#  Kinetic theory formulation of the $P$\- and $CP$-odd terms in the photon
self-energy in a medium

José F<EMAIL_ADDRESS>John D<EMAIL_ADDRESS>
Laboratory of Theoretical Physics, Department of Physics
University of Puerto Rico, Río Piedras, Puerto Rico 00936

(Jan 2024)

###### Abstract

In an optically active medium, such as a plasma that contains a neutrino
background, the left-handed and right-handed polarization photon modes acquire
different dispersion relations. We study the propagation of photons in such a
medium, which is otherwise isotropic, within the framework of the covariant
collissionless Boltzmann equation incorporating a term that parametrizes the
optical activity. Using the linear response approximation, we obtain the
formulas for the components of the photon polarization tensor, expressed in
terms of integrals over the momentum distribution function of the background
particles. The main result here is the formula for the $P$\- and $CP$-breaking
component of the photon polarization tensor in terms of the parameter involved
in the new term we consider in the Boltzmann equation to describe the effects
of optical activity. We discuss the results for some particular cases, such as
long-wavelength and nonrelativistic limits, for illustrative purposes. We also
discuss the generalizations of the $P$\- and $CP$-breaking term we included in
the Boltzmann equation. In particular we consider the application to a plasma
with a neutrino background and establish contact with calculations of the
photon self-energy in those systems in the framework of thermal field theory.

##  1 Introduction and outline

It was shown sometime ago, in the context of _thermal field theory_ (TFT),
that the general expression for the photon self-energy in an isotropic medium,
consistent with gauge and Lorentz invariance in four dimensions, may contain a
term which signals $P$\- and $CP$\- symmetry breaking, either in the
Lagrangian, or in the background, or both[1, 2]. The effect of this term is
that the dispersion relation of the photon transverse modes, which would
otherwise be degenerate, are split according to the polarization of the
propagating mode. A consequence of this is the rotation of the linear
polarization of an electromagnetic wave traveling in such media, or
birefringence. These effects arise, for example, when photons propagate in a
medium that contains a neutrino background[3, 4, 5].

On the other hand it has also been shown that the effect mentioned can be
described in the context of the classical Maxwell equations in terms of an
additional electromagnetic _constant_ $\zeta$, besides the standard dielectric
($\epsilon$) and magnetic ($\mu$) constants[6]. A covariant version of this
formulation has also been given[7]. Following Ref. [6] we refer to $\zeta$ as
the _activity constant_ , and to this kind of medium as an _optically active
medium_.

Our objective in the present work is to revisit the study the propagation of
photons in such media, in the framework of the covariant, collisionless,
Boltzmann equation, but incorporating a term that gives rise to the optical
activity effects mentioned above. This approach, which lies somewhere in the
middle between the two approaches mentioned, that is, TFT on one hand, and a
purely phenomenological description on the other, could be more suitable than
those two approaches in certain situations.

Recently, the optical activity effects on the cosmic microwave photons as they
travel through the medium of the cosmic neutrino background have been
considered in the context of the cosmic birefringence[8]. Independently of
whether or not the photon-neutrino interactions are responsible for the
observed effects in this context, or whether another source of the optical
activity may be required as suggested in Ref. [8], our work may be useful for
further development in this area, and other astrophysical contexts in which
optical activity plays a role.

While in the present work we restrict ourselves to the collisionless Boltzman
equation, modified to incorporate the optical activity effects, the method we
use can be extended to include the effect of collisions in an optically active
plasma, for example by adapting the techniques used for this purpose in the
framework of the Boltzmann equation for an ordinary plasma (see, e.g., Ref.
[9] and references therein). For example, in the case of photons propagating
in a plasma medium with a neutrino background, it may be important to include
also the effects of collisional interactions. While such interactions
typically lead to _damping effects_ , under the appropriate circumstances can
also lead to growth effects. An example of a growth effect was provided in
Ref. [4], with regard to the evolution of a magnetic field perturbation in
such a medium. This particular phenomena has been considered by Semikoz and
Sokoloff[10] as a mechanism for the generation of large-scale magnetic fields
in the early Universe as a consequence of the neutrino-plasma interactions. In
this sense the present work provides the mechanism for considering the effects
of collisions in an optically active medium on a firm ground.

In short, our proposal is to consider the relativistic collisionless Boltzmann
equation for a given particle specie “$a$”, of charge $q_{a}$,

$p\cdot\partial^{(x)}f_{a}=-q_{a}F^{\mu\nu}p_{\nu}\partial^{(p)}_{\mu}f_{a}\,,$
(1.1)

modified by adding a term proportional to the dual electromagnetic tensor
$\tilde{F}_{\mu\nu}$, as we state precisely below, including the explanation
of the various symbols that enter in this equation. The strategy is to use the
linear response approach to obtain the expression for the induced current, and
thereby the photon polarization tensor, in terms of the particle distribution
functions, and study various aspects of the results that could be useful for
applications, such as the corresponding dispersion relations and the
interpretation in terms of the activity constant. In the present work we
restrict ourselves to implement this program for an isotropic system, that is,
the momentum distribution functions of all the particle species are isotropic
in the rest frame of the system. The generalization of the approach to other
cases, for example a two-stream component plasma is straightforward from a
conceptual point of view, although of course the details will be different and
they can be important in specific physical contexts.

The outline of the rest of the paper is as follows. In Section 2 we present
the covariant Boltzmann equation, that includes the $P$\- and $CP$-breaking
term to describe the effects of an optically active medium. For consistency,
we discuss some particular features and consequences of the equation,
including current conservation and the role of the discrete symmetries of the
new term. In Section 3, we consider the solution of the equation, using the
standard linearization method. The expression for the induced current is
determined, in terms of integrals of the momentum distribution functions of
the particles. There we establish contact with the photon polarization tensor,
or equivalently the photon self-energy, in the TFT language, specifically as
used in Ref. [1], and the formulas for the components of the polarization
tensor are obtained. The results for the longitudinal and transverse
components of the photon self-energy are the familiar ones. The formula for
the $P$\- and $CP$-breaking term is the new result here. In Section 4 we
discuss some details of the results obtained, and consider specifically some
particular cases (e.g., the long-wavelength and the nonrelativistic limit)
that are useful in many applications and can serve as benchmark references for
more general situations. We also point out possible generalizations of the
$P$\- and $CP$-breaking term we included in the Boltzmann equation, in
particular how it applies in the context of a plasma with a neutrino gas as a
background, and establish contact with calculations of the photon self-energy
in such backgrounds in the framework of TFT[3]. Possible avenues for
extensions and exploration of the present work are mentioned in Section 5.

