiments may be converted to weakly-interacting
integer quantum Hall states with the aid of CF theory. In 1995, Davis
Davis_1995  and others BEC  became the first to produce the Bose-Einstein
condensates (BEC) by magnetically confined alkali atoms. Following that,
Cooper and Wilkin CF_on_BEC  investigated the first FQHE of a charged Bose
system using the Jain’s CF model. Recent developments in rotating BEC FQHE_BEC
have provided an intriguing framework for exploring FQHE. Rapidly rotating
BEC, trapped in a a two-dimensional (2D) harmonic potential creates a
fictitious magnetic field ($B$), perpendicular to the 2D-plane that resembles
like 2DES in perpendicular magnetic field. Landau levels (LL) are formed and
then there should be a high chance to observe FQHE in fast-rotating BEC of
diluted ultra-cold Bose atoms CF_on_BEC . As the bosonic system is applied
high rotation, so most of the particles fly out, which makes the atomic
density of the system very low. In this situation of extremely low atomic
density, bosons may exclusively dwell in the lowest LL (LLL). Since the
kinetic energy becomes constant, the problem will be accurately determined by
only the interactions potential. At low energy limit only binary collisions
are significant in a diluted cold gas and the s-wave scattering length s_wave
is the only characteristic used to describe these collisions. In this
instance, a suitable expression for the interaction term is difficult to
formulate. So, short-ranged interactions like $\delta$-function type potential
Delta_pot  between the atoms can be considered. Jain, Regnault, and other
theoretician QH_spinlessBOSON  studied the interacting bosons using CF theory
and considered short-ranged $\delta$-function like interaction between the
Bose particles. The correlated FQH-states produced in rotating BEC may be
explained by the weakly-interacting CFs of Bose atoms, which are the bound
states of Bose atoms and the odd number of flux quanta (or quantized vortices)
CF_on_BEC . Since the vortices tied to the atoms create Berry phases that
partially cancel the Aharonov-Bohm phase produced by applied magnetic field;
this bound state (Bose-CF) is governed by fermionic statistics. The Bose-CFs
experience a reduced magnetic field

$\displaystyle B^{*}=B-p\rho\phi_{0}\;;$ (1)

here, $B$ represents fictitious magnetic field, $\phi_{0}$ is the magnetic
flux- quantum; while the number density of the Bose atoms is $\rho$, and
$p=1,3,5,$ is the odd number of flux quanta attached. This Bose-CF creates a
new type of LLs called $\Lambda$-levels in this weaker magnetic field. The LL
filling fraction of the Bose atoms ($\nu$) can be related with the filling
fraction of Bose-CF $\Lambda$-level (assumed to be an integer $n$) from the
above relation (1) as

$\displaystyle\nu=\frac{n}{np+1}$ (2)

Using this mapping, we can demonstrate that interacting bosons can be treated
like fermions CP_statistics  and that the filling fractions precisely match
the Jain’s sequence Jain_CF . If we consider the particles are in LLL i.e.
$n=1$ then from the above relation (2) we can have $\nu=1/2,1/4,1/6$ fractions
using odd values flux quanta $p=1,3,5$ respectively.
For a particular filling fraction, different partially spin-polarized states
can be found depending upon the values of Zeeman energy ($E_{z}$) and
cyclotron energy ($E_{c}$) of the system partial_spin ; partial_spin1 ;
partial_spin2 . Landau-level mixing LL_mix  becomes crucial in case of
partially spin-polarized FQH-states. To explain the partially spin-polarized
states CF theory is not sufficient then we need to apply other theoretical
techniques polarized ; polarization_MI . Here we are considering fully spin-
polarized state in presence of strong magnetic field i.e. Zeeman energy is
much higher than the cyclotron energy.

In this work, spin-conserving collective excitation has been explored for the
filling fractions $\nu=1/4,1/6$ in bosonic FQH-states. We obtained double
roton-minima for both the two fractions. For $\nu=1/4$ two roton-minima are
obtained at the wave-vector $k=1.26$ and $2.38$; whereas for $\nu=1/6$ those
are shifted to lower values of wave-vector $k=1.08$ and $2.06$. Such shift of
roton-minima has been explained by the strong correlation between the
particles in rotating diluted ultra-cold Bose atoms.

##  I Diluted ultra-cold rotating Bose atoms confined in a harmonic trap

We have considered $N$ weakly interacting Bose atoms of mass $m$, rotating
with angular velocity $\Omega\hat{z}$ trapped in a 2D isotropic harmonic trap
(XY-plane). Thus the Hamiltonian for the system can be written as,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{p_{i}^{2}}{2m}+\frac{1}{2}m\omega^{2}r_{i}^{2}-\Omega
L_{z}+V$ (3) $\displaystyle=$
$\displaystyle\sum_{i}\frac{1}{2m}(\vec{p}_{i}-\vec{A})^{2}+(\omega-\Omega)L_{z}+V\;;$

where $\vec{r_{i}}$, $\vec{p_{i}}$ is the position, momentum of i-th atom
respectively, $L_{z}$ is the projection of angular momentum along Z-axis, $V$
is the interaction Hamiltonian between the atoms and the trap energy is
$\hbar\omega$. Corresponding vector potential so obtained is
$\vec{A}=m\omega(-y,x)$. This indicates the equivalence to the Hamiltonian of
a particle experiencing an effective magnetic field Hamiltonian . Since the
system is sufficiently dilute so that all the atoms are confined in the LLL.
If we can assume $\omega=\Omega$, then the problem becomes identical with the
FQHE in 2DES.

##  II Interacting potential between diluted Bose atoms

### 1\. Short ranged contact interaction

At the low energy limit, the effective interactions between the Bose atoms can
be denoted by a constant in momentum space, since the atoms scattered by the
s-wave s_wave  only. Consequently, an instantaneous $\delta$-function may be
used to represent the interaction in real space. It gives rise to the Gross-
Pitaevski (GP) equation GP_equn ; GP_equn1  for rotating BEC and super-fluid
systems. This interaction in 2D system has the form

$\displaystyle V=g\sum_{i<j}\delta(\vec{r}_{i}-\vec{r}_{j})\;;$ (4)

with $g$ as the strength of the interaction exact_12  and $\vec{r}_{i}$,
$\vec{r}_{j}$ is the position coordinate of $i$-th and $j$-th particle.

#### P$\ddot{o}$schl-Teller interaction

In quantum Monte Carlo calculations, handling the $\delta$-function potential
is particularly challenging and needs a significant amount of computational
resources. Regnault and Jolicoeur QH_spinlessBOSON  used exact diagonalization
method to obtain low-lying states of limited number of bosons interacting with
$\delta$-function potential. Chang et. al. FQHE_BEC  and other group studied
the low-energy excitation spectra by Monte Carlo method using
$\delta$-function interaction but for very small number of bosons up to 12.
The energy spectra of a large number of particles cannot be calculated using
the $\delta$-function potential. So, in our study, we considered the
P$\ddot{o}$schl-Teller (PT) interaction potential ($V_{PT}$) Poschl-Teller  to
get over this issue. PT-interaction has the form

$\displaystyle V_{PT}=U\sum_{i<j}\frac{2\mu}{cosh^{2}\;(\mu r_{ij})}\;;$ (5)

where, the parameter $U$ represents interaction strength and
$r_{ij}=\mid\vec{r}_{i}-\vec{r}_{j}\mid$ is the separation distance between
two particles. Here $\mu$ denotes a parameter having unit of inverse of the
magnetic length $l_{B}$; where $1/\mu$ measures the width of the interaction.
In order to explore the $\mu$-dependent nature of excitation a range of
$\mu$-values have been taken into consideration. As we raise the $\mu$-value,
then the essence of PT-interaction goes well with $\delta$-function. But for
smaller values of $\mu$ the nature of PT potential differs from the contact
interaction, as the range of interaction increases with decrease of $\mu$
SDE_MI ; Physica_B . That’s why we have considered $\mu$-values between a
specific range in our calculation. However, the PT interaction helps us to get
our estimated result in thermodynamic limit ($N\rightarrow\infty$) i.e. finite
but large number of particles.

