Positivity bounds on electromagnetic properties of media

Paolo<EMAIL_ADDRESS>Oliver
<EMAIL_ADDRESS>Borna<EMAIL_ADDRESS>and
Leonardo<EMAIL_ADDRESS>

1ICTP, International Centre for Theoretical Physics, 34151 Trieste, Italy

2IFPU, Institute for Fundamental Physics of the Universe, 34014 Trieste, Italy

3Laboratory for Theoretical Fundamental Physics, EPFL, 1015 Lausanne,
Switzerland

4Institut für Theoretische Physik, ETH Zürich, 8093 Zürich, Switzerland

Abstract
We study the constraints imposed on the electromagnetic response of general
media by microcausality (commutators of local fields vanish outside the light
cone) and positivity of the imaginary parts (the medium can only absorb energy
from the external field). The equations of motion for the average
electromagnetic field in a medium – the macroscopic Maxwell equations – can be
derived from the in-in effective action and the effect of the medium is
encoded in the electric and magnetic permeabilities
$\varepsilon(\omega,|\boldsymbol{k}|)$ and $\mu(\omega,|\boldsymbol{k}|)$.
Microcausality implies analyticity of the retarded Green’s functions when the
imaginary part of the $4$-vector $(\omega,\boldsymbol{k})$ lies in forward
light cone. With appropriate assumptions about the behavior of the medium at
high frequencies one derives dispersion relations, originally studied by
Leontovich. In the case of dielectrics these relations, combined with the
positivity of the imaginary parts, imply bounds on the low-energy values of
the response, $\varepsilon(0,0)$ and $\mu(0,0)$. In particular the quantities
$\varepsilon(0,0)-1$ and $\varepsilon(0,0)-1/\mu(0,0)$ are constrained to be
positive and equal to integrals over the imaginary parts of the response. We
discuss various improvements of these bounds in the case of non-relativistic
media and with additional assumptions about the UV behavior.

###### Contents

  1. 1 Introduction
  2. 2 Effective Maxwell equations in matter
  3. 3 Positivity and dissipation
  4. 4 Causality and analyticity
  5. 5 High-energy behavior
  6. 6 Bounds on the low-energy behavior
  7. 7 Analyticity of $\Pi^{\mu\nu}$
  8. 8 Examples and improvements
  9. 9 Conclusions and future directions
  10. A Review of the Closed Time Path formalism
  11. B Linear response
  12. C Lindhard function
  13. D The absence of zeros
  14. E Damped wave response

##  1 Introduction

The coefficients of operators in an effective field theory (EFT) are dictated
by symmetries and by the requirement of a healthy low-energy theory. More
interestingly, other constraints, not readily discernible from the low-energy
regime, can be obtained by making general assumptions about the theory and its
UV behavior. Under the mild assumptions that the UV completion is Lorentz-
invariant, local and unitary, one can derive various inequalities that the
low-energy coefficients must satisfy. The crucial link between low-energy
properties and the UV completion lies in the analyticity of the $S$-matrix,
which ultimately stems from microcausality – the property that local operators
commute outside the light cone. Cauchy’s theorem allows one to relate low-
energy and high-energy contours in the complex plane. This, together with the
positivity of the imaginary parts of the amplitude implied by the optical
theorem, establishes bounds on the low-energy coefficients. Ref. [1] put
forward these so-called “positivity bounds” and the general idea that “not
everything goes” once one has made assumptions about the UV. This has been
generalized in many subsequent papers that aim at separating EFT’s that have a
conventional UV completion from those that do not (see the recent [2, 3, 4, 5,
6, 7] and references therein). These ideas have been used to constrain, for
instance, higher-dimension operators in the Standard Model EFT (e.g. [8]),
massive gravity (e.g. [9]) and operators that correct General Relativity (e.g.
[10]).

While this program has progressed under the assumption of (linearly realized)
Lorentz invariance – a natural assumption in particle physics – a significant
portion of physics explores systems in which Lorentz invariance is
spontaneously broken. In cosmology, a preferred reference frame always exists
– the one comoving with the energy density – resulting in the breaking of
Lorentz invariance in EFT’s describing phenomena such as inflation or dark
energy. Condensed matter, almost by definition, constitutes a field of study
characterized by the spontaneous breaking of Lorentz invariance. Other
examples include field theories at finite temperature and/or chemical
potential, like the LHC quark-gluon plasma and ordinary fluids, or EFT’s
describing localized objects such as black holes or defects. Clearly, it would
be valuable to derive bounds on EFT’s without relying on Lorentz invariance.

Without Lorentz invariance, generalizing the arguments above is not
straightforward (attempts in this direction include [11, 12, 13]). The issue
is that, in general, the $S$-matrix cannot serve as the bridge between UV and
IR. The $S$-matrix that describes the scattering of low-energy excitations
might not even exist at high energy [14, 15, 16]. In the absence of Lorentz
invariance states cannot be boosted and, generally, low-energy excitations
only exist up to a certain energy. Additionally, excitations are typically
unstable so that the $S$-matrix is, even at low energies, an approximate
concept at best. One can restrict to situations in which the low-energy
excitations survive in the UV and are sufficiently stable so that the
$S$-matrix exists at low and high energies. A simple example of this situation
has been studied in [15, 16]. Unfortunately, the conclusion is that the
$S$-matrix, in the absence of Lorentz invariance, does not enjoy the analytic
properties that allowed one to connect UV and IR in the presence of Lorentz
invariance. The relation between the $S$-matrix and correlation functions is
non-local and microcausality is not sufficient to derive nice properties for
the scattering amplitude, like in the Lorentz-invariant case. Without
analyticity connecting the UV and the IR becomes challenging, and drawing
general conclusions seems difficult.

These difficulties with the $S$-matrix suggest to go back and study a simpler
object, in which microcausality is manifest: the (retarded) two-point
function. In the absence of Lorentz invariance, but preserving rotations and
spacetime translations, the two-point function is a rich object; a function of
two variables, $\omega$ and $|\boldsymbol{k}|$, similar to the $S$-matrix in
the Lorentz-invariant case. Using the analytic properties of the two-point
function of conserved currents and assuming the theory reaches a conformal
fixed point in the UV, in [14] it was possible to derive positivity bounds on
the operators that describe a conformal superfluid.

The possibility of deriving positivity bounds in the absence of Lorentz
invariance should not come as a surprise. Indeed the use of analyticity and
dispersion relations to connect UV with the IR goes back to Kramers and Kronig
[17, 18] in their study of the electromagnetic response of media, in which
Lorentz invariance is clearly broken. In this paper we wish to revisit this
old problem, focussing on the case of dielectrics, from the more modern point
of view of setting bounds on non-Lorentz-invariant EFT’s.

Returning to this well-studied problem (see [19] for an extended review)
requires humility and a clarification of motivation. Firstly, to the best of
our knowledge, some of the results, notably the dispersion relations (6.6) and
(7.9) on the dielectric permittivity $\varepsilon$ and magnetic permeability
$\mu$ at low frequency and momentum, as well as the summary in Fig. 4, are
new. Secondly, we wish to describe the problem as an example of settings
bounds on Lorentz-breaking EFT’s: the assumptions and the language used in the
condensed matter community are not immediately extendable to more general
setups. For instance, the assumptions about the UV behavior of the system –
and even the very definition of UV – are markedly different for a normal
medium and, say, an EFT that describes cosmic inflation. The derivation of
“macroscopic” Maxwell equations in terms of the in-in effective action should
facilitate the extension to other contexts. We sense a similarity with the
Lorentz-invariant case: while dispersion relations for the $S$-matrix were
routinely used since the ‘60s, their application to constrain EFT’s is much
more recent and required a change in perspective.

The Maxwell equations in a medium describe the evolution of the average
electromagnetic field in a material and they can be derived using the in-in
effective action formalism. We present this derivation in §2, deferring some
background material to Apps. A and B. The response of the medium can be
parametrized in terms of $\varepsilon(\omega,|\boldsymbol{k}|)$ and
$\mu(\omega,|\boldsymbol{k}|)$, where both the electric and magnetic responses
depend both on frequency and wave-vector.111In specific cases it may be a good
approximation to neglect the dependence of $\varepsilon$ and $\mu$ on
$|\boldsymbol{k}|$, keeping only the one on $\omega$. For instance, in the
interaction of visible light with standard matter, the wavelength of the
photon is $\alpha^{-1}\simeq 137$ times longer than the atomic size, so that
one can approximate the response as local in space. $S$-matrix positivity
bounds always stem from the fact that the imaginary part of the amplitude has
a definite sign. In the present context, the analogous statement is that the
medium can only absorb and not emit energy when perturbed by an external
electromagnetic field; we discuss this assumption in §3. As we argued above,
the crucial ingredient to relate UV and IR is analyticity. In §4, we study the
domain of analyticity of the photon propagator in the medium: as a consequence
of microcausality, the propagator is analytic when the imaginary part of the
four-vector $(\omega,\boldsymbol{k})$ lies in the forward light cone. This
generalizes the textbook statement that a retarded Green’s function must be
analytic in the upper half-plane of complex $\omega$, which is a consequence
of retardation only. Microcausality is clearly more restrictive and,
correspondingly, one can generalize the celebrated Kramers-Kronig relations to
a one-parameter family of relations, first derived by M. Leontovich [20].
These relations are a general property of linear response theory (see for
example [21] for an introduction to the subject) and, as such, they should
perhaps receive more attention. They are crucial to constrain the
electromagnetic response of a medium: the Kramers-Kronig relations only
reflect causation, allowing for immediate response at a distance, while, as we
will see, bounds on the magnetic response require microcausality.

Leontovich relations, like the Kramers-Kronig ones, require assumptions about
the UV behavior. This is the topic of §5. For a concrete model that describes
the UV limit of the material, we study in App. C the case of degenerate
fermions: the so-called Lindhard function and its relativistic generalization.
Notice that our study is not confined to condensed matter: for instance one
could have a medium of nuclear matter in which the charged constituents are in
relativistic motion. Microcausality (Leontovich relations) together with the
assumption of a “passive” medium (sign-definiteness of imaginary parts) can be
combined to give bounds on the low-energy and momentum limit of $\varepsilon$
and $\mu$ (§6). (Notice that this low-energy limit exists since we confine our
analysis to dielectric materials; one would have divergences when studying
conductors or superconductors.) We are unable to prove that the one-particle
irreducible (1PI) self-energy, the object that directly appears in the
macroscopic Maxwell equations, is analytic in the same region as the photon
propagator (§7 and App. D), unless some further physical assumption is added.
We are however able to prove a more limited result regarding its domain of
analyticity, which is sufficient to re-derive the low-energy bounds on
$\varepsilon$ and $\mu$ via another route. Stronger bounds can be derived by
making further assumptions (§8). One way is to set a lower bound on
dissipation, in the same way one does in the $S$-matrix program where some
knowledge on the total cross-section sharpens the low-energy bounds. Another
possible assumption is that the medium is non-relativistic, so that its
response is actually confined in a smaller region compared to what is allowed
by relativistic microcausality. We discuss the higher order terms in the low
energy expansion of the response functions in §9, where we conclude the paper
with many open generalizations of these methods.

### Notation

We work with mostly plus metric signature $(-+++)$. Our convention for Fourier
transform is

$f(p)=\int\differential[4]{x}e^{-ip\cdot x}f(x)\,,\qquad
f(x)=\int\frac{\differential[4]{p}}{(2\pi)^{4}}e^{ip\cdot x}f(p)\,.$

##  2 Effective Maxwell equations in matter

The electromagnetic dynamics in matter is governed by the following action,

$S=S_{\gamma}[a]+S_{M}\big{[}a;\psi\big{]}\,.$ (2.1)

The first part is the free photon action

$S_{\gamma}=-\frac{1}{4g^{2}}\int\differential[4]{x}f_{\mu\nu}^{2}\,,$ (2.2)

with $a_{\mu}$ the vector field and
$f_{\mu\nu}={\partial}_{\mu}a_{\nu}-{\partial}_{\nu}a_{\mu}$ the field
strength, and we have pulled out the dimensionless coupling $g$. The second
term in Eq. 2.1 represents the dynamics of matter in which all matter fields
are collectively denoted by $\psi$. In what follows, we will not need to
specify the precise form of $S_{M}$ apart from some minimal assumptions (for
instance on the high-energy behavior of response functions, see §5). For
massless spin-one fields we must have gauge symmetry, i.e. $a_{\mu}\to
a_{\mu}+{\partial}_{\mu}\Lambda$ and $\psi\to\psi\,e^{iq\Lambda}$ for a charge
$q$. Most of the time we will suppress dependences on the matter fields.

The action Eq. 2.1 specifies the dynamics but we still need to determine the
state. Unlike the usual situation in high energy physics where the vacuum
state and its excitations are studied, here we are interested in a many-body
state described by a density matrix $\rho$. We assume that the system without
external perturbations is in equilibrium, therefore $\rho$ commutes with the
Hamiltonian.222In some cases, for example superfluids, a combination of time
translation and an internal symmetry is broken into a diagonal subgroup. In
these situations, we call the unbroken symmetry “time translation”. We thank
A. Podo for this comment. Moreover, we assume that the system is homogeneous
therefore it also commutes with the spatial components of the momentum
operator. Notice that a generic density matrix, with finite average energy
density, breaks Lorentz boosts.333In fact, Lorentz boosts mix different energy
states in the expansion $\rho=\sum_{n}c_{n}\ket{n}\bra{n}$ given below and
does not commute with the density matrix. By contrast, we will assume that
rotation is a good symmetry of the system.

A generic density matrix of this type can be written as
$\rho=\sum_{n}c_{n}\ket{n}\bra{n}$ in terms of the eigenstates of the four-
momentum operator $P^{\mu}\ket{n}=p_{n}^{\mu}\ket{n}$ and $c_{n}$ non-negative
numbers satisfying $\sum_{n}c_{n}=1$. The discussion will not crucially depend
on the form of the density matrix but an example to have in mind is the grand
canonical ensemble $\rho=\exp(-\beta(H-\mu Q))/Z$, with
$Z=\Tr\exp(-\beta(H-\mu Q))$, $\beta$ the inverse temperature, $H$ the
Hamiltonian, $\mu$ the chemical potential and $Q$ a $U(1)$ charge. In
covariant form $\rho\propto\exp(\beta u_{\mu}(P^{\mu}-\mu J^{\mu}))$ with
$u^{\mu}$ the normalized ($u_{\mu}u^{\mu}=-1$) velocity of the medium and
$J^{\mu}$ the conserved current. In the rest frame of the medium
$u^{\mu}=(1,\boldsymbol{0})$.

