# Boundary conditions and infrared divergences

Lissa de Souza<EMAIL_ADDRESS>Claudio
<EMAIL_ADDRESS>Luca
<EMAIL_ADDRESS>†Dipartimento di Fisica,
Università degli Studi di Pavia, Via Bassi, 6, 27100 Pavia, Italy  ⋆Istituto
Nazionale di Fisica Nucleare – Sezione di Pavia, Via Bassi 6, 27100 Pavia,
Italy

###### Abstract

We review the procedure to construct quasi-free ground states, for real scalar
fields whose dynamics is dictated by the Klein-Gordon equation, on standard
static Lorentzian manifolds with a time-like boundary. We observe that,
depending on the assigned boundary condition of Robin type, this procedure
does not always lead to the existence of a suitable bi-distribution
$w_{2}\in\mathcal{D}^{\prime}(M\times M)$ due to the presence of infrared
divergences. As a concrete example we consider a Bertotti-Robinson spacetime
in two different coordinate patches. In one case we show that infrared
divergences do not occur only for Dirichlet boundary conditions as one might
expect a priori, while, in the other case, we prove that they occur only when
Neumann boundary conditions are imposed at the time-like boundary.

###### keywords:

quantum field theory on curved spacetimes, infrared divergences, Bertotti-
Robinson spacetime

††journal: …

##  1 Introduction

In the past decade, we have witnessed a steadily growing interest towards the
interplay between boundary conditions and the construction of free and
interacting quantum field theories living on a large class of backgrounds
possessing a (conformal) timelike boundary. From a physical viewpoint the
reasons are several ranging from the desire of a better understanding of the
structural aspects of a quantum field theory on asymptotically AdS spacetimes
to the necessity of a thorough investigation of newly discovered phenomena,
among which remarkable are the anti-Unruh and the anti-Hawking effect, see
e.g. [1, 2, 3, 4] .

In this endeavor, even when focusing on the simple scenario of a free, scalar
field theory, one realizes from the very beginning that the presence of a
timelike boundary entails already at a classical level a sharp difference from
the standard formulation when the underlying background, say $(M,g)$, is
globally hyperbolic. As a matter of fact, if the dynamics is ruled by a Klein-
Gordon operator $P=\Box_{g}-m^{2}-\xi R$, see Equation (1), the associated
initial value problem has to be supplemented with the assignment of a suitable
boundary condition at $\partial M$. For several years, it has been customary
to focus the attention only on that of Dirichlet type, since it has the
distinguished property of being always admissible, in the sense that it
entails not only that the underlying Cauchy problem is well-posed, but also
that $P$ admits unique advanced and retarded fundamental solutions, which are
in turn the building blocks to implement the canonical commutation relations
at a quantum level.

Yet, in the past few years it has been highlighted that under suitable
constraints on the parameters of the theory, in particular $m$ and $\xi$, one
has the freedom of assigning to a scalar field theory a much larger class of
admissible boundary conditions, among which noteworthy are those of Robin
type, see e.g. [5, 6, 7, 8, 9, 10, 11, 12, 13]. This realization has prompted
the beginning of several research projects aimed at scrutinizing both the
structural aspects and the physical properties of the underlying quantum
counterpart. In this respect one of the main questions which needs to be
addressed concerns the identification of a class of physically admissible
quantum states. When the underlying spacetime is globally hyperbolic, it is
widely accepted that one needs to restrict the attention to the so-called
Hadamard states [14]. Once more denoting by $(M,g)$ the underlying background
this entails that one needs to identify a bi-distribution
$\omega_{2}\in\mathcal{D}^{\prime}(M\times M)$, which is a solution to the
underlying equation of motion and whose antisymmetric part coincides up to a
multiplicative constant to the difference between the advanced and the
retarded fundamental solutions and it abides by a suitable constraint on its
singular structure.

At a physical level this establishes a condition on the ultraviolet behaviour
of the quantum state which has far reaching consequences, among which it is
worth recalling the existence of a covariant notion of Wick polynomials and
the finiteness of the quantum fluctuations of all observables, see e.g. [15].