Figure 1: The schematic diagram of ground state for the two-component bosonic
filling fraction $\nu=1/2$: The two-component spin states have their own
$\Lambda$-levels which are represented by blue straight lines (left panels
denotes one-spin states and right panel denotes another spin-states). The
number ($0,1,2,3,...$) written beside the lines represents $\Lambda$-level
index. Bose-CFs are represented by the green colored solid-dots with pink
arrow-lines (Bose atom by solid-dots and magnetic flux attachment by arrow-
lines). Figure 2: Pair-correlation function $g(r_{ij})$ for the filling
fractions $\nu=1/2,1/4,1/6$ are plotted as a function of two particle’s
separation distance $r_{ij}=\mid\vec{r}_{i}-\vec{r}_{j}\mid$. For this result
sufficiently large number of particles are taken but the nature is independent
of the number of particles. Figure 3: Spin-conserving collective excitation:
In each block blue straight lines represent one-component $\Lambda$-levels (CF
Landau levels) only. Bosonic FQH-filling fractions of $\nu=1/2,1/4,$ and $1/6$
translate into one filled $\Lambda$-level with flux attachment $1,3,5$
respectively. Therefore, in this figure just one flux quanta is attached, so
it represents $1/2$ filling fraction; if there are $3$, $5$ fluxes, it would
represent $1/4,1/6$ fractions respectively. Lowest-order possible excitations
for $\nu=1/2$ are presented here, which follows same for other two fractions.
These three blocks are the possible spin-conserving excitations that are most
likely to occur. The transitions can occur as: (A) 0 $\rightarrow$ 1; (B) 0
$\rightarrow$ 2; (C) 0 $\rightarrow$ 3\. In each of the three blocks, there
are particle-hole pair i.e. CF-exciton. Missing of a Bose-CF is shown by the
dotted-circle and the transition of Bose-CF from $n_{i}$-th $\Lambda$-level to
the $n_{f}$-th $\Lambda$-level is shown by dotted arrow-lines.

###  II.1 2\. Long range Coulomb potential

Additionally, in 2DES, FQHE occurs in the presence of a robust Coulomb
interaction. We are thrilled to observe collective excitation in a bosonic
system for long-range Coulomb interaction ($1/r$) too, despite the fact that
such systems are uncommon in nature. It has the form

$\displaystyle V_{Cou}=\sum_{i<j}\frac{C}{\mid\vec{r}_{i}-\vec{r}_{j}\mid};$
(6)

where, $C$ is the strength of the interaction containing all the information
about charge of particles and coefficients of the medium.

## Wave function & calculation procedures

It is advantageous to investigate the bulk characteristics of FQHE in
spherical geometry since it has no edges. Thus, we can define the composite
fermion wave function in the typical spherical geometry Haldane ; Jain_book ;
Dirac  for our numerical computations. It is hypothesised that $N$ correlated
atoms are present on a sphere’s surface of radius $R$ under the influence of a
radial magnetic field. A ’magnetic monopole’ of strength $Q$ that is
positioned at the centre of the sphere is assumed to be the source of the
magnetic field. This monopole generates a total magnetic flux of $2Q\phi_{0}$
throughout the sphere’s surface. So the radius of the sphere can be expressed
as $R=\sqrt{Q}l_{B}$ ($l_{B}=\sqrt{\frac{\hbar c}{eB}}$ is magnetic length).
This translates into a system of CFs with an effective flux $2q=2Q-p(N-1)$
Dev_Jain . Here $Q$ has to be chosen to ensure that the state at $q$ is an
integral quantum Hall state with an integer filling $n$; resulting in the same
filling fraction $\nu$, same as equation (2). The value of angular momentum
($L$) for a particle in the $n$-th LL is $n+Q-1$ monopole_harmonics . For a
single-particle the basis state is given by monopole harmonics Jain_kamilla .
Thus the ground-state wave function at filling fraction $\nu$ for $N$ Bose
atoms is given by wave_funcn ,

$\Psi^{0}=J^{-1}P_{LLL}J^{2}\;\Phi_{1}(\Omega_{1},\Omega_{2},\cdots\Omega_{N})\;.$
(7)

Here, the position of $i$-th particle on the spherical surface is
$\Omega_{i}$, and $\Phi_{1}$ denotes the Slater determinant of fully filled
$n$-number of $\Lambda$-levels of Bose-CFs. $P_{LLL}$ projects all the
particles into the LLL PLL  and $J$ denotes the Jastrow factor DM_2014  having
the form

$\displaystyle J=\prod_{i<j}^{N}(u_{i}v_{j}-u_{j}v_{i})^{p}.$

We have used the spherical spinor-variables,

$\displaystyle u=cos(\theta/2)e^{-i\phi/2}\mbox{~{}~{}~{} and
~{}~{}~{}~{}}v=sin(\theta/2)e^{i\phi/2}\;;$

where, $\theta$ and $\phi$ can vary from $0$ to $\pi$ and $0$ to $2\pi$
respectively. Following the CF transformation, the system’s overall wave-
function remains symmetric; since both $\Phi_{1}$ and $J$ are odd in this
case. The ground state for filling fraction $\nu=1/2$ is shown in FIG. 1,
where we considered that the Zeeman energy ($E_{z}$) between the two spin-
states is larger than the 3-times of cyclotron energy gap ($E_{c}$) then only
we can expect collective excitations between same spin-component only (Other
spin $\Lambda$-levels has no role here). At this condition the system behaves
as a fully spin-polarized state.

A very significant physical quantity for an interacting system is it’s pair-
correlation function $g(r_{ij})$, which is basically the probability of having
two particles at a separation distance $r_{ij}$ PC . Let for a $N$-particle
system with particle density $\rho$, the pair correlation function between
particle $1$ and $2$ can be calculated as a function of relative coordinates
between those two particles

$\displaystyle g(r_{12})=\frac{N(N-1)}{\rho}\int d^{2}r_{3}d^{2}r_{4}\cdots
d^{2}r_{N}\mid\Psi^{0}(r_{1},r_{2},\cdots,r_{N})\mid^{2}$

In FQHE the many-body ground state wave function ($\Psi^{0}$) is normalized so
that the pair-correlation function approaches to unity at a large separation
distance. Generally, $g(r_{ij})$ stables to a constant value for large
separation distance $r_{ij}$, which confirms that the wave-function describe a
strongly interacting liquid-state. For long-range order crystalline state
$g(r_{ij})$ oscillates throughout the large distance polarization_MI . We have
calculated the pair-correlation function for those three fractions
$\nu=1/2,1/4,1/6$ and plotted the result in FIG. 2, which indicates that those
three states are pure liquid-like.

Figure 4: Spin-conserving excitation energy spectra for the filling fractions
$\nu=1/4,1/6$. In the two right-most boxes, energy spectra (in unit of $U$)
are shown using PT-interaction ($V_{PT}$) for different values of $\mu$. These
$\mu$-values are increasing downwards (indicating range of interaction narrows
down) in both of the filling fractions. Energy spectra (in unit of $C$) using
long-range Coulomb interaction ($V_{cou}$) are depicted in the left-most box.
Wave-vector ($k$) times $l_{B}$ is plotted along X-axis, which is connected to
the total angular momentum as $k=L/R$.