### Effective action

We are interested in studying the evolution of the average electromagnetic
field in the system, i.e.
$F_{\mu\nu}\equiv\expectationvalue{f_{\mu\nu}}=\Tr(\rho f_{\mu\nu})$. We
emphasize that average fields are defined as an ensemble average rather than
spatial average (as for instance in [22]) since the latter are not well-
defined at arbitrarily high energies. We assume that the amplitude of
macroscopic fields is small and therefore that it is sufficient to consider
the equation of motion linear in the fields – equivalently the action is
quadratic. However, the coupling to matter is not necessarily weak and
therefore for most of the discussion we avoid performing any perturbative
expansion in the matter sector.

For the purpose of studying the evolution of the average electromagnetic
fields taking into account the presence of matter, the natural object to study
is the Closed Time Path (CTP) effective action (also called Schwinger-Keldysh
or in-in effective action). Varying the effective action gives the equation of
motion for the average fields. We require the in-in, as opposed to in-out,
effective action since it is only the former that describes a causal equation
of motion. For instance, the equation of motion following from the in-out
effective action can have complex solutions for real boundary conditions [23].
Moreover, dissipation cannot be described in the usual in-out formalism. We
review these concepts in App. A.

In the following we obtain an expression for the CTP effective action
(hereafter simply called effective action) up to quadratic order using the
background field method. As explained in App. A, this is a straightforward
extension of the standard background field method to the CTP formalism. The
purpose of this calculation is to obtain Maxwell’s equations in matter from a
top-down approach. The reader who is not interested in the details can jump
directly to Eq. 2.13.

A path integral representation of the effective action $\Gamma[A_{1},A_{2}]$
is obtained by combining Eq. A.18 and Eq. A.21 as follows:

$e^{i\Gamma[A_{1},A_{2}]}=\int^{\rm CTP}_{\rho,{\rm
1PI}}\mathcal{D}a_{1}\mathcal{D}a_{2}\mathcal{D}\psi_{1}\mathcal{D}\psi_{2}\,\exp\bigg(iS[a_{1}+A_{1},\psi_{1}]-iS[a_{2}+A_{2},\psi_{2}]\bigg{missing})\,.$
(2.3)

Here is an explanation of the relation: in the CTP formalism we calculate the
path integral on a time contour going from the initial time to the final time,
taken to be $-\infty$ and $+\infty$ respectively, and then back to the initial
time (see the figure in Eq. A.6). The fields that live on the forward and
backward contours are labeled by indices 1 and 2 respectively. We integrate
over all the field configurations (for the gauge and matter fields) subject to
the boundary condition that at the final time the forward and backward fields
match, i.e. $a_{1}(t_{f})=a_{2}(t_{f})$ and $\psi_{1}(t_{f})=\psi_{2}(t_{f})$
(writing “CTP” over the integral in Eq. 2.3 is a reminder of this final
boundary condition). The initial condition is fixed by the density matrix
$\rho$. To implement the initial condition we multiply the integrand by
$\bra{a_{1}(t_{i}),\psi_{1}(t_{i})}\rho\ket{a_{2}(t_{i}),\psi_{2}(t_{i})}$.
$S[a,\psi]$ is the total action in Eq. 2.1 for the photon and matter. It is
deformed by adding a background $A_{1}$ and $A_{2}$ for the photon.

Finally, the effective action, by construction, only produces 1PI diagrams in
a perturbative expansion, which we are reminded of by the “1PI” index in Eq.
2.3. It means that we can drop from the beginning terms that can only generate
non-1PI diagrams. That is why we have dropped the external current terms
$K_{1,2}$ present in Eq. A.18. We emphasize that by 1PI we mean diagrams that
remain connected after cutting an _internal photon_ line: it could be that the
matter itself has self-interactions for which we must keep all of them even
non-1PI ones.

A few comments about gauge invariance are in order. As is usual in the path
integral approach to gauge theories, we add a gauge fixing term to the action
in Eq. 2.3 and integrate over all components of the gauge fields, i.e.
$\mathcal{D}a_{1}\mathcal{D}a_{2}=\mathcal{D}a_{1}^{\mu}\mathcal{D}a_{2}^{\mu}$.
We choose the gauge fixing term to be $S_{\rm
gf}[a]=-\frac{1}{2}\alpha\int({\partial}_{\mu}a^{\mu})^{2}$ for an arbitrary
coefficient $\alpha$. The exponent in Eq. 2.3 is then
$S_{\gamma}[a_{1}+A_{1}]+S_{M}[a_{1}+A_{1},\psi_{1}]+S_{\rm gf}[a_{1}]$,
similarly for the $a_{2}$ fields, without shifting the gauge fixing part. Any
dependence on the parameter $\alpha$ must drop out in a physical quantity. It
is easy to see that the effective action is invariant under the separate gauge
transformations of the background fields

$\Gamma[A_{1}+{\partial}\Lambda_{1},A_{2}+{\partial}\Lambda_{2}]=\Gamma[A_{1},A_{2}]\,,$
(2.4)

for arbitrary functions $\Lambda_{1}$ and $\Lambda_{2}$ (we assume they vanish
at infinity to make sure the boundary conditions are not altered).

We would like to have an expression for the effective action
$\Gamma[A_{1},A_{2}]$ up to quadratic order in the background fields (but
otherwise non-perturbative). Dropping vertices generating non-1PI terms, we
obtain

$S[a+A,\psi]=S[a,\psi]+S_{\gamma}[A]+\int\differential[4]{x}J^{\mu}(a,\psi)A_{\mu}+\frac{1}{2}\int\differential[4]{x}N^{\mu\nu}(a,\psi)A_{\mu}A_{\nu}+\dots\,,$
(2.5)

in which we have defined

$J^{\mu}\equiv\functionalderivative{S_{M}[a,\psi]}{a_{\mu}(x)}\,,\qquad\qquad
N^{\mu\nu}\delta(x-y)\equiv\functionalderivative{J^{\mu}[a(x),\psi(x)]}{a_{\nu}(y)}\,.$
(2.6)

Both terms can depend on the internal photon fields $a_{\mu}$ and the matter
fields $\psi$. In the definition of $N^{\mu\nu}$ we have assumed that the
current is a local function of the fields. We provide some examples for
clarification. In scalar QED, the relevant part of the matter action is
$S_{M}=\int-|D_{\mu}\psi|^{2}$ with $\psi$ a complex scalar and
$D_{\mu}={\partial}_{\mu}-ia_{\mu}$. Then we have
$J^{\mu}=-i(\psi^{\dagger}D^{\mu}\psi-\psi(D^{\mu}\psi)^{\dagger})$ and
$N^{\mu\nu}=-2|\psi|^{2}\eta^{\mu\nu}$. For fermions, $S_{M}=\int
i\bar{\psi}\not{D}\psi$ with $\psi$ a Dirac spinor. Then we obtain
$J^{\mu}=\bar{\psi}\gamma^{\mu}\psi$ and $N^{\mu\nu}=0$.

Since the $S_{\gamma}[A]$ term only depends on the background field we factor
it out in Eq. 2.3 and write the effective action as follows,

$\Gamma[A_{1},A_{2}]=S_{\gamma}[A_{1}]-S_{\gamma}[A_{2}]+\Gamma_{M}[A_{1},A_{2}]\,,$
(2.7)

in which the last term is the contribution of matter and is given by

$e^{i\Gamma_{M}[A_{1},A_{2}]}=\int_{\rho,{\rm 1PI}}^{\rm
CTP}\mathcal{D}a_{1}\mathcal{D}a_{2}\mathcal{D}\psi_{1}\mathcal{D}\psi_{2}\,\exp(iS[a_{1},\psi_{1}]+i\int
J_{1}^{\mu}A_{1\mu}+\frac{i}{2}\int N_{1}^{\mu\nu}A_{1\mu}A_{1\nu}-\\{1\to
2\\})\,.$ (2.8)

The leading term, corresponding to $A_{1,2}=0$, vanishes due to the
normalization of the effective action. The linear term would be

$i\int^{\rm CTP}_{\rho,{\rm
1PI}}e^{i(S_{1}-S_{2})}\int\differential[4]{x}(J_{1}^{\mu}A_{1\mu}-J_{2}^{\mu}A_{2\mu})=i\int\differential[4]{x}\expectationvalue{J^{\mu}(x)}_{\rm
1PI}(A_{1\mu}-A_{2\mu})\,,$ (2.9)

where we have suppressed the measure of the path integral and
$S_{1}=S[a_{1},\psi_{1}]$, etc. By translation invariance
$\expectationvalue{J^{\mu}}_{\rm 1PI}\propto u^{\mu}$ is independent of the
coordinates and can only be proportional to the $u^{\mu}$, the four-velocity
of the medium. This term is either zero, e.g. for neutral fluids, or it is
canceled by a homogeneous background with opposite charge, e.g. electrons in a
solid.444For electrons in a solid the assumption of homogeneity is not
completely correct since the presence of a lattice of ions breaks spatial
translations to a discrete subgroup. Most often this effect is ignored by
assuming a homogeneous background with the opposite charge without any
dynamics, known as the Jellium model. For a discussion of possible granularity
effects see [19]. We must add a similar term to the effective action

$\int\differential[4]{x}(A_{1\mu}-A_{2\mu})J_{\rm ext}^{\mu}$ (2.10)

to model the external current controlled by the experimentalist, e.g. charges
on a capacitor. The quadratic terms can be written as follows:

$\Gamma_{M}[A_{1},A_{2}]=\frac{1}{2}\int\differential[4]{x}\differential[4]{y}\begin{bmatrix}A_{1\mu}(x)\,\,A_{2\mu}(x)\end{bmatrix}S^{\mu\nu}(x,y)\begin{bmatrix}A_{1\nu}(y)\\\
A_{2\nu}(y)\end{bmatrix}+\dots$ (2.11)

in which

$S^{\mu\nu}(x,y)=i\begin{bmatrix}\expectationvalue{TJ^{\mu}(x)J^{\nu}(y)}&-\expectationvalue{J^{\nu}(y)J^{\mu}(x)}\\\
-\expectationvalue{J^{\mu}(x)J^{\nu}(y)}&\expectationvalue{\bar{T}J^{\mu}(x)J^{\nu}(y)}\end{bmatrix}_{\rm
1PI}+\expectationvalue{N^{\mu\nu}}_{\rm 1PI}\delta(x-y)\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\,,$ (2.12)

where we have used Eq. A.8 to write the coefficient matrix $S^{\mu\nu}$ in
terms of the correlation functions of the current $J^{\mu}$ plus contact
terms.555It is useful to compare Eq. 2.11 with similar expressions in App. A.
First of all, in both Eq. A.24 and Eq. A.27, unlike in Eq. 2.11, we have used
the $r/a$ representation. More importantly, the reader should note that Eq.
2.11 has similarities and differences with both Eq. A.24 and Eq. A.27. It is
similar to Eq. A.24 because there is an external field $A_{\mu}$ (similar to
$K$ in Eq. A.24) in which we are expanding. The difference is that in Eq. A.24
there are external currents, needed to perform the Legendre transform. That is
why instead of simple correlation functions of the current in Eq. 2.12 we end
up having only the 1PI part. The presence of these external currents is the
similarity to Eq. A.27 while the difference is that in Eq. 2.11 we have not
included the free part of the action. Diagrammatically this corresponds to the
matter corrections to the effective action with two external legs shown in
Fig. 1.

Figure 1: Schematic diagrams associated to the contribution from matter to the
quadratic effective action. The blobs correspond to correlation functions of
$J$ (left) and the contact term (right). External propagators are not included
in the diagrams.

The effective Maxwell equation is obtained by varying $\Gamma[A_{1},A_{2}]$
with respect to $A_{1}$ (or equivalently $A_{2}$) and then setting
$A_{1}=A_{2}=A$ (see also Eq. A.17), which will be the value of the average
field. Doing so we obtain

$\frac{1}{g^{2}}{\partial}_{\alpha}F^{\alpha\mu}+\int\differential[4]{y}\Pi^{\mu\nu}(x,y)A_{\nu}(y)=-J_{\rm
ext}^{\mu}(x)\,,$ (2.13)

where we have also added the external current as explained in Eq. 2.10.666The
quadratic CTP effective action contains the information about all the two-
point functions including the fluctuations. In fact a standard approach is to
solve for the variable $A_{a}\equiv A_{1}-A_{2}$ and obtain a Langevin
equation sourced by a noise term. In this work we focus on the average fields
and neglect fluctuations. An interesting question would be to study properties
of the fluctuations through the fluctuation-dissipation theorem. See §9. The
influence of matter is captured by the second term,

$\Pi^{\mu\nu}(x,y)=i\theta(x^{0}-y^{0})\expectationvalue{[J^{\mu}(x),J^{\nu}(y)]}_{\rm
1PI}+\expectationvalue{N^{\mu\nu}}_{\rm 1PI}\delta(x-y)\,.$ (2.14)

Eq. (2.13) is the Maxwell equation in matter and its solution gives the
average fields. As mentioned above, here we have restricted to the quadratic
effective action which is a good approximation in situations with weak
electromagnetic field compared to, for instance, typical interatomic
electromagnetic fields.777For real materials, the interatomic electric field
could be estimated as $\sim e^{2}/a_{0}\sim 10^{2}$ eV in which $e$ is the
charge and $a_{0}$ is the Bohr radius. This turns out to be large compared to
what can be produced experimentally. By considering cubic or higher-order
terms in the effective action one can study nonlinear effects (in the average
field) as in Eq. A.28. We emphasize that by construction, the effective action
contains the information at all-loop orders in a perturbative expansion.

Comparison of Eq. 2.13 to the general form of the effective equation of motion
in Eq. A.28 shows that the retarded Green’s function of the photon,

$G_{\gamma}^{\mu\nu}\equiv
i\theta(x^{0}-y^{0})\expectationvalue{[a^{\mu}(x),a^{\nu}(y)]}\,,$

is expressed as

$(G_{\gamma}^{-1})^{\mu\nu}=\frac{1}{g^{2}}(\Delta^{-1})^{\mu\nu}-\Pi^{\mu\nu}\,,$
(2.15)

with $\Delta^{\mu\nu}$ the free retarded photon propagator. $\Pi^{\mu\nu}$ is
called the photon self-energy tensor.