In view of these remarks, in the past few years, much effort has been devoted
to finding a suitable generalization of the Hadamard condition whenever the
underlying background $(M,g)$ admits a timelike boundary. Thanks to the effort
of several research groups, it is nowadays understood what it is a good notion
of Hadamard states in this framework and several explicit examples has been
constructed, see e.g. [5, 16, 17, 18]. Yet, in all these works, the focus has
always been on the one hand on the control of the ultraviolet behaviour of the
underlying two-point correlation function. On the other hand, barring some
very specific examples, it has always been assumed that $(M,g)$ is stationary,
since the existence of a timelike Killing field, entails that one can focus
the attention either on ground or on thermal/KMS states as natural candidates
for being of Hadamard form.

In almost all scenarios considered, a great deal of attention has always been
given to controlling the underlying ultraviolet behaviour. Yet almost no
attention has been given to the possibility that infrared divergences can
occur and that they are connected to the choice of an underlying boundary
condition. The main goal of this paper is to warn about this possibility
which, from a technical viewpoint, is tantamount to the existence of an
obstruction in finding in the first place a suitable bi-distribution
$\omega_{2}\in\mathcal{D}^{\prime}(M\times M)$ out of which one can define a
desired, quantum state.

More precisely, we shall proceed as follows. In the next section we shall
recall succinctly and in great generality how one can construct on a globally
hyperbolic spacetime with a timelike boundary $(M,g)$ a two-point correlation
function associated to a ground state for a real, massive scalar field. This
part of the paper is based on existing literature and its main goal is mainly
to highlight were a potential infrared divergence can occur in the most
general scenario, so to emphasize that this obstruction is always a potential
threat. Subsequently in Section 3 we focus the attention on a specific
scenario, namely a real, massive scalar field on the four-dimensional
Bertotti-Robinson spacetime [19, 20], here presented in two different
coordinate patches. On the one hand this background is relevant at a physical
level since it is both an exact solution of the Einstein-Maxwell’s equations
and it represents a good approximation of the near horizon geometry of
Reissner-Nordstrøm black hole, see e.g. [21]. On the other hand, the
background isometries are such to allow us to give a detailed, analytic
construction both of the underlying fundamental solutions with Robin boundary
conditions. Subsequently we shall turn our attention to the construction of
the two-point correlation functions of an underlying ground state and we shall
show explicitly that, in one of the coordinate patches considered, only with
Dirichlet boundary conditions, no infrared singularity is present. On the
contrary, in the second coordinate patch, this pathology occurs only if we
consider Neumann of boundary conditions, hence allowing in this case also
those of Robin type.

##  2 General Construction

In this section we review succinctly the construction of the two-point
correlation function associated to the ground state of a real, massive scalar
field, so to highlight at which stage a potential infrared singularity might
occur and the relation between its insurgence and the underlying boundary
condition. Henceforth with $(M,g)$ we refer to a globally hyperbolic, oriented
and time-oriented Lorentzian manifold with timelike boundary of dimension
$\dim M=n\geq 2$, as formalized in [23].

In addition, we assume that $(M,g)$ is standard static, namely it is isometric
to $\mathbb{R}\times\Sigma$ with line element

$ds^{2}=-\beta dt^{2}+h,$

where $t\in\mathbb{R}$ plays the rôle of time coordinate along the
$\mathbb{R}$-direction, while $\beta$ is a positive, time-independent, smooth
function. In addition $h$ is a time-independent Riemannian metric on $\Sigma$.
As a consequence the boundary $\partial M\simeq\mathbb{R}\times\partial\Sigma$
is also naturally endowed with a static metric. On top of this class of
backgrounds we consider a real scalar field $\Phi:M\to\mathbb{R}$ whose
dynamics is ruled by

$P\Phi=\left(\Box_{g}-V\right)\Phi=0,$ (1)

where $\Box_{g}$ is the d’Alembert wave operator built out of the underlying
metric $g$, while $V$ is a time-independent potential, which we assume to be
smooth in the interior of $M$. A rather common choice for such potential is
$V=m^{2}+\xi R$ where $m\geq 0$ plays the rôle of mass parameter while
$\xi\in\mathbb{R}$ denotes an arbitrary coupling to the scalar curvature $R$.
Since $M$ possesses a timelike boundary, the solutions to Equation (1) cannot
be constructed only by assigning initial data on a Cauchy surface
$\Sigma_{t_{0}}\simeq\\{t_{0}\\}\times\Sigma$, $t_{0}\in\mathbb{R}$, but it is
necessary to supplement suitable boundary conditions on $\partial M$. Among
the plethora of possible choices, we are mainly interested in those that
entail the existence of unique advanced and retarded fundamental solutions
associated to $P$ as well as of a two-point correlation function
$\omega_{2}\in\mathcal{D}^{\prime}(M\times M)$, which allows to define a
quantum state for the field $\Phi$ whose ultraviolet singular behaviour is of
Hadamard form.