When a Bose-CF excites from the filled $0$-th $\Lambda$-level to an empty
$n_{f}\;$-th $\Lambda$-level having same spin, then the excited state wave-
function is given by DM_SSM

$\Psi(L)=J^{-1}P_{LLL}J^{2}\sum_{m_{h}}|m_{h}>\;<q,m_{h};n_{f}+q,m_{p}|L,M>\;.$
(8)

Here $|m_{h}>$ represents the Slater determinant, where $N-1$ number of Bose-
CF residing with a quasi-hole in the lowest $\Lambda$-level and one quasi-
particle in the $n_{f}$-th $\Lambda$-level. The projection of angular momentum
of the quasi-particle is $m_{p}$ and that of the quasi-hole is $m_{h}$.
$<q,m_{h};n_{f}+q,m_{p}|L,M>$ represent the well-known Clebsch-Gordan
coefficients. The sum of all angular momenta is $L$. To avoid the numerical
complexity, the Z-component of total angular momentum ($M=0$) is taken as
zero, without losing generality.

The superposition of all probable excitons yields the actual collective
excitation rather than a single CF-exciton state. For spin-conserving
excitations, three exciton states are taken into account 0 $\rightarrow$ 1; 0
$\rightarrow$ 2 and 0 $\rightarrow$ 3 i.e. one CF can experience this
stimulation by hopping from the $0$-th $\Lambda$-level to a higher one. For
$\nu=1/2$ filling fraction, the possible collective spin-conserving
excitations are depicted in FIG. 3. Similar collective excitations are taken
into account for the other two filling fractions. Nevertheless, for a given
$L$, the wave functions for those three exciton states are not always
orthogonal to one another. Since it is important to diagonalize the
Hamiltonian in an orthonormal basis Mandal_Jain  in order to get significant
results, we need to use a technique to orthogonalize the low-energy excited
states. A well established technique known as Gram-Schmid orthonormalization
procedure GS_ortho  can be used in the subspace of three states to derive the
orthonormal basis. Then we have calculated the effective Hamiltonian matrix in
the new orthogonal basis and diagonalize the effective Hamiltonian matrix to
obtain the energies of the excited states. The energy spectra i.e. the
difference between the excited-state energy and the ground-state is obtained
by

$\Delta(L)=\frac{<\Psi(L)|H|\Psi(L)>}{<\Psi(L)|\Psi(L)>}-\frac{<\Psi^{0}|H|\Psi^{0}>}{<\Psi^{0}|\Psi^{0}>}$
(9)

The first term of the R.H.S. of the above equation (9) is the excitation
energy and the latter one is the ground state energy. The Hamiltonian of the
system in equation (3) will only be the interaction potential ($V$); as we
have assumed that the particles are restricted into the LLL and their kinetic
energies are quantized. Quantum Monte-Carlo approach was used for the
numerical computation of the multidimensional integration. In a previous
study, Moumita et. al. SDE_MI  demonstrated the variation of ground state
energies for the three filling fractions using PT-interaction potential. Here
we calculated the excitation spectra ($\Delta$) for $\nu=1/4,1/6$ for both
long-range and short-range interaction potential considering variation of
range of interaction. The energy spectra are computed by Monte Carlo
integration with large number of particles upto $N=160$ and then the fitted
results are plotted in FIG. 4. Two roton-minima are found for both the two
fractions in case of long-range interaction (leftbox in FIG. 4). The roton-
minima for $\nu=1/6$ are obtained at lower wave-vector region ($1.08$ and
$2.06$) and those are shifted to higher wave-vector region ($1.26$ and $2.38$)
for $\nu=1/4$. In case of short-range PT-interaction, we observed that when
the value of $\mu$ is low, there is a long-range interaction, the roton-minima
is highly sharp; however, when we raise the $\mu$-value, i.e. decrease the
range of interaction, the roton-minima gets shallow and eventually disappears.
Two roton-minima are prominent only for $\nu=1/6$ not for $\nu=1/4$ (right-
most boxes in FIG. 4). But for electronic FQH-system, we observed only one
roton-minima for the filling fraction $\nu=1/2p+1$ in both the lowest order
and next higher order excitation mode JPTC , that is not followed by bosonic
FQH-system. From this fact, we can conclude that collective excitation in
bosonic FQH-system strongly depends on the range of interaction between the
particles.

## Conclusion

Two roton-minima are obtained for both the two fractions in case of long-range
Coulomb interaction. The roton-minima are obtained at the wave-vectors
($kl_{B}$) $1.26$ and $2.38$ for $\nu=1/4$ and those are shifted to $1.08$ and
$2.06$ for $\nu=1/6$ in case of long-range interaction. But for short-range
PT-interaction two roton-minima are more prominent for $\nu=1/6$ only.
Increase of flux attachment adds up the strong correlation between the
particles to obtain two roton-minima even for short-range interaction in the
case of $\nu=1/6$. So the nature of the spectra in bosonic FQH-system,
strongly depends upon the range of interaction as well as the number of
attached flux quanta.

## Acknowledgement

We thank the Department of Physics, IIEST Shibpur, Howrah, West Bengal for the
computational facilities.

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# Conservation of energy and momentum for an electromagnetic field propagating
into a linear medium from the vacuum

Michael E. Crenshaw  US Army Combat Capabilities Development Command Aviation
and Missile Center (DEVCOM AvMC), Redstone Arsenal, AL 35898, USA

###### Abstract

The form of the energy–momentum tensor when a quasimonochromatic field
propagates into and through an antireflection-coated, sourceless, transparent,
continuous, linear magneto-dielectric medium, initially at rest in the local
frame, remains controversial. The Minkowski energy–momentum tensor is the main
component of the electromagnetic conservation law. It has been known for over
a century that the electromagnetic conservation law is unsound as evidenced by
alternative energy–momentum tensors that have been proposed to ameliorate
known physical deficiencies (violation of conservation of angular and linear
momentum) and by the various material energy–momentum tensors and coupling
forces that have been introduced to repair or complete the law. The extant
resolution is to treat the continuum electromagnetic system as a subsystem and
add a phenomenological material subsystem energy–momentum tensor. We show that
the four-divergence of the total, electromagnetic plus material,
energy–momentum tensor produces an energy continuity theorem in which the two
non-zero terms depend on different powers of the refractive index $n$. Then
the extant resolution of the Abraham–Minkowski controversy is self-
inconsistent.

##  I Introduction

A large region of otherwise empty space that completely contains, for times of
interest between an initial time $t_{i}$ and a final time $t_{f}$
($-\infty<t_{i}\leq t\leq t_{f}<\infty$), a finite quasimonochromatic field
propagating toward and through a sourceless, transparent, continuous, linear
magneto-dielectric medium that is initially at rest in the local frame is a
thermodynamically closed system $\Sigma$, definitively, regardless of what
subsystems that one chooses to identify [1]. In particular, any forces and
whatever material motion that is imparted by the interaction with the field
are part of the closed system along with the incident, refracted, transmitted,
and reflected fields. While conservation principles can be unambiguously
applied [2] to the thermodynamically closed system, just described, continuum
electrodynamics has traditionally been formulated solely in terms of the
macroscopic Maxwell field equations (see Eqs. (8d)) and the constitutive
relations, ${\bf D}={\bf E}+{\bf P}=\varepsilon{\bf E}$ and ${\bf B}={\bf
H}+{\bf M}=\mu{\bf H}$.

The total energy–momentum tensor of the system, when the field is in the
medium, has come to be viewed as being composed of an electromagnetic part and
a material part [3, 4]. Presenting the emerging viewpoint in 1979, Brevik [5]
commented that there exists “no unique prescription for the separation of this
total energy–momentum tensor into a field part and a matter part.” In their
2007 review of the Abraham–Minkowski controversy, Pfeifer, Nieminen,
Heckenberg, and Rubinsztein-Dunlop [6] present the extant viewpoint that “any
electromagnetic energy–momentum tensor must always be accompanied by a
counterpart material energy–momentum tensor, and that the division of the
total energy–momentum tensor into these two components is entirely arbitrary.”
If the separation is truly arbitrary, then the allowable electromagnetic and
material components of the total momentum extend well beyond the two specific
field and material pairs prescribed by Barnett [7] and Barnett and Loudon [8].
Arbitrary quantities cannot be verified experimentally and there are examples,
in the experimental record [1, 5, 7], of experiments that prove the Minkowski
electromagnetic momentum and experiments that prove the Abraham momentum
although experiments are unable to discriminate between the two momentums [1,
2].