### Self-energy tensor

The self-energy can compactly be expressed as the second variation of the
matter effective action,

$\Pi^{\mu\nu}(x,y)=\frac{\delta^{2}\Gamma_{M}[A_{r},A_{a}]}{\delta
A_{a,\mu}(x)\delta A_{r,\nu}(y)}\Bigg{|}_{A_{a}=0,A_{r}=0}\,,$ (2.16)

where we have defined $A_{r}\equiv(A_{1}+A_{2})/2$ and $A_{a}\equiv
A_{1}-A_{2}$ (known as the $r/a$ or physical representation, discussed also in
App. A). Reality of the effective action,
$\Gamma_{M}[A_{r},A_{a}]^{*}=-\Gamma_{M}[A_{r},-A_{a}]$, implies that the
self-energy is real, $\Pi^{\mu\nu}{}^{*}=\Pi^{\mu\nu}$. It ensures that
solutions to the effective Maxwell equation Eq. 2.13 with real sources and
boundary conditions are real. Gauge invariance of the effective action implies
that the self-energy is transverse,

${\partial}_{x^{\mu}}\Pi^{\mu\nu}(x,y)={\partial}_{y^{\nu}}\Pi^{\mu\nu}(x,y)=0\,.$
(2.17)

This ensures that if $A_{\mu}$ is a solution of Eq. 2.13 then
$A_{\mu}+{\partial}_{\mu}\Lambda$ is also a solution. Moreover, by translation
symmetry self-energy is only a function of the distance, i.e.
$\Pi^{\mu\nu}(x,y)=\Pi^{\mu\nu}(x-y)$. In Fourier space

$\Pi^{\mu\nu}(p)=\int\differential[4]{x}e^{-ip\cdot x}\Pi^{\mu\nu}(x)\,,$
(2.18)

with $p^{\mu}$ the four-momentum vector. As discussed above, Lorentz boosts
are broken by the medium so $\Pi^{\mu\nu}$ can be a function of Lorentz-
invariant combinations made out of $p^{\mu}$ and the four velocity $u^{\mu}$
of the medium. Useful combinations are

$\omega\equiv-p^{\mu}u_{\mu}\,,\qquad k^{2}\equiv
p^{\mu}p_{\mu}+(p^{\mu}u_{\mu})^{2}\,.$ (2.19)

In the rest frame of the medium $u^{\mu}=(1,\boldsymbol{0})$ and then
$\omega=p^{0}$ is the energy and $k^{2}=\boldsymbol{p}^{2}$ is the magnitude
of spatial momentum-squared. For this reason, most often we will simply write
the four-momentum as $p^{\mu}=(\omega,\boldsymbol{k})$ and the self-energy as
$\Pi^{\mu\nu}(\omega,k)$.

The condition Eq. 2.17 in Fourier space implies that $p_{\mu}\Pi^{\mu\nu}=0$.
In the Lorentz-invariant case, this condition fixes the whole tensor structure
up to a function known as the vacuum polarization. In the presence of a medium
we will have two functions as given below. The projection matrix onto the
subspace transverse to $p^{\mu}$ is defined as

$\mathcal{P}^{\mu\nu}\equiv\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\,.$
(2.20)

We denote the projection of the medium four-velocity onto this subspace by
$\bar{u}^{\mu}\equiv\mathcal{P}^{\mu\nu}u_{\nu}=u^{\mu}+\frac{\omega}{p^{2}}p^{\mu}$.
We further decompose this subspace into longitudinal and transverse parts,
with respect to $\bar{u}^{\mu}$, using the projection matrices

$\mathcal{P}_{L}^{\mu\nu}\equiv\frac{\bar{u}^{\mu}\bar{u}^{\nu}}{\bar{u}^{2}}\,,\qquad\mathcal{P}_{T}^{\mu\nu}\equiv\mathcal{P}^{\mu\nu}-\mathcal{P}_{L}^{\mu\nu}\,.$
(2.21)

In the rest frame of the medium these projection matrices have the following
components888Another way to think about the decomposition is as follows. The
condition of gauge invariance fixes $\Pi^{00}$ in terms of $\Pi^{0i}$, and
$\Pi^{0i}$ in terms of the $\Pi^{ij}$ components. Then the spatial part of the
tensor is decomposed into pieces in the direction of $\boldsymbol{k}$
(longitudinal part) and orthogonal to it (transverse part).

$\displaystyle\mathcal{P}_{L}^{00}=-\frac{k^{2}}{p^{2}}\,,\qquad\mathcal{P}_{L}^{0i}=-\frac{\omega
k^{i}}{p^{2}}\,,\qquad\mathcal{P}_{L}^{ij}=-\frac{\omega^{2}}{p^{2}}\frac{k^{i}k^{j}}{k^{2}}\,,$
(2.22)
$\displaystyle\mathcal{P}_{T}^{00}=\mathcal{P}_{T}^{0i}=0\,,\qquad\mathcal{P}_{T}^{ij}=\delta^{ij}-\frac{k^{i}k^{j}}{k^{2}}\,.$
(2.23)

The generic form of the self-energy tensor is then999Another possible term is
a projector constructed out of $\varepsilon^{\mu\nu\rho\sigma}$. The presence
of this term signals that the medium breaks parity, for instance because of
the presence of chiral molecules (e.g. sugar), and distinguishes between
right-handed and left-handed polarizations of the photon. See [24]. Materials
with this effect are called optically active. In real materials this effect
does not survive in the limit $\omega\to 0$, which will be the main focus of
this paper. Therefore, we do not consider such terms here. An analogous effect
is cosmic birefringence with the coupling $\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$
studied for instance in [25, 26]. The main difference is that in this case one
of the photon polarizations becomes unstable (which is harmless in cosmology
since the instability gets regularized by the Hubble scale).

$\Pi^{\mu\nu}=-\pi_{L}(\omega,k)p^{2}\mathcal{P}_{L}^{\mu\nu}+\pi_{T}(\omega,k)k^{2}\mathcal{P}_{T}^{\mu\nu}\,,$
(2.24)

where the prefactors are for later convenience. The two functions
$\pi_{L}(\omega,k)$ and $\pi_{T}(\omega,k)$ model the response of the medium
to an external current according to Eq. 2.13 at frequency $\omega$ and
momentum $k$.

### Phenomenological definitions

The discussion of the previous section shows that the effect of the medium can
be described by two functions $\pi_{L}$ and $\pi_{T}$ which are related to the
correlation function of the current as given by Eq. 2.14. Phenomenologically
the medium is usually modeled through the electric permittivity and magnetic
permeability. In this section we relate these two descriptions. See for
instance [19]. The electric and the magnetic field are defined
as101010Alternatively, one can define four-vectors $\tilde{E}^{\mu}\equiv-
u_{\alpha}F^{\alpha\mu}$ and
$\tilde{B}^{\mu}\equiv-\varepsilon^{\alpha\beta\lambda\mu}F_{\alpha\beta}u_{\lambda}/2$
which in the rest frame reduce to Eq. 2.25. However, in a boosted frame
$\tilde{E}$ and $\tilde{B}$ depend on both $\boldsymbol{E}$ and
$\boldsymbol{B}$. One can show that
$F^{\mu\nu}=(u^{\mu}\tilde{E}^{\nu}-u^{\nu}\tilde{E}^{\mu})+\varepsilon^{\mu\nu\alpha\beta}\tilde{B}_{\alpha}u_{\beta}$.

$E^{i}=F^{0i}\,,\qquad B^{i}=\frac{1}{2}\varepsilon^{ijk}F_{jk}\,.$ (2.25)

It is easy to check that the electric field has transverse and longitudinal
components but the magnetic field is only transverse. The effect of the medium
is modeled in terms of an induced current. This is compatible with the
effective Maxwell equation obtained in Eq. 2.13 re-written as

$\frac{1}{g^{2}}{\partial}_{\nu}F^{\nu\mu}=-(J_{\rm in}^{\mu}+J_{\rm
ext}^{\mu})\,,\qquad J_{\rm
in}^{\mu}(x)\equiv\int\differential[4]{y}\Pi^{\mu\nu}(x,y)A_{\nu}(y)\,,$
(2.26)

where $J_{\rm in}^{\mu}=(\rho_{{\rm in}},\boldsymbol{J}_{{\rm in}})$ is the
induced current. Since $p_{\mu}\Pi^{\mu\nu}=0$, the induced current is
conserved. The conventional approach is to relate $J^{\mu}_{\rm in}$ to the
electric and magnetic fields, written in the rest frame, as follows111111This
is the most general definition assuming invariance under parity. See footnote
9.

$\displaystyle g^{2}\rho_{{\rm
in}}\equiv(1-\varepsilon){\partial}_{i}E^{i}\,,$ (2.27) $\displaystyle
g^{2}J_{{\rm
in},T}^{i}\equiv(\tilde{\varepsilon}-1){\partial}_{t}E_{T}^{i}+\left(1-\frac{1}{\tilde{\mu}}\right)\varepsilon^{ijk}{\partial}_{j}B_{k}\,,$
(2.28)

in which the index $T$ refers to the transverse part of the vector. Notice
that there is no need to have an expression for $J_{{\rm in},L}^{i}$ since it
is given by Eq. 2.27 using the current conservation. Moreover, $\varepsilon$,
$\tilde{\varepsilon}$ and $\tilde{\mu}$ must be considered as (non-local)
operators acting on the right-hand side; in Fourier space they will be general
functions of $\omega$ and $k$. We obtain the familiar Maxwell equations in the
presence of matter,

$\varepsilon\,\boldsymbol{\nabla}\cdot\boldsymbol{E}=g^{2}\rho_{\rm
ext}\,,\qquad\qquad\frac{1}{\tilde{\mu}}\boldsymbol{\nabla}\times\boldsymbol{B}-\tilde{\varepsilon}\,{\partial}_{t}\boldsymbol{E}_{T}=g^{2}\boldsymbol{J}_{{\rm
ext},T}\,.$ (2.29)

It is important to note that the definition Eq. 2.28 is ambiguous. In fact one
can show, using the source-independent Maxwell equations, that the
transformation $\tilde{\varepsilon}\to\tilde{\varepsilon}+\delta f$ and
$1/\tilde{\mu}\to 1/\tilde{\mu}+\delta
f{\partial}_{t}^{2}/\boldsymbol{\nabla}^{2}$ for any $\delta f$ leaves the
equation invariant.121212The change in the equation will be $\delta
f{\partial}_{t}({\partial}_{t}\boldsymbol{\nabla}\times\boldsymbol{B}/\boldsymbol{\nabla}^{2}-\boldsymbol{E}_{T})$
which vanishes using
$\boldsymbol{\nabla}\times\boldsymbol{E}+{\partial}_{t}\boldsymbol{B}=0$.
Therefore one has to pick a convention. The two most widely used ones are:

  * •

Set $\tilde{\mu}=1$ and only keep $\tilde{\varepsilon}$ which in this context
is usually called $\varepsilon_{T}$. In this case, the entire induced current
is written in terms of the electric field. The two quantities $\varepsilon$
and $\varepsilon_{T}$ are sometimes packed into a tensor called the dielectric
tensor.

  * •

Set $\tilde{\varepsilon}=\varepsilon$. In this context we simply write
$\tilde{\mu}$ as $\mu$.

In the following we will mainly work with the second convention. Using the
definition in Eq. 2.26, after a bit of algebra, one can relate the above
quantities to $\pi_{L}$ and $\pi_{T}$ as follows131313In the first convention,
the transverse part of the dielectric tensor $\varepsilon_{T}$ can be obtained
from Eq. 2.30 by using the transformation rule discussed above. More precisely
by choosing $\delta f=k^{2}/\omega^{2}(1-\mu^{-1})$ we force $\tilde{\mu}=1$
and get $\varepsilon_{T}=1+g^{2}k^{2}\pi_{T}/\omega^{2}$.:

$\varepsilon-1=g^{2}\pi_{L}\,,\qquad
1-\frac{1}{\mu}=g^{2}\left(\pi_{T}-\frac{\omega^{2}}{k^{2}}\pi_{L}\right)\,.$
(2.30)

For later usage, we note that
$g^{2}k^{2}\pi_{T}=p^{2}+\omega^{2}\varepsilon-k^{2}/\mu$. By using the above
relations we can re-write the effective action defined in Eq. 2.7 with matter
part given in Eq. 2.11. After a bit of algebra, using the $r/a$ representation
defined below Eq. 2.16, we obtain

$\Gamma[A_{r},A_{a}]=\frac{1}{g^{2}}\int\frac{\differential[4]{p}}{(2\pi)^{4}}\left[\varepsilon(\omega,k)\boldsymbol{E}_{a}(-p)\cdot\boldsymbol{E}_{r}(p)-\frac{1}{\mu(\omega,k)}\boldsymbol{B}_{a}(-p)\cdot\boldsymbol{B}_{r}(p)+\dots\right]\,,$
(2.31)

in which we used the definition of electric and magnetic field given in Eq.
2.25 for both $A_{a}$ and $A_{r}$ accordingly. We have also resorted to
Fourier space for simplicity. The dots in Eq. 2.31 stand for a bunch of terms.
First, terms of the form $\sim A_{a}A_{a}$ which capture fluctuations (see
footnote 6). Second, we should add a coupling to the external current which we
have suppressed here. Finally, there are also higher-order terms, e.g.
$\propto A^{3}$, which we neglect. We emphasize that we define $\varepsilon$
and $\mu$ as the coefficients appearing in the effective action that are in
principle well-defined at all energies. It may happen that in some situations
$\varepsilon$ and $\mu$ are not the best variables to work with. For instance,
in a conductor it is more convenient to define a conductivity tensor. In those
cases, one can easily translate the discussion of this paper in terms of the
more appropriate variables.

### Photon propagator

We can write an expression for the retarded photon propagator in the medium.
The free photon propagator, taking into account the gauge fixing term, is
given by

$\Delta^{\mu\nu}=\frac{1}{p^{2}}\left(\mathcal{P}^{\mu\nu}+\frac{1}{\alpha}\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\,,$
(2.32)

where $\alpha$ depends on the gauge choice and we must use the correct
$i\epsilon$ prescription for the retarded Green’s function, i.e.
$p^{2}=-(\omega+i\epsilon)^{2}+k^{2}$. From Eq. 2.15 we obtain the photon
retarded Green’s function in the medium141414The inverse of a tensor
$a\mathcal{P}_{T}^{\mu\nu}+b\mathcal{P}_{L}^{\mu\nu}+cp^{\mu}p^{\nu}/p^{2}$ is
$\mathcal{P}_{T}^{\mu\nu}/a+\mathcal{P}_{L}^{\mu\nu}/b+p^{\mu}p^{\nu}/cp^{2}$
because different projectors are orthogonal to each other.

