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&\qquad\qquad=\int_{\rho}\mathcal{D}\phi_{1}\,\mathcal{D}\phi_{2}\,e^{i(S[\phi_{1}]-S[\phi_{2}])}\,\phi_{2}(x^{\prime}_{1})\dots\phi_{2}(x^{\prime}_{n})\phi_{1}(x_{1})\dots\phi_{1}(x_{m})\,.\end{split}$
(A.8)

For brevity we have suppressed the initial and final boundary conditions of
the path integral.

### Generating functional

A convenient way to deal with this type of observable is to define a
generating functional393939We used $K_{1,2}$ as the argument of the generating
function to save $J$ as the electromagnetic current in the main text.,

$\begin{split}Z[K_{1},K_{2}]&\equiv\int\mathcal{D}\phi_{1}\,\mathcal{D}\phi_{2}\,e^{i(S[\phi_{1}]-S[\phi_{2}])}\,e^{i\int_{x}(K_{1}(x)\phi_{1}(x)-K_{2}(x)\phi_{2}(x))}\\\
&=\expectationvalue{\bar{T}\Big{(}e^{-i\int_{x}K_{2}(x)\phi(x)}\Big{)}T\Big{(}e^{i\int_{x}K_{1}(x)\phi(x)}\Big{)}}\,,\end{split}$
(A.9)

where in the second line we have used Eq. A.8 (from now on we suppress the
dependence on the density matrix). Assuming $K$ is real,
$\mathcal{O}_{K}\equiv T\Big{(}e^{i\int_{x}K(x)\phi(x)}\Big{)}$ is unitary and
it follows that $Z[K,K]=\Tr[\rho\,\mathcal{O}_{K}^{\dagger}\mathcal{O}_{K}]=1$
is normalized. Taking variations of $Z$ with respect to the arguments gives
various correlation functions. More specifically, we have

$\begin{split}&\expectationvalue{\bar{T}(\phi(x^{\prime}_{1})\dots\phi(x^{\prime}_{n})){T}(\phi(x_{1})\dots\phi(x_{m}))}\\\
&\qquad\qquad\qquad=i\functionalderivative{K_{2}(x_{1}^{\prime})}\dots
i\functionalderivative{K_{2}(x_{n}^{\prime})}\frac{1}{i}\functionalderivative{K_{1}(x_{1})}\dots\frac{1}{i}\functionalderivative{K_{1}(x_{m})}Z[K_{1},K_{2}]\big{|}_{K_{1}=K_{2}=0}\,.\end{split}$
(A.10)

We can interpret the current term as a deformation of the action, i.e.
$S_{K}\equiv S+\int_{x}K(x)\phi(x)$ and therefore
$Z[K_{1},K_{2}]=\Tr[U_{K_{2}}^{\dagger}U_{K_{1}}\rho]$ in which $U_{K}$ is the
time evolution with the deformed action. As a result, we can obtain
correlation functions in the deformed theory by varying $Z$ and setting
$K_{1}=K_{2}=K$ rather than zero. For instance

$\frac{1}{i}\functionalderivative{K_{1}(x)}Z\big{|}_{K_{1}=K_{2}=K}=i\functionalderivative{K_{2}(x)}Z\big{|}_{K_{1}=K_{2}=K}=\expectationvalue{\phi(x)}_{K}\,,$
(A.11)

similarly for higher orders.

Another useful object is the logarithm of the generating function, i.e.
$Z[K_{1},K_{2}]\equiv\exp(iW[K_{1},K_{2}])$. Similar to the standard case,
$W[K_{1},K_{2}]$ generates connected correlations functions. Notice that

$Z[K_{1},K_{2}]^{*}=\expectationvalue{\mathcal{O}_{K_{2}}^{\dagger}\mathcal{O}_{K_{1}}}^{*}=\expectationvalue{\mathcal{O}_{K_{1}}^{\dagger}\mathcal{O}_{K_{2}}}=Z[K_{2},K_{1}]\,,$
(A.12)

which implies $W[K_{1},K_{2}]^{*}=-W[K_{2},K_{1}]$. More generally we can
consider complex currents. In that case we have
$W[K_{1},K_{2}]^{*}=-W[K_{2}^{*},K_{1}^{*}]$.

### Effective action

Another useful object is the effective action. As is well-known, the effective
action is the generating functional for 1PI diagrams. As a result, at tree
level it may be used to give all loop corrections to an observable. In
particular, it yields the quantum equation of motion. (For a review in
standard QFT see [53, 54] and in the context of CTP see [55, 56, 23].) The
effective action is defined as the Legendre transform of $W$, as follows:

$\Gamma[\phi_{1},\phi_{2}]\equiv
W[K_{1},K_{2}]-\int(K_{1}\phi_{1}-K_{2}\phi_{2})\,,$ (A.13)

when the currents satisfy the following;

$\functionalderivative{W}{K_{1}(x)}-\phi_{1}(x)=0\qquad\text{and}\qquad\functionalderivative{W}{K_{2}(x)}+\phi_{2}(x)=0\,.$
(A.14)

It follows that

$\functionalderivative{\Gamma}{\phi_{1}(x)}=-K_{1}(x)\qquad\text{and}\qquad\functionalderivative{\Gamma}{\phi_{2}(x)}=K_{2}(x)\,.$
(A.15)

Moreover, the second variation of $\Gamma$ gives the inverse of the full
connected propagator as follows:

$\frac{\delta^{2}\Gamma}{\delta\phi^{i}\delta\phi^{j}}=-\functionalderivative{K_{i}}{\phi^{j}}=-\left(\functionalderivative{\phi^{j}}{K_{i}}\right)^{-1}=-\left(\frac{\delta^{2}W}{\delta
K_{i}\delta K_{j}}\right)^{-1}\,,$ (A.16)

where we have used Eq. A.14 and Eq. A.15. We also used the notation
$\phi^{1}=\phi_{1}$ and $\phi^{2}=-\phi_{2}$ to account for minus signs.
Taking more derivatives reveals that any connected correlation function can be
constructed out of tree diagrams with edges corresponding to the full
propagator and vertices corresponding to $\Gamma$ as the action, hence the
name quantum effective action.

Taking the complex conjugate of Eq. A.14 and using the relation below Eq. A.12
we conclude that if the pair $(\phi_{1},\phi_{2})$ corresponds to
$(K_{1},K_{2})$ then $(\phi_{2},\phi_{1})$ is associated to
$(K_{2}^{*},K_{1}^{*})$. Therefore, from the definition we have
$\Gamma[\phi_{1},\phi_{2}]^{*}=-\Gamma[\phi_{2},\phi_{1}]$. In addition, since
$\phi_{1}=\phi_{2}$ corresponds to $K_{1}=K_{2}$ we have $\Gamma[\phi,\phi]=0$
(note that by definition $W[K,K]=0$). Finally, notice that

$\functionalderivative{\Gamma}{\phi_{1}}\Big{|}_{\phi_{1}=\phi_{2}=\phi}=-\functionalderivative{\Gamma}{\phi_{2}}\Big{|}_{\phi_{1}=\phi_{2}=\phi}=-K\,,$
(A.17)

is the quantum-corrected equation of motion for the field expectation value in
the presence of a current $K$, i.e. $\expectationvalue{\phi}_{K}=\phi$. It is
worthwhile to contrast this with the expectation value we calculate in the
standard in-out formalism. In the latter case, the assumption is that
interactions turn on and off adiabatically in the far past and the far future.
Therefore, the initial and final states only are equal up to a phase. This is
a good approximation for particle physics but it is incapable of taking into
account out-of-equilibrium processes, e.g. dissipation.

For perturbative calculations one has to add all 1PI diagrams for a given
number of external legs. Another approach, more useful for non-perturbative
results, is the background field method. Consider the original theory in the
presence of a given background. We have

$\exp(iW[K_{1},K_{2}]_{\bar{\phi}_{1},\bar{\phi}_{2}})\equiv\int\mathcal{D}\phi_{1}\,\mathcal{D}\phi_{2}\,e^{i(S[\phi_{1}+\bar{\phi}_{1}]-S[\phi_{2}+\bar{\phi}_{2}])}\,e^{i\int_{x}(K_{1}(x)\phi_{1}(x)-K_{2}(x)\phi_{2}(x))}\,.$
(A.18)

From the generating function in the presence of the background we can define
the effective action as before,

$\Gamma[\phi_{1},\phi_{2}]_{\bar{\phi}_{1},\bar{\phi}_{2}}=W[K_{1},K_{2}]_{\bar{\phi}_{1},\bar{\phi}_{2}}-\int(K_{1}\phi_{1}-K_{2}\phi_{2})\,,$
(A.19)

with $K_{1}$ and $K_{2}$ defined by the same equations as in Eq. A.14 with $W$
replaced by $W_{\bar{\phi}_{1},\bar{\phi}_{2}}$. Now it is a straightforward
exersise to show that

$\Gamma[\phi_{1},\phi_{2}]_{\bar{\phi}_{1},\bar{\phi}_{2}}=\Gamma[\phi_{1}+\bar{\phi}_{1},\phi_{2}+\bar{\phi}_{2}]\,,$
(A.20)

where the right-hand side is the effective action of the theory without the
background field as defined in Eq. A.13. An important consequence of this is
that we can use the theory with the background to calculate the effective
action. In other words we have

$\Gamma[\phi_{1},\phi_{2}]=\Gamma[0,0]_{\phi_{1},\phi_{2}}=W[K_{1},K_{2}]_{\phi_{1},\phi_{2}}\,,$
(A.21)

where the last equality follows from Eq. A.19. Notice that by means of a field
redefinition in Eq. A.18 we have
$W[K_{1},K_{2}]_{\phi_{1},\phi_{2}}=W[K_{1},K_{2}]-\int(K_{1}\phi_{1}-K_{2}\phi_{2})$,
and therefore Eq. A.21 is consistent with Eq. A.13.