##  2 $P$\- and $CP$-breaking kinetic equation

Without further preamble, the equation we consider is,

$p\cdot\partial^{(x)}f_{a}=[-q_{a}F^{\mu\nu}+\gamma_{a}\tilde{F}^{\mu\nu}]p_{\nu}\partial^{(p)}_{\mu}f_{a}\,.$
(2.1)

We will discuss some generalizations of this equation in Section 4.2. In this
expression, $f_{a}(x,p)$ is the number density of the particles specie “$a$”,
of charge $q_{a}$, in the plasma, expressed as a function of the four-vectors
$x^{\mu}$ and $p^{\mu}$, and normalized as specified below [e.g., see Eq.
(2.10)]. Further, $\tilde{F}^{\mu\nu}$ is the dual electromagnetic field
tensor,

$\tilde{F}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}\,,$
(2.2)

and we are using the shorthand symbols

$\displaystyle\partial^{(x)}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial x^{\mu}}\,,$
$\displaystyle\partial^{(p)}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial p^{\mu}}\,.$ (2.3)

We use the conventions such that $g^{\mu\nu}$ has diagonal elements
$(1,-1,-1,-1)$ and $\epsilon^{0123}=+1$.

The parameter $\gamma_{a}$ is a phenomenological parameter, which in this
approach is unknown. However Eq. (2.1), or its generalizations, as we will
discuss, parametrize effectively the results of the calculations of optical
activity in some systems, such as those considered in the references already
cited (e.g., Refs, [3, 4, 5]). In this sense this approach serves as a bridge
between those calculations on one hand, and a pure phenomenological
description in terms of the electrodynamics equations on the other (e.g.,
Refs. [1, 6, 7]).

Before entering in the practical calculations, there are various aspects of
these equations that are worth discussing. We consider them below.

###  2.1 Current conservation

The electromagnetic current density is

$j_{\mu}=\sum_{a}q_{a}J_{a\mu}\,,$ (2.4)

where

$J_{a\mu}=2\int\frac{d^{4}p}{(2\pi)^{3}}\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)p_{\mu}f_{a}\,.$ (2.5)

We have introduced the velocity four-vector of the medium, $u^{\mu}$, which
has components

$u^{\mu}=(1,\vec{0})\,,$ (2.6)

in the medium’s own rest frame. Notice that

$\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)=\frac{1}{2E_{ap}}\delta(p^{0}-E_{ap})\,,$ (2.7)

with

$E_{ap}=\sqrt{|\vec{p}|^{2}+m^{2}_{a}}\,,$ (2.8)

since $\theta(p\cdot u)=0$ for the negative solution $p^{0}=-E_{ap}$.
Therefore Eq. (2.5) reduces to

$J_{a\mu}=\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}p_{\mu}f_{a}\,,$ (2.9)

which is the conventional expression for the current density four-vector of
each specie. In particular, its zeroth component is the particle number
density of each specie,

$n_{a}=J^{0}_{a}=\int\frac{d^{3}p}{(2\pi)^{3}}f_{a}\,.$ (2.10)

We now consider the divergence of the current density. From Eq. (2.5),

$\partial^{(x)}\cdot
J_{a}=2\int\frac{d^{4}p}{(2\pi)^{3}}\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)(p\cdot\partial^{(x)}f_{a})\,,$ (2.11)

and using Eq. (2.1)

$\partial^{(x)}\cdot J_{a}=A^{\mu\nu}_{a}J_{a\mu\nu}\,,$ (2.12)

where we have defined

$A^{\mu\nu}_{a}=-q_{a}F^{\mu\nu}+\gamma_{a}\tilde{F}^{\mu\nu}\,,$ (2.13)

and

$\displaystyle J_{a\mu\nu}$ $\displaystyle\equiv$ $\displaystyle\int
d^{4}p\,\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)p_{\nu}\partial^{(p)}_{\mu}f_{a}$ (2.14) $\displaystyle=$
$\displaystyle-\int
d^{4}p\,f_{a}\,\partial^{(p)}_{\mu}\left[p_{\nu}\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)\right]\,.$

In the second equality in Eq. (2.14) we have integrated by parts, dropping the
surface term since $f_{a}$ vanishes for infinite momentum. Current
conservation is a consequence of the fact that, while $A^{\mu\nu}_{a}$ is
antisymmetric,

$A^{\mu\nu}_{a}=-A^{\nu\mu}_{a}\,,$ (2.15)

$J_{a\mu\nu}$ is, as we show below, symmetric,

$J_{a\mu\nu}=J_{a\nu\mu}\,.$ (2.16)

Therefore,

$A^{\mu\nu}_{a}J_{a\mu\nu}=0\,,$ (2.17)

which together with Eq. (2.12) imply that

$\partial^{(x)}\cdot J_{a}=0\,.$ (2.18)

The proof of Eq. (2.16) is by straightforward algebra. Taking derivative that
appears in the integrand of Eq. (2.14) we obtain,

$\displaystyle\partial^{(p)}_{\mu}\left[p_{\nu}\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)\right]$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}\delta(p^{2}-m^{2}_{a})\theta(p\cdot u)$ (2.19)
$\displaystyle\mbox{}+p_{\nu}\partial^{(p)}_{\mu}\left[\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)\right]\,,$

and for the second term in Eq. (2.19)

$\partial^{(p)}_{\mu}\left[\delta(p^{2}-m^{2}_{a})\theta(p\cdot
u)\right]=p_{\mu}\delta^{\prime}(p^{2}-m^{2}_{a})\theta(p\cdot
u)+u_{\mu}\delta(p^{2}-m^{2}_{a})\delta(p\cdot u)\,.$ (2.20)

In the last formula, the product of the two _delta_ functions give zero
because the two conditions,

$p^{0}=\pm\sqrt{|\vec{p}|^{2}+m^{2}_{a}}\,,$ (2.21)

and

$p^{0}u^{0}=\vec{p}\cdot\vec{u}\,,$ (2.22)

cannot be satisfied simultaneously. Therefore we get

$J_{a\mu\nu}=-\int d^{4}p\,f\,\theta(p\cdot
u)\left[g_{\mu\nu}\delta(p^{2}-m^{2}_{a})+p_{\mu}p_{\nu}\delta^{\prime}(p^{2}-m^{2}_{a})\right]\,,$
(2.23)

which explicitly verifies Eq. (2.16).