$\frac{1}{g^{2}}G_{\gamma}^{\mu\nu}=\frac{\mathcal{P}_{T}^{\mu\nu}}{p^{2}-g^{2}k^{2}\pi_{T}}+\frac{\mathcal{P}_{L}^{\mu\nu}}{p^{2}(1+g^{2}\pi_{L})}+\frac{1}{\alpha}\frac{p^{\mu}p^{\nu}/p^{2}}{p^{2}}\,.$
(2.33)

Since the electromagnetic field is always coupled to a conserved current the
last term in Eq. 2.33 drops out and therefore the physical part of the
propagator is independent of $\alpha$. Dispersion relations for propagating
degrees of freedom correspond to poles of the propagator or equivalently,
zeros of $(G_{\gamma}^{\mu\nu})^{-1}$. Calculating the inverse we obtain the
following dispersion relations151515Notice that, naively, the longitudinal
dispersion relation must read $p^{2}\varepsilon=0$. However, $p^{2}$ cancels
out by the same factor in the longitudinal projector as given in Eq. 2.22.

$\varepsilon(\omega,k)\omega^{2}-\frac{1}{\mu(\omega,k)}k^{2}=0\,,\qquad\qquad\varepsilon(\omega,k)=0\,,$
(2.34)

corresponding to the transverse and longitudinal parts respectively. In the
weak coupling limit the latter corresponds to a collective behavior of the
charged particles known as plasma oscillations [27].

### Linear response

It is useful to discuss a related but slightly different situation. Let us
assume that we put the system in a given external electromagnetic field. This
induces a current in the system. The question is how this current is related
to the applied field at linear order. More formally, we are looking at the
action $S_{\gamma}[a]+S_{M}\big{[}a+A_{\rm ext};\psi\big{]}$ for the applied
electromagnetic field $A^{\mu}_{\rm ext}$ (we drop the kinetic term for
$A^{\mu}_{\rm ext}$ since its dynamics is controlled by the experimentalist)
and we are after the response of the system. As discussed in App. B we have

$J_{\rm in}^{\mu}(x)=\int\differential[4]{y}G_{J}^{\mu\nu}(x,y)A_{{\rm
ext},\nu}(y)\,,$ (2.35)

with the response function

$G_{J}^{\mu\nu}(x,y)=i\theta(x^{0}-y^{0})\expectationvalue{[J^{\mu}(x),J^{\nu}(y)]}+\expectationvalue{N^{\mu\nu}}\delta(x-y)\,,$
(2.36)

in which $J^{\mu}$ is the current operator and $N^{\mu\nu}$ captures the
dependence of the current on the electromagnetic field as defined in Eq. 2.6.
The response function $G_{J}^{\mu\nu}$ looks similar to the self-energy
$\Pi^{\mu\nu}$ as given in Eq. 2.14 with the crucial difference that the
latter includes only 1PI terms. In fact comparison between Eq. 2.35 and Eq.
2.26 reveals that $G_{J}^{\mu\nu}$ relates the induced current to the applied
field, while $\Pi^{\mu\nu}$ relates it to the total field. A relation between
the two is easily obtained by noting that the total field is given by the
external field plus the field produced by the induced current,

$A^{\mu}(x)=A_{\rm
ext}^{\mu}(x)+g^{2}\int\differential[4]{y}\Delta^{\mu\nu}(x-y)J_{{\rm
in},\nu}(y)\,.$ (2.37)

Combination of Eq. 2.35, Eq. 2.37 and Eq. 2.26 shows that161616This relation
can also be derived more rigorously using the fact that
$\Pi\sim\frac{\delta^{2}\Gamma_{M}}{\delta A^{2}}$ and actually going through
the Legendre transform from the generating function $W$ (in the notation of
App. A) and expanding the currents.

$(\Pi^{-1})^{\mu\nu}=(G_{J}^{-1})^{\mu\nu}+g^{2}\Delta^{\mu\nu}\,,$ (2.38)

where the $\alpha$-dependent part of the photon propagator is dropped because
it cancels from Eq. 2.37 due to current conservation. In terms of the
transverse and longitudinal parts we have

$G_{J}^{\mu\nu}=\frac{p^{2}k^{2}\pi_{T}}{p^{2}-g^{2}k^{2}\pi_{T}}\mathcal{P}_{T}^{\mu\nu}-\frac{p^{2}\pi_{L}}{1+g^{2}\pi_{L}}\mathcal{P}_{L}^{\mu\nu}\,.$
(2.39)

$\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\scalebox{0.85}{\includegraphics{resum}}}}}}}}$}+\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set<EMAIL_ADDRESS>0.0pt\hbox{\set<EMAIL_ADDRESS>0.0pt\hbox{\set@color\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics{frac2}}}}}}}$}}}}}$
Figure 2: The retarded Green’s function of the current $G_{J}^{\mu\nu}$ is
given by the resummation of the self-energy tensor $\Pi^{\mu\nu}$ with the
free photon propagator $\Delta^{\mu\nu}$. External propagators (with lower
opacity) are for shown for clarity and are not included in the expression. The
blob corresponds to $\Pi^{\mu\nu}$ (including the contact term).

In fact, as indicated in Fig. 2, this relation is the familiar resummation of
the bubble diagrams.

##  3 Positivity and dissipation

Applying an external electromagnetic field to the system changes its energy.
We have derived the general expression for the change in the energy of the
system as a consequence of an applied background field in Eq. B.7. For the
background field $A_{\rm ext}^{\mu}$, this gives171717Equivalently we obtain
the same result for $\imaginary G_{\gamma}^{\mu\nu}$ using $J_{\rm ext}^{\mu}$
as the external source and $G_{\gamma}^{\mu\nu}$ as the linear response.

$\Delta H=\int\frac{\differential[4]{p}}{(2\pi)^{4}}\omega A_{{\rm
ext},\mu}(-p)\imaginary G_{J}^{\mu\nu}(p)A_{{\rm ext},\nu}(p)\,,$ (3.1)

where $G_{J}^{\mu\nu}$ is the response function given in Eq. 2.36. Here we are
assuming that the system is invariant under spacetime inversion, i.e. parity
and time reversal, and therefore the expression appearing in the integrand of
Eq. B.7, known as the dissipative part of the response function, is equal to
its imaginary part.181818As discussed below Eq. B.8, this requires some
knowledge about the phase factor under spacetime inversion of the operator
$J^{\mu}=\delta S_{M}/\delta A_{\mu}$. Since $A^{\mu}$ describes a spin-one
particle which is its own anti-particle, all phases associated to spacetime
inversion must be real, i.e. $\pm 1$ (see [28]). For the photon it is +1.
Generally $\Delta H$ could be both positive or negative. However, we restrict
ourselves to situations in which the system only absorbs energy from the
source and therefore $\Delta H\geq 0$. A system with this property is known as
passive. This implies that $\omega\imaginary G_{J}^{\mu\nu}(p)$ is a
positive-(semi)definite matrix. Notice that by reality of the response
function, we conclude that $G_{J}^{\mu\nu}(p)^{*}=G_{J}^{\mu\nu}(-p)$ and
therefore $\imaginary G_{J}^{\mu\nu}(p)$ is odd in $\omega$.

It is instructive to have a microscopic understanding of this result. First of
all notice that the contact term in Eq. 2.36 does not contribute to the
imaginary part. Moreover, we have

$\begin{split}\imaginary
G^{\mu\nu}_{J}(p)&=\frac{1}{2i}\int\differential[4]{x}e^{-ip\cdot
x}\bigg{(}G^{\mu\nu}_{J}(x)-G^{\mu\nu}_{J}(-x)\bigg{)}\\\
&=\frac{1}{2}\int\differential[4]{x}e^{-ip\cdot
x}\bigg{(}\theta(t)\expectationvalue{[J^{\mu}(x),J^{\nu}(0)]}-\theta(-t)\expectationvalue{[J^{\nu}(-x),J^{\mu}(0)]}\bigg{)}\\\
&=\frac{1}{2}\int\differential[4]{x}e^{-ip\cdot
x}\expectationvalue{[J^{\mu}(x),J^{\nu}(0)]}\,.\end{split}$ (3.2)

In the second line we have used the fact that the response function is
symmetric under $\mu\leftrightarrow\nu$ (see Eq. B.8) and the contact term
does not contribute. To obtain the last line we have used translation symmetry
and $\theta(t)+\theta(-t)=1$. In other words, the imaginary part of the linear
response function in Fourier space is given by the Fourier transform of the
commutator of the operators without the theta function. By inserting a
resolution of the identity we obtain

$\begin{split}2\imaginary
G^{\mu\nu}_{J}(p)&=\int\differential[4]{x}e^{-ip\cdot
x}\Tr(\rho\big{(}J^{\mu}(x)J^{\nu}(0)-J^{\nu}(0)J^{\mu}(x)\big{)})\\\
&=\sum_{n}\int\differential[4]{x}e^{-ip\cdot x}\bra{n}\rho\big{(}e^{-iP\cdot
x}J^{\mu}(0)e^{iP\cdot x}J^{\nu}(0)-J^{\nu}(0)e^{-iP\cdot
x}J^{\mu}(0)e^{iP\cdot x}\big{)}\ket{n}\\\
&=\sum_{n,m}c_{n}\bigg{(}\int\differential[4]{x}e^{-i(p+p_{n}-p_{m})\cdot
x}\bra{n}J^{\mu}(0)\ket{m}\bra{m}J^{\nu}(0)\ket{n}-(\mu\leftrightarrow\nu,p_{n}\leftrightarrow
p_{m})\bigg{)}\\\
&=\sum_{n,m}(2\pi)^{4}\delta(p+p_{n}-p_{m})\bra{n}J^{\mu}(0)\ket{m}\bra{m}J^{\nu}(0)\ket{n}(c_{n}-c_{m})\,.\end{split}$
(3.3)

In the second line we have used the definition of the trace and the spacetime
translation operator $J^{\mu}(x)=e^{-iP\cdot x}J^{\mu}(0)e^{iP\cdot x}$. In
the third line we have introduced a resolution of the identity between the two
currents and used $\rho\ket{n}=c_{n}\ket{n}$. For the last line we have
integrated over $x$ and changed $m\leftrightarrow n$ in the second term. We
construct a symmetric object by contracting with a set of real vectors
$V^{\mu}(p)$ and obtain

$V_{\mu}(p)V_{\nu}(p)\imaginary
G^{\mu\nu}_{J}(p)=\frac{1}{2}\sum_{n,m}(2\pi)^{4}\delta(p+p_{n}-p_{m})\,|\bra{n}J^{\mu}(0)\ket{m}V_{\mu}(p)|^{2}(c_{n}-c_{m})\,.$
(3.4)

We can see that for a generic choice of the $c_{n}$ coefficients $\imaginary
G^{\mu\nu}_{J}(p)$ does not have a definite sign. However, if we assume that
the $c_{n}$ are monotonically decreasing functions of the energies $E_{n}$
then we can argue that the above quantity has a definite sign as follows. For
$\omega>0$ the argument of the delta function allows only for terms with
$E_{n}<E_{m}$. Therefore the right-hand side is positive (similarly negative
for $\omega<0$). This condition holds for most interesting cases such as
finite temperature191919In the presence of a conserved quantity we expand in
terms of the eigenstates of $P^{\mu}-\mu J^{\mu}$. with $c_{n}=e^{-\beta
E_{n}}$.202020Violations of this condition are possible if the system is
prepared such that high-energy levels are more populated than low-energy ones:
in this case the $c_{n}$ coefficients will not be monotonic. Such population
inversion phenomena happen, for instance, in lasers. We should note that
unlike the positivity condition in the Lorentz-invariant case, which is a
consequence of positive norms in the Hilbert space, here it requires certain
conditions on the many-body state as well.

Remember that what appears in the effective Maxwell equations, Eq. 2.13, is
the self-energy $\Pi^{\mu\nu}$, which is the 1PI part of $G_{J}^{\mu\nu}$.
Here we will argue that $\imaginary\Pi^{\mu\nu}$ has the same positivity
property. First, we take $V^{\mu}$ to be transverse with respect to $u^{\mu}$,
i.e. $\mathcal{P}_{T}^{\mu\nu}V_{\nu}=V^{\mu}$. Therefore, from Eq. 2.39 we
have

$V_{\mu}(p)V_{\nu}(p)\imaginary
G^{\mu\nu}_{J}(p)=\boldsymbol{V}^{2}\frac{p^{4}k^{2}\imaginary\pi_{T}}{|p^{2}-g^{2}k^{2}\pi_{T}|^{2}}\,,$
(3.5)

where without loss of generality we have taken $V^{\mu}=(0,\boldsymbol{V})$.
The positivity condition then implies that $\imaginary\pi_{T}>0$ for
$\omega>0$ (and negative for $\omega<0$). If instead we take
$V^{\mu}=\bar{u}^{\mu}$, i.e. along the longitudinal direction, we obtain

$V_{\mu}(p)V_{\nu}(p)\imaginary
G^{\mu\nu}_{J}(p)=\frac{k^{2}\imaginary\pi_{L}}{|1+g^{2}\pi_{L}|^{2}}\,,$
(3.6)

where we have used $\bar{u}^{2}=-k^{2}/p^{2}$. Therefore we conclude that
$\imaginary\pi_{L}>0$ for $\omega>0$ (and negative for $\omega<0$). We
emphasize that the same result could be obtained working with
$G_{\gamma}^{\mu\nu}$.

While the imaginary parts of $\pi_{L}$ and $\pi_{T}$ are sign-definite, the
imaginary part of the magnetic permeability $\mu$ is not.212121This last
statement, while very well-known in the older literature (for instance [29]),
appears to be a controversy in some recent papers [30, 31, 32]. This can
easily be seen from Eq. 2.30: the combination
$\text{Im}(\pi_{T}-\omega^{2}\pi_{L}/k^{2})$ does not appear to have a
definite sign. In contrast, the imaginary part of electric permittivity has
the same sign as $\imaginary\pi_{L}$.