### Perturbation in the background field

It is useful to write a perturbative expansion, in the background field, for
the effective action. As discussed in the main text, this is useful for
situations in which the average field is weak (compared to some microscopic
scale in the problem). We start by expanding $W[K_{1},K_{2}]$ perturbatively
in the currents. It is very convenient to work, instead of $K_{i}$ for
$i=1,2$, with linear combinations

$K_{r}\equiv\frac{1}{2}(K_{1}+K_{2})\,,\qquad K_{a}\equiv K_{1}-K_{2}\,.$
(A.22)

(Working with these new variables is sometimes referred to as the $r/a$ or
physical representation.) Normalization and reality conditions imply
$W[K_{r},K_{a}=0]=0$ and $W[K_{r},K_{a}]^{*}=-W[K_{r},-K_{a}]$. Expanding
$W[K_{r},K_{a}]$ gives

$W[K_{r},K_{a}]=\frac{1}{2}\int\differential{x}\differential{y}\frac{\delta^{2}W}{\delta
K_{I}(x)\delta K_{J}(y)}\Big{|}_{K=0}K_{I}(x)K_{J}(y)+\dots\,,$ (A.23)

in which the indices $I,J,\dots$ are either $r$ or $a$ as introduced above.
Notice that the zeroth order term vanishes by normalization. The coefficient
of the linear term, i.e.
$\functionalderivative{W}{K_{I}}\propto\functionalderivative{Z}{K_{I}}$, is
related to the expected value of the field in the absence of any external
sources, which we assume vanishes (or is compensated for by a counter-term).
Therefore, the leading term is quadratic as indicated above. It requires a
little algebra to express the coefficients of the expansion in terms of the
correlation functions of the fields:

$W[K_{r},K_{a}]=\frac{1}{2}\int\differential{x}\differential{y}\begin{bmatrix}K_{r}(x)K_{a}(x)\end{bmatrix}\begin{bmatrix}0&G_{A}(x,y)\\\
G_{R}(x,y)&iG_{S}(x,y)\end{bmatrix}\begin{bmatrix}K_{r}(y)\\\
K_{a}(y)\end{bmatrix}+\dots\,,$ (A.24)

where we have defined

$G_{R}(x,y)=i\theta(t_{x}-t_{y})\expectationvalue{[\phi_{x},\phi_{y}]}\,,\quad
G_{A}(x,y)=-i\theta(t_{y}-t_{x})\expectationvalue{[\phi_{x},\phi_{y}]}\,,\quad
G_{S}(x,y)=\frac{1}{2}\expectationvalue{\\{\phi_{x},\phi_{y}\\}}\,,$ (A.25)

referred to as retarded, advanced and symmetric Green’s functions. Note that
the $\theta$-functions come from the (anti-)time ordering in the closed time
path. The fact that the coefficient of $K_{r}^{2}$ vanishes is a consequence
of the normalization condition. More generally the coefficient of $K_{r}^{n}$
is identically zero. The coefficient of $K_{a}^{2}$ is purely imaginary which
is a consequence of the reality condition. More generally, the coefficient of
$K_{r}^{n}K_{a}^{m}$ is purely imaginary for even $m$. It is possible to show
that all the coefficients in the above expansion can be written in terms of
nested commutators and/or anti-commutators. For details we refer the reader to
[57, 55].

Similar to the (connected graphs) generating function $W[K_{r},K_{r}]$, we can
expand $\Gamma[\phi_{r},\phi_{a}]$ in terms of the fields,

$\phi_{r}\equiv\frac{\phi_{1}+\phi_{2}}{2}\,,\qquad\phi_{a}\equiv\phi_{1}-\phi_{2}\,.$
(A.26)

In the $r/a$ representation, the normalization and reality condition read
$\Gamma[\phi_{r},\phi_{a}=0]=0$ and
$\Gamma[\phi_{r},\phi_{a}]^{*}=-\Gamma[\phi_{r},-\phi_{a}]$. Coefficients of
the expansion can be obtained by using the same trick as in Eq. A.16. Up to
quadratic order, by using Eq. A.24 and Eq. A.16, we have

$\Gamma[\phi_{r},\phi_{a}]=\frac{1}{2}\int\differential{x}\differential{y}\begin{bmatrix}\phi_{r}(x)\phi_{a}(x)\end{bmatrix}\begin{bmatrix}0&-G_{A}^{-1}(x,y)\\\
-G_{R}^{-1}(x,y)&iG_{R}^{-1}G_{S}G_{A}^{-1}(x,y)\end{bmatrix}\begin{bmatrix}\phi_{r}(y)\\\
\phi_{a}(y)\end{bmatrix}+\dots\,,$ (A.27)

in which it is was used that
$\int\differential{y}G_{R}(x,y)G_{R}^{-1}(y,z)=\delta(x-z)$. Once again, the
vanishing of $\phi_{r}^{2}$ term results from normalization and the
coefficient of $\phi_{a}^{2}$ is purely imaginary because of reality, with
similar generalizations to higher orders. The effective equation of motion in
the presence of an external current $K$, given in Eq. A.17, is now obtained:

$\functionalderivative{\Gamma}{\phi_{a}(x)}\Big{|}_{\phi_{a}=0,\phi_{r}=\phi}=\frac{\delta^{2}\Gamma}{\delta\phi_{a}(x)\delta\phi_{r}(y)}\Big{|}_{\phi=0}\phi(y)+\frac{1}{2}\frac{\delta^{3}\Gamma}{\delta\phi_{a}(x)\delta\phi_{r}(y)\delta\phi_{r}(z)}\Big{|}_{\phi=0}\phi(y)\phi(z)+\dots=-K(x)\,,$
(A.28)

where we have included the next-to-leading order for illustration. The
solution is the expectation value of the field in the presence of the external
current $K$, that is

$\expectationvalue{\phi}_{K}(x)=\functionalderivative{W}{K_{a}(x)}\Big{|}_{K_{a}=0,K_{r}=K}=\int\differential{y}G_{R}(x,y)K(y)+\int\differential{y}\differential{z}C_{R}(x,y,z)K(y)K(z)+\dots\,,$
(A.29)

with $G_{R}$ given in Eq. A.25 and
$C_{R}(x,y,z)=-\theta(t_{x}-t_{y})\theta(t_{y}-t_{z})\expectationvalue{[[\phi_{x},\phi_{y}],\phi_{z}]}$.
More generally, all the higher-order terms involve nested commutators [57,
55]. This result can also be obtained from a direct in-in calculation [58]. In
this paper, we focus on the leading term which is linear in the external
current. In the statistical mechanics literature, the linear approximation is
usually referred to as the Kubo formula [59].

##  Appendix B Linear response

In this section we review some standard results in linear response theory.
Consider an action $S$ which has been deformed by adding a set of background
fields $K_{n}$ to be called $S_{K}$. We do not restrict the way the action is
deformed but we assume that it remains local. We define a set of composite
operators as

$\mathcal{O}_{n}(x)\equiv\functionalderivative{S_{K}}{K_{n}(x)}\,.$ (B.1)

A simple example is the addition of external currents $S[\phi]\to S[\phi]+\int
K(x)\phi(x)$ as discussed below Eq. A.10 in which case the operator
$\mathcal{O}$ is the dynamical field $\phi$. In general $\mathcal{O}_{n}$ can
be a function of the dynamical fields and their (spatial and time) derivatives
as well as the background fields. For real background fields, the operators
are Hermitian. We would like to study the evolution of the expectation value
$\expectationvalue{\mathcal{O}_{n}(x)}$, in some given density matrix, up to
linear order in the background fields $K_{n}$. Notice that
$\mathcal{O}_{n}(x)$ is the Heisenberg operator evolved with the full
Hamiltonian including the effect of background fields. From Eq. A.8,
$\expectationvalue{\mathcal{O}_{n}(x)}_{K}$ is given by

$\expectationvalue{\mathcal{O}_{n}(x)}_{K}=\int\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\,e^{i(S_{K}[\phi_{1}]-S_{K}[\phi_{2}])}\,\mathcal{O}_{n}(x)\,,$
(B.2)

in which $\phi$ collectively represents the dynamical degrees of freedom in
the system. We notice that up to linear order in the background fields

$\mathcal{O}_{n}(x)=\mathcal{O}_{n}^{(0)}(x)+\int\differential[4]{y}\mathcal{O}_{nm}^{(1)}(x,y)K_{m}(y)+\dots$
(B.3)

in which
$\mathcal{O}_{n}^{(0)}(x)=\functionalderivative{S_{K}}{K_{n}(x)}\big{|}_{K=0}$
and $\mathcal{O}_{nm}^{(1)}(x,y)=\frac{\delta^{2}S_{K}}{\delta K_{n}(x)\delta
K_{m}(y)}\big{|}_{K=0}$. Although we have used a more general notation here,
there is an implicit delta function (or its derivatives) hidden in the second
term of Eq. B.3 resulting from the locality assumption. In addition, up to
linear order, the deformed action is
$S_{K}=S+\int\differential[4]{y}\mathcal{O}_{n}^{(0)}(y)K_{n}(y)+\dots$.
Therefore, expansion of Eq. B.2 up to linear order gives

$\expectationvalue{\mathcal{O}_{n}(x)}_{K}-\expectationvalue{\mathcal{O}_{n}(x)}_{K=0}=\int\differential[4]{x^{\prime}}G_{nm}(x,x^{\prime})K_{m}(x^{\prime})\,,$
(B.4)

with

$G_{nm}(x,x^{\prime})\equiv
i\theta(t-t^{\prime})\expectationvalue{[\mathcal{O}_{n}^{(0)}(x),\mathcal{O}_{m}^{(0)}(x^{\prime})]}+\expectationvalue{\mathcal{O}_{nm}^{(1)}(x,x^{\prime})}\,.$
(B.5)

Equivalently, this result can be derived from the standard in-in formalism
using the interaction picture. The linear term in Eq. A.29 is a special case
of Eq. B.4 when the deformation is an external current. The relevant example
in this work is matter coupled to an external electromagnetic field
$A^{\mu}_{\rm ext}$. This is equivalent to considering the following
deformation of the action we started with in Eq. 2.1:
$S_{\gamma}[a]+S_{M}[a+A_{\rm ext},\psi]$. Then it is simply seen that
$\mathcal{O}^{(0)}$ and $\mathcal{O}^{(1)}$ corresponds to $J^{\mu}$ and
$N^{\mu\nu}$ respectively defined in Eq. 2.6 and the analogue of Eq. B.5 will
be Eq. 2.36.