###  2.2 Modification of the Boltzmann equation

If the distribution function is treated as a function only of $\vec{p}$ rather
than $\vec{p}$ and $p^{0}$ separately, in other words we explicitly set

$f^{(s)}_{a}(t,\vec{x},\vec{p})\equiv\left.f_{a}(p)\right|_{p^{0}=E_{p}}\,,$
(2.24)

it is well known that the covariant equation given in Eq. (1.1) is equivalent
to (see, e.g., Ref. [11])

$\partial_{t}f^{(s)}_{a}+\vec{v}_{a}\cdot\nabla_{x}f^{(s)}_{a}=-\vec{F}_{a}\cdot\nabla_{p}f^{(s)}_{a}\,,$
(2.25)

with

$\vec{F}_{a}=q_{a}(\vec{E}+\vec{v}_{a}\times\vec{B})\,.$ (2.26)

Equation (2.25) is the standard form of the Boltzmann equation for charged
particles in an electromagnetic field.

In the case that $f_{a}$ satisfies Eq. (2.1) the corresponding equation,
analogous to Eq. (2.25), is

$\partial_{t}f^{(s)}_{a}+\vec{v}_{a}\cdot\nabla_{x}f^{(s)}_{a}=(-\vec{F}_{a}+\vec{G}_{a})\cdot\nabla_{p}f^{(s)}_{a}\,,$
(2.27)

where $\vec{F}_{a}$ is given above, and

$\vec{G}_{a}=\gamma_{a}(\vec{B}-\vec{v}_{a}\times\vec{E})\,.$ (2.28)

The result given in Eq. (2.27) follows easily from the fact that the elements
of $\tilde{F}_{\mu\nu}$ are obtained from $F_{\mu\nu}$ by making the
replacement

$\displaystyle\vec{E}$ $\displaystyle\rightarrow$ $\displaystyle\vec{B}\,,$
$\displaystyle\vec{B}$ $\displaystyle\rightarrow$ $\displaystyle-\vec{E}\,,$
(2.29)

which implies that $\vec{G}_{a}$ is obtained from $\vec{F}_{a}$ by making the
same replacement. Thus, for example, in the presence of only a magnetic field
$\vec{B}$, the equation is

$\partial_{t}f^{(s)}_{a}+\vec{v}_{a}\cdot\nabla_{x}f^{(s)}_{a}=(-q_{a}\vec{v}_{a}\times\vec{B}+\gamma_{a}\vec{B})\cdot\nabla_{p}f^{(s)}_{a}\,.$
(2.30)

A quick observation that follows from Eq. (2.27) is that some discrete space-
time symmetries are broken in the system when the $\tilde{F}_{\mu\nu}$ term is
present in the Boltzmann equation. This is obvious, for example for parity
($P$), from the fact that $\vec{E}$ and $\vec{B}$ have opposite phase under a
$P$ transformation, and therefore the same holds for $\vec{F}$ and $\vec{G}$.

##  3 Linearization of the kinetic equation and $\pi_{\mu\nu}$

###  3.1 Linearization and the induced current

The dispersion relations for the propagating photons are obtained by
linearizing the kinetic equation. We put

$f_{a}=f_{a0}+f_{a1}+\cdots\,,$ (3.1)

where $f_{a0}$ is the equilibrium distribution, and $f_{a1}$ is linear in
$F^{\mu\nu}$. Substituting Eq. (3.1) in Eq. (2.1), and retaining only terms
that are linear in $F^{\mu\nu}$, gives

$p\cdot\partial^{(x)}f_{a1}=[-q_{a}F^{\mu\nu}+\gamma_{a}\tilde{F}^{\mu\nu}]p_{\nu}\partial^{(p)}_{\mu}f_{a0}\,.$
(3.2)

The next step is to consider the momentum space equation corresponding to Eq.
(3.2). Denoting the wave vector by $k^{\mu}$, the momentum space equation is
obtained from Eq. (3.2) by making the replacements

$\displaystyle F_{\mu\nu}$ $\displaystyle\rightarrow$ $\displaystyle
f_{\mu\nu}\,,$ $\displaystyle\tilde{F}_{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{f}_{\mu\nu}\,,$ $\displaystyle f_{a1}$
$\displaystyle\rightarrow$ $\displaystyle\hat{f}_{a1}\,,$
$\displaystyle\partial^{(x)}_{\mu}f_{a1}$ $\displaystyle\rightarrow$
$\displaystyle-ik_{\mu}\hat{f}_{a1}\,,$ (3.3)

with the understanding that the functions on the right-hand side are the
Fourier transforms of those on the left. In particular,

$\tilde{f}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}f_{\alpha\beta}\,,$
(3.4)

in correspondence with Eq. (2.2). Furthermore, remembering that we are
considering an isotropic system, $f_{a0}$ is a function only of

${\cal E}\equiv p\cdot u\,,$ (3.5)

in which case

$\partial^{(p)}_{\mu}f_{a0}=u_{\mu}f_{a0}^{\prime}\,,$ (3.6)

where

$f_{a0}^{\prime}\equiv\frac{\partial f_{a0}}{\partial{\cal E}}\,.$ (3.7)

With these substitutions, the momentum space equation corresponding to Eq.
(3.2) is,

$(-ik\cdot
p)\hat{f}_{a1}=\left[-q_{a}f^{\mu\nu}+\gamma_{a}\tilde{f}^{\mu\nu}\right]u_{\mu}p_{\nu}f_{a0}^{\prime}\,,$
(3.8)

which gives

$\hat{f}_{a1}=\left(\frac{i}{k\cdot
p}\right)\left[-q_{a}f^{\mu\nu}+\gamma_{a}\tilde{f}^{\mu\nu}\right]u_{\mu}p_{\nu}f_{a0}^{\prime}\,.$
(3.9)