As a final comment, we conclude from Eq. 3.4 that generally $\imaginary
G_{J}^{\mu\nu}$ does not have a gap even if the spectrum of the theory is
gapped. By gap in the imaginary part we mean an energy scale below which the
imaginary part vanishes. The reason is that the delta function in Eq. 3.4 has
support, focusing on the energy part of the delta function, on $E_{n}-E_{m}$
as long as $c_{n}-c_{m}$ is nonzero. Even if all energy eigenstates are larger
than a mass gap, i.e. $E_{n}>m$ for a mass scale $m$, the differences $\delta
E_{mn}\equiv E_{m}-E_{n}$ could in principle be arbitrarily small. Consider,
for instance, a thermal ensemble $c_{n}=e^{-\beta E_{n}}$ with temperature
much larger than the gap so $m\beta\ll 1$. Then there is a nonvanishing
contribution to the imaginary part at frequency $\omega=\delta E_{mn}$ with
magnitude proportional to $c_{n}-c_{m}\simeq\beta\delta E_{mn}e^{-\beta
E_{n}}$ which is not necessarily exponentially suppressed since $\beta E_{n}$
could be small.

In contrast, when the temperature is much smaller than the gap, i.e.
$m\beta\gg 1$, only the vacuum state will be relevant and therefore
$c_{n}\simeq\delta_{n,0}$. Then the expression Eq. 3.4 becomes

$V_{\mu}(p)V_{\nu}(p)\imaginary
G^{\mu\nu}_{J}(p)=\frac{1}{2}\sum_{n}(2\pi)^{4}\bigg{(}\delta(p-p_{n})-\delta(p+p_{n})\bigg{)}|\bra{0}J^{\mu}(0)\ket{n}V_{\mu}(p)|^{2}\,.$
(3.7)

Therefore, one recovers the usual statement that in a gapped theory the
imaginary part of the response function vanishes below the gap.

##  4 Causality and analyticity

The principles of quantum mechanics and relativity state that measurements at
spacelike separation must not interfere. As a result, the commutator of any
two local (bosonic) operators
$[\mathcal{O}_{a}(x),\mathcal{O}_{b}(x^{\prime})]$ must vanish for
$(x-x^{\prime})^{2}>0$, a fact known as microcausality. See also [33, 34].
This implies that the current response function $G_{J}^{\mu\nu}(x)$ and the
retarded photon propagator $G_{\gamma}^{\mu\nu}(x)$ vanish for $t<0$ (because
of the theta function) and $x^{2}>0$. Notice that the contact term in Eq. 2.36
is relevant only in the coincident limit and does not affect the conclusion.
Moreover, for the photon we must project out the gauge dependent part
(equivalently choose $\alpha\to\infty$ in Eq. 2.33) since it is not a physical
observable.

By using retardation and microcausality one can argue for the analyticity of
the Fourier transform of $G_{J}^{\mu\nu}(x)$ and $G_{\gamma}^{\mu\nu}(x)$ in
the complex space of $p^{\mu}=(\omega,\boldsymbol{k})$. More precisely,

$G_{J}^{\mu\nu}(p)=\int\differential[4]{x}e^{-ip\cdot x}G_{J}^{\mu\nu}(x)\,,$
(4.1)

is a holomorphic function in the complex space of the argument,
$p^{\mu}\in\mathbb{C}^{4}$, if $\omega_{I}>0$ and $p_{I}^{2}<0$, where we have
defined $p^{\mu}=p^{\mu}_{R}+ip^{\mu}_{I}$. We refer to this condition as
$p_{I}^{\mu}\in{\rm FLC}$ in which FLC is short for the forward light cone.
This is simple to argue: the integral of Eq. 4.1 is exponentially converging
for $p_{I}^{\mu}x_{\mu}=-\omega_{I}t+\boldsymbol{k}_{I}\cdot\boldsymbol{x}<0$.
Due to causality the integrand has support only for $t>0$ and $x^{2}<0$ which
implies that $p_{I}^{\mu}x_{\mu}\leq-t(\omega_{I}-|\boldsymbol{k}_{I}|)$. As a
result, a sufficient condition for convergence is that
$\omega_{I}>|\boldsymbol{k}_{I}|\geq 0$ or $p_{I}^{\mu}\in{\rm FLC}$. An
expression like Eq. 4.1 in the region of convergence defines an analytic
function simply because complex derivatives are well-defined. An important
assumption here is the polynomial boundedness of the response function at
infinity, i.e. the Green’s function does not grow exponentially fast at
infinity.222222If the system is unstable such that $G(t)\sim e^{+\gamma t}$
for $t\to+\infty$ then the region of analyticity shrinks to
$\omega_{I}\geq\gamma$. Similarly, if the response of the system goes to zero
exponentially fast $G(t)\sim e^{-\gamma t}$ for $t\to+\infty$ then it is
analytic in a larger region $\omega_{I}\geq-\gamma$. We emphasize that the
region of analyticity does not depend on the real part of the four-momentum
$p_{R}^{\mu}$. In other words the analyticity region is
$\mathbb{R}^{4}\times{\rm FLC}\subset\mathbb{C}^{4}$, however most of the time
we just refer to it as the FLC. Needless to say that same results hold for the
physical part of $G_{\gamma}^{\mu\nu}(p)$. In passing, we mention that the
reality condition implies that $G_{J}^{\mu\nu}(p)^{*}=G_{J}^{\mu\nu}(-p^{*})$.
Notice that if $p_{I}^{\mu}\in{\rm FLC}$ then
$\text{Im}(-p^{\mu}{}^{*})\in{\rm FLC}$ and therefore both sides are in the
region of analyticity.

Since the tensor structures of $G_{\gamma}^{\mu\nu}$ and $G_{J}^{\mu\nu}$ are
known as in Eq. 2.33 and Eq. 2.39, it is appropriate to study the analyticity
of the coefficients. Let us consider a general tensor of the form

$f^{\mu\nu}=A(\omega,k)\mathcal{P}_{T}^{\mu\nu}+B(\omega,k)\mathcal{P}_{L}^{\mu\nu}\,,$
(4.2)

for some functions $A$ and $B$ of the frequency and momentum. If $f^{\mu\nu}$
is analytic in $p_{I}^{\mu}\in{\rm FLC}$, one can show that $A$ and $B$ must
be analytic in the FLC. Indeed, from Eq. 2.22 and Eq. 2.23 we have
$B=-p^{2}k_{i}k_{j}f^{ij}/\omega^{2}$ which implies that $B$ is analytic in
the FLC. Notice that the factor $1/\omega^{2}$ is harmless since the pole at
$\omega=0$ lies outside the FLC after the appropriate $i\epsilon$
prescription. By looking at the trace of the spatial part we get
$\delta_{ij}f^{ij}=2A-\omega^{2}B/p^{2}$. Once again, the factor $1/p^{2}$
with appropriate $i\epsilon$ prescription is harmless for analyticity: in fact
this is the propagator for a massless free particle (see also App. E).
Therefore, we conclude that $A$ must be analytic in the FLC. However, the
analyticity of $A$ and $B$ is not sufficient to guarantee that $f^{\mu\nu}$ is
analytic. If we look at $f^{\mu\nu}$ for $i\neq j$ we have

$f^{ij}=-\frac{k^{i}k^{j}}{k^{2}}\left(A+\frac{\omega^{2}}{p^{2}}B\right)\,.$
(4.3)

The factor $1/k^{2}$ introduces singularities in the FLC. The reason is that
the equation
$k^{2}=k_{R}^{2}-k_{I}^{2}+2i\boldsymbol{k}_{R}\cdot\boldsymbol{k}_{I}=0$ has
nontrivial, i.e. $k^{i}\neq 0$, solutions which are independent of
$\omega_{I}$ and therefore one can find points
$(\omega_{R}+i\omega_{I},\boldsymbol{k}_{R}+i\boldsymbol{k}_{I})$ with
$k^{2}=0$ that lie in the forward light cone. (In fact, $1/k^{2}$ is the
Fourier transform of $-1/\boldsymbol{\nabla}^{2}$ which describes
instantaneous action at a distance.) These poles must be removed by zeros of
the numerator to ensure analyticity of $f^{ij}$. We conclude that the
combination $A+\omega^{2}B/p^{2}$ must be zero at $k^{2}=0$.

Application of the above results to $G_{\gamma}^{\mu\nu}$ given in Eq. 2.33
shows that the combinations $1/(1+g^{2}\pi_{L})$ and
$1/(p^{2}-g^{2}k^{2}\pi_{T})$ are analytic in the FLC. We will use this fact
in §6 to bound the low-energy limits of $\varepsilon$ and $\mu$. Moreover,
from the discussion below Eq. 4.3 we require that the combination

$\frac{1}{k^{2}}\left[\frac{1}{p^{2}-g^{2}k^{2}\pi_{T}}+\frac{\omega^{2}/p^{4}}{1+g^{2}\pi_{L}}\right]\stackrel{{\scriptstyle
k^{2}=0}}{{\sim}}\frac{1}{k^{2}}\frac{g^{2}(k^{2}\pi_{T}-\omega^{2}\pi_{L})}{\omega^{2}(1+g^{2}\pi_{L})(\omega^{2}+g^{2}k^{2}\pi_{T})}\,,$
(4.4)

is regular near $k^{2}=0$. To obtain the second expression we have used
$p^{2}=-\omega^{2}$ in the limit $k^{2}\to 0$. We conclude that the
combination $k^{2}\pi_{T}-\omega^{2}\pi_{L}$ must go to zero at least like
$k^{2}$ when $k^{2}\to 0$.232323The reader may wonder how one can exclude the
possibility of the denominator of Eq. 4.4 having a singularity. For instance,
if $\pi_{L}\sim 1/k^{2}$ as $k^{2}\to 0$ then it removes the unwanted
$1/k^{2}$ factor in front of Eq. 4.4. Notice that, apparently, this
possibility is harmless for the analyticity of the combination
$1/(1+g^{2}\pi_{L})$ required by causality. However, as we will argue in §7
and App. D, $\pi_{L}(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ is
analytic in the upper-half plane (UHP) of complex $\omega$ as long as
$\boldsymbol{q}\cdot\boldsymbol{\xi}=0$ and $|\boldsymbol{\xi}|<1$. This is a
consequence of both causality and positivity. In this parametrization
$k^{2}=q^{2}+\omega^{2}\xi^{2}$, and therefore it is not possible to have
$\pi_{L}\sim 1/k^{2}$ since it implies a pole at $\omega=+iq/\xi$ in the UHP.
A similar discussion applies to $k^{2}\pi_{T}$. This indicates that, as a
consequence of microcausality, the two functions $\pi_{L}$ and $\pi_{T}$
cannot be completely unrelated. Similarly, the coefficients in the expression
for $G_{J}^{\mu\nu}$ given in Eq. 2.39 are analytic; since these are slightly
more complicated and do not give new information we do not report them here.
We will deal with the analyticity of $\Pi^{\mu\nu}$ in §7.

In order to avoid dealing with several complex variables at the same time we
parametrize the FLC by several single-complex-variable subspaces. More
precisely, we write

$p^{\mu}=(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ (4.5)

for complex $\omega$ and real vectors $\boldsymbol{q}$ and $\boldsymbol{\xi}$
with the condition that $\xi\equiv|\boldsymbol{\xi}|<1$. It is easy to see
that every point $p^{\mu}=(\omega,\boldsymbol{k})$ in the FLC can be written
in the form (4.5) for appropriate values of $\boldsymbol{q}$ and
$\boldsymbol{\xi}$, i.e.
$\boldsymbol{\xi}=\frac{\boldsymbol{k}_{I}}{\omega_{I}}$ and
$\boldsymbol{q}=\boldsymbol{k}_{R}-\omega_{R}\boldsymbol{\xi}$. As discussed
in [14], the form (4.5) is the most general single-variable parametrization.
Any other choice for $\boldsymbol{k}(\omega)$ would either change the
analyticity region (by mixing the real and imaginary parts) or spoil the
behavior at infinity. Then we see that
$G_{\gamma}^{\mu\nu}(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ and
$G_{J}^{\mu\nu}(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$, now regarded
as functions of $\omega$, are analytic in the UHP of $\omega\in\mathbb{C}$,
i.e. $\omega_{I}>0$. The same holds for the coefficients appearing in Eq. 2.33
and Eq. 2.39 as discussed above.

For a generic function $\chi(z)$ that is analytic in the UHP (and continuous
onto the real line) and decays at infinity one can use Cauchy’s theorem,
applied to $\chi(z)/(z-z_{R})$, to prove the following relation

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}}}}}}$}\qquad\qquad\chi(z_{R})=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-z_{R}}\,\,\chi(z)\,,$
(4.6)

in which $z_{R}$ is real, and ${\rm PV}$ means the principal value of the
integral for the pole at $z=z_{R}$. An important assumption here is that
$\chi(z)\to 0$ when $|z|\to\infty$, which implies that we can neglect the
contour at infinity. Taking the real part of Eq. 4.6 gives an expression for
$\real\chi(z_{R})$ in terms of an integral over the $\imaginary\chi(z)$ over
the real axis. We can apply this general result to the analytic functions that
we found above by using the parametrization of Eq. 4.5. Let us denote by
$\chi(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ such an analytic
function, e.g. $1/(1+g^{2}\pi_{L})$. Then Eq. 4.6 gives242424One can also
derive this relation by noting that
$\chi(x)=\chi(x)\theta(t-\boldsymbol{\xi}\cdot\boldsymbol{x})$ for any $\xi<1$
as required by microcausality. Taking the Fourier transform of both sides
gives Eq. 4.7. See [20, 35] for details.

$\chi(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\,\,\chi(z,\boldsymbol{q}+z\boldsymbol{\xi})\,,$
(4.7)

for real $\omega$ as well as real $\boldsymbol{q}$ and $\boldsymbol{\xi}$ with
$\xi<1$ as required by Eq. 4.5. By shifting the spatial momentum
$\boldsymbol{q}\to\boldsymbol{q}-\omega\boldsymbol{\xi}$ and relabeling it we
can rewrite Eq. 4.7 as

$\chi(\omega,\boldsymbol{k})=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\,\,\chi(z,\boldsymbol{k}+(z-\omega)\boldsymbol{\xi})\,.$
(4.8)

We emphasize that in this relation all independent variables are real. Eq.
(4.8) was initially written by M. Leontovich [20] and we will call it (as well
as Eq. 4.7) Leontovich’s relation. This relation is the generalization of the
famous Kramers-Kronig formula, derived assuming only retardation, taking into
account the condition of microcausality. In fact, for
$\boldsymbol{\xi}=\boldsymbol{0}$ one recovers the Kramers-Kronig relations.
Taking the real part of the both sides in Eq. 4.8, as mentioned above, gives
an expression for $\real\chi(\omega,\boldsymbol{k})$ in terms of an integral
of $\imaginary\chi(z,\boldsymbol{k}+(z-\omega)\boldsymbol{\xi})$ for any
$\boldsymbol{\xi}$. In §6 we will apply Leontovich’s relation to the various
analytic combinations of $\pi_{L}$ and $\pi_{T}$.