Introducing background fields changes the energy of the system. At each moment
in time, the total energy of the system is the expectation value of the
Hamiltonian $\Tr(\rho H_{K})$ where $H_{K}$ is the Hamiltonian modified by the
background fields $K_{n}$. Let us assume that the background fields vanish at
$t=\pm\infty$ and are turned on for some period of time. The change in the
energy of the system is calculated by $\Delta
H=\int_{-\infty}^{+\infty}\differential{t}\derivative{t}\Tr(\rho H_{K})$. The
time derivative of the density matrix, switching to the Schrödinger picture,
is $i\dot{\rho}=[H_{K},\rho]$, so that its contribution vanishes after taking
the trace. The remaining term comes from the explicit time dependence in
$H_{K}$ associated to the background fields $K_{n}$. We note that

$\partialderivative{\mathcal{H}_{K}}{K_{n}}=\pi\partialderivative{\dot{\phi}}{K_{n}}-\left[\partialderivative{\mathcal{L}_{K}}{\dot{\phi}}\partialderivative{\dot{\phi}}{K_{n}}+\partialderivative{\mathcal{L}_{K}}{K_{n}}\right]=-\partialderivative{\mathcal{L}_{K}}{K_{n}}\,,$
(B.6)

in which $\mathcal{H}_{K}$ and $\mathcal{L}_{K}$ are Hamiltonian and
Lagrangian densities and
$\pi\equiv{\partial}\mathcal{L}_{K}/{\partial}\dot{\phi}$ is the momentum
conjugate to $\phi$. For simplicity of notation we have only used one
dynamical field $\phi$, with an obvious generalization to more complicated
systems. We should note that for constructing the Hamiltonian we regard
$\dot{\phi}(\phi,\pi,K)$ as a function of $\pi$ as well as the background
fields. Therefore, we can write

$\begin{split}\Delta
H&=\int\differential[4]{x}\Tr(\rho\dot{\mathcal{H}}_{K})=\int\differential[4]{x}\Tr(\rho\partialderivative{\mathcal{H}_{K}}{K_{n}})\dot{K}_{n}=-\int\differential[4]{x}\expectationvalue{\mathcal{O}_{n}(x)}_{K}\dot{K}_{n}\\\
&=-\int\frac{\differential[4]{p}}{(2\pi)^{4}}i\omega
K_{n}(-p)G_{nm}(p)K_{m}(p)\\\
&=\int\frac{\differential[4]{p}}{(2\pi)^{4}}\omega
K_{n}(-p)\left(\frac{G_{nm}(p)-G_{mn}(-p)}{2i}\right)K_{m}(p)\,.\end{split}$
(B.7)

In the first line, we have used Eq. B.6 and Eq. B.1. In the second line, we
have restricted to the linear response formula Eq. B.4 assuming, as it is
relevant for the context of this paper, that
$\expectationvalue{\mathcal{O}_{n}(x)}_{K=0}=0$ and switched to Fourier space.
In the last line, we have used the reality of the integral to replace $G_{nm}$
with the expression in the parenthesis. Notice that for Hermitian
$\mathcal{O}_{n}$, the Green’s function is real implying that
$G_{nm}(-p)=G_{nm}(p)^{*}$. The expression inside the parenthesis is called
the dissipative part of the response function. This can be simplified assuming
spacetime inversion (parity and time reversal) symmetry of the state. First of
all, note that the contribution of the contact term in Eq. B.5 is symmetric
under $n\leftrightarrow m$. The reason is that it is symmetric under
$(n,x)\leftrightarrow(m,x^{\prime})$ but because of the delta function it is
also symmetric under $x\leftrightarrow x^{\prime}$. Focusing on the first term
of Eq. B.5 we have (see also [60])

$\begin{split}G_{mn}(x)&=i\theta(t)\Tr(\rho[\mathcal{O}^{(0)}_{m}(x),\mathcal{O}^{(0)}_{n}(0)])=-i\theta(t)\Tr(\rho[\mathcal{O}^{(0)}_{n}(-x),\mathcal{O}^{(0)}_{m}(0)])\\\
&=-i\theta(t)(\eta_{n}\eta_{m})^{*}\Tr(\Theta\rho[\mathcal{O}^{(0)}_{n}(x),\mathcal{O}^{(0)}_{m}(0)]\Theta^{-1})\\\
&=-i\theta(t)(\eta_{n}\eta_{m})^{*}\Tr((\rho[\mathcal{O}^{(0)}_{n}(x),\mathcal{O}^{(0)}_{m}(0)])^{\dagger})\\\
&=(\eta_{n}\eta_{m})^{*}G_{nm}(x)\,.\end{split}$ (B.8)

In the first line we have used spacetime translation invariance to shift the
argument inside the commutator. In the second line we have used the spacetime
inversion operator $\Theta$ and that the operator transforms as
$\Theta\mathcal{O}^{(0)}_{m}(x)\Theta^{-1}=\eta_{m}\mathcal{O}^{(0)}_{m}(-x)$
with some phase $\eta_{m}$ which depends on the operator. Moreover, we have
used invariance of the density matrix under spacetime inversion, i.e.
$\Theta\rho\Theta^{-1}=\rho$.404040If the set of operators are in the trivial
representation under spacetime transformations, i.e. scalar operators, then it
is enough that the system is invariant under time-reversal and the role of
parity is played by rotation symmetry. Otherwise, we also need invariance
under parity to rewrite Eq. B.7 in terms of $\imaginary G_{mn}$. For instance,
in the case of the current operator $J^{\mu}$, it is possible to have a parity
violating term in the two point function (see Footnote 9). This term, while
contributing to the expression for the imaginary part, does not contribute to
the dissipative part in Eq. B.7. Finally, we use the fact that $\Tr(\Theta
A\Theta^{-1})=\Tr(A^{\dagger})$ for any linear operator $A$.414141Remember
that for an antiunitary operator $\Theta$, if we define
$\ket{\tilde{a}}\equiv\Theta\ket{a}$ and
$|\tilde{b}\rangle\equiv\Theta\ket{b}$, we have
$\innerproduct{a}{b}=\langle\tilde{b}|\tilde{a}\rangle$. Then it is easy to
see that for any linear operator $A$, we have
$\bra{b}A^{\dagger}\ket{a}=\bra{\tilde{a}}\Theta
A\Theta^{-1}|\tilde{b}\rangle$. The trace $\Tr
A^{\dagger}=\sum_{n}\bra{n}A^{\dagger}\ket{n}$ can be written as
$\sum_{\tilde{n}}\bra{\tilde{n}}\Theta A\Theta^{-1}\ket{\tilde{n}}$, which is
$\Tr(\Theta A\Theta^{-1})$. See also [61]. Hence we conclude that the response
function is symmetric in its indices up to some phase factors. In cases in
which the phase factors are unity, the expression for the dissipative part of
the response function in Eq. B.7 is simply $\imaginary G_{nm}(p)$.

##  Appendix C Lindhard function

In this section we study the electromagnetic properties of a gas of fermions,
e.g. electrons, at finite density and zero temperature. For finite temperature
calculations see for example [62]. The condition of finite density of charged
particles can be modeled by adding a chemical potential to the Lagrangian. See
also [63, 64, 65]. Therefore a Dirac fermion $\psi$ at finite chemical
potential $\mu$ can be described by

$\mathcal{L}=i\bar{\psi}(\not{{\partial}}+i\mu\not{u})\psi-m\bar{\psi}\psi$
(C.1)

in which $u^{\mu}=(1,0)$ is the rest frame velocity of the fermions and $m$
their mass. We assume $\mu>m>0$. Then the ground state of the system consists
of fermions filling up to energy $E_{F}=\sqrt{m^{2}+k_{F}^{2}}=\mu$, known as
the Fermi energy, where $k_{F}$ is called the Fermi momentum.

The photon coupling is similar to the Lorentz-invariant case, i.e.
${\partial}_{\mu}\to{\partial}_{\mu}-iA_{\mu}$. We want to calculate the
photon (retarded) self-energy which is given by
$\expectationvalue{[J^{\mu},J^{\nu}]}_{\rm 1PI}$ with
$J^{\mu}=\bar{\psi}\gamma^{\mu}\psi$, perturbatively in the coupling to the
photon. At leading order we can can drop the 1PI index, corresponding to the
diagram shown in Fig. 6. Moreover, it is easier to first calculate the diagram
using the usual Feynman rules for the $S$-matrix (which gives the time ordered
correlation function) and then obtain the retarded response function by moving
the singularities with an appropriate $i\epsilon$ prescription [66].

$q^{0}$$\mu$

Figure 6: Left: Leading order correction to the photon self-energy from the
gas of fermions at finite chemical potential. Right: The contour of
integration in the presence of the chemical potential.