In this article, we derive the well-known electromagnetic conservation law [2,
9, 3, 6]

$\partial_{\beta}{\sf T}_{M}^{\alpha\beta}=0$ (1)

as a formal (axiomatic) theorem of the Maxwell–Minkowski equations
(macroscopic Maxwell equations) and constitutive relations for the
electromagnetic field in a gradient-index antireflection-coated, sourceless,
transparent, continuous, linear magneto-dielectric medium in the limit that
the gradient Minkowski force ${\bf f}_{M}=(-{\bf E}^{2}\nabla\varepsilon-{\bf
H}^{2}\nabla\mu)/2$ and the concomitant reflection can be neglected. (We use
the conventions of summing over repeated indices on the same side of the equal
sign, that Greek indices are elements of $(0,1,2,3)$, and that Roman indices
from the middle of the alphabet are in $(1,2,3)$.)

The electromagnetic Minkowski [10] energy–momentum tensor [9, 6]

${\sf T}_{M}^{\alpha\beta}\;=\;\left[\begin{matrix}\frac{1}{2}({\bf
D}\cdot{\bf E}+{\bf B}\cdot{\bf H})&({\bf E}\times{\bf H})^{1}&({\bf
E}\times{\bf H})^{2}&({\bf E}\times{\bf H})^{3}\cr({\bf D}\times{\bf
B})^{1}&{\sf W}^{11}&{\sf W}^{12}&{\sf W}^{13}\cr({\bf D}\times{\bf
B})^{2}&{\sf W}^{21}&{\sf W}^{22}&{\sf W}^{23}\cr({\bf D}\times{\bf
B})^{3}&{\sf W}^{31}&{\sf W}^{32}&{\sf W}^{33}\cr\end{matrix}\right]\;,$ (2a)
where ${\sf W}^{ij}=-D^{i}E^{j}-B^{i}H^{j}+\frac{1}{2}({\bf D}\cdot{\bf
E}+{\bf B}\cdot{\bf H})\;\delta^{ij}\;,$ (2b)

is formally derived as the main component of the electromagnetic conservation
law, Eq. (1). The electromagnetic energy

$U_{em}=\int_{\Sigma}{\sf T}_{M}^{00}\;dv=\frac{1}{2}\int_{\Sigma}\left({\bf
D}\cdot{\bf E}+{\bf B}\cdot{\bf H}\right)\;dv$ (3)

and the electromagnetic Minkowski momentum

${\bf G}_{M}=\frac{1}{c}\int_{\Sigma}({\sf T}_{M}^{10},{\sf T}_{M}^{20},{\sf
T}_{M}^{30})\;dv=\int_{\Sigma}\frac{{\bf D}\times{\bf B}}{c}\;dv$ (4)

form a Lorentz four-vector $(U_{em}/c,-{\bf G}_{M})$ [2, 3, 6, 11]. The
Minkowski energy–momentum tensor has a well-defined formal relationship to the
electromagnetic conservation law and to its axioms, the Maxwell–Minkowski
equations and constitutive relations.

If there were no problems with the electromagnetic conservation law then the
Minkowski momentum and the Minkowski energy–momentum tensor, which are
formally derived from the Maxwell–Minkowski equations and constitutive
relations, would be settled physics. Instead, the Minkowski energy–momentum
tensor remains enmeshed in an ancient controversy.

In 1909, Abraham [12] noted that the Minkowski energy–momentum tensor, Eq.
(2a), lacks transpose symmetry indicating violation of conservation of angular
momentum. Abraham proposed a transpose symmetric energy–momentum tensor and an
accompanying linear momentum ${\bf G}_{A}={\bf G}_{M}/n^{2}$ for the field in
a simple linear dielectric. Transpose symmetry is typically considered to be a
necessary characteristic of the total energy–momentum tensor and the Abraham
tensor is extensively supported in the scientific literature [1, 5, 13].
Nowadays, the Abraham energy–momentum tensor is often treated as being
incomplete because the Abraham linear momentum is not globally conserved. For
now, we concentrate on the Minkowski energy–momentum tensor, which has its own
historical imperative.

The lack of transpose symmetry in the Minkowski energy–momentum tensor has
come to be deemed as acceptable based on the provenance of the electromagnetic
conservation law, Eq. (1), as a theorem of the Maxwell–Minkowski equations and
constitutive relations or based on appealing to the existence of
transformations that diagonalize certain energy–momentum tensors [6, 14].
Alternatively, the Minkowski energy–momentum tensor is assumed to be
incomplete, but fixable with the addition of a phenomenological material
momentum energy–momentum tensor [6]. Both of these contradictory viewpoints
are supported in the current scientific literature [6, 11, 14].

The lack of transpose symmetry in the Minkowski energy–momentum tensor is not
the only problem with the electromagnetic conservation law, Eq. (1). The
finite quasimonochromatic field, initially in the free space portion of the
thermodynamically closed system, propagates toward and then enters the
sourceless, transparent, continuous, linear magneto-dielectric medium at
normal incidence through a gradient-index antireflection coating. (The
quasimonochromatic field has a constant amplitude throughout its duration
except for a short smooth ramp up in amplitude at turn-on and a short smooth
ramp down in amplitude at turn-off.) The field re-enters the vacuum through
the gradient-index antireflection coating on the opposite side of the medium.
The usual textbook [9, 15, 16, 17] constitutive relations for a
quasimonochromatic field with center frequency $\omega_{p}$ propagating
through an anti-reflection coated, sourceless, transparent, continuous, linear
magneto-dielectric medium, at rest in the local frame, (with dispersion
treated in lowest-order [18]) are ${\bf D}=\varepsilon(\omega_{p},{\bf r}){\bf
E}$, ${\bf B}=\mu(\omega_{p},{\bf r}){\bf H}$, and
$n=\sqrt{\varepsilon(\omega_{p},{\bf r})\mu(\omega_{p},{\bf r})}$.

In the Fresnel drag experiments [19, 20], motion of the medium in the local
frame is a specified condition of the system and the velocity in the local
frame has an effect on the constitutive relations. Here, the medium is
initially at rest in the local frame and unless the radiation pressure and
duration are extraordinary, the velocity of the medium in the local frame will
be quite small. Then the effect of the material velocity on the parameters,
$\varepsilon$, $\mu$, and $n$, can be treated as negligible, as is done in the
textbook derivation of Fresnel reflection [9, 15, 16, 17]. Even more so for
the current case because the gradient-index antireflection coating makes the
acceleration and velocity of the material in the local frame negligible.

While the field is inside the medium, the electric field ${\bf E}$ is reduced
in amplitude ($\mu<\varepsilon$ at optical frequencies) compared to the
incident electric field ${\bf E}_{0}$ in vacuum with ${\bf
E}=\sqrt{n/\varepsilon}{\bf E}_{0}$. Similarly, the magnetic field ${\bf H}$
is enhanced in amplitude compared to the amplitude in the vacuum with ${\bf
H}=\sqrt{\varepsilon/n}{\bf H}_{0}$. The longitudinal width $w=w_{0}/n$ of the
field (in the direction of propagation) is reduced by a factor of $n$ due to
the reduced speed of light in a linear medium. Then the linear momentum

${\bf G}_{M}=\int_{\Sigma}\left(\varepsilon\sqrt{\frac{n}{\varepsilon}}{\bf
E}_{0}\times\mu\sqrt{\frac{\varepsilon}{n}}{\bf H}_{0}\right)\frac{dv}{c}$
(5a) ${\bf G}_{M}\;\sim\;\frac{n|{\bf E}_{0}||{\bf H}_{0}|w_{0}A_{0}}{c}$ (5b)

that is obtained by evaluating the Minkowski linear momentum, Eq. (4), while
the quasimonochromatic field, with cross-sectional area $A_{0}$ is inside the
medium, is a nominal factor of $n$ [6, 11, 21, 22, 23, 24, 18] greater than
the total linear momentum of the incident field. (The Abraham momentum ${\bf
G}_{A}={\bf G}_{M}/n^{2}$ is a factor of $n$ smaller than the incident linear
momentum.)