Before closing this section let us mention two more points regarding
Leontovich’s relation. First, as we said, Eq. 4.7 gives the real part of the
function in terms of its imaginary part and vice versa. Therefore, we can
express the whole function in terms of the imaginary part as follows

$\begin{split}\chi(\omega,\boldsymbol{k}+\omega\boldsymbol{\xi})&=\frac{1}{\pi}{\rm
PV}\int\frac{\differential{z}}{z-\omega}\,\,\imaginary\chi(z,\boldsymbol{k}+z\boldsymbol{\xi})+i\int\differential{z}\delta(z-\omega)\imaginary\chi(z,\boldsymbol{k}+z\boldsymbol{\xi})\\\
&=\frac{1}{\pi}\int\frac{\differential{z}}{z-\omega-i\epsilon}\,\,\imaginary\chi(z,\boldsymbol{k}+z\boldsymbol{\xi})\,,\end{split}$
(4.9)

where in the first line we have replaced the real part using Eq. 4.7 and to
obtain the second line we have used Sokhotski-Plemelj. Although we have
derived Eq. 4.9 for real $\omega$, it remains valid for complex $\omega$ as
well in the domain of analyticity, i.e. in the UHP.252525We should note that
we can write the analog of Eq. 4.9 for Eq. 4.8, i.e. $\xi$ only appears on the
right-hand side, assuming $\omega$ is real. However, this form does not
continue to hold for complex $\omega$ simply because the right-hand side
involves an integral over
$\imaginary\chi(z,\boldsymbol{k}+(z-\omega)\boldsymbol{\xi})$ which is not an
analytic function of $\omega$. The reason is that the right-hand side of Eq.
4.9 is a sum over functions $1/[\omega-(z-i\epsilon)]$ which do not have poles
in the UHP.262626More rigorously one can start from $\chi(\omega)$ for complex
$\omega$ by writing it in terms of an integral over the real axis using
Cauchy’s theorem. The function over the real axis can then be written in terms
of an integral over the imaginary part as given in Eq. 4.9. After some algebra
one recovers Eq. 4.9 now with $\omega$ complex. Eq. (4.9) will be used in §7
and App. D to study the analyticity of the self-energy tensor $\Pi^{\mu\nu}$.

Finally, we note that the condition of microcausality, equivalently
Leontovich’s relation, restricts the form of the imaginary part of the
function. We have already seen in the previous section that $\omega\imaginary
G^{\mu\nu}_{\gamma}$ and $\omega\imaginary G^{\mu\nu}_{J}$ are positive
definite (for passive materials). However, one can ask: does any positive
matrix give an eligible imaginary part of a causal function? Interestingly,
the answer is no. This can be seen from Eq. 4.8. The fact that the left-hand
side in Eq. 4.8 is independent of $\boldsymbol{\xi}$ is a restriction on the
imaginary part from causality. More precisely, the integral over the imaginary
part along two different lines in the $(\omega,\boldsymbol{k})$ space
associated with two different vectors $\boldsymbol{\xi}_{1}$ and
$\boldsymbol{\xi}_{2}$, as shown below, must give the same
result272727Equivalently, one can show that Eq. 4.10 is a direct consequence
of the fact that the imaginary part is given by the Fourier transform of a
commutator which vanishes outside the full light cone as given in Eq. 3.2.

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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}}}}$}\qquad{\rm
PV}\int\differential{z}\frac{\imaginary\chi(z,\boldsymbol{k}+(z-\omega)\boldsymbol{\xi}_{1})}{z-\omega}={\rm
PV}\int\differential{z}\frac{\imaginary\chi(z,\boldsymbol{k}+(z-\omega)\boldsymbol{\xi}_{2})}{z-\omega}\,.$
(4.10)

Notice that this restriction does not exist if we only consider retardation
($\boldsymbol{\xi}=\boldsymbol{0}$) as we do for Kramers-Kronig relations. A
more detailed study of this set of constraints is beyond the scope of this
paper and will be explored elsewhere.

##  5 High-energy behavior

One of the assumptions required for the Leontovich relation Eq. 4.8 to be
valid is that the arc at infinity can be neglected. This requires some
knowledge about the high-energy behavior of the two-point function of
$J^{\mu}$, or equivalently of $A^{\mu}$. We stress that we require the high-
energy behavior for complex $p^{\mu}$ in the domain of analyticity. As was
argued in [14], the behavior at high energies in the complex plane is dictated
by the short distance behavior of the two-point function in position space.

Unfortunately, as far as we know, there is no universal bound on the
asymptotic behavior of the two-point function. By contrast, for the $S$-matrix
such a bound exists [36, 37], at least for gapped theories, only by requiring
unitarity, causality and Lorentz invariance. One possible assumption, as
studied in [14], is that at high energies the system is well-described by a
conformal field theory which implies that the two-point function of a
conserved current behaves as $\sim p^{d-2}$ in $d$ spacetime dimensions (and
as $\sim p^{2}\log p^{2}$ in $d=4$).

In condensed matter systems one can usually assume that the Green’s functions
vanish at high energies [22, 38, 27], i.e. that the medium becomes irrelevant
at high frequencies. The main idea is that at high energies one can think of
the system as a collection of charged particles (say, electrons). In this
limit, i.e. $\omega\gg\nu_{c}$ in which $\nu_{c}$ is the average collision
frequency [27], one can also neglect interactions among the particles.
Therefore, the system is described as a gas of free electrons. The dielectric
response function of this system has been calculated long ago by Lindhard [39]
in the non-relativistic limit. We provide a first-principle derivation,
including relativistic effects, in App. C; the calculation is basically the
one-loop correction to the photon propagator (see Fig. 6) assuming a finite
chemical potential for electrons. As explained in Eq. C.6, there are two
contributions to the response function: one is the standard quantum
electrodynamics correction and the other is the contribution from the finite
electron density. The finite density contribution, as given in Eq. C.20, at
high energies $\omega,k\to\infty$, behaves as

$g^{2}\pi_{L}\to-\frac{\omega_{p}^{2}}{\omega^{2}}+\dots\,,\qquad
g^{2}\pi_{T}\to-\frac{\omega_{p}^{2}}{k^{2}}+\dots\,,$ (5.1)

in which $\omega_{p}^{2}\equiv g^{2}n/m$ is called the plasma frequency with
$n$ the number density and $m$ the mass of the electrons. Notice that from Eq.
5.1 we conclude that the imaginary part decays even faster, as $1/\omega^{3}$.
Eq. (5.1) implies that $\varepsilon_{(T)}=1-\omega_{p}^{2}/\omega^{2}+\dots$
at high frequencies both for the longitudinal and transverse parts of the
dielectric tensor (see footnote 13) and we will refer to it as “plasma
behavior”. We should note that in general there are different components
present in the medium that have different plasma frequencies. In Eq. 5.1 we
take $\omega_{p}^{2}$ to be the largest one, usually associated with the
lightest particles.

What about the Lorentz-invariant contribution? As given in Eq. C.8, at
energies below the mass $\omega,k\ll m$, its value is very small, $\pi_{L,{\rm
qed}}\approx-p^{2}/(60\pi^{2}m^{2})$ and negligible. However, for energies
above the mass it grows as $\log p^{2}$. Therefore we are forced to close the
contour at moderately high energies in which the description by means of a
non-relativistic plasma is a good approximation, and avoid extremely high
energies in which relativistic effects become important. This is shown in Fig.
3 for the real parts of $\pi_{L}$ and $\pi_{T}$. There is a window of energies
in which relativistic corrections are not important and we can trust the
plasma behavior of non-relativistic fermions.

Figure 3: Real (left) and imaginary (right) parts of the functions $\pi_{L}$
and $\pi_{T}$ obtained from Eq. C.6 and Eq. C.7 for the system of non-
interacting fermions at leading order in the coupling to the photon. We have
chosen $\boldsymbol{k}=\omega\boldsymbol{\xi}$ and set
$|\boldsymbol{\xi}|=0.5$ but other values result in qualitatively similar
behavior. We have chosen $m/k_{F}=100$ which is a typical value in materials.
We observe that the relativistic growth starts from energies $\omega\sim m$
which is much larger than the non-relativistic plasma regime. The region above
the pair production threshold $p^{2}=-4m^{2}$ is shaded. There is a guaranteed
range of energies where Eq. 5.1 applies since
$\omega_{p}^{2}/m^{2}=g^{2}k_{F}^{3}/3\pi^{2}m^{3}\ll 1$ for $k_{F}/m\ll 1$.

Moreover, the error in neglecting the relativistic corrections is controlled
by the ratio $\omega_{p}^{2}/m^{2}$ which is of order $10^{-12}$ for typical
materials. If we close the contour of integration at, say,
$\sim\order{10^{2}}\omega_{p}$ the contribution from Eq. 5.1 is negligible.

For media of “higher energies”, for example nuclear matter, one cannot close
the contour below the electron mass. One can still make use of the Kramers-
Kronig and Leontovich relations, however, as we will discuss in §8.

##  6 Bounds on the low-energy behavior

In this section we bound $\varepsilon(0,0)$ and $\mu(0,0)$ using the
Leontovich relation Eq. 4.8 and positivity conditions on the imaginary parts.
Notice that both $\varepsilon$ and $\mu$ can in principle have singularities
like $1/\omega^{n}$ as $\omega\to 0$ without violating causality. For
instance, in the case of conductors we have $\varepsilon\sim i\sigma/\omega$
where $\sigma$ denotes the conductivity. For superconductors, we have
$\varepsilon,\mu\sim 1/(\omega^{2}-c_{s}^{2}k^{2})$ associated to the
propagation of the Goldstone mode with speed of sound $c_{s}$ (see [40, 41]).
In this paper, we restrict ourselves to cases in which there is no singularity
at the origin. In other words, our results will be applicable only to
insulators (dielectrics) where at low energies there is no other dynamical
degree of freedom than photons and leave other – perhaps more interesting –
cases to future studies.282828See [42] for a discussion of the classification
of low-energy degrees of freedom based on the derivative expansion. From Eq.
2.30 we see that $g^{2}\pi_{L}=\varepsilon(0,0)-1$ and $g^{2}k^{2}\pi_{T}=0$
at low energies. Moreover, the combination of Eq. 2.30 and the causality
condition Eq. 4.4 implies that
$g^{2}(k^{2}\pi_{T}-\omega^{2}\pi_{L})=(1-\mu(0,0)^{-1})k^{2}$ at low
energies.

In the following we use the longitudinal and transverse parts of
$G_{\gamma}^{\mu\nu}$ to write dispersion relations involving
$\varepsilon(0,0)$ and $\mu(0,0)$. It is possible to obtain similar (and
equivalent) relations using $G_{J}^{\mu\nu}$. We do not apply the Leontovich
relation directly to $\Pi^{\mu\nu}$ since we still need to argue for its
analyticity. This will be discussed in §7. We note that many of these results
have been derived in the literature on this subject [43, 44, 19].

### Longitudinal part

Applying the discussion around Eq. 4.2 to the longitudinal part of
$G_{\gamma}^{\mu\nu}$ given in Eq. 2.33 implies that
$1/(p^{2}\varepsilon(\omega,\boldsymbol{k}))$ is analytic when $\text{Im }p$
lies in the FLC, where, recall, $\varepsilon=1+g^{2}\pi_{L}$. Let us consider
the function $1/\varepsilon-1$. This function is analytic in the same region
and, according to Eq. 5.1, goes to zero at infinity. Eq. (4.7) implies

$\frac{1}{\varepsilon(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})}-1=-\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\,\,\frac{g^{2}\pi_{L}(z,\boldsymbol{q}+z\boldsymbol{\xi})}{1+g^{2}\pi_{L}(z,\boldsymbol{q}+z\boldsymbol{\xi})}\,.$
(6.1)

Setting $\omega=\boldsymbol{q}=0$ we find

$\frac{1}{\varepsilon(0,0)}-1=-\frac{2g^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}\,\,\frac{\imaginary\pi_{L}(z,z\boldsymbol{\xi})}{|1+g^{2}\pi_{L}(z,z\boldsymbol{\xi})|^{2}}\,.$
(6.2)

To obtain Eq. 6.2 we used the reality condition
$\pi_{L}(-z,-z\boldsymbol{\xi})=\pi_{L}(z,z\boldsymbol{\xi})^{*}$, from which
it follows that
$\imaginary\pi_{L}(z,z\boldsymbol{\xi})=\imaginary\varepsilon(z,z\boldsymbol{\xi})$
is odd in $z$ and so also that $\varepsilon(0,0)$ is real. In this way we have
written $1/\varepsilon$ in terms of $\imaginary\pi_{L}$ which has a positivity
property as discussed in §3. A similar relation can be obtained for nonzero
$\boldsymbol{q}$.292929Notice that $\varepsilon(0,\boldsymbol{q})$ is also
real due to rotational invariance,
$\varepsilon(0,\boldsymbol{q})^{*}=\varepsilon(0,-\boldsymbol{q})=\varepsilon(0,\boldsymbol{q})$.
We should note that for our purposes $\boldsymbol{\xi}$ on the right-hand side
is not necessary and can be set to zero; as discussed in Eq. 4.10, by
causality the integral turns out to be independent of $\boldsymbol{\xi}$ (for
$\xi<1$). It may be useful, however, to keep $\boldsymbol{\xi}$ in case
something is known about the imaginary part at finite momentum. Finally, from
Eq. 6.2 one can argue that $1/\varepsilon(0,0)\leq 1$ which implies

$\varepsilon(0,0)\geq 1\,.$ (6.3)

The latter possibility of $\varepsilon(0,0)<0$ is ruled out following the
discussion of §7. Noticeably, since $\varepsilon(0,0)\geq 1$, there is no
electric analog of paramagnetism.