The Feynman propagator of the modified Lagrangian Eq. C.1 is

$S(x-y)\equiv
i\expectationvalue{T\psi(x)\bar{\psi}(y)}=\int\frac{\differential[4]{p}}{(2\pi)^{4}}e^{ip\cdot(x-y)}\frac{-\not{p}^{\prime}+m}{p^{\prime}{}^{2}+m^{2}-i\epsilon\,{\rm
sgn}(\omega(\omega+\mu))}\,,$ (C.2)

in which $p^{\prime}{}^{\mu}=p^{\mu}+\mu u^{\mu}=(\omega+\mu,\boldsymbol{k})$.
Notice that in the absence of the chemical potential we recover the Lorentz-
invariant $i\epsilon$ prescription for the propagator. Therefore, by using the
standard Feynman rules of quantum electrodynamics, for the diagram of Fig. 6
we get

$i\Pi^{\mu\nu}=-\int\frac{\differential[4]{q}}{(2\pi)^{4}}\Tr(\gamma^{\mu}S(q)\gamma^{\nu}S(q+p))\,.$
(C.3)

After some algebra one finds that

$i\Pi^{\mu\nu}=-\int\frac{\differential[4]{q}}{(2\pi)^{4}}\frac{4I^{\mu\nu}(q)}{[-(q^{0}+i\epsilon\,{\rm
sgn}(q^{0}-\mu))^{2}+E_{\boldsymbol{q}}^{2}][-(q^{0}+\omega+i\epsilon\,{\rm
sgn}(q^{0}+\omega-\mu))^{2}+E_{\boldsymbol{q}+\boldsymbol{k}}^{2}]}\,,$ (C.4)

with $E_{\boldsymbol{q}}=\sqrt{m^{2}+\boldsymbol{q}^{2}}$. To obtain Eq. C.4
we have changed the variable of integration $q^{0}\to q^{0}-\mu$ and used
properties of the gamma matrices, e.g.
$\\{\gamma^{\mu},\gamma^{\nu}\\}=-2\eta^{\mu\nu}$ in our metric signature, to
simplify the numerator as

$I^{\mu\nu}(q)=q^{\mu}(p+q)^{\nu}+q^{\nu}(p+q)^{\mu}-(q\cdot(p+q)+m^{2})\eta^{\mu\nu}\,.$
(C.5)

The denominator has poles at $q^{0}=\pm E_{\boldsymbol{q}}-i\epsilon\,{\rm
sgn}(q^{0}-\mu)$ and $q^{0}+\omega=\pm
E_{\boldsymbol{q}+\boldsymbol{k}}-i\epsilon\,{\rm sgn}(q^{0}+\omega-\mu)$. One
can think of the effect of the nonzero chemical potential in terms of
deforming the $q^{0}-$integration contour. This is shown in Fig. 6 with the
solid line. An illustrative way to perform the integral is to decompose it
into a contour used in the Lorentz-invariant case with $\mu=0$ (dashed line in
Fig. 6) plus a closed curve which captures the effect of the chemical
potential (dash-dotted line). Thus we can write the result as follows

$\Pi^{\mu\nu}=\Pi^{\mu\nu}_{\rm qed}+\Pi^{\mu\nu}_{F}\,.$ (C.6)

To extract the longitudinal and transverse parts we use definition Eq. 2.24
which implies that

$\pi_{L}=\frac{1}{k^{2}}\Pi^{00}\,,\qquad\pi_{T}=\frac{1}{2k^{2}}(\delta_{ij}\Pi^{ij}-\omega^{2}\pi_{L})\,.$
(C.7)

The longitudinal and transverse parts can be decomposed similarly to Eq. C.6.
The first term is the Lorentz-invariant result which can be found in any
quantum field theory textbook. The condition of Lorentz invariance implies
that there is no distinction between longitudinal and transverse directions,
i.e. $\pi_{L}=-k^{2}\pi_{T}/p^{2}$, therefore $\Pi^{\mu\nu}_{\rm
qed}=\pi_{L,{\rm qed}}(p^{\mu}p^{\nu}-p^{2}\eta^{\mu\nu})$, with424242This is
the result of on-shell renormalization scheme, $\Pi^{\mu\nu}_{\rm qed}(0)=0$.

$\pi_{L,{\rm qed}}=-\frac{k^{2}}{p^{2}}\pi_{T,{\rm
qed}}=-\frac{1}{2\pi^{2}}\int_{0}^{1}\differential{x}x(1-x)\log(\frac{x(1-x)p^{2}+m^{2}}{m^{2}})\,.$
(C.8)

The next term comes from the finite chemical potential and can be calculated
as

$\Pi^{\mu\nu}_{F}(\omega,k)=-2\int\frac{\differential[3]{\boldsymbol{q}}}{(2\pi)^{3}}\frac{\theta(E_{F}-E_{\boldsymbol{q}})}{E_{\boldsymbol{q}}}\left[\frac{I^{\mu\nu}(q^{0}=E_{\boldsymbol{q}},\boldsymbol{q})}{(E_{\boldsymbol{q}}+\omega)^{2}-E_{\boldsymbol{q}+\boldsymbol{k}}^{2}}+\big{(}\omega\to-\omega,\boldsymbol{k}\to-\boldsymbol{k}\big{)}\right]\,.$
(C.9)

We can now obtain the retarded correlation functions by imposing the
appropriate $i\epsilon$ prescription; in both expressions Eq. C.8 and Eq. C.9,
it is understood that $\omega\to\omega+i\epsilon$ to have the correct
analyticity properties.

For the longitudinal part we can perform the angular part of the integral and
obtain

$\pi_{L,F}=\int_{0}^{k_{F}}\frac{q^{2}\differential{q}}{2\pi^{2}k^{2}E_{q}}\left[1-\frac{(\omega+2E_{q})^{2}-k^{2}}{4kq}\log(\frac{\omega(\omega+2E_{q})+2kq-k^{2}}{\omega(\omega+2E_{q})-2kq-k^{2}})\right]+\big{(}\omega\to-\omega\big{)}\,.$
(C.10)

Similarly, for the transverse component we obtain

$\begin{split}\pi_{T,F}=\int_{0}^{k_{F}}\frac{q^{2}\differential{q}}{4\pi^{2}k^{2}E_{q}}\Bigg{[}-\left(1+\frac{\omega^{2}}{k^{2}}\right)+&\frac{\omega^{2}(\omega+2E_{q})^{2}-k^{2}(k^{2}+4q^{2}+4\omega
E_{q})}{4k^{3}q}\times\\\
&\log(\frac{\omega(\omega+2E_{q})+2kq-k^{2}}{\omega(\omega+2E_{q})-2kq-k^{2}})\Bigg{]}+\big{(}\omega\to-\omega\big{)}\,.\end{split}$
(C.11)

The above integrals can also be performed analytically but the expressions are
not very illuminating so we do not report them here.

It is instructive to check the conditions of analyticity and positivity for
these functions. Let us first focus on the Lorentz-invariant piece in Eq. C.8.
Since this is the outcome of a perturbative calculation, following the
discussion in §7, $\pi_{L,{\rm qed}}$ (and similarly $k^{2}\pi_{T,{\rm qed}}$)
must be analytic in the FLC. The only source of non-analyticity is the branch
cut of the $\log$ function which for the principal branch locates along the
negative real axis of the argument. Therefore, from Eq. C.8 it is easy to see
that the location of the branch cut is $p^{2}+m^{2}/(x(1-x))\leq 0$ or
equivalently $-(\omega+i\epsilon)^{2}+k^{2}+M^{2}=0$ for any $M^{2}\geq
4m^{2}$. As we will argue in detail in App. E, solutions to this equation
always lie outside the FLC and therefore $\pi_{L,{\rm qed}}$ is analytic in
the FLC. We can calculate the imaginary part of Eq. C.8 for real $\omega$ and
$k$ as follows

$\begin{split}\imaginary\pi_{L,{\rm
qed}}&=-\frac{1}{2\pi^{2}}\int_{0}^{1}\differential{x}x(1-x)\theta\big{[}-x(1-x)p^{2}-m^{2}\big{]}\pi{\rm
sgn}(-\omega)\\\
&=\frac{1}{24\pi}\left[1+\left(1+\frac{2m^{2}}{-p^{2}}\right)\sqrt{1-\frac{4m^{2}}{-p^{2}}}\right]{\rm
sgn}(\omega)\theta(-p^{2}-4m^{2})\,.\end{split}$ (C.12)

In the first line we have used $\log(-|x|\pm i\epsilon)=\log(|x|)\pm i\pi$ and
the fact that the imaginary part of the argument of the $\log$ in Eq. C.8
comes with the combination $\sim-i\epsilon\,\omega$ which gives the sign
function. As is well-known, the imaginary part is nonzero only for timelike
momenta larger than the pair production threshold, i.e. $-p^{2}>4m^{2}$, after
which it grows positive and saturates to $1/12\pi$ for $\omega>0$.