The existence of a difference between the linear momentum of the incident
field and the Minkowski linear momentum in the medium has been known for a
long time and has been extensively documented in the scientific literature. In
the old days, the difference in linear momentum was attributed to the action
of a Minkowski pull-force by the field entering the medium [11]. More
recently, it has been attributed to a material pseudomomentum [25] and a
canonical material momentum [7, 8],

Nevertheless, it continues to be reported that the Minkowski linear momentum
is provably conserved [6] (or ‘almost’ conserved [6]) or that the energy, Eq.
(3), and Minkowski momentum, Eq. (4), form a Lorentz four-vector [2, 3, 6, 11]
based on the electromagnetic conservation law, Eq. (1), despite the
longstanding and widespread knowledge that the Minkowski linear momentum is
contradicted by the momentum of the incident field by a non-negligible factor
of $n$, Eq. (5b) [6, 21, 22, 23, 24, 18]. The Minkowski linear momentum is not
globally conserved.

The electromagnetic conservation law, Eq. (1), is a formal (axiomatic) theorem
of the Maxwell–Minkowski equations and constitutive relations for an
antireflection-coated, sourceless, transparent, linear magneto-dielectric
medium so that the contradiction with global conservation of linear momentum,
by a factor of $n$, as well as the contradiction with conservation of angular
momentum, explicitly disproves the axioms. Then it has been known for over a
century, or should have been known, that the Maxwell–Minkowski equations are
formally false. Instead, it has been surmised that the total system is
comprised of an electromagnetic subsystem and a material subsystem. Then the
macroscopic Maxwell equations for the electromagnetic field in a simple linear
medium are incomplete and the Maxwell model is incomplete.

Section II of this article presents the axioms and theorems that are necessary
for a rigorous derivation of the energy and momentum continuity equations of
classical continuum electrodynamics, including the electromagnetic
conservation law, as formal (axiomatic) theorems of the Maxwell–Minkowski
equations and constitutive relations. The Minkowski tensor is explicitly
identified as the energy–momentum tensor component of the formally derived
electromagnetic conservation law. We emphasize the formal theory aspect
because if a theorem of a formal theory is proven false then the axioms of the
formal theory are proven false.

In Section III, we discuss the simultaneous conservation of energy and
momentum. Whereas the kinematic energy $mv^{2}/2$ and momentum $mv$ of a
particle are linearly independent in the dynamical variable, the energy and
linear momentum in a linear medium are both quadratic in the fields. The
kinetic energy of the material is second-order in smallness and can be
neglected in the limiting case under consideration such that the
electromagnetic energy, Eq. (3), is equal to the total energy

$U_{tot}=U_{em}\;.$ (6)

In order to be simultaneously conserved, the total energy and the total
momentum in a linear medium must have the same dependence on the material
permeability $\varepsilon$, permittivity $\mu$, and refractive index $n$
because the total energy and momentum are not linearly independent. We find
that the total momentum formula [25]

${\bf G}_{tot}=\int_{\Sigma}\frac{n{\bf E}\times{\bf H}}{c}dv$ (7)

has the same dependence on the refractive index as the total energy, Eq. (6),
(Eq. (3)).

In Sec. IV, we review the procedure of adding a material subsystem
energy–momentum tensor to the Minkowski energy–momentum tensor, where the
latter is now viewed as an electromagnetic subsystem tensor. The total
energy–momentum tensor ${\sf T}_{tot}^{\alpha\beta}$ and/or its main elements
appear extensively in the scientific literature [6, 7]. We prove that the
conservation law $\partial_{\beta}{\sf T}_{tot}^{\alpha\beta}=0$ that is
constructed from the total, electromagnetic plus material, energy–momentum
tensor is false because the four-divergence of the total energy–momentum
tensor produces an energy continuity equation in which the two non-zero terms
depend on different powers of the refractive index $n$. The energy continuity
equation $\partial_{\beta}{\sf T}_{tot}^{0\beta}=0$ is self-inconsistent and
also violates Poynting’s theorem.

We conclude by reiterating that the electromagnetic conservation law, Eq. (1),
is a formal theorem of the Maxwell–Minkowski equations and constitutive
relations for a quasimonochromatic field traversing a sourceless, transparent,
linear magneto-dielectric medium in the limit that the Minkowski force ${\bf
f}_{M}=(-{\bf E}^{2}\nabla\varepsilon-{\bf H}^{2}\nabla\mu)/2$ on the
gradient-index antireflection coating can be neglected. We show that the
incident linear momentum and the globally conserved total linear momentum, Eq.
(7), contradict the Minkowski momentum, Eq. (4), and thereby disproves the
electromagnetic conservation law, Eq. (1), in which the Minkowski momentum
appears. Because the electromagnetic conservation law is proven to be false by
global conservation, the axioms of the law, the Maxwell–Minkowski equations
and constitutive relations, are proven to be false.

##  II Formal theory of the Minkowski energy–momentum tensor

The microscopic Maxwell equations for electromagnetic fields in the vacuum of
free space are fundamental. When these laws are applied to the propagation of
light in a medium, one typically obtains the Maxwell–Minkowski equations [9,
15, 16, 17]

$\nabla\times{\bf H}-\frac{\partial{\bf D}}{\partial(ct)}=\frac{{\bf
J}_{f}}{c}$ (8a) $\nabla\times{\bf E}+\frac{\partial{\bf B}}{\partial(ct)}=0$
(8b) $\nabla\cdot{\bf D}=\rho_{f}$ (8c) $\nabla\cdot{\bf B}=0$ (8d)

for the macroscopic fields ${\bf E}$, ${\bf B}$, ${\bf D}$, and ${\bf H}$.
Here, $\rho_{f}$ is the free charge density and ${\bf J}_{f}$ is the free
current density. Maxwell–Minkowski is the usual textbook [9, 15, 16, 17]
representation of the macroscopic Maxwell field equations, although there are
other representations [26, 27, 4, 28, 29].

It is straightforward to construct the electromagnetic continuity equations
and the electromagnetic conservation law as formal identities of the
Maxwell–Minkowski equations using the rules of algebra and calculus for
scalars, vectors, matrices and tensors. We take the scalar product of Eq. (8b)
with ${\bf H}$, the scalar product of Eq. (8a) with ${\bf E}$, subtract the
results, and apply a common vector identity $\nabla\cdot({\bf X}\times{\bf
Y})={\bf Y}\cdot(\nabla\times{\bf X})-{\bf X}\cdot(\nabla\times{\bf Y})$ to
produce a continuity equation

$\frac{1}{c}\left({\bf E}\cdot\frac{\partial{\bf D}}{\partial t}+{\bf
H}\cdot\frac{\partial{\bf B}}{\partial t}\right)+\nabla\cdot({\bf E}\times{\bf
H})=-\frac{{\bf J}_{f}}{c}\cdot{\bf E}$ (9)

that is a formal theorem, Poynting’s theorem, of the Maxwell-Minkowski
equations. The quantity

${\bf S}=c({\bf E}\times{\bf H})$ (10)

is identified as the Poynting energy flux vector.