The induced current is obtained from Eqs. (2.4) and (2.9) using Eqs. (3.1) and
(3.9). The terms containing the equilibrium distributions $f_{a0}$ do not
contribute. This is most easily seen by going to the medium’s own rest frame.
Due to the isotropy condition, the vector current density $\vec{j}_{a}$ is
zero, while the total $j^{0}$ is zero assuming that the charge density is zero
in equilibrium. Thus,

$j_{\mu}=i\sum_{a}q_{a}\left[-q_{a}f^{\lambda\nu}+\gamma_{a}\tilde{f}^{\lambda\nu}\right]u_{\lambda}I^{(1)}_{a\mu\nu}\,,$
(3.10)

where

$I^{(1)}_{a\mu\nu}=\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}\frac{f_{a0}^{\prime}}{k\cdot
p}p_{\mu}p_{\nu}\,.$ (3.11)

The next step is to write Eq. (3.10) in terms of the vector potential rather
than the field. We consider separately the two terms in Eq. (3.10).

####  3.1.1 $f^{\mu\nu}$ term

For the term with $f^{\mu\nu}$, using

$f_{\alpha\beta}=-i[k_{\alpha}A_{\beta}-k_{\beta}A_{\alpha}]\,,$ (3.12)

we have

$u^{\alpha}p^{\beta}f_{\alpha\beta}=-i[(k\cdot u)p_{\nu}-(k\cdot
p)u_{\nu}]A^{\nu}\,,$ (3.13)

which gives

$f^{\lambda\nu}u_{\lambda}I^{(1)}_{a\mu\nu}=-i[(k\cdot
u)I^{(1)}_{a\mu\nu}-I^{(2)}_{a}u_{\mu}u_{\nu}]A^{\nu}\,,$ (3.14)

where $I^{(1)}_{a\mu\nu}$ is given in Eq. (3.11) and

$I^{(2)}_{a}=\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}(p\cdot u)f_{a0}^{\prime}\,.$
(3.15)

To arrive at Eq. (3.14) we have used again the fact that we are considering an
isotropic system so $f_{a0}$ is a function of $p\cdot u$, and therefore

$\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}f_{a0}^{\prime}p_{\mu}=I^{(2)}_{a}u_{\mu}\,.$
(3.16)

For the sake of completeness, we mention that the contribution to the current
from the term given in Eq. (3.14) is transverse by itself, as can be easily
verified explicitly by multiplying by $k^{\mu}$ and using Eq. (3.16).

####  3.1.2 $\tilde{f}^{\mu\nu}$ term

We now derive the relation analogous to Eq. (3.14) for the
$\tilde{f}^{\mu\nu}$ term. From Eqs. (3.4) and (3.12), relabeling some of the
Lorentz indices, it follows that

$\tilde{f}^{\lambda\nu}u_{\lambda}I^{(1)}_{a\mu\nu}=-i\epsilon^{\lambda\nu\alpha\beta}k_{\alpha}u_{\beta}A_{\nu}I^{(1)}_{a\mu\lambda}\,,$
(3.17)

where $I^{(1)}_{a\mu\nu}$ is defined in Eq. (3.11). Since $I^{(1)}_{a\mu\nu}$
is a symmetric in $\mu,\nu$, and depends only on $k^{\mu}$ and $u^{\mu}$, it
is of the form

$I^{(1)}_{a\mu\nu}=Cg_{\mu\nu}+X_{1}u_{\mu}u_{\nu}+X_{2}k_{\mu}k_{\nu}+X_{3}(k_{\mu}u_{\nu}+u_{\mu}k_{\nu})\,.$
(3.18)

Only the $C$ term contributes in Eq. (3.17) and therefore

$\tilde{f}^{\lambda\nu}u_{\lambda}I^{(1)}_{a\mu\nu}=-iC_{a}\epsilon_{\mu\nu\alpha\beta}k^{\alpha}u^{\beta}A^{\nu}\,.$
(3.19)

$C_{a}$ can in turn be written in terms of the tensor $R^{\lambda\rho}$
defined in Eq. (A.1) as

$C_{a}=\frac{1}{2}R^{\lambda\rho}I^{(1)}_{a\lambda\rho}\,.$ (3.20)

###  3.2 $\pi_{\mu\nu}$ and $\pi_{T,L,P}$

Using Eqs. (3.14) and (3.19) in Eq. (3.10), the induced current is then
expressed in the form $j_{\mu}=-\pi_{\mu\nu}A^{\nu}$ [Eq. (A.1)], with

$\pi_{\mu\nu}=\sum_{a}\left\\{q_{a}^{2}[(k\cdot
u)I^{(1)}_{a\mu\nu}-I^{(2)}_{a}u_{\mu}u_{\nu}]-q_{a}\gamma_{a}C_{a}\epsilon_{\mu\nu\alpha\beta}k^{\alpha}u^{\beta}\right\\}\,.$
(3.21)

This expression for $\pi_{\mu\nu}$ can be decomposed in terms of $\pi_{T,L,P}$
as given in Eq. (A.2). $\pi_{P}$ can be written by inspection, while
$\pi_{T,L}$ can be obtained by projecting the term in square brackets in Eq.
(3.21), which is symmetric and transverse, with $R^{\mu\nu}$ and $Q^{\mu\nu}$,
using Eqs. (A.10) and (A.9). Thus,

$\displaystyle\pi_{L}$ $\displaystyle=$
$\displaystyle-\sum_{a}q_{a}^{2}\left(\frac{k^{2}D_{a}}{\kappa^{2}}\right)\,,$
$\displaystyle\pi_{T}$ $\displaystyle=$ $\displaystyle\sum_{a}q_{a}^{2}(k\cdot
u)C_{a}\,,$ $\displaystyle\pi_{P}$ $\displaystyle=$
$\displaystyle\sum_{a}iq_{a}\gamma_{a}\kappa C_{a}\,,$ (3.22)

where $C_{a}$ has been defined in Eq. (3.20), while

$\displaystyle D_{a}$ $\displaystyle=$ $\displaystyle u^{\mu}u^{\nu}[(k\cdot
u)I^{(1)}_{a\mu\nu}-I^{(2)}_{a}u_{\mu}u_{\nu}]\,.$ (3.23)