From a mathematical viewpoint the first problem has been addressed in [6],
while the second one has been mainly discussed in [24, 25], although only on
asymptotically anti-de Sitter spacetimes. It is important to stress that, in
these works, the main task has always been the control of the ultraviolet
behaviour of $\omega_{2}$, while no attention has been given to the
possibility that infrared divergences might occur. This translates in the
presence of an obstruction to the existence of $\omega_{2}$ as a well-defined
distribution and, although this is abstractly known as a lurking possibility,
to the best of our knowledge, it has not been sufficiently stressed that this
might be directly connected to the choice of an underlying boundary condition.
As mentioned in Section 1 this is the main goal of this paper and, to this end
we start by recalling succinctly the procedure leading to the construction of
ground states in the case in hand.

In order to make more crystal clear our message, we shall focus the attention
only on boundary conditions of Robin type, although our conclusions can be
drawn also for more general choices. Hence, starting from Equation (1), we
consider the Fourier transform along the time-direction, namely

$\widehat{\Phi}(\omega,x)\doteq\int\limits_{\mathbb{R}}dt\,e^{i\omega
t}\Phi(t,x),$

where $x$ stands for a choices of coordinates on $\Sigma$. Therefore Equation
(1) can be equivalently written as an eigenvalue problem

$K\widehat{\Phi}=\omega^{2}\widehat{\Phi},$ (2)

where, denoting by $\Delta_{h}$ the Laplace-Beltrami operator built out of the
Riemannian metric $h$, the operator $K$ reads

$K=\beta\Delta_{h}-\frac{1}{2}h^{ij}(\partial_{i}\beta)\partial_{j}+\beta V.$
(3)

In studying Equation (1), the first step consists of establishing whether the
Klein-Gordon operator $P$ admits unique advanced and retarded fundamental
solutions $E_{adv}/E_{ret}\in\mathcal{D}^{\prime}(M\times M)$. As already
mentioned, this is not the case unless one assigns suitable boundary
conditions for the field $\Phi$ on $\partial
M\simeq\mathbb{R}\times\partial\Sigma$. To this end, as discussed thoroughly
in [6], it is convenient to read $K$ as a second order partial differential
operator on $L^{2}(\Sigma)$, the space of square-integrable functions on a
Cauchy surface $\Sigma$ with respect to the metric-induced induced volume form
$\beta^{-1}d\mu_{h}$. In this way, it is possible to prove that admissible
boundary conditions are in 1:1 correspondence with the self-adjoint extension
of $K$ on $L^{2}(\Sigma)$. To make our case clear, among the plethora of
possible choices, we restrict our attention only to those scenarios for which
it is admissible a boundary condition that is static and of Robin type, namely
there exists $\alpha\neq\alpha(t)\in C^{\infty}(\partial M)$ such that

$\left.\Phi\right|_{\partial M}=\alpha\,\partial_{n}\Phi|_{\partial M},$ (4)

where $\partial_{n}$ denotes the derivative along the direction normal to the
boundary.

Observe that in Equation (4), the symbol $|_{\partial M}$ denotes the
restriction both of $\Phi$ and of $\partial_{n}\Phi$ on the boundary. This
operation is legitimate only if both $\Phi$ and $\partial_{n}\Phi$ are
continuous on $\partial M$, a feature which is strongly dependent both on the
choice of $V$ and of $g$. When the solution $\Phi$ to Equation (1) is not
regular enough, then all restrictions needs to be replaced by suitable trace
maps, which we avoid defining in this paper in order to avoid unnecessary
technical details. Hence we refer an interested reader to [6] for the analysis
of the general scenario.