Adding the vector product of ${\bf B}$ with Eq. (8a), the vector product of
${\bf D}$ with Eq. (8b), the product of Eq. (8d) with $-{\bf H}$, and the
product of Eq. (8c) with $-{\bf E}$ produces the well-known continuity
equation [9]

$\frac{\partial}{\partial t}\frac{{\bf D}\times{\bf B}}{c}+{\bf
D}\times(\nabla\times{\bf E})+{\bf B}\times(\nabla\times{\bf H})$
$-(\nabla\cdot{\bf D}){\bf E}-(\nabla\cdot{\bf B}){\bf H}=-\rho_{f}{\bf
E}-\frac{1}{c}{\bf J}_{f}\times{\bf B}$ (11)

that is also a formal theorem of the Maxwell–Minkowski equations. The
Minkowski momentum density

${\bf g}_{M}=\frac{{\bf D}\times{\bf B}}{c}$ (12)

is identified in the first term of Eq. (11). The Minkowski momentum

${\bf G}_{M}=\int_{\Sigma}\frac{{\bf D}\times{\bf B}}{c}dv$ (13)

is the Minkowski momentum density integrated over the volume of the system
${\Sigma}$. The energy

$U=\frac{1}{2}\int_{\Sigma}\left({\bf E}\cdot{\bf D}+{\bf H}\cdot{\bf
B}\right)dv$ (14)

is the energy density

$u=({\bf E}\cdot{\bf D}+{\bf H}\cdot{\bf B})/2$ (15)

integrated over the volume.

The rate at which the fields do work on a continuous distribution of charge
and current

$w_{mech}={\bf J}_{f}\cdot{\bf E}$ (16)

and the Lorentz force density law

$\frac{d{\bf p}_{mech}}{dt}={\bf f}_{L}=\rho_{f}{\bf E}+\frac{{\bf
J}_{f}}{c}\times{\bf B}$ (17)

are physical interpretations of terms that are formally derived in the energy
and momentum continuity theorems, Eqs. (9) and (11).

We derived the electromagnetic energy and momentum continuity equations as
identities of the Maxwell–Minkowski equations, Eqs. (8d). In contrast,
derivations of the energy and momentum continuity equations in textbooks [9,
15, 16, 17] typically start with postulating the work rate $w_{mech}$, Eq.
(16), and the Lorentz force density ${\bf f}_{L}$, Eq. (17). Then the
postulated sources, Eqs. (16) and (17), are expressed in terms of the fields
by substitution of the Maxwell–Minkowski equations, Eqs. (8d), to derive the
energy and momentum continuity equations, Eqs. (9) and (11). Consequently,
there is scientific inertia for the presence of the sources, $\rho_{f}$ and
${\bf J}_{f}$, and for treating continuum electrodynamics as an open system
[2].

We seek to apply the axiomatic formal theory to a thermodynamically closed
system that consists of a large finite volume ${\Sigma}$ of otherwise-empty
space that contains, as an initial condition at time $t_{i}$, the finite
quasimonochromatic electromagnetic field in the vacuum and a gradient-index
antireflection-coated block of sourceless, transparent, linear magneto-
dielectric material at rest in the local frame. In the absence of sources and
sinks the total system is thermodynamically closed, definitively, regardless
of what field and material subsystems that one might choose to identify.
Because global conservation principles can be applied, without ambiguity, to
thermodynamically closed systems, we specify that

$\rho_{f}=0$ (18a) ${\bf J}_{f}=0$ (18b)
$[\varepsilon,\;\mu,\;n=\sqrt{\varepsilon\mu}]\in\mathbb{R}\geq 1\;,$ (18c)

are additional axioms of the formal theory for our sourceless transparent
system. Then the Maxwell–Ampère law, Eq. (8a), and the Gauss law, Eq. (8c),
become homogeneous with a right-hand side of zero.

It is straightforward to derive the homogeneous energy and momentum continuity
equations

$\frac{1}{c}\left({\bf E}\cdot\frac{\partial{\bf D}}{\partial t}+{\bf
H}\cdot\frac{\partial{\bf B}}{\partial t}\right)+\nabla\cdot({\bf E}\times{\bf
H})=0$ (19a) $\frac{\partial}{\partial t}\frac{{\bf D}\times{\bf B}}{c}+{\bf
D}\times(\nabla\times{\bf E})+{\bf B}\times(\nabla\times{\bf H})$
$-(\nabla\cdot{\bf D}){\bf E}-(\nabla\cdot{\bf B}){\bf H}=0$ (19b)

as theorems of the homogeneous Maxwell–Minkowski equations

$\nabla\times{\bf H}-\frac{\partial{\bf D}}{\partial(ct)}=0$ (20a)
$\nabla\times{\bf E}+\frac{\partial{\bf B}}{\partial(ct)}=0$ (20b)
$\nabla\cdot{\bf D}=0$ (20c) $\nabla\cdot{\bf B}=0$ (20d)

using the same procedure that was used to derive Eqs (9) and (11).

The usual textbook [9, 15, 16, 17] derivation of the continuity laws by
postulating Eqs. (16) and (17) is not applicable to a neutral linear medium in
which $\rho_{f}$ and ${\bf J}_{f}$, and the associated work rate $w_{mech}$
and Lorentz force density ${\bf f}_{L}$, do not exist. Deriving the energy and
momentum continuity theorems by formally combining the Maxwell equations as
axioms in the manner described is arguably more fundamental than the usual
textbook procedure because the formal axiomatic procedure works for a medium
without, as well as with, sources $\rho_{f}$ and ${\bf J}_{f}$.

We define a simple linear medium as a sourceless, transparent, isotropic,
homogeneous, continuous linear magneto-dielectric medium. Here, $\varepsilon$
is a continuum abstraction of the electric permittivity, $\mu$ is a continuum
abstraction of the magnetic permeability, and $n$ is the macroscopic
refractive index. Dispersion is treated in lowest-order [18] such that
$\varepsilon$, $\mu$, and $n$ depend on the center frequency $\omega_{p}$ of
the incident quasimonochromatic field. We do not include additional orders of
dispersion [7] because that is an exercise in complexity for a second-order
consequence. Unless the radiation is of extraordinary intensity and duration,
the velocity of the antireflection-coated material that is initially at rest
in the Laboratory Frame of Reference will be minimal and neglecting the
effects of the material motion on the permittivity, permeability, and
refractive index is an “extremely accurate approximation indeed” [1].

Treating the continuum electrodynamic system in lowest order, the
permittivity, permeability, and refractive index are constant in time and a
simple continuous function of location in space. The usual constitutive
relations [18], in lowest order,

${\bf D}=\varepsilon({\omega_{p},{\bf r}}){\bf E}$ (21a) ${\bf
B}=\mu({\omega_{p}},{\bf r}){\bf H}$ (21b)
$n=\sqrt{\varepsilon({\omega_{p}},{\bf r})\mu({\omega_{p}},{\bf r})}\,,$ (21c)

where the permittivity, permeability, and refractive index are evaluated at
the center frequency $\omega_{p}$ of the quasimonochromatic field, are
additional axioms for the theoretical model of our system. The material
parameters are piecewise homogeneous (homogeneous material and homogeneous
vacuum) except for a short smooth transition modeled as a gradient-index
antireflection coating.

Although it is always possible to start with a fully microscopic model of the
field and material, any microstructure of the field and medium has been
eliminated in the continuum limit. The continuous material and continuous
field cannot be reliably un-averaged or re-quantized at the microscopic scale
once the continuum limit is invoked.