Using Eqs. (3.11) and (3.15), the integral formulas for $C_{a}$ and $D_{a}$
can be written in the form

$\displaystyle C_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}\left\\{p^{2}-(p\cdot
u)^{2}+\frac{1}{\kappa^{2}}[p\cdot k-(k\cdot u)(p\cdot
u)]^{2}\right\\}\frac{f_{a0}^{\prime}}{k\cdot p}\,,$ $\displaystyle D_{a}$
$\displaystyle=$ $\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}\\{(k\cdot
u)(p\cdot u)^{2}-(p\cdot u)(k\cdot p)\\}\frac{f_{a0}^{\prime}}{k\cdot p}\,.$
(3.24)

We consider the evaluation of the integrals for $C_{a}$ and $D_{a}$ in the
medium’s rest frame, and thus we set

$u^{\mu}=(1,\vec{0})\,.$ (3.25)

We decompose $k^{\mu}$ in the form

$k^{\mu}=(\omega,\vec{\kappa})\,,$ (3.26)

and in the integrands we will write

$p^{\mu}=(E_{ap},\vec{p})\,,$ (3.27)

with the understanding that we are working in the rest frame of the medium. In
particular $f_{a0}$ is a function only of $E_{ap}$, and

$f_{a0}^{\prime}=\frac{\partial f_{a0}}{\partial E_{ap}}\,.$ (3.28)

Thus, for example, the numerator of the integrand for $D_{a}$ in Eq. (3.2)

$(k\cdot u)(p\cdot u)^{2}-(p\cdot u)(k\cdot p)=\omega E^{2}_{ap}-E_{ap}(\omega
E_{ap}-\vec{\kappa}\cdot\vec{p})=E_{ap}\vec{\kappa}\cdot\vec{p}\,,$ (3.29)

while the denominator can be written as

$E_{ap}(\omega-\vec{\kappa}\cdot\vec{v}_{a})\,,$ (3.30)

with

$\vec{v}_{a}=\frac{\vec{p}}{E_{ap}}\,.$ (3.31)

Substituting these in the expression for $D_{a}$ in Eq. (3.2),

$D_{a}=\int\frac{d^{3}p}{(2\pi)^{3}}\left[\frac{\vec{\kappa}\cdot\vec{v}_{a}}{\omega-\vec{\kappa}\cdot\vec{v}_{a}}\right]f_{a0}^{\prime}\,,$
(3.32)

and similarly,

$C_{a}=-\frac{1}{2}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\vec{v}^{\,2}_{a\perp}}{\omega-\vec{\kappa}\cdot\vec{v}_{a}}f_{a0}^{\prime}\,,$
(3.33)

where

$\vec{v}_{a\perp}=\vec{v}_{a}-\frac{1}{\kappa^{2}}(\vec{\kappa}\cdot\vec{v}_{a})\vec{\kappa}\,.$
(3.34)

Although we do not indicate it explicitly, it is understood that in these
formulas for $C_{a}$ and $D_{a}$, as well as in the formulas for $\pi_{L,T,P}$
given below, the singularity of the integrand at
$\omega=\vec{\kappa}\cdot\vec{v}_{a}$ is to be handled by making the
replacement

$\omega\rightarrow\omega+i0^{+}\,,$ (3.35)

as usual.

From Eq. (3.2) we then obtain

$\displaystyle\pi_{L}$ $\displaystyle=$
$\displaystyle-\frac{k^{2}}{\kappa^{2}}\sum_{a}q_{a}^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\left[\frac{\vec{\kappa}\cdot\vec{v}_{a}}{\omega-\vec{\kappa}\cdot\vec{v}_{a}}\right]f_{a0}^{\prime}\,,$
$\displaystyle\pi_{T}$ $\displaystyle=$
$\displaystyle-\frac{\omega}{2}\sum_{a}q^{2}_{a}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\vec{v}^{\,2}_{a\perp}}{\omega-\vec{\kappa}\cdot\vec{v}_{a}}f_{a0}^{\prime}\,,$
$\displaystyle\pi_{P}$ $\displaystyle=$
$\displaystyle-\frac{i\kappa}{2}\sum_{a}q_{a}\gamma_{a}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\vec{v}^{\,2}_{a\perp}}{\omega-\vec{\kappa}\cdot\vec{v}_{a}}f_{a0}^{\prime}\,.$
(3.36)

The corresponding expressions for $\epsilon_{\ell,t}$ obtained from Eq. (A.23)
reproduce the standard classic results, e.g.,

$\epsilon_{\ell}-1=\frac{1}{\kappa^{2}}\sum_{a}q_{a}^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\left[\frac{\vec{\kappa}\cdot\vec{v}_{a}}{\omega-\vec{\kappa}\cdot\vec{v}_{a}}\right]f_{a0}^{\prime}\,,$
(3.37)

and similarly for $\epsilon_{t}$. On the other hand, the formula for
$\pi_{P}$, and the corresponding formula for $\epsilon_{p}$ obtained from Eq.
(A.23), are new.

##  4 Discussion

###  4.1 Dispersion relations in the long-wavelength limit

The longitudinal dispersion relation is not affected by the $\gamma_{a}$
terms, as already indicated. Therefore, we focus on the transverse ones (i.e.,
polarizations perpendicular to $\vec{\kappa}$) which involve both $\pi_{T}$
and $\pi_{P}$. Moreover, we consider specifically the long-wavelength limit,

$\omega\gg\kappa v_{a}\,,$ (4.1)

which is a particularly useful and representative of more general situations.