### Transverse part

The coefficient of the transverse part of $G_{\gamma}^{\mu\nu}$ in Eq. 2.33 is
$1/(p^{2}-g^{2}k^{2}\pi_{T})$, which is analytic following the discussion in
§4. Since this is divergent at low energies when
$\boldsymbol{k}=\omega\boldsymbol{\xi}$, we instead consider the combination

$\chi_{1}\equiv\frac{\omega^{2}}{p^{2}-g^{2}k^{2}\pi_{T}}=\frac{1}{-\varepsilon+\frac{k^{2}}{\omega^{2}}\frac{1}{\mu}}\,.$
(6.4)

Next, we parametrize the momentum according to Eq. 4.5,
$\chi_{1}(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$. By using Eq. 5.1,
which implies that $k^{2}\pi_{T}\to\text{const.}$ at infinity, we see that in
the limit $\omega\to\infty$ the function approaches $-1/(1-\xi^{2})$.
Therefore, for the combination $\chi_{1}+1/(1-\xi^{2})$, we are able to
neglect the contour at infinity in the Leontovich relation. Using Eq. 4.7 we
obtain

$\frac{1}{-\varepsilon(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})+\frac{(\boldsymbol{q}+\omega\boldsymbol{\xi})^{2}}{\omega^{2}\mu(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})}}+\frac{1}{1-\xi^{2}}=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\left[\frac{z^{2}}{-z^{2}+(\boldsymbol{q}+z\boldsymbol{\xi})^{2}(1-g^{2}\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi}))}+\frac{1}{1-\xi^{2}}\right]\,.$
(6.5)

Setting $\boldsymbol{q}=\boldsymbol{0}$ and taking the limit $\omega\to 0$,
from the real part of Eq. 6.5 we obtain

$\frac{1}{-\varepsilon(0,0)+\frac{\xi^{2}}{\mu(0,0)}}+\frac{1}{1-\xi^{2}}=\frac{2g^{2}\xi^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}\,\,\frac{\imaginary\pi_{T}(z,z\boldsymbol{\xi})}{|-1+\xi^{2}-g^{2}\xi^{2}\pi_{T}(z,z\boldsymbol{\xi})|^{2}}\,.$
(6.6)

Like before we can restrict the integration range to positive $z$ because
$\pi_{T}(-z,-z\boldsymbol{\xi})=\pi_{T}(z,z\boldsymbol{\xi})^{*}$. In the
limit $\xi\to 0$, this equation reproduces the longitudinal dispersion
relation. The reason is that, as argued below Eq. 4.4, by causality the
combination $k^{2}\pi_{T}-\omega^{2}\pi_{L}\sim k^{2}$ as $k\to 0$. Therefore,
$\xi^{2}\pi_{T}(z,z\boldsymbol{\xi})\sim\pi_{L}(z,z\boldsymbol{\xi})$ as
$\xi\to 0$ at fixed $z$. However, for finite $\xi$ this equation has new
information. By the positivity condition on the imaginary part of $\pi_{T}$
the right-hand side must be positive. Therefore we obtain

$\varepsilon(0,0)\geq\frac{\xi^{2}}{\mu(0,0)}+1-\xi^{2}\,.$ (6.7)

Another possibility would be $\varepsilon(0,0)<\xi^{2}\mu(0,0)$ but this
cannot happen as we will argue in §7. Optimal bounds are found by setting
$\xi=0$ and $\xi=1$. Again we observe that $\boldsymbol{\xi}=\boldsymbol{0}$
yields the longitudinal bound given in Eq. 6.3. On the other hand at $\xi=1$
we obtain

$\varepsilon(0,0)\geq\frac{1}{\mu(0,0)}\,.$ (6.8)

One can also consider, instead of $\chi_{1}$ in Eq. 6.4, another function
defined as

$\chi_{2}\equiv\frac{k^{2}}{p^{2}-g^{2}k^{2}\pi_{T}}=\frac{1}{\frac{1}{\mu}-\frac{\omega^{2}}{k^{2}}\varepsilon}\,,$
(6.9)

which is analytic in the FLC since $1/(p^{2}-g^{2}k^{2}\pi_{T})$ is analytic
in that region. Once again we parametrize it as
$\chi_{2}(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ and use the
Leontovich relation in the symmetric form given in Eq. 4.7. In the limit
$\omega\to\infty$, $\chi_{2}\to-\xi^{2}/(1-\xi^{2})$. Therefore, we use
Leontovich’s relation for the combination $\chi_{2}+\xi^{2}/(1-\xi^{2})$ and
get

$\frac{1}{\frac{1}{\mu(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})}-\frac{\omega^{2}\varepsilon(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})}{(\boldsymbol{q}+\omega\boldsymbol{\xi})^{2}}}+\frac{\xi^{2}}{1-\xi^{2}}=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\left[\frac{(\boldsymbol{q}+z\boldsymbol{\xi})^{2}}{-z^{2}+(\boldsymbol{q}+z\boldsymbol{\xi})^{2}(1-g^{2}\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi}))}+\frac{\xi^{2}}{1-\xi^{2}}\right]\,.$
(6.10)

Setting $\boldsymbol{q}=\boldsymbol{0}$ and then taking $\omega\to 0$ limit
gives Eq. 6.6 where both sides are multiplied by $\xi^{2}$. However, we can
also take the limit $\omega\to 0$ but keep $\boldsymbol{q}$ finite:

$\mu(0,\boldsymbol{q})+\frac{\xi^{2}}{1-\xi^{2}}=\frac{g^{2}}{\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z}\,\,\frac{(\boldsymbol{q}+z\boldsymbol{\xi})^{4}\imaginary\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi})}{|-z^{2}+(\boldsymbol{q}+z\boldsymbol{\xi})^{2}-g^{2}(\boldsymbol{q}+z\boldsymbol{\xi})^{2}\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi})|^{2}}\,.$
(6.11)

Since the right-hand side is positive, this gives a lower bound on the
magnetic permeability at finite momentum $\boldsymbol{q}$ in the static limit
($\omega=0$). Setting $\xi=0$ and then taking303030The limit $q\to 0$ of Eq.
6.11 is subtle, since the RHS is discontinuous at
$\boldsymbol{q}=\boldsymbol{0}$. The correct procedure is to take the limit
after integration. $\boldsymbol{q}\rightarrow\boldsymbol{0}$ we conclude

$\mu(0,0)\geq 0\,.$ (6.12)

The bounds obtained above on $\varepsilon(0,0)$ and $\mu(0,0)$ are summarized
in Fig. 4 in which the shaded region is allowed. The horizontal and vertical
boundaries correspond to the analyticity and positivity of the longitudinal
and transverse parts of the Green’s function respectively.

Figure 4: The allowed region for $\varepsilon$ and $\mu$ from positivity and
analyticity. The horizontal boundary results from Eq. 6.2 (or equivalently Eq.
7.4) and the positivity of $\imaginary\pi_{L}$ while the vertical boundary
results from Eq. 6.6 (or equivalently Eq. 7.9) and the positivity of
$\imaginary\pi_{T}$ in the limit $\xi\to 1$. While $\varepsilon$ is forced to
be larger than unity, $\mu$ could be both less than 1, in which the medium is
called diamagnetic, or larger than 1, for a paramagnetic medium. Notice that
only diamagnetism is constrained by causality and positivity.

Notice that the inequality $\varepsilon(0,0)\mu(0,0)\geq 1$ is nothing but the
condition of subluminal speed of propagation of photons inside the medium,
since the transverse photon propagator at low energies can be written as
$1/(-\varepsilon(0,0)\omega^{2}+k^{2}/\mu(0,0))$. Moreover,
$\varepsilon(0,0)>0$ is required to avoid a ghost, and $\mu(0,0)>0$ to avoid a
gradient instability. The fact that one can recover bounds from subluminality
condition has been observed in the Lorentz-invariant context as well [1].
However, we emphasize that here we obtain dispersion relations which in
principle could be estimated or measured, and thereby, the bounds could get
stronger. We will see an example below. Moreover, it should be pointed out
that the vertical boundary cannot be obtained only by using retardation, i.e.
the Kramers-Kronig relation, since that corresponds to setting
$\boldsymbol{\xi}=\boldsymbol{0}$ while to obtain the vertical boundary we
needed to take $\xi\rightarrow 1$. Therefore, while Lorentz symmetry is broken
by the presence of the medium, Lorentz invariance of the theory has nontrivial
consequences.

##  7 Analyticity of $\Pi^{\mu\nu}$

In the previous section we used the photon Green’s function to bound the
values of $\varepsilon$ and $\mu$ at low energies. Since these quantities are
defined through the self-energy tensor $\Pi^{\mu\nu}$, one may wonder why we
have not used $\Pi^{\mu\nu}$ directly. The reason is that we do not have a
general argument for the analyticity of $\Pi^{\mu\nu}$, and we will elaborate
on this issue here.

The argument for the analyticity of $G_{J}^{\mu\nu}(p)$ and
$G_{\gamma}^{\mu\nu}(p)$ is that microcausality requires commutators to vanish
for spacelike separated points. However, there is no representation of
$\Pi^{\mu\nu}$ in terms of a commutator of local operators. We emphasize that
Eq. 2.14 is not such an expression, because we restrict to the 1PI part of the
commutator (up to a contact term) and it is not clear that this has the same
property. In fact, we remind the reader that 1PI correlation functions are
calculated from the effective action, which requires adding a background
external current which is in principle a non-local function of the fields. As
a result, it is not obvious that the $\expectationvalue{[J(x),J(y)]}_{\rm
1PI}$ vanishes for spacelike separated points $x,y$. More physically,
$\Pi^{\mu\nu}$, in contrast to $G_{\gamma}^{\mu\nu}$ and $G_{J}^{\mu\nu}$,
does not describe the reaction of the system to an external source and as such
its microcausality is not guaranteed (cf. the contribution of Kirzhnitz in
[19]).

In fact, by definition, the self-energy tensor is given by the difference of
the inverse of the photon propagator in the full and the free theory as given
in Eq. 2.15. From this relation any pole of $\Pi^{\mu\nu}$ corresponds to a
zero of $G_{\gamma}^{\mu\nu}$ which seems to be harmless. On the other hand, a
pole in $\Pi^{\mu\nu}$ might imply a singularity in $G_{\gamma}^{\mu\nu}$ at a
different point. The reason is that near a pole $\Pi^{\mu\nu}$ diverges and
therefore the equation $\frac{1}{g^{2}}(\Delta^{-1})^{\mu\nu}-\Pi^{\mu\nu}=0$
likely has a solution. By Eq. 2.15 this means that $G_{\gamma}^{\mu\nu}$ is
singular. However, we could not find a reason for this singularity to
necessarily lie in the FLC (and therefore not be allowed).313131As an example
take $\Pi^{\mu\nu}$ of the form
$\Pi^{\mu\nu}=\frac{1}{g^{2}}(\Delta^{-1})^{\mu\nu}+{\rm pole}$. Then
everything is perfectly consistent for $G_{\gamma}^{\mu\nu}$.

However, in the perturbative limit this additional pole is always near the
singularity of $\Pi^{\mu\nu}$ and is thus not allowed. More precisely, in the
limit that one can expand perturbatively in the coupling, it easy to see that
the self-energy tensor must have the same analyticity region as the photon
propagator by the following argument. Manipulating Eq. 2.15 yields

$g^{2}\Pi=\Delta^{-1}-\Bigg{[}1+\sum_{n=1}g^{2n}\Delta^{-1}\cdot
G_{(n)}\Bigg{]}^{-1}\cdot\Delta^{-1}\;,$ (7.1)

where we have written $G_{\gamma}=g^{2}\Delta+\sum_{n=1}g^{2(n+1)}G_{(n)}$ and
for brevity we have resorted to matrix notation for the spacetime indices with
dot being matrix multiplication. By expansion we get

$\displaystyle\Pi_{(0)}$ $\displaystyle=\Delta^{-1}\cdot
G_{(1)}\cdot\Delta^{-1}\,,$ (7.2) $\displaystyle\Pi_{(1)}$
$\displaystyle=\Delta^{-1}\cdot G_{(2)}\cdot\Delta^{-1}-\Delta^{-1}\cdot
G_{(1)}\cdot\Delta^{-1}\cdot G_{(1)}\cdot\Delta^{-1}\,,\quad\text{etc.}\,,$
(7.3)

where we have written $\Pi=\sum_{n=0}g^{2n}\Pi_{(n)}$. Notice that since the
full photon propagator $G^{\mu\nu}_{\gamma}$ vanishes outside the light cone
in real space, and so is analytic when its imaginary part lies in the FLC in
Fourier space, all the terms in the perturbative expansion are separately
causal. This must be true since in the limit that the coupling goes to zero
each term is parametrically smaller than the previous order, i.e. if one of
them is acausal this cannot be removed by the addition of other terms. As a
result, by looking at Eq. 7.3 and higher orders, we see that at each order
$\Pi_{(i)}$ is constructed out of causal contributions and therefore it must
be causal.

So a singularity of $\Pi^{\mu\nu}$ in the FLC, if it exists, must be non-
perturbative in the coupling. Notice that this singularity cannot be of the
form of a branch point or essential singularity. The presence of a branch
point or essential singularity in $\Pi^{\mu\nu}$ spoils the analyticity of
$G_{\gamma}^{\mu\nu}$. As a result, if there is a singularity in the FLC it
must be a pole.

The situation is different if we take into account the condition of
positivity. In App. D we show that a response function, parametrized as
$(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ with the condition
$\boldsymbol{q}\cdot\boldsymbol{\xi}=0$, cannot have any zeros in the region
of analyticity, i.e. $\omega\in$ UHP. Applying this to $G_{\gamma}^{\mu\nu}$,
we conclude that $G_{\gamma}^{-1}$ is analytic and therefore, by using Eq.
2.15, so is $\Pi^{\mu\nu}(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ with
$\boldsymbol{q}\cdot\boldsymbol{\xi}=0$. Unfortunately, the theorem is not
conclusive beyond the condition $\boldsymbol{q}\cdot\boldsymbol{\xi}=0$. In
the following, we will exploit this fact to write other forms of dispersion
relations directly using $\Pi^{\mu\nu}$.

### Self-energy dispersion

Although we are not able to prove the analyticity of $\Pi^{\mu\nu}$ in ${\rm
FLC}\subset\mathbb{C}^{4}$, even taking into account positivity, one can still
write dispersion relations directly using $\Pi^{\mu\nu}$. For instance, we
concluded above that the coefficient of the longitudinal part, $p^{2}\pi_{L}$,
is analytic in the UHP of $\omega$ assuming the parametrization
$(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ with
$\boldsymbol{q}\cdot\boldsymbol{\xi}=0$. Therefore, we can apply Leontovich’s
relation Eq. 4.7 to the combination $g^{2}\pi_{L}=\varepsilon-1$, and get

$\varepsilon(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})-1=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\,\,g^{2}\pi_{L}(z,\boldsymbol{q}+z\boldsymbol{\xi})\,,\qquad(\boldsymbol{q}\cdot\boldsymbol{\xi}=0)\,.$
(7.4)

Therefore, setting $\boldsymbol{q}=\boldsymbol{0}$ and $\omega=0$, we obtain

$\varepsilon(0,0)-1=\frac{2g^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}\,\,\imaginary\pi_{L}(z,z\boldsymbol{\xi})\,.$
(7.5)

This relation is analogous to Eq. 6.2 which was obtained from the analyticity
of the photon propagator. In particular, from the positivity of the right-hand
side we conclude that $\varepsilon(0,0)\geq 1$; this rules out the other
possibility $\varepsilon(0,0)<0$ discussed after Eq. 6.2.