The electromagnetic continuity equations, Eqs. (19b), can be written row-wise
as a single differential equation [9, 32] using the constitutive relations,
Eqs. (21c). We can write Eq. (19b) in component form as [9]

$\frac{\partial({\bf D}\times{\bf
B})^{i}}{\partial(ct)}+\sum_{j}\frac{\partial}{\partial x^{j}}{\sf
W}^{ij}=-\frac{\varepsilon{\bf
E}^{2}}{2}\frac{\nabla_{i}\varepsilon}{\varepsilon}-\frac{\mu{\bf
H}^{2}}{2}\frac{\nabla_{i}\mu}{\mu}\,.$ (22)

Then Eqs. (19a) and (22) can be written as a single equation [9],

$\partial_{\beta}{\sf T}_{M}^{\alpha\beta}={\bf f}_{M}^{\alpha}\;,$ (23)

where

${\sf T}_{M}^{\alpha\beta}=\left[\begin{matrix}\frac{1}{2}({\bf D}\cdot{\bf
E}+{\bf B}\cdot{\bf H})&({\bf E}\times{\bf H})^{1}&({\bf E}\times{\bf
H})^{2}&({\bf E}\times{\bf H})^{3}\cr({\bf D}\times{\bf B})^{1}&{\sf
W}^{11}&{\sf W}^{12}&{\sf W}^{13}\cr({\bf D}\times{\bf B})^{2}&{\sf
W}^{21}&{\sf W}^{22}&{\sf W}^{23}\cr({\bf D}\times{\bf B})^{3}&{\sf
W}^{31}&{\sf W}^{32}&{\sf W}^{33}\cr\end{matrix}\right]$ (24a) ${\sf
W}^{ij}=-D^{i}E^{j}-B^{i}H^{j}+\frac{1}{2}({\bf D}\cdot{\bf E}+{\bf
B}\cdot{\bf H})\delta^{ij}$ (24b) ${\bf
f}^{\alpha}_{M}=\left(0,-\frac{\varepsilon{\bf
E}^{2}}{2}\frac{\nabla\varepsilon}{\varepsilon}-\frac{\mu{\bf
H}^{2}}{2}\frac{\nabla\mu}{\mu}\right)$ (24c)
$\partial_{\beta}=\left(\frac{\partial}{\partial(ct)},\frac{\partial}{\partial
x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\;.$ (24d)

Here, Eq. (23) is explicitly proved to be a formal (axiomatic) theorem of the
sourceless electromagnetic continuity equations, Eqs. (19b), and the
constitutive relations, Eqs. (21c). Moreover, Eq. (23) has been derived as a
formal theorem of the Maxwell–Minkowski equations, Eqs. (8d), constitutive
relations, Eqs. (21c), and sources (18c), by way of the electromagnetic
continuity equations, Eqs. (19b). The Minkowski energy–momentum tensor, Eq.
(24a), has been formally derived for a neutral simple linear medium from the
Maxwell–Minkowski equations and constitutive relations as the main component
of the theorem, Eq. (23).

In the limit that the gradient-index antireflection coating is sufficiently
smooth that we can neglect reflection, then the Minkowski force ${\bf
f}^{\alpha}_{M}$ becomes negligible such that Eq. (23) becomes

$\partial_{\beta}{\sf T}_{M}^{\alpha\beta}=0\;,$ (25)

which is widely known as the electromagnetic conservation law [9]. The
vanishing right-hand side of Eq. (25) proves that there is no macroscopic
force from the macroscopic electromagnetic system acting on the medium. The
macroscopic medium remains kinematically stationary in the local frame of
reference. Absent reflection, which is the case here, the electromagnetic
energy and the Minkowski momentum constitute a Lorentz four-vector
$(U_{em}/c,-{\bf G}_{M})$ [2, 3, 6, 11]. and the electromagnetic subsystem is
thermodynamically closed.

Except, the Minkowski momentum, Eq. (4), that is obtained from the Minkowski
energy–momentum tensor in the electromagnetic conservation law for the finite
quasimonochromatic field in a linear medium is a factor of $n$ greater than
the momentum of the same field that is incident from the vacuum, Eq. (5b).
Then Minkowski linear momentum is not globally conserved, by a non-negligible
factor of $n$ [6, 11, 21, 22, 23, 24, 18], and the electromagnetic energy and
the Minkowski momentum do not constitute a Lorentz four-vector.

##  III Energy and momentum conservation

In the continuum limit, the fields are all macroscopic fields. The total
energy and total momentum are both quadratic in the macroscopic fields and
both must have the same dependence on the macroscopic parameters,
$\varepsilon$, $\mu$, and $n$, in order to be simultaneously conserved.

The electromagnetic fields can be written in terms of the vector potential
${\bf A}$ as

${\bf E}=-\nabla\phi-\frac{\partial{\bf A}}{\partial(ct)}$ ${\bf
B}=\nabla\times{\bf A}\;.$

In the absence of sources, $\rho_{f}$ and ${\bf J}_{f}$, we can use the
Coulomb gauge in which $\phi=0$. We then apply the envelope function [18]

${\bf A}({\bf r},t)=\frac{1}{2}\left({\bf{\tilde{A}}}({\bf
r},t)e^{-i(\omega_{p}t-{\bf k}_{p}\cdot{\bf r})}+c.c.\right)$ (26)

to solve

$\frac{1}{2c}\int_{\Sigma}\left({\bf D}\cdot{\bf E}+{\bf B}\cdot{\bf
H}\right)dv=\int_{\Sigma}\frac{\zeta|{\bf E}\times{\bf H}|}{c}dv$ (27)

for the unknown factor $\zeta$.

We obtain the nominal factor

$\zeta=n$ (28)

for fields with slowly varying envelopes, i.e., quasimonochromatic fields,
propagating through an anti-reflection coated, sourceless, transparent,
continuous, linear magneto-dielectric medium that is initially at rest in the
local frame.

As discussed above Eq. (5b), the amplitudes of the electric and magnetic
fields in the medium are different from their amplitudes in the vacuum, but
the changes offset such that the product $EH$ remains constant. Using the
solution $\zeta=n$ of Eq. (26), the momentum

${\bf G}_{tot}=\int_{\Sigma}\frac{n{\bf E}\times{\bf H}}{c}dv$ (29)

is the globally conserved momentum counterpart of the globally conserved
energy for quasimonochromatic fields.

The momentum density of the field inside the medium

${\bf g}_{tot}=\frac{n{\bf E}\times{\bf H}}{c}$ (30)

is a factor of $n$ greater than the momentum density of the incident field in
the vacuum [6, 11, 21, 22, 23, 24, 18]. Integrating the enhanced momentum
density, Eq. (30), over the narrower pulse when the field is inside the medium
proves that the momentum, Eq. (29), is globally conserved.

Global conservation of the momentum, Eq. (29), in the absence of any
significant reflection, was previously demonstrated by the current author
using a finite-difference time-domain numerical solution of the wave equation
with numerical integration of the electromagnetic momentum, Eq. (29), of the
propagated field at various points in time [21].

Global conservation of momentum for an antireflection coated linear medium is
consistent with global conservation of energy. The electromagnetic energy
density in the linear medium, Eq. (15), is well-known to be a factor of $n$
greater than the energy density in the vacuum. Integrating the enhanced energy
density over the narrower pulse demonstrates global conservation of the
electromagnetic energy, Eq. (14). The enhanced energy density is due to the
narrower pulse, not to a material energy density. Similarly, the enhanced
momentum density, Eq. (30), is due to the narrower pulse, not to a material-
motion momentum density. Experiments, which are typically performed with cw
fields, and the definition of photons in the medium must take the reduced
volume of the field in the medium into account.