Using the fact that we are considering the case that the distribution
functions $f_{a0}$ are isotropic, by straightforward manipulation of the
integrand,

$\displaystyle C_{a}(\omega,\kappa\rightarrow 0)$ $\displaystyle=$
$\displaystyle\frac{\Omega^{2}_{a}}{\omega}\,,$ (4.2)

where

$\Omega^{2}_{a}=\int\frac{d^{3}p}{(2\pi)^{3}E_{ap}}\left(1-\frac{v^{2}_{a}}{3}\right)f_{a0}\,.$
(4.3)

For reference, recall that the plasma frequency ($\omega_{pl,a})$ of each
specie is given by

$\omega^{2}_{pl,a}=q^{2}_{a}\Omega^{2}_{a}\,.$ (4.4)

For example, in the nonrelativistic limit,

$\Omega^{2}_{a}=\frac{n_{a0}}{m_{a}}\,,$ (4.5)

where $n_{a0}$ is the equilibrium particle number density of the specie [see
Eq. (2.10)]. From Eq. (3.2) we then obtain in the long-wavelength limit

$\pi_{P}(\omega,\kappa\rightarrow 0)=\frac{i\kappa\gamma_{P}}{\omega}\,,$
(4.6)

and the well-known result

$\displaystyle\pi_{T}(\omega,\kappa\rightarrow 0)$ $\displaystyle=$
$\displaystyle\Omega^{2}_{0}\,,$ (4.7)

where

$\Omega^{2}_{0}=\sum_{a}q^{2}_{a}\Omega^{2}_{a}\,,$ (4.8)

and

$\gamma_{P}=\sum_{a}q_{a}\gamma_{a}\Omega^{2}_{a}\,.$ (4.9)

As reviewed in Appendix A, the transverse dispersion relations are determined
as the solutions of

$k^{2}-(\pi_{T}+\lambda\pi_{P})=0\,,\qquad(\lambda=\pm)\,.$ (4.10)

Substituting in this equation the results for $\pi_{P}$ and $\pi_{T}$ given in
Eqs. (4.6) and (4.7), respectively, in the long-wavelength limit the equation
becomes

$\omega^{2}-(\kappa^{2}+\Omega^{2}_{0})-\frac{i\lambda\kappa\gamma_{P}}{\omega}=0\,.$
(4.11)

In the limit $\gamma_{P}\rightarrow 0$ we obtain the standard transverse
solutions,

$\omega=\omega_{T}(\kappa)\equiv\sqrt{\kappa^{2}+\Omega^{2}_{0}}\,,$ (4.12)

for either polarization. In the more general situation, assuming

$|\gamma_{P}|\ll\frac{2\omega^{3}_{T}(\kappa)}{\kappa}\,,$ (4.13)

the solutions are

$\omega(\kappa)=\omega_{T}(\kappa)+i\frac{\lambda\kappa\gamma_{P}}{2\omega^{2}_{T}(\kappa)}\,.$
(4.14)

The dispersion relations are such that one polarization mode is damped
(absorption by the medium) while the other one grows (emission by the medium).
Which is one or the other depends on the sign of $\gamma_{P}$, which in turn
depends on the relative signs and values of the $q_{a}\gamma_{a}$ terms in Eq.
(4.9).

###  4.2 Generalization

Equation (2.1) is probably the simplest equation of the kind we are
discussing, but there are some possible generalizations. Here we mention some
of them. We will write them in the generic form

$p\cdot\partial^{(x)}f_{a}=[-q_{a}F^{\mu\nu}+\Gamma_{a}^{\mu\nu}]p_{\nu}\partial^{(p)}_{\mu}f_{a}\,.$
(4.15)

The requirement is that $\Gamma^{\mu\nu}$ must contain, in some form, the dual
tensor $\tilde{F}^{\alpha\beta}$, and that it is antisymmetric in $\mu,\nu$ so
that the proof of current conservation given in Section 2.1 applies in this
case as well, with the identification of
$A^{\mu\nu}=-q_{a}F^{\mu\nu}+\Gamma^{\mu\nu}$ in place of Eq. (2.13).

One possibility is

$\Gamma_{a}^{\mu\nu}=\gamma^{(1)}_{a}(u\cdot\partial^{(x)})\tilde{F}^{\mu\nu}\,,$
(4.16)

where $\gamma^{(1)}_{a}$ is a constant parameter. The steps to arrive at Eq.
(3.9) for $\hat{f}_{a1}$ apply also in this case, with the identification

$\gamma_{a}=-i(k\cdot u)\gamma^{(1)}_{a}\,.$ (4.17)

Thus for example, from Eqs. (4.6) and (4.9), we can see that in the long
wavelength limit this gives a contribution to $\pi_{P}$ of the form

$\kappa q_{a}\gamma^{(1)}_{a}\Omega^{2}_{a}\,.$ (4.18)

The main qualitative difference, relative to the case in which $\gamma_{a}$ is
independent of $\omega$, is that in the present case the corresponding
contribution to $\pi_{P}$ is real, which in turn produces a real term in the
dispersion relations of opposite sign for the $(\pm)$ polarizations.

More generally, we can consider the equation

$\Gamma_{a}^{\mu\nu}=\int\,d^{4}x^{\prime}\Gamma_{a}(x-x^{\prime})\tilde{F}^{\mu\nu}(x^{\prime})\,.$
(4.19)

The momentum space equation in the linear approximation for $\hat{f}_{a1}$ is
again Eq. (3.9), but with $\gamma_{a}$ being the Fourier transform of
$\Gamma_{a}(x-x^{\prime})$, that is, $\gamma_{a}$ defined by writing

$\Gamma_{a}(x-x^{\prime})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik\cdot(x-x^{\prime})}\gamma_{a}(k)\,.$
(4.20)

The fact that $\Gamma_{a}(x-x^{\prime})$ is real implies that

$\gamma_{a}^{\ast}(k)=\left.\gamma_{a}(k)\right|_{k\rightarrow-k}\,.$ (4.21)

Being a scalar, $\gamma_{a}$ is a function of the scalar variables $\omega$
and $\kappa$ defined in Eq. (A.3), a fact that we indicate by writing it as
$\gamma_{a}(\omega,\kappa)$ when needed. In particular, Eq. (4.21) actually
implies the condition

$\gamma_{a}^{\ast}(\omega,\kappa)=\gamma_{a}(-\omega,\kappa)\,.$ (4.22)

This case includes the original Eq. (2.1) (constant $\gamma_{a}$) as well as
Eq. (4.16) as special cases, and of course Eq. (4.17) is consistent with Eq.
(4.22), as it should be.