The contribution of the finite chemical potential in Eq. C.10 and Eq. C.11 is
slightly more involved. First we deal with the $\log$ which has a similar
structure both for the longitudinal and transverse parts. Notice that one can
re-write the argument of the logarithm, in a more suggestive way, as
$(\omega\pm E_{q}+i\epsilon)^{2}-(k\pm q)^{2}-m^{2}$. Therefore, the location
of the branch cut is similar to the Lorentz-invariant case and therefore
outside the FLC. Notice that the shifts $\omega\to\omega\pm E_{q}$ and $k\to
k\pm q$ do not affect this conclusion as $E_{q}$ and $q$ are real. Another
source of non-analyticity are the factors $1/k^{2}$. However, it is
straightforward to take the $k^{2}\to 0$ limit of the integrand

$\displaystyle\pi_{L,F}$
$\displaystyle=\int_{0}^{k_{F}}q^{2}\differential{q}\left[\frac{4(-3E_{q}^{2}+q^{2})}{3\pi^{2}E_{q}(4E_{q}^{2}-\omega^{2})\omega^{2}}+\order{k^{2}}\right]$
(C.13) $\displaystyle k^{2}\pi_{T,F}$
$\displaystyle=\int_{0}^{k_{F}}q^{2}\differential{q}\left[\frac{4(-3E_{q}^{2}+q^{2})}{3\pi^{2}E_{q}(4E_{q}^{2}-\omega^{2})}+\order{k^{2}}\right]\,,$
(C.14)

which is regular. Further, one could also worry about the appearance of
$k=\sqrt{k^{2}}$ in Eq. C.10 and Eq. C.11. The absence of an $\order{k}$ term
in the expansion above shows that $k=0$ is not a branch point. Therefore, both
$\pi_{L,F}$ and $k^{2}\pi_{T,F}$ are analytic in the FLC as required by
causality, as discussed in §4. In addition, we see from Eq. C.13 and Eq. C.14
that the combination $k^{2}\pi_{T,F}-\omega^{2}\pi_{T,F}=\order{k^{2}}$, a
consequence of causality. The imaginary parts of $\pi_{L,F}$ and $\pi_{T,F}$
can be obtained by a similar calculation presented in Eq. C.12, although the
result will be more complicated. We will calculate the imaginary part in the
non-relativistic limit below.

### Non-relativistic limit

We note that if all the energies are much smaller than the mass, i.e.
$\omega,k\ll m$, one can neglect the Lorentz-invariant piece and the integrand
of the second term can be expanded as follows

$\Pi_{F}^{\mu\nu}\simeq-\int\frac{\differential[3]{\boldsymbol{q}}}{(2\pi)^{3}}\frac{\theta(\tilde{E}_{F}-\tilde{E}_{\boldsymbol{q}})}{m^{2}}\left[\frac{I^{\mu\nu}(m+\tilde{E}_{\boldsymbol{q}},\boldsymbol{q})}{\tilde{E}_{\boldsymbol{q}}-\tilde{E}_{\boldsymbol{q}+\boldsymbol{k}}+\omega}+\big{(}\omega\to-\omega,\boldsymbol{k}\to-\boldsymbol{k}\big{)}\right]\,,$
(C.15)

where we have defined $E_{\boldsymbol{q}}=m+\tilde{E}_{\boldsymbol{q}}$ with
$\tilde{E}_{\boldsymbol{q}}\approx q^{2}/2m$ in the non-relativistic limit and
$\tilde{E}_{F}\equiv E_{F}-m\approx k_{F}^{2}/2m\ll m$. For the longitudinal
component we have $I^{00}\simeq 2m^{2}$ and therefore we can do the integrals
and obtain

$\pi_{L}=\frac{mk_{F}}{2\pi^{2}k^{2}}\left[\frac{1}{2}-\frac{(k^{2}-2m\omega)^{2}-4k_{F}^{2}k^{2}}{8k_{F}k^{3}}\log(\frac{k^{2}+2k_{F}k-2m\omega}{k^{2}-2k_{F}k-2m\omega})\right]+\big{(}\omega\to-\omega\big{)}\,.$
(C.16)

This result has famously been obtained by Lindhard [39] and is known as
Lindhard function. The prefactor is sometimes written in terms of the plasma
frequency $\omega_{p}^{2}\equiv g^{2}n/m$ and the Fermi velocity $v_{F}\equiv
k_{F}/m$, i.e.
$\frac{mk_{F}}{2\pi^{2}}=\frac{3\omega^{2}_{p}}{2g^{2}v_{F}^{2}}$. To derive
this we use the fact that the number density of fermions is
$n=k_{F}^{3}/3\pi^{2}$. Similarly, for the transverse piece we obtain
$\delta_{ij}I^{ij}\simeq
3m\omega+\boldsymbol{q}\cdot(2\boldsymbol{q}-\boldsymbol{k})$. Notice that in
the non-relativistic approximation we keep terms like $m\omega\sim k^{2}\sim
kk_{F}$. The rest of the calculation is very similar to the longitudinal case.
The final answer is

$\begin{split}\pi_{T}=\frac{k_{F}}{32\pi^{2}mk^{4}}\Bigg{[}&\frac{1}{2}(3k^{4}-12m^{2}\omega^{2}-4k_{F}^{2}k^{2})\\\
&-\frac{(4k^{4}-[(k^{2}-2m\omega)^{2}-4k_{F}^{2}k^{2}])[(k^{2}-2m\omega)^{2}-4k_{F}^{2}k^{2}]}{8k_{F}k^{3}}\log(\frac{k^{2}+2k_{F}k-2m\omega}{k^{2}-2k_{F}k-2m\omega})\Bigg{]}\\\
&+\big{(}\omega\to-\omega\big{)}\,.\end{split}$ (C.17)

The above expression differs slightly from what was given by Lindhard [39];
the reason is likely due to the fact that the spin of fermions has been
neglected in his calculation. It is useful to look at the above results in the
low-energy and high-energy limits. We are mainly interested in the functions
parametrized as $\pi_{L}(\omega,\omega\xi)$ and $\pi_{T}(\omega,\omega\xi)$
for $\xi<1$ (see §4). Taking $\omega\to 0$ in this parametrization we obtain

$g^{2}\pi_{L}(\omega,\omega\xi)=-\frac{\omega_{p}^{2}}{\omega^{2}}+\dots\,,\qquad
g^{2}\pi_{T}(\omega,\omega\xi)=-\frac{\omega_{p}^{2}}{\omega^{2}\xi^{2}}+\dots\,.$
(C.18)

Equivalently, in terms of electric permittivity and magnetic permeability we
get

$\varepsilon=1-\frac{\omega_{p}^{2}}{\omega^{2}}+\dots\,,\qquad\frac{1}{\mu}=1-\left(\frac{2k_{F}^{2}}{5m^{2}}\right)\frac{\omega_{p}^{2}}{\omega^{2}}+\dots\,,\quad(\omega\to
0)\,.$ (C.19)

To obtain the expansion for $\mu$ one needs to be careful about the expansion
of the combination $\pi_{T}-\omega^{2}\pi_{L}/k^{2}$ as naively the leading
order terms given in Eq. C.19 vanish. Moreover, the transverse part of
dielectric tensor, as defined in footnote 13, is
$\varepsilon_{T}=1-\omega_{p}^{2}/\omega^{2}+\dots$. One can understand this
result by noting that all the charged particles in this system are freely
moving. Therefore, at low energies it behaves like a conductor which in this
case is very well modeled by the simple Drude model (see the discussion in
§8). We remind the reader that the $i\epsilon$ prescription is understood here
as $(\omega+i\epsilon)^{2}=\omega^{2}+i\epsilon\omega$. The effect of
interaction with a background lattice or with the other fermions can be
modeled by adding a finite decay width, i.e. $\omega^{2}+i\gamma\omega$. In
that case, a useful quantity is conductivity $\sigma_{(T)}\equiv
i\omega(1-\varepsilon_{(T)})$ which in the present case gives, at low
frequencies, $\sigma_{(T)}=\omega_{p}^{2}/\gamma$.

For high energies, we note that the main interest is in the range
$k_{F}\ll\omega\ll m$ for the non-relativistic limit. In that case one obtains

$g^{2}\pi_{L}(\omega,\omega\xi)=-\frac{\omega_{p}^{2}}{\omega^{2}}+\dots\,,\qquad
g^{2}\pi_{T}(\omega,\omega\xi)=-\frac{\omega_{p}^{2}}{\omega^{2}\xi^{2}}+\dots\,,\quad(\omega\to\infty)\,.$
(C.20)

Moreover, we have
$g^{2}(\pi_{T}-\omega^{2}\pi_{L}/k^{2})=2k_{F}^{2}\omega_{p}^{2}/(5m^{2}\omega^{2})+\dots$
relevant for $\mu$. We will use these relations in §5 and §6. The fact that
the low and high energy limits in Eq. C.18 and Eq. C.20 turn out to be the
same is a consequence of taking the non-relativistic limit, i.e. $k_{F}\ll m$
or $v_{F}\ll 1$. Setting $k=\omega\xi$ for $\xi\ll 1/v_{F}$ and for energies
$\omega\ll m$ is effectively equivalent to taking $k\to 0$ limit for all
$\omega$ in this range.

Finally, let us calculate the imaginary part of the Lindhard function in the
non-relativistic limit. A nonzero imaginary part comes from the logs in Eq.
C.16 and Eq. C.17. We can see that

$\begin{split}\imaginary\log(\frac{k^{2}+2k_{F}k-2m\omega}{k^{2}-2k_{F}k-2m\omega})&=-\pi\Big{[}\theta\big{(}2m\omega-k^{2}-2k_{F}k\big{)}-\theta\big{(}2m\omega-k^{2}+2k_{F}k\big{)}\Big{]}\\\
&=\pi\theta\big{(}2k_{F}k-|2m\omega-k^{2}|\big{)}\,.\end{split}$ (C.21)

To obtain the first line we have used $\log(\cdots)\to\log(\cdots-i\epsilon)$.
To obtain the second line we note that if $|2m\omega+k^{2}|>2k_{F}k$ both of
the theta functions are either zero or one. Notice that for the part
$(\omega\to-\omega)$ in both Eq. C.16 and Eq. C.17, we have
$\log(\dots+i\epsilon)$ and therefore the imaginary part has a minus sign
compared to Eq. C.21. After some algebra we obtain