The difference between global conservation of energy and global conservation
of momentum is that reflection changes the sign of momentum, but not energy.
If part of the field is reflected, we must impute a momentum to the material
that is twice the magnitude of the momentum of the reflected field and in the
direction of propagation. Then,

${\bf G}_{forward}=\int_{\Sigma^{+}}\frac{n{\bf E}\times{\bf H}}{c}dv$ (31a)
${\bf G}_{backward}=-\int_{\Sigma^{-}}\frac{{\bf E}\times{\bf H}}{c}dv$ (31b)
${\bf G}_{mat}=2\int_{\Sigma^{-}}\frac{{\bf E}\times{\bf H}}{c}dv$ (31c)

where ${\Sigma^{-}}$ is the vacuum side of the plane of incidence ($n=1$) and
${\Sigma^{+}}$ is the space on the material side of the plane of incidence
($n$). If any of the field has exited the medium, then there is additional
reflection and additional material momentum. As long as the refracted field
remains in the medium, the total, field plus material, momentum is

${\bf G}_{tot}={\bf G}_{forward}+{\bf G}_{backward}+{\bf G}_{mat}\;.$ (32)

If reflection is negligible, as is the main case here, there is no need to
assign a kinematic momentum to the material and the kinetic energy is nil. The
electromagnetic energy and electromagnetic momentum are simultaneously
conserved and the electromagnetic system is thermodynamically closed such that

${\bf G}_{tot}={\bf G}_{forward}={\bf G}_{em}=\int_{\Sigma}\frac{n{\bf
E}\times{\bf H}}{c}dv\;.$ (33)

##  IV Material momentum approach

In this section, we review the basic details of the way that the total
energy–momentum tensor is presented in the scientific literature. The extant
procedure is to add a phenomenological material momentum ${\bf G}_{mat}$ to
the Minkowski momentum and write the tautology [7, 30]

${\bf G}_{tot}={\bf G}_{M}+{\bf G}_{mat}$ (34)

such that the whole is the sum of the parts. This is part of a larger view of
the total system as consisting of an electromagnetic subsystem and a material
subsystem. The conservation law for the total system, Ref. [6] Eq. (10),

$\partial_{\beta}{\sf T}_{tot}^{\alpha\beta}=\partial_{\beta}\left({\sf
T}_{M}^{\alpha\beta}+{\sf T}_{mat}^{\alpha\beta}\right)=0\;,$ (35)

is the tautological sum of the conservation law for the electromagnetic
conservation law for the electromagnetic subsystem and the conservation law
for the material motion subsystem. In this scenario, the two subsystems are
assumed to be coupled by equal and opposite forces, $\pm{\bf f}_{M}$, in term
of the gradients of the permittivity and permeability.

The usual model for the material is the flow of particles of dust in the
continuum limit. In their comprehensive review article, Pfeifer, Nieminen,
Heckenberg, and Rubinsztein-Dunlop [6] present the total, electromagnetic plus
dust, energy–momentum tensor

${\sf T}_{tot}^{\alpha\beta}=\left[\begin{matrix}\frac{1}{2}({\bf D}\cdot{\bf
E}+{\bf B}\cdot{\bf H})+\rho_{0}c^{2}&({\bf E}\times{\bf H})+\rho_{0}c{\bf
v}\cr({\bf E}\times{\bf H})+\rho_{0}c{\bf v}&{\sf
W}_{tot}\cr\end{matrix}\right]\;,$ (36)

where $\rho_{0}$ is the density of the dust and ${\sf W}_{tot}^{ij}={\sf
W}^{ij}+\rho_{0}v^{i}v^{j}$.

We apply the four-divergence operator to the total energy–momentum tensor, as
shown in Eq. (35). The first row, $\partial_{\beta}{\sf T}_{tot}^{0\beta}$, is
the energy continuity equation

$\frac{1}{c}\left({\bf E}\cdot\frac{\partial{\bf D}}{\partial t}+{\bf
H}\cdot\frac{\partial{\bf B}}{\partial t}\right)+\nabla\cdot({\bf E}\times{\bf
H})+\nabla\cdot\rho_{0}c{\bf v}=0\;.$ (37)

Pfeifer, Nieminen, Heckenberg, and Rubinsztein-Dunlop, Ref. [6] Eq. (44), use
global conservation of total momentum to show that

$\rho_{0}{\bf v}=(n-1)\frac{{\bf E}\times{\bf H}}{c}\;.$ (38)

Substituting Eq. (38) into Eqs. (36) and (37) produces

${\sf T}_{tot}^{\alpha\beta}=\left[\begin{matrix}\frac{1}{2}({\bf D}\cdot{\bf
E}+{\bf B}\cdot{\bf H})+\rho_{0}c^{2}&n({\bf E}\times{\bf H})\cr n({\bf
E}\times{\bf H})&{\sf W}_{tot}\cr\end{matrix}\right]\;,$ (39)

and

$\frac{1}{c}\left({\bf E}\cdot\frac{\partial{\bf D}}{\partial t}+{\bf
H}\cdot\frac{\partial{\bf B}}{\partial t}\right)+\nabla\cdot n({\bf
E}\times{\bf H})=0\;.$ (40)

The total, field plus matter, momentum ${\bf G}_{tot}=\int_{\Sigma}c^{-1}({\sf
T}_{tot}^{10},{\sf T}_{tot}^{20},{\sf T}_{tot}^{20})dv$, Eq. (28), is globally
conserved. Note that the energy continuity equation, Eq. (37), obviously
violates Poynting’s theorem. The next step makes it clear how much of a
problem that is because the two nonzero terms in Eq. (40) depend on different
powers of $n$. Equation (40) is manifestly false.

Partanen, Häyryen, Oksanen, and Tulkki [31] propose that photons couple to the
material system to form mass-polariton quasiparticles. Assuming that the
Minkowski momentum is conserved, they write the total, field plus matter,
energy–momentum tensor as

${\sf T}_{MP}^{\alpha\beta}=\left[\begin{matrix}\frac{n^{2}}{2}({\bf
D}\cdot{\bf E}+{\bf B}\cdot{\bf H})&n^{2}({\bf E}\times{\bf H})\cr n^{2}({\bf
E}\times{\bf H})&{\sf W}\cr\end{matrix}\right]\;.$ (41)

In this case, the energy conservation law $\partial_{0}{\sf T}_{MP}^{\alpha
0}=0$ is self-consistent and is apparently consistent with Poynting’s theorem
in a linear medium because the common factor $n^{2}$ can be factored out of
the energy continuity equation $\partial_{\beta}{\sf T}_{MP}^{0\beta}=0$:

$\frac{n^{2}}{c}\left({\bf E}\cdot\frac{\partial{\bf D}}{\partial t}+{\bf
H}\cdot\frac{\partial{\bf B}}{\partial t}\right)+\nabla\cdot n^{2}({\bf
E}\times{\bf H})=0\;.$ (42)

However, the mass-polariton energy–momentum tensor, Eq. (41), is false because
the total, field plus matter, momentum ${\bf G}_{MP}=\int_{\Sigma}c^{-1}({\sf
T}_{MP}^{10},{\sf T}_{MP}^{20},{\sf T}_{MP}^{20})dv$ is equal to the Minkowski
momentum and is not globally conserved, Eq. (5b). The total energy
${U}_{tot}=\int_{\Sigma}{\sf T}_{MP}^{00}dv$ is not globally conserved either.

The electromagnetic plus dust tensor treatment of the total energy–momentum
tensor and the mass polariton model are illustrative of the many theories that
are based on adding a material subsystem energy–momentum tensor to the
electromagnetic subsystem energy–momentum tensor (or adding a material
momentum to the linear momentum). However, the two subsystem energy–momentum
tensors do not necessarily add in the fashion of Eq. (35) if the dynamical
laws of each subsystem are different. This can be illustrated by considering
two non-interacting systems, one electromagnetic and one kinematic. The two
conservation laws can be combined in accordance with Eq. (35) with zero
external forces on the two systems, but the four-divergence of the sum
energy–momentum tensor mixes the closed systems together.

##  V Conclusion

The Maxwell–Minkowski equations are not fundamental laws of physics, they are
macroscopic equations that are phenomenologically extrapolated from a
perturbation of the fundamental microscopic Maxwell equations. The non-
negligible factor of $n$ difference between the Minkowski linear momentum and
the globally conserved total momentum in this simplest-case thermodynamically
closed system proves that there is an essential contradiction between global
conservation of the total momentum and the electromagnetic conservation law
that is derived as an identity of the Maxwell–Minkowski equations and
constitutive relations in a closed system that consists of a
quasimonochromatic field, initially in vacuum, incident on a gradient-index
antireflection-coated, sourceless, simple linear magneto-dielectric material,
initially at rest in the vacuum local frame. The disproof of the
electromagnetic conservation law by global conservation of energy and momentum
in a thermodynamically closed system proves that the axioms, the
Maxwell–Minkowski equations and constitutive relations, are formally false.

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