###  4.3 Neutrino background

An example case in which $\gamma_{a}$ has the form given in Eq. (4.17) is
afforded by an electron plasma with a neutrino background. The calculation of
$\pi_{P}$ in that case gives[3]

$\pi_{P}(\omega,\kappa\rightarrow
0)=\frac{\sqrt{2}G_{F}\alpha}{3\pi}\left(\frac{\omega^{2}_{pl,e}}{m^{2}_{e}}\right)(n_{\nu_{e}}-n_{\bar{\nu}_{e}})\kappa\,,$
(4.23)

where $n_{\nu_{e}}$ and $n_{\bar{\nu}_{e}}$ stand for the number densities of
the electron neutrinos and antineutrinos, and $\omega_{pl,e}$ is the electron
plasma frequency. For simplicity let us consider the nonrelativistic limit, so
that only the electrons (no positrons) are present, in which case (i.e., Eqs.
(4.4) and (4.5))

$\omega^{2}_{pl,e}=q^{2}_{e}\Omega^{2}_{e}=\frac{q^{2}_{e}n_{e0}}{m_{e}}\,,$
(4.24)

where $q_{e}$ is the electron charge and $n_{e0}$ is the equilibrium electron
number density. On the other hand, for this case that we are considering, in
the framework of the kinetic equation

$\pi_{P}=\frac{i\kappa q_{e}\gamma_{e}\Omega^{2}_{e}}{\omega}\,.$ (4.25)

Therefore, in the framework of the kinetic equation, the effects of the
neutrino background can be parametrized in terms of a $\gamma_{e}$ parameter
for the electron of the form

$\gamma_{e}=-i\omega\gamma^{(1)}_{e}\,,$ (4.26)

with

$\gamma^{(1)}_{e}=q_{e}\frac{\sqrt{2}G_{F}\alpha}{3\pi
m^{2}_{e}}(n_{\nu_{e}}-n_{\bar{\nu}_{e}})\,.$ (4.27)

The main lesson here is that the kinetic approach allows us to parametrize the
effects produced by the $\pi_{P}$ term in the photon polarization tensor in
terms of the parameter $\gamma_{a}$. In this framework, $\gamma_{a}$ is a
phenomenological parameter that must be determined by other means, e.g.,
thermal field theory in the case of an electron plasma with neutrino
background, as we have seen. Nevertheless, the kinetic approach allows us to
study further the consequences of the presence of the $\pi_{P}$ term, such as
the effects of external fields[10], streaming neutrino background[5] or
collisional plasmas[9], among others.

##  5 Conclusions and Outlook

In this work we have proposed a method to study the propagation of photons in
an optically active isotropic medium, based on the covariant collisionless
Boltzmann equation. As shown in Section 2, the covariant Boltzmann equation
can be modified by adding a term that gives rise to the optical activity
effects, in a way that is consistent with the general requirements of current
conservation and symmetry considerations. In Section 3, using the linear
response method, we obtained an expression for the induced current, expressed
in terms of integrals over the momentum distribution function of the
background particles. There we established contact with the photon
polarization tensor, or equivalently the photon self-energy, in the TFT
language, specifically as used in Ref. [1], and the formulas for the
components of the polarization tensor were obtained. The results for the
longitudinal and transverse components of the photon self-energy, $\pi_{L,T}$
respectively, are the familiar ones. The new result here is the formula for
the $P$\- and $CP$-breaking component $\pi_{P}$ due to the new term we
considered in the Boltzmann equation to describe the effects of optical
activity. In Section 4 we discussed some details of the results obtained, and
considered specifically some particular cases (e.g., the long-wavelength and
the nonrelativistic limit) that are useful in practical applications and
representative of more general situations. To emphasize the usefulness of the
method, we pointed out how the $P$\- and $CP$-breaking term we included in the
Boltzmann equation can be generalized, in particular how it applies to a
plasma with a neutrino gas as a background, and established contact with
calculations of the photon self-energy in such contexts in the framework of
TFT[3]. The strength and advantages of the method here presented comes from
its semiclassical standpoint, which in many circumstances is more suitable
than the thermal field theory approach for incorporating other potentially
important effects such as collisions, external fields, stream backgrounds and
multicomponent plasmas.

##  Appendix A Notation and conventions

We use the notation and conventions used in Ref. [1], which we briefly review
here for convenience. The momentum of the propagating photon is denoted by
$k^{\mu}$, and $u^{\mu}$ is the velocity four-vector of the medium, already
introduced in Eq. (2.5).

###  A.1 Photon polarization tensor

In the context of TFT, the photon self-energy, $\pi_{\mu\nu}$ gives rise to a
contribution to the effective Lagrangian of the photon and the corresponding
field equation that leads to identify

$j_{\mu}=-\pi_{\mu\nu}A^{\nu}\,,$ (A.1)

as the induced current in the presence in the external field, and whence
$\pi_{\mu\nu}$ as the polarization tensor. As discussed in that reference, the
most general form of $\pi_{\mu\nu}$ in an isotropic medium is

$\pi_{\mu\nu}(k,u)=\pi_{T}(\omega,\kappa)R_{\mu\nu}(k,u)+\pi_{L}(\omega,\kappa)Q_{\mu\nu}(k,u)+\pi_{P}(\omega,\kappa)P_{\mu\nu}(k,u)\,,$
(A.2)

where $\omega$ and $\kappa$ are the scalar variables

$\omega=k\cdot u\,,\qquad\kappa=(\omega^{2}-k^{2})^{\frac{1}{2}}\,,$ (A.3)

which have the interpretation of the energy and the magnitude of the momentum
of the photon, in the rest frame of the medium. The tensors $R,Q,P$ are
defined as follows. First, the component of $u^{\mu}$ transverse to $k^{\mu}$
is

$\tilde{u}_{\mu}=\tilde{g}_{\mu\nu}u^{\nu}\,,$ (A.4)

where

$\tilde{g}_{\mu\nu}=g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\,.$ (A.5)

Then,

$\displaystyle Q_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\tilde{u}_{\mu}\tilde{u}_{\nu}}{\tilde{u}^{2}}\,,$
$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{g}_{\mu\nu}-Q_{\mu\nu}\,,$ $\displaystyle P_{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{i}{\kappa}\epsilon_{\mu\nu\alpha\beta}k^{\alpha}u^{\beta}\,.$
(A.6)

It is useful to remember that all three tensors are transverse to $k^{\mu}$,
that is

$k^{\mu}T_{\mu\nu}=0=k^{\nu}T_{\mu\nu}\,;\quad(T=R,Q,P)\,,$ (A.7)

and also that $R$ and $P$ are transverse to $u^{\mu}$ as well,

$u^{\mu}T_{\mu\nu}=0=u^{\nu}T_{\mu\nu}\,;\quad(T=R,P)\,.$ (A.8)