$\imaginary\pi_{L}=\frac{m}{16\pi
k^{5}}\Big{[}[4k_{F}^{2}k^{2}-(k^{2}-2m\omega)^{2}]\theta\big{(}2k_{F}k-|2m\omega-k^{2}|\big{)}-[4k_{F}^{2}k^{2}-(k^{2}+2m\omega)^{2}]\theta\big{(}2k_{F}k-|2m\omega+k^{2}|\big{)}\Big{]}\,,$
(C.22)

and for the transverse part

$\begin{split}\imaginary\pi_{T}=\frac{1}{256\pi
mk^{7}}\Big{[}&(4k^{4}+[4k_{F}^{2}k^{2}-(k^{2}-2m\omega)^{2}])[4k_{F}^{2}k^{2}-(k^{2}-2m\omega)^{2}]\theta\big{(}2k_{F}k-|2m\omega-k^{2}|\big{)}\\\
&-(4k^{4}+[4k_{F}^{2}k^{2}-(k^{2}+2m\omega)^{2}])[4k_{F}^{2}k^{2}-(k^{2}+2m\omega)^{2}]\theta\big{(}2k_{F}k-|2m\omega+k^{2}|\big{)}\Big{]}\,.\end{split}$
(C.23)

### Gapped Lindhard

As mentioned above, at low energies, the Lindhard function describes a
conductor rather than an insulator. However, starting from the Lindhard
function, it is possible to build a model for an insulator by introducing a
gap. Following [67], we modify the imaginary part of the Lindhard function as
follows

$\imaginary\tilde{\pi}_{L}(\omega,\boldsymbol{k})=\begin{cases}\imaginary\pi_{L}\Big{(}{\rm
sgn}(\omega)\sqrt{\omega^{2}-E_{g}^{2}},\boldsymbol{k}\Big{)}\,,\qquad&|\omega|\geq
E_{g}\,,\\\ 0\,,\qquad&|\omega|<E_{g}\,,\end{cases}$ (C.24)

with $E_{g}$ the energy gap. A similar modification can be done for the
transverse part. Next, one can construct the modified real part from Eq. C.24
by using the Kramers-Kronig relations (or equivalently Leontovich’s relation
with $\boldsymbol{\xi}=\boldsymbol{0}$). One can verify that, for instance, we
obtain $\varepsilon(0,0)=1+\omega_{p}^{2}/E_{g}^{2}$. See [67] for details.
Notice that this construction is non-relativistic and it is not obvious how it
can be generalized for the relativistic cases Eq. C.10 and Eq. C.11. The
modification must be such that the modified imaginary part satisfies the
condition Eq. 4.10.

##  Appendix D The absence of zeros

In this section we argue that a response function $\chi(\omega)$ cannot have
zeros in the UHP of complex $\omega$. Notice that by causality, the response
$\chi(t)$ must vanish for $t\leq 0$ which implies that $\chi(\omega)$ is
analytic in the upper half $\omega$-plane. We also assume that it vanishes in
the high-energy limit, $\chi(\omega)\to 0$ for $|\omega|\to\infty$. Moreover,
$\chi$ must satisfy a positivity condition, that is, $\omega\imaginary\chi\geq
0$ for $\omega\in\mathbb{R}$. Finally, by reality of the response function
$\chi(t)$ we have $\chi(\omega)^{*}=\chi(-\omega^{*})$. In particular, this
implies that $\chi$ is real on the imaginary axis. Using the above assumptions
it is possible to argue that $\chi(\omega)$ does not have zeros in the UHP. A
stronger version of this was provided by Landau [68] (apparently it was first
proposed by N. N. Meǐman): the equation $\chi(\omega)=a$ for real $a$ has, at
most, one solution in the UHP. Therefore, $\chi(\omega)$ along the imaginary
axis is monotonically decreasing from $\chi(0)$ to zero and elsewhere is
always complex. This argument is very well explained in [68] (§123) and we do
not repeat it here. Instead we give a simpler version of the proof and discuss
a possible extension when we also have nonzero momentum.

It has been argued in Eq. 4.9 that we can always write the response function
in the domain of analyticity as an integral over its imaginary part.
Restricting to the single-variable case we get

$\chi(\omega)=\frac{1}{\pi}\int_{0}^{\infty}\frac{\differential{z^{2}}}{z^{2}-(\omega+i\epsilon)^{2}}\,\,\imaginary\chi(z)\,,$
(D.1)

where we have used the fact that $\imaginary\chi(\omega)$ is odd. The above
relation is valid for complex $\omega$ in the UHP. Restricting to the
imaginary axis $\omega=i\omega_{I}$ we obtain

$\chi(i\omega_{I})=\frac{1}{\pi}\int_{0}^{\infty}\frac{\differential{z^{2}}}{z^{2}+\omega_{I}^{2}}\,\,\imaginary\chi(z)>0\,,$
(D.2)

which is strictly positive (otherwise it means that the imaginary part is
always zero and therefore the function is everywhere zero) and monotonically
decreasing. On the other hand, away from the imaginary axis we can calculate
the imaginary part from Eq. D.1

$\imaginary\chi(\omega)=\frac{\imaginary\omega^{2}}{\pi}\int_{0}^{\infty}\frac{\differential{z^{2}}}{|z^{2}-\omega^{2}|^{2}}\,\,\imaginary\chi(z)\,,$
(D.3)

which means that $\imaginary\chi(\omega)$ has the same sign as
$\imaginary\omega^{2}=2\omega_{I}\omega_{R}$ and in particular it is nonzero.
We conclude that $\chi(\omega)$ cannot have any zeros in the UHP.

When we have momentum dependence, as in the main text, one can use the
parameterization
$\tilde{\chi}(\omega)\equiv\chi(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$.
Then for any real $\boldsymbol{q}$ and $\boldsymbol{\xi}$ with $\xi<1$ the
function $\tilde{\chi}(\omega)$ is analytic in the UHP for complex $\omega$.
Moreover, the positivity condition for $\chi$ implies the same also for
$\tilde{\chi}$. However, the reality property is slightly different for
$\tilde{\chi}$. Remember that for a generic four-momentum
$p^{\mu}=(\omega,\boldsymbol{k})$ we have $\chi(p)^{*}=\chi(-p^{*})$ and, in
particular, for $p^{\mu}=ip^{\mu}_{I}$ then $\chi(p)$ is real. But in terms of
$\tilde{\chi}$, assuming rotational invariance, we have

$\tilde{\chi}(\omega)^{*}=\chi(\omega,(\boldsymbol{q}+\omega\boldsymbol{\xi})^{2})^{*}=\chi(-\omega^{*},(-\boldsymbol{q}-\omega^{*}\boldsymbol{\xi})^{2})\neq\tilde{\chi}(-\omega^{*})\,.$
(D.4)

Therefore, the reality condition does not imply that
$\tilde{\chi}(\omega)^{*}$ and $\tilde{\chi}(-\omega^{*})$ are equal for
general $\boldsymbol{q}$ and $\boldsymbol{\xi}$. However, if we restrict
ourselves to the points for which $\boldsymbol{q}\cdot\boldsymbol{\xi}=0$ then
$\tilde{\chi}(\omega)^{*}=\tilde{\chi}(-\omega^{*})$. With this condition the
function $\tilde{\chi}(\omega)$ will be real on the imaginary axis. By using
Eq. 4.9, we write

$\chi(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})=\frac{1}{\pi}\int_{0}^{\infty}\frac{\differential{z^{2}}}{z^{2}-(\omega+i\epsilon)^{2}}\,\,\imaginary\chi(z,\boldsymbol{q}+z\boldsymbol{\xi})\,,\qquad(\boldsymbol{q}\cdot\boldsymbol{\xi}=0)\,,$
(D.5)

using $\imaginary\tilde{\chi}(-z)=-\imaginary\tilde{\chi}(z)$. Then the analog
of Eq. D.2 and Eq. D.3 can be easily derived. We conclude that
$\chi(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ cannot have any zeros in
the UHP of complex $\omega$ when $\boldsymbol{q}\cdot\boldsymbol{\xi}=0$.

One may wonder how restrictive this is. Is it possible to find appropriate
$\boldsymbol{q}$ and $\boldsymbol{\xi}$ such that
$k^{2}=(\boldsymbol{q}+\omega\boldsymbol{\xi})^{2}$, with
$\boldsymbol{q}\cdot\boldsymbol{\xi}=0$, for any point in the analyticity
region? If this is true then the function cannot have zeros in the whole
region of analyticity. Unfortunately the answer is no. A little algebra shows
that in order to relate any point $(\omega,\boldsymbol{k})$ to a form
$(\omega,\boldsymbol{q}+\omega\boldsymbol{\xi})$ with
$\boldsymbol{q}\cdot\boldsymbol{\xi}=0$, keeping $k^{2}$ fixed, we need

$\xi^{2}=\frac{\boldsymbol{k}_{R}\cdot\boldsymbol{k}_{I}}{\omega_{R}\omega_{I}}\,,\qquad
q^{2}=k_{R}^{2}-k_{I}^{2}-(\omega_{R}^{2}-\omega_{I}^{2})\xi^{2}\,.$ (D.6)

From the above expression it is clear that the conditions $0<\xi^{2}<1$ and
$q^{2}>0$ are not satisfied for many points in the analyticity region.
Therefore, we cannot exclude the possibility of having zeros, at least from
the arguments given above, for $\chi(\omega,\boldsymbol{k})$ in the whole
analyticity region.

Notice that the condition of positivity is crucial for the above argument. For
instance, let us consider the following function

$\frac{\log(-(\omega+i\epsilon)^{2}+c_{1}^{2}k^{2}+\omega_{0}^{2})-\log(\omega_{1}^{2})}{-(\omega+i\epsilon)^{2}+c_{2}^{2}k^{2}+\omega_{2}^{2}}\,.$
(D.7)

This function goes to zero at infinity and it is analytic in the FLC provided
that $0<c_{i}<1$ and $\omega_{i}^{2}>0$. This ensures that the poles of the
denominator and the branch cut of the $\log$ function lie outside the FLC.
However, we see that it has zeros at
$-(\omega+i\epsilon)^{2}+c_{1}^{2}k^{2}+(\omega_{0}^{2}-\omega_{1}^{2})=0$
which for $\omega_{0}^{2}-\omega_{1}^{2}<0$ lie in the FLC (see App. E for
details). However, if we calculate the imaginary part of this function we see
that in order to have positivity we must have $c_{1}>c_{2}$ and
$\omega_{0}^{2}-\omega_{2}^{2}>\omega_{1}^{2}>0$, i.e.
$\omega_{0}^{2}>\omega_{1}^{2}$. Therefore, the condition of positivity
forbids zeros for the $\log$ function in the FLC, consistent with the above
result.