Similarly, from the coefficient of the transverse part of $\Pi^{\mu\nu}$ we
conclude that $k^{2}\pi_{T}$ is analytic with the same parametrization as
above. Therefore, we can consider the combination323232This combination is
actually $\varepsilon_{T}-1$.

$\chi_{3}\equiv\frac{g^{2}k^{2}\pi_{T}}{\omega^{2}}=(\varepsilon-1)+\frac{k^{2}}{\omega^{2}}\left(1-\frac{1}{\mu}\right)\,.$
(7.6)

Assuming $\varepsilon\to 1$ and $\mu\to 1$ at high energy (see Eq. 5.1) this
function goes to zero at infinity. By using Leontovich’s relation we obtain

$(\varepsilon(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})-1)+\frac{q^{2}+\omega^{2}\xi^{2}}{\omega^{2}}\left(1-\frac{1}{\mu(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})}\right)=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\,\,\frac{g^{2}(q^{2}+z^{2}\xi^{2})}{z^{2}}\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi})\,,$
(7.7)

where we have used $\boldsymbol{q}\cdot\boldsymbol{\xi}=0$. For nonzero $q$
the limit $\omega\to 0$ is not well-defined. Setting
$\boldsymbol{q}=\boldsymbol{0}$ we obtain

$(\varepsilon(0,0)-1)+\xi^{2}\left(1-\frac{1}{\mu(0,0)}\right)=\frac{2g^{2}\xi^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}\,\,\imaginary\pi_{T}(z,z\boldsymbol{\xi})\,.$
(7.8)

This is analogous to Eq. 6.6. Once again, taking the limit $\xi\to 0$ gives
the same relation as Eq. 7.5. However, sending $\xi\to 1$ we obtain

$\varepsilon(0,0)-\frac{1}{\mu(0,0)}=\frac{2g^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}\,\,\imaginary\pi_{T}(z,z)\,.$
(7.9)

Using the positivity of $\imaginary\pi_{T}$, we find $\varepsilon(0,0)\geq
1/\mu(0,0)$ which is compatible with Eq. 6.6.

##  8 Examples and improvements

In this section we discuss further the results derived above, focussing on the
possibility of saturating the bounds and improving them under some stronger
assumptions.

### Models living on the boundaries

One can ask: is it possible, at least conceptually, to lie on the boundaries
of the allowed region shown in Fig. 4? The horizontal boundary with
$\varepsilon(0,0)=1$ corresponds to a medium without any electric response at
low energies. Therefore, it would suffice to set $\pi_{L}\equiv 0$. Different
points along the boundary $\varepsilon(0,0)=1$ then correspond to different
limiting values of $\pi_{T}$. To have a mathematically consistent example let
us consider the following form:

$g^{2}\pi_{T}(\omega,\boldsymbol{k})=\frac{\omega_{T}^{2}}{-\omega^{2}-i\gamma\omega+c_{s}^{2}k^{2}+\omega_{0}^{2}}\,.$
(8.1)

As discussed in App. E, as long as $\gamma>0$, $0\leq c_{s}^{2}\leq 1$,
$\omega_{0}^{2}>0$ and $\omega_{T}^{2}>0$ this is an analytic function when
$\text{Im}(\omega,\boldsymbol{k})$ lies in the FLC, with the correct
positivity condition. Moreover, from $\mu^{-1}=1-g^{2}\pi_{T}$ we see that the
condition $\mu(0,0)>0$ implies that $\omega_{T}^{2}<\omega_{0}^{2}$. More
physically, an ensemble of magnetic dipoles does not contribute to
$\varepsilon$ and results in paramagnetism. The paramagnetic response can be
arbitrarily large as we approach the Curie point, $\mu\to\infty$.

The vertical boundary of Fig. 4 corresponds to $\varepsilon(0,0)\mu(0,0)=1$.
Therefore, we need both an electric and magnetic response to lie on the
vertical boundary. Since it implies that the speed of light is unity, perhaps
the easiest way is to start from imposing Lorentz invariance on $\pi_{L}$ and
$\pi_{T}$: $\pi_{L}=-\frac{k^{2}}{p^{2}}\pi_{T}$. Therefore, we can consider

$g^{2}\pi_{L}(\omega,k)=-\frac{k^{2}}{p^{2}}g^{2}\pi_{T}=\frac{\omega_{L}^{2}}{-\omega^{2}-i\varepsilon\omega+k^{2}+\omega_{0}^{2}}\,,$
(8.2)

which for $\omega_{0}^{2}>0$ and $\omega_{L}^{2}>0$ has the correct
analyticity and positivity properties. In the Lorentz-invariant case we have
$\varepsilon=\mu^{-1}=1+g^{2}\pi_{L}$ at all energies. One may wonder whether
in Eq. 8.2 any other function, with the correct properties, will also work.
However, one must be careful about the positivity of
$\pi_{T}=-p^{2}\pi_{L}/k^{2}$ since the prefactor $p^{2}$ changes sign. For
instance, adding a subluminal speed of sound $c_{s}$ or finite decay width
$\gamma$ to Eq. 8.2, while being consistent for $\pi_{L}$, is not consistent
for $\pi_{T}$. It is, in fact, possible to provide a more physical example.
Consider the theory

$\mathcal{L}=-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}({\partial}_{\mu}\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}+\frac{\alpha}{\Lambda^{2}}\phi^{2}F_{\mu\nu}F^{\mu\nu}\,,$
(8.3)

in which $\phi$ is a massive scalar field. Integrating it out, say in some
nontrivial background, gives corrections to the photon kinetic term that are
always proportional to $F_{\mu\nu}F^{\mu\nu}$. As a result, the condition
$\varepsilon=\mu^{-1}$ is satisfied. The Lagrangian above describes a scalar
particle with electric and magnetic polarizabilities which are equal and
opposite, since $F_{\mu\nu}F^{\mu\nu}\propto E^{2}-B^{2}$. This situation is
rather common, since it corresponds to the dimension-6 operator above. In
order to deviate from this relation one has to consider higher-order operators
like $F^{\mu\alpha}F_{\mu}^{\beta}\partial_{\alpha}\phi\partial_{\beta}\phi$,
which are naturally suppressed. For instance pions and kaons have
polariazabilities which are approximately equal and opposite, see for
example[45].

### Lower bound on dissipation

The bounds of Fig. 4 can be improved if one has some knowledge about the
right-hand side of Eq. 7.5 and Eq. 7.9 (equivalently (6.2) and (6.6)). In
particular one needs a lower bound on dissipation. One possibility is to
assume the “plasma” behavior at large $\omega$ as discussed in Sec. 5. At high
energy one has $\pi_{L}\sim-\omega_{p}^{2}/\omega^{2}$. This implies that the
function $\omega^{2}\pi_{L}(\omega,0)+\omega_{p}^{2}$ is analytic in the UHP
of $\omega$ and decays at infinity. Writing the dispersion relation (4.6) for
$z_{R}=0$ gives the well-known sum rule (see for instance [22]):

$\omega_{p}^{2}=\frac{2g^{2}}{\pi}\int_{0}^{\infty}\omega\differential\omega\imaginary\pi_{L}(\omega,0)\,.$
(8.4)

One can use this equality to give a lower bound on the right-hand side of Eq.
7.5. If dissipation happens at higher and higher frequencies, one can have an
arbitrarily small right-hand side of Eq. 7.5 while preserving the constraint
Eq. 8.4. If we assume that the imaginary part is zero above some $\omega_{\rm
UV}$ then Eq. 7.5, for $\boldsymbol{\xi}=\boldsymbol{0}$, becomes

$\varepsilon(0,0)-1=\frac{2g^{2}}{\pi}\int_{0}^{\omega_{\rm
UV}}\frac{\differential{z}}{z}\,\,\imaginary\pi_{L}(z,0)\geq\frac{2g^{2}}{\pi}\int_{0}^{\omega_{\rm
UV}}\frac{z\differential{z}}{\omega_{\rm
UV}^{2}}\,\,\imaginary\pi_{L}(z,0)=\frac{\omega_{p}^{2}}{\omega_{\rm
UV}^{2}}\,.$ (8.5)

In reality the imaginary part will not vanish exactly above the frequency
$\omega_{\rm UV}$, but it will decay typically as $\omega^{-3}$. Anyway the
integrals over frequency are convergent, so the bound above will only receive
relative corrections of order unity. In real materials this ratio is roughly
of order unity. One can follow the same logic for $\pi_{T}$, which at high
energy goes as $\pi_{T}\sim-\omega_{p}^{2}/(\xi^{2}\omega^{2})$, see Eq. 5.1.
One obtains another sum rule for $\omega_{p}^{2}$, viz.

$\omega_{p}^{2}=\frac{2g^{2}\xi^{2}}{\pi}\int_{0}^{+\infty}z\differential{z}\imaginary\pi_{T}(z,z\boldsymbol{\xi})\;.$
(8.6)

This can be used in Eq. 7.9 to give

$(\varepsilon(0,0)-1)+\xi^{2}\left(1-\frac{1}{\mu(0,0)}\right)\geq\frac{\omega_{p}^{2}}{\omega_{\rm
UV}^{2}}\,.$ (8.7)

Notice this does not give a bound on the low-frequency speed of light in the
medium, given by $1/\sqrt{\varepsilon(0,0)\mu(0,0)}$.

In passing it is worthwhile to mention another relation for
$\mu(0,\boldsymbol{q})$ (other than Eq. 6.11) which involves the plasma
frequency. Let us consider the function

$\chi_{4}\equiv
g^{2}k^{2}\pi_{T}=\omega^{2}(\varepsilon-1)+k^{2}\left(1-\frac{1}{\mu}\right)\,.$
(8.8)

From Eq. 5.1, in the limit $\omega\to\infty$ we have
$\chi_{4}\to-\omega_{p}^{2}$. Therefore, we can apply Leontovich’s relation to
$\chi_{4}+\omega_{p}^{2}$ which gives

$\omega^{2}(\varepsilon(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})-1)+(q^{2}+\omega^{2}\xi^{2})\left(1-\frac{1}{\mu(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})}\right)+\omega_{p}^{2}=\frac{1}{i\pi}{\rm
PV}\int_{-\infty}^{+\infty}\frac{\differential{z}}{z-\omega}\left[g^{2}(q^{2}+z^{2}\xi^{2})\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi})+\omega_{p}^{2}\right]\,,$
(8.9)

assuming $\boldsymbol{q}\cdot\boldsymbol{\xi}=0$. For nonzero $q$, taking the
$\omega\to 0$ limit gives

$q^{2}\left(1-\frac{1}{\mu(0,\boldsymbol{q})}\right)+\omega_{p}^{2}=\frac{2g^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}(q^{2}+z^{2}\xi^{2})\imaginary\pi_{T}(z,\boldsymbol{q}+z\boldsymbol{\xi})\,.$
(8.10)

Restricting to $\boldsymbol{\xi}=\boldsymbol{0}$ we obtain

$\frac{q^{2}+\omega_{p}^{2}}{q^{2}}-\frac{1}{\mu(0,\boldsymbol{q})}=\frac{2g^{2}}{\pi}\int_{0}^{+\infty}\frac{\differential{z}}{z}\imaginary\pi_{T}(z,q)\,.$
(8.11)

This relation was derived in [19]. In particular, assuming the positivity
condition, it implies $\mu(0,\boldsymbol{q})>q^{2}/(q^{2}+\omega_{p}^{2})$.

### Wider “light”cone

Most materials consist of non-relativistic particles, i.e. their typical
velocity is much smaller than the speed of light. Does this fact have any
implications for the allowed values for electric and magnetic response of the
medium? The answer is probably yes. Let us consider a system of charged
particles. In certain limits, the behavior of the system is described, in
kinetic theory, by a single-particle distribution function $f(\boldsymbol{V})$
(see [27] for details). In this limit one can calculate the dielectric tensor,
defined above Eq. 2.30, as follows

$\varepsilon^{ij}(\omega,\boldsymbol{k})\equiv
1+\frac{g^{2}}{\omega^{2}}\Pi^{ij}=\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)\delta^{ij}+\frac{\omega_{p}^{2}}{\omega^{2}}\int\differential[3]{\boldsymbol{V}}\frac{V^{i}V^{j}}{\omega+i\epsilon-\boldsymbol{V}\cdot\boldsymbol{k}}\,k^{\ell}\partialderivative{f(\boldsymbol{V})}{V^{\ell}}\,.$
(8.12)

Notice that the dielectric tensor contains all the information about the
electromagnetic properties of the medium. Eq. (8.12) has the remarkable
property that it is analytic in a larger region than the FLC; poles of Eq.
8.12 are located at $\omega_{I}=\boldsymbol{V}\cdot\boldsymbol{k}_{I}$ which
are far from the FLC for non-relativistic velocities $|\boldsymbol{V}|\ll 1$.
Intuitively, the reason is that a change in the total electromagnetic field at
a point $\boldsymbol{x}$ can induce a current in a different point
$\boldsymbol{x}^{\prime}$, mainly due to the movement of charged particles
from $\boldsymbol{x}$ to $\boldsymbol{x}^{\prime}$, and this occurs at low
velocity. More precisely, in the weak field limit, one has to solve the
linearized equation of motion for $f(\boldsymbol{V})$ perturbations, i.e. the
Vlasov equation, to obtain Eq. 8.12. This equation predicts a slower
propagation of information; the factor
$(\omega+i\epsilon-\boldsymbol{V}\cdot\boldsymbol{k})^{-1}$ is in fact the
propagator of the linearized Vlasov equation.

The above description is only valid if one can neglect (thermal or quantum)
fluctuations in the system. Indeed in order to be able to solve for the
single-particle distribution function, $f(\boldsymbol{V})$, one generally
requires the knowledge of multi-particle correlation functions. It turns out
that for sufficiently dense systems, all the multi-particle correlation
functions are expressible in terms of $f(\boldsymbol{V})$, as for instance in
Eq. 8.12, which is the two-point function of the current. In particular the
process of exchanging a photon from $\boldsymbol{x}$ to