They satisfy various product relations, among them

$R^{\mu\nu}R_{\mu\nu}={R^{\mu}}_{\mu}=2\,,\quad
Q^{\mu\nu}Q_{\mu\nu}={Q^{\mu}}_{\mu}=1\,,\quad P^{\mu\nu}P_{\mu\nu}=-2\,,$
(A.9)

and

$R^{\mu\lambda}Q_{\lambda\nu}=0\,,\quad Q^{\mu\lambda}P_{\lambda\nu}=0\,,\quad
P^{\mu\lambda}R_{\lambda\nu}={P^{\mu}}_{\nu}\,,\quad
P^{\mu\lambda}P_{\lambda\nu}={R^{\mu}}_{\nu}\,.$ (A.10)

In the rest frame of the medium, the components of $R_{\mu\nu}$ and
$P_{\mu\nu}$ are,

$\displaystyle R_{00}=R_{0i}=R_{i0}=0\,,$ $\displaystyle\qquad
R_{ij}=\delta_{ij}+\frac{\kappa_{i}\kappa_{j}}{\kappa^{2}}\,,$ $\displaystyle
P_{00}=P_{0i}=P_{i0}=0\,,$ $\displaystyle\qquad
P_{ij}=\frac{i}{\kappa}\epsilon_{ijk}\kappa^{k}\,.$ (A.11)

###  A.2 Dispersion relations

The equation $\partial^{\mu}F_{\mu\nu}=j_{\nu}$, in momentum space becomes

$[(k^{2}-\pi_{T})R_{\mu\nu}+(k^{2}-\pi_{L})Q_{\mu\nu}-\pi_{P}P_{\mu\nu}]A^{\nu}=0\,,$
(A.12)

which determines the dispersion relations and polarization vectors of the
propagating modes. To discuss them we recall the definition of the transverse
vectors $e^{\mu}_{1,2}$, which in the rest frame of the medium have components

$e^{\mu}_{1,2}=(0,\vec{e}_{1,2})\,,$ (A.13)

where $\vec{e}_{1,2}$ are unit vectors with

$\vec{e}_{1,2}\cdot\hat{\kappa}=0\,,\quad\vec{e}_{2}=\hat{\kappa}\times\vec{e}_{1}\,.$
(A.14)

In covariant form, they satisfy

$R_{\mu\nu}e^{\nu}_{a}=e_{a\mu}\,,\quad
Q_{\mu\nu}e^{\nu}_{a}=0\,,\qquad(a=1,2)\,,$ (A.15)

and

$e^{\mu}_{2}=-iP^{\mu\nu}e_{1\nu}\,,\quad e^{\mu}_{1}=iP^{\mu\nu}e_{2\nu}\,.$
(A.16)

In addition it is useful to introduce

$e^{\mu}_{3}=\frac{\tilde{u}^{\mu}}{\sqrt{-\tilde{u}^{2}}}\,,$ (A.17)

which together with $e^{\mu}_{1,2}$ form a basis in the subspace orthogonal to
$k^{\mu}$.

From the fact that $R^{\mu\nu}$ and $P^{\mu\nu}$ acting on $\tilde{u}_{\nu}$
give zero, it follows that $A^{\mu}\sim e^{\mu}_{3}$ is a solution of Eq.
(A.12) provided

$k^{2}-\pi_{L}=0\,,$ (A.18)

which is the equation for the dispersion relation $\omega_{L}(\kappa)$ for the
longitudinal mode. Since the presence of the $\gamma_{a}$ term does not affect
$\pi_{L}$, the dispersion relation for the longitudinal mode is not affected.

In the absence of the $\pi_{P}$ term, Eq. (A.15) implies that $A^{\mu}\sim
e^{\mu}_{1,2}$, or any combination of them, give a solution of Eq. (A.12) if

$k^{2}-\pi_{T}=0\,,$ (A.19)

which gives the dispersion relation $\omega_{T}(\kappa)$, The transverse
modes, corresponding to the polarization vectors $e^{\mu}_{1,2}$ are therefore
degenerate, with the same dispersion relation $\omega_{T}(\kappa)$.

As a consequence of the relations in Eq. (A.16), neither $A^{\mu}\sim
e^{\mu}_{1}$ nor $A^{\mu}\sim e^{\mu}_{2}$ are separately solutions of the
equation. At this point it is useful to introduce the circular polarization
vectors

$e^{(\pm)\mu}=\frac{1}{\sqrt{2}}(e^{\mu}_{1}\pm ie^{\mu}_{2})\,,$ (A.20)

which satisfy

$P^{\mu\nu}e^{(\lambda)}_{\nu}=\lambda e^{(\lambda)\mu}\,,\quad\lambda=\pm\,,$
(A.21)

in addition to identities analogous to Eq. (A.15). It then follows that
$A^{\mu}\sim e^{(\pm)\mu}$ are each a solution of the equation, with the
corresponding dispersion relation being the solution of

$k^{2}-(\pi_{T}+\lambda\pi_{P})=0\,.$ (A.22)

###  A.3 Dielectric tensor

An equivalent way to express the presence of the $\pi_{P}$ term in the photon
self-energy is in terms of the components of the dielectric tensor, which are
given by[12]

$1-\epsilon_{t}=\pi_{T}/\omega^{2},\quad
1-\epsilon_{\ell}=\pi_{L}/k^{2},\quad\epsilon_{p}=\pi_{P}/\omega^{2}\,.$
(A.23)

The interpretation is that, in the rest frame of the medium, the induced
current vector is given by

$\vec{j}=i\omega[(1-\epsilon_{\ell})\vec{E}_{\ell}+(1-\epsilon_{t})\vec{E}_{t}-i\epsilon_{p}\hat{\kappa}\times\vec{E}]\,,$
(A.24)

where we are writing, in that frame,

$k^{\mu}=(\omega,\vec{\kappa})\,,$ (A.25)

while $\vec{E}_{\ell}$ and $\vec{E}_{t}$ denote the components of the electric
field parallel and transverse to $\vec{\kappa}$, respectively. The
$\epsilon_{p}$ term breaks the degeneracy between the two transverse
polarization states of the propagating photon.

## References

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