##  Appendix E Damped wave response

The simplest possible retarded Green’s function is the one associated to the
wave equation with damping,

$\ddot{\psi}+\gamma\dot{\psi}-c_{s}^{2}\nabla^{2}\psi+\omega_{0}^{2}\psi=F\,.$
(E.1)

The Green’s function in Fourier space is given by

$\begin{split}G(\omega,k)=\frac{F}{\psi}&=\frac{1}{-\omega^{2}-i\gamma\omega+c_{s}^{2}k^{2}+\omega_{0}^{2}}\\\
&=\frac{1}{-(\omega+i\gamma/2)^{2}+c_{s}^{2}k^{2}+(\omega_{0}^{2}-\gamma^{2}/4)}\equiv\frac{1}{P^{2}+m^{2}}\,,\end{split}$
(E.2)

where in the second line we have rearranged the terms to write it similarly to
the Lorentz-invariant case. We have defined
$P^{\mu}\equiv(\Omega,\boldsymbol{k})$ with $\Omega=\omega+i\gamma/2$,
$\boldsymbol{k}=c_{s}\boldsymbol{k}$ and $m^{2}=\omega_{0}^{2}-\gamma^{2}/4$.
We would like to find the conditions on the parameters $\gamma$, $c_{s}^{2}$
and $\omega_{0}^{2}$ (assuming they are real numbers) such that it corresponds
to a physical Green’s function, i.e. with no singularities in the FLC of the
imaginary part of $p^{\mu}=(\omega,\boldsymbol{k})$. Singularities of Eq. E.2
satisfy

$P_{1}^{2}-P_{2}^{2}+m^{2}=0\,,\qquad P_{1}^{\mu}P_{2\mu}=0\,,$ (E.3)

where we have decomposed $P^{\mu}=P^{\mu}_{1}+iP^{\mu}_{2}$ into real and
imaginary parts. Next, we consider different possibilities for $P_{1}^{\mu}$.
If it is spacelike then it can be written as
$P_{1}^{\mu}=(0,\boldsymbol{k}_{1})$ and therefore we have

$P_{2}^{2}=m^{2}+k_{1}^{2}\,,\qquad\boldsymbol{k}_{1}\cdot\boldsymbol{k}_{2}=0\,.$
(E.4)

If it is timelike we can write it as $P_{1}^{\mu}=(\Omega_{1},0)$ and
therefore the location of the singularities must satisfy

$P_{2}^{2}=m^{2}-\Omega_{1}^{2}\,,\qquad\Omega_{2}=0\,,$ (E.5)

which, in particular, means that $P_{2}$ is necessarily spacelike. Finally,
for null vectors we write $P_{1}^{\mu}=\Omega_{1}(1,\hat{{n}})$, where
$\hat{n}^{2}=1$, and we have

$P_{2}^{2}=m^{2}\,,\qquad-\Omega_{2}+(\hat{{n}}\cdot\boldsymbol{k}_{2})=0\,.$
(E.6)

In the cases above we have assumed nonzero $P_{1}^{\mu}$; otherwise we would
get $P_{2}^{2}=m^{2}$ without any further restriction. Let us for the moment
assume that $c_{s}^{2}\geq 0$ then in components
$P^{\mu}_{1}=(\omega_{R},c_{s}\boldsymbol{k}_{R})$ and
$P^{\mu}_{2}=(\omega_{I}+\gamma/2,c_{s}\boldsymbol{k}_{I})$.

### Case 1: $m^{2}>0$.

This means $\omega_{0}^{2}>\gamma^{2}/4$. In all the cases discussed above we
can see that if a solution exists (for some choice of $P_{1}^{\mu}$) then
$P_{2}^{\mu}$ must be spacelike. Therefore, we must have

$-c_{s}k_{I}<\omega_{I}+\frac{\gamma}{2}<c_{s}k_{I}\,.$ (E.7)

Notice also that all $P_{2}^{\mu}$ that satisfy (E.7) correspond to poles: it
is indeed straightforward to see that for any $P_{2}^{\mu}$ one can find a
solution for $P_{1}^{\mu}$. But then it means that we must require $\gamma>0$
and $c_{s}\leq 1$ to avoid singularities in the FLC in the plane of
$(\omega_{I},\boldsymbol{k}_{I})$.

### Case 2: $m^{2}<0$.

This means that $\omega_{0}^{2}<\gamma^{2}/4$. Then solutions for
$P_{2}^{\mu}$ can be spacelike, timelike or null. We have already discussed
the spacelike solutions above, see Eq. (E.7), so there is nothing new here.
For timelike or null solutions we note that they always satisfy $0\geq
P_{2}^{2}\geq m^{2}$ and therefore we have

$c_{s}k_{I}\leq\absolutevalue{\omega_{I}+\frac{\gamma}{2}}\leq\sqrt{c_{s}^{2}k^{2}_{I}+\frac{\gamma^{2}}{4}-\omega_{0}^{2}}\,.$
(E.8)

It is straightforward to check that all timelike $P_{2}^{\mu}$ that satisfy
this correspond to poles. But then it means that we must require $\gamma>0$,
$\omega_{0}^{2}\geq 0$ and $c_{s}\leq 1$ to avoid singularities in the FLC in
the plane of $(\omega_{I},\boldsymbol{k}_{I})$. See Fig. 7. The case $m^{2}=0$
does not give further constraints.

### Case 3: $c_{s}^{2}<0$.

In this case we write $c_{s}=i\tilde{c}_{s}$ which means that in components
$P^{\mu}_{1}=(\omega_{R},-\tilde{c}_{s}\boldsymbol{k}_{I})$ and
$P^{\mu}_{2}=(\omega_{I}+\gamma/2,\tilde{c}_{s}\boldsymbol{k}_{R})$. Then the
location of poles will be the same as Eq. E.7 and Eq. E.8 with
$\boldsymbol{k}_{I}$ replaced with $\boldsymbol{k}_{R}$. It means that the
singularities always appear in the FLC in the plane of
$(\omega_{I},\boldsymbol{k}_{I})$. Therefore, $c_{s}^{2}<0$ is not a valid
possibility.

In summary, we need $\gamma>0$, $0\leq c_{s}^{2}\leq 1$ and
$\omega_{0}^{2}\geq 0$ for a consistent Green’s function of the form Eq. E.2.
The locations of the poles are shown in Fig. 7.

Figure 7: Singularities of $G(\omega,k)$ defined in Eq. E.2 are located in the
grey region. The solid gray lines correspond to the upper limit in Eq. E.8
while the dashed lines are associated to Eq. E.7 (or lower limit of Eq. E.8).
For a physical Green’s function singularities must be outside the FLC (pink
shade).

Before closing this section we write the imaginary part of Eq. E.2 for later
usage. If the damping parameter is finite we have

$\imaginary
G(\omega,k)=\frac{\gamma\omega}{(-\omega^{2}+c_{s}^{2}k^{2}+\omega_{0}^{2})^{2}+\gamma^{2}\omega^{2}}\,.$
(E.9)

Notice that the positivity condition on the imaginary part implies $\gamma>0$.
For infinitesimal damping we must replace $\gamma\to\epsilon$. Then the
imaginary part becomes

$\imaginary
G(\omega,k)=\pi\frac{|\omega|}{\omega}\delta(-\omega^{2}+c_{s}^{2}k^{2}+\omega_{0}^{2})\,.$
(E.10)

## References

  * [1] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, _Causality, analyticity and an IR obstruction to UV completion_ , _JHEP_ 10 (2006) 014 [hep-th/0602178].
  * [2] B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, _Positive moments for scattering amplitudes_ , _Phys. Rev. D_ 104 (2021) 036006 [2011.00037].
  * [3] S. Caron-Huot and V. Van Duong, _Extremal Effective Field Theories_ , _JHEP_ 05 (2021) 280 [2011.02957].
  * [4] A. J. Tolley, Z.-Y. Wang and S.-Y. Zhou, _New positivity bounds from full crossing symmetry_ , _JHEP_ 05 (2021) 255 [2011.02400].
  * [5] N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, _The EFT-Hedron_ , _JHEP_ 05 (2021) 259 [2012.15849].
  * [6] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, _The S-matrix bootstrap. Part I: QFT in AdS_ , _JHEP_ 11 (2017) 133 [1607.06109].
  * [7] J. Elias Miro, A. Guerrieri and M. A. Gumus, _Bridging positivity and S-matrix bootstrap bounds_ , _JHEP_ 05 (2023) 001 [2210.01502].
  * [8] J. Elias Miro, A. Guerrieri and M. A. Gumus, _Extremal Higgs couplings_ , 2311.09283.
  * [9] B. Bellazzini, G. Isabella, S. Ricossa and F. Riva, _Massive gravity is not positive_ , _Phys. Rev. D_ 109 (2024) 024051 [2304.02550].
  * [10] A. Guerrieri, H. Murali, J. Penedones and P. Vieira, _Where is M-theory in the space of scattering amplitudes?_ , _JHEP_ 06 (2023) 064 [2212.00151].
  * [11] D. Baumann, D. Green, H. Lee and R. A. Porto, _Signs of Analyticity in Single-Field Inflation_ , _Phys. Rev. D_ 93 (2016) 023523 [1502.07304].
  * [12] T. Grall and S. Melville, _Positivity bounds without boosts: New constraints on low energy effective field theories from the UV_ , _Phys. Rev. D_ 105 (2022) L121301 [2102.05683].
  * [13] M. Freytsis, S. Kumar, G. N. Remmen and N. L. Rodd, _Multifield positivity bounds for inflation_ , _JHEP_ 09 (2023) 041 [2210.10791].
  * [14] P. Creminelli, O. Janssen and L. Senatore, _Positivity bounds on effective field theories with spontaneously broken Lorentz invariance_ , _JHEP_ 09 (2022) 201 [2207.14224].
  * [15] L. Hui, I. Kourkoulou, A. Nicolis, A. Podo and S. Zhou, _$S$ -matrix positivity without Lorentz invariance: a case study_, 2312.08440.
  * [16] P. Creminelli, M. Delladio, O. Janssen, A. Longo and L. Senatore, _Non-analyticity of the $S$-matrix with spontaneously broken Lorentz invariance_, 2312.08441.
  * [17] R. Kronig, _On the theory of the dispersion of X-rays_ , _J. Opt. Soc. Am._ 12 (1926) 547.
  * [18] H. Kramers, _La diffusion de la lumière par les atomes_ , _Atti Cong. Intern. Fisici, (Transactions of Volta Centenary Congress) Como._ 2 (1927) 545.
  * [19] L. V. Keldysh, D. A. Kirzhnitz and A. A. Maradudin, _The dielectric function of condensed systems_ , 1989, https://api.semanticscholar.org/CorpusID:118667756.
  * [20] M. Leontovich, _Generalization of the Kramers-Kronig formulas to media with spatial dispersion_ , _J. Exptl. Theoret. Phys. (U.S.S.R.)_ 40 (1961) 907.
  * [21] P. M. Chaikin and T. C. Lubensky, _Principles of Condensed Matter Physics_. Cambridge University Press, 1995.
  * [22] J. D. Jackson, _Classical Electrodynamics_. Wiley, 1998.
  * [23] E. A. Calzetta and B.-L. B. Hu, _Nonequilibrium Quantum Field Theory_. Oxford University Press, 2009, 10.1017/9781009290036.
  * [24] J. F. Nieves and P. B. Pal, _$P$ and CP Odd Terms in the Photon Selfenergy Within a Medium_, _Phys. Rev. D_ 39 (1989) 652.
  * [25] S. M. Carroll, _Quintessence and the rest of the world_ , _Phys. Rev. Lett._ 81 (1998) 3067 [astro-ph/9806099].
  * [26] Y. Nakai, R. Namba, I. Obata, Y.-C. Qiu and R. Saito, _Can we explain cosmic birefringence without a new light field beyond Standard Model?_ , _JHEP_ 01 (2024) 057 [2310.09152].
  * [27] S. Ichimaru, _Basic Principles Of Plasma Physics: A Statistical Approach Mathematics_. CRC Press, 1973.
  * [28] S. Weinberg, _The Quantum theory of fields. Vol. 1: Foundations_. Cambridge University Press, 6, 2005, 10.1017/CBO9781139644167.
  * [29] A. A. Rukhadze and V. P. Silin, _Electrodynamics of media with spatial dispersion_ , _Soviet Physics Uspekhi_ 4 (1961) 459.
  * [30] V. A. Markel, _Can the imaginary part of permeability be negative?_ , _Phys. Rev. E_ 78 (2008) 026608.
  * [31] R. A. Depine and A. Lakhtakia, _Comment i on “resonant and antiresonant frequency dependence of the effective parameters of metamaterials”_ , _Phys. Rev. E_ 70 (2004) 048601.
  * [32] A. L. Efros, _Comment ii on “resonant and antiresonant frequency dependence of the effective parameters of metamaterials”_ , _Phys. Rev. E_ 70 (2004) 048602.
  * [33] S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, _Microcausality in curved space-time_ , _Phys. Rev. D_ 77 (2008) 084016 [0709.1483].
  * [34] T. Hartman, S. Jain and S. Kundu, _Causality Constraints in Conformal Field Theory_ , _JHEP_ 05 (2016) 099 [1509.00014].
  * [35] D. Melrose and R. Stoneham, _Generalised kramers-kronig formula for spatially dispersive media_ , _Journal of Physics A: Mathematical and General_ 10 (2001) L17.
  * [36] M. Froissart, _Asymptotic behavior and subtractions in the Mandelstam representation_ , _Phys. Rev._ 123 (1961) 1053.
  * [37] A. Martin, _Unitarity and high-energy behavior of scattering amplitudes_ , _Phys. Rev._ 129 (1963) 1432.
  * [38] L. D. Landau and E. M. Lifshitz, _Electrodynamics of Continuous Media_. Pergamon, New York, 1984.
  * [39] J. Lindhard, _On the properties of a gas of charged particles_ , _Kgl. Danske Videnskab. Selskab Mat.-fys. Medd._ Vol: 28, No. 8 (1954) .
  * [40] S. Weinberg, _Superconductivity for Particular Theorists_ , _Progress of Theoretical Physics Supplement_ 86 (1986) 43.
  * [41] A. M. J. Schakel, _Boulevard of broken symmetries_ , 1998.
  * [42] M. J. Landry and H. Liu, _A systematic formulation of chiral anomalous magnetohydrodynamics_ , 2212.09757.
  * [43] O. Dolgov, D. Kirzhnits and V. Losyakov, _Admissible values of permittivity and magnetic permeability of matter_ , _J. Exp. Theor. Phys_ 56 (1982) .
  * [44] D. A. Kirzhnits, _General properties of electromagnetic response functions_ , _Mod. Probl. Condens. Matter Sci._ 24 (1989) 41.
  * [45] B. R. Holstein, _Pion polarizability and chiral symmetry_ , _Comments Nucl. Part. Phys._ 19 (1990) 221.
  * [46] M. E. Peskin and D. V. Schroeder, _An Introduction to quantum field theory_. Addison-Wesley, Reading, USA, 1995.
  * [47] P. Raman and A. Sinha, _QFT, EFT and GFT_ , _JHEP_ 12 (2021) 203 [2107.06559].
  * [48] G. Sommer, _Present state of rigorous analytic properties of scattering amplitudes_ , _Fortsch. Phys._ 18 (1970) 577.
  * [49] T. Hartman, S. A. Hartnoll and R. Mahajan, _Upper Bound on Diffusivity_ , _Phys. Rev. Lett._ 119 (2017) 141601 [1706.00019].
  * [50] L. V. Delacretaz, A. L. Fitzpatrick, E. Katz and M. T. Walters, _Thermalization and hydrodynamics of two-dimensional quantum field theories_ , _SciPost Phys._ 12 (2022) 119 [2105.02229].
  * [51] M. P. Heller, A. Serantes, M. Spaliński and B. Withers, _Rigorous Bounds on Transport from Causality_ , _Phys. Rev. Lett._ 130 (2023) 261601 [2212.07434].
  * [52] M. P. Heller, A. Serantes, M. Spaliński and B. Withers, _The Hydrohedron: Bootstrapping Relativistic Hydrodynamics_ , 2305.07703.
  * [53] R. Jackiw, _Functional evaluation of the effective potential_ , _Phys. Rev. D_ 9 (1974) 1686.
  * [54] L. F. Abbott, _Introduction to the Background Field Method_ , _Acta Phys. Polon. B_ 13 (1982) 33.
  * [55] K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu, _Equilibrium and Nonequilibrium Formalisms Made Unified_ , _Phys. Rept._ 118 (1985) 1.
  * [56] E. Wang and U. W. Heinz, _A Generalized fluctuation dissipation theorem for nonlinear response functions_ , _Phys. Rev. D_ 66 (2002) 025008 [hep-th/9809016].
  * [57] G.-z. Zhou, Z.-b. Su, B.-l. Hao and L. Yu, _Closed time path green’s functions and critical dynamics_ , _Phys. Rev. B_ 22 (1980) 3385.
  * [58] S. Weinberg, _Quantum contributions to cosmological correlations_ , _Phys. Rev. D_ 72 (2005) 043514 [hep-th/0506236].
  * [59] R. Kubo, _Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems_ , _Journal of the Physical Society of Japan_ 12 (1957) 570.
  * [60] S. A. Hartnoll, _Lectures on holographic methods for condensed matter physics_ , _Class. Quant. Grav._ 26 (2009) 224002 [0903.3246].
  * [61] A. Uhlmann, _Anti- (conjugate) linearity_ , _Science China Physics, Mechanics & Astronomy_ 59 (2016) .
  * [62] H. A. Weldon, _Covariant Calculations at Finite Temperature: The Relativistic Plasma_ , _Phys. Rev. D_ 26 (1982) 1394.
  * [63] E. V. Shuryak, _Quantum Chromodynamics and the Theory of Superdense Matter_ , _Phys. Rept._ 61 (1980) 71.
  * [64] A. Nicolis, A. Podo and L. Santoni, _The connection between nonzero density and spontaneous symmetry breaking for interacting scalars_ , _JHEP_ 09 (2023) 200 [2305.08896].
  * [65] A. Podo and L. Santoni, _Fermions at finite density in the path integral approach_ , _JHEP_ 02 (2024) 182 [2312.14753].
  * [66] P. Coleman, _Introduction to Many-Body Physics_. 2015.
  * [67] Z. H. Levine and S. G. Louie, _New model dielectric function and exchange-correlation potential for semiconductors and insulators_ , _Phys. Rev. B_ 25 (1982) 6310.
  * [68] L. D. Landau and E. M. Lifshitz, _Statistical Physics, Part 1_ , vol. 5 of _Course of Theoretical Physics_. Butterworth-Heinemann, Oxford, 1980.