As a consequence of [6, Prop. 19], for each boundary condition as in Equation
(4) there exists a corresponding self-adjoint extension on $L^{2}(\Sigma)$ of
the operator $K$ as in Equation (3) which we denote by $K_{\alpha}$. In turn,
this result combined with [6, Thm. 30], entails that the operator $P$ admits
unique advanced and retarded Green’s operators, $E^{\pm}_{\alpha}$ whose
associated integral kernels read
$\mathcal{E}^{-}_{\alpha}(x,x^{\prime})=\Theta(t-t^{\prime})\mathcal{E}_{\alpha}(x,x^{\prime})$
and
$\mathcal{E}^{+}_{\alpha}(x,x^{\prime})=-\Theta(t^{\prime}-t)\mathcal{E}_{\alpha}(x,x^{\prime})$,
where $\mathcal{E_{\alpha}}\in\mathcal{D}^{\prime}(M\times M)$ is such that,
for all $f,f^{\prime}\in C^{\infty}_{0}(M)$,

$\mathcal{E}_{\alpha}(f,f^{\prime})=\int\limits_{\mathbb{R}^{2}}dtdt^{\prime}\,\left(f(t),K^{-\frac{1}{2}}_{\alpha}\sin[K^{\frac{1}{2}}_{\alpha}(t-t^{\prime})]f^{\prime}(t)\right)_{L^{2}(\Sigma)},$
(5)

where $(,)_{L^{2}(\Sigma)}$ denotes the inner product of $L^{2}(\Sigma)$,
while $K^{-\frac{1}{2}}_{\alpha}\sin[K^{\frac{1}{2}}_{\alpha}(t-t^{\prime})]$
is defined in terms of the functional calculus for $K_{\alpha}$. Having
established the existence of the advanced and retarded Green’s operators
associated to $P$, the next step in the construction of a quantum field theory
consists of individuating $\omega_{2}\in\mathcal{D}^{\prime}(M\times M)$, the
two-point correlation function of a quantum state. In the case in hand, and
considering a boundary condition as per Equation (4), this amounts to solving
the system

$\left\\{\begin{array}[]{l}\left(P\otimes\mathbb{I}\right)\omega_{2,\alpha}=0,\quad\left(\mathbb{I}\otimes
P\right)\omega_{2,\alpha}=0,\\\
\omega_{2,\alpha}(f,f^{\prime})-\omega_{2,\alpha}(f^{\prime},f)=i\mathcal{E}_{\alpha}(f,f^{\prime})\quad\forall
f,f^{\prime}\in\mathcal{D}(M),\end{array}\right.$ (6)

where $\mathcal{E}_{\alpha}$ is as per Equation (5) and where the subscript
$\alpha$ in $\omega_{2,\alpha}$ highlights the dependence of the two-point
function from the boundary condition. Still exploiting the functional calculus
for $K_{\alpha}$, Proposition 5.5. in [24] entails that a solution to Equation
(6), corresponding to the two-point function of a ground state of Hadamard
form exists if there exists in turn
$\omega_{2,\alpha}\in\mathcal{D}^{\prime}(M\times M)$ which reads

$\omega_{2,\alpha}(f,f^{\prime})=\int\limits_{\mathbb{R}^{2}}dtdt^{\prime}\,\left(f(t),K^{-\frac{1}{2}}_{\alpha}\exp[iK^{\frac{1}{2}}_{\alpha}(t-t^{\prime})]f^{\prime}(t)\right)_{L^{2}(\Sigma)}.$
(7)

While such expression appears to be innocuous, observe that the symmetric part
of $\omega_{2,\alpha}$ reads formally

$\omega^{S}_{2,\alpha}(f,f^{\prime})=\int\limits_{\mathbb{R}^{2}}dtdt^{\prime}\,\left(f(t),K^{-\frac{1}{2}}_{\alpha}\cos\left[K^{\frac{1}{2}}_{\alpha}(t-t^{\prime})\right]f^{\prime}(t)\right)_{L^{2}(\Sigma)}.$
(8)

Yet, contrary to the antisymmetric part of $\omega_{2}$ which coincides up to
multiplicative constants with Equation (5), Equation (8) is potentially ill-
defined due to the $0$-modes of the operator $K_{\alpha}$ which entail that
the integrand might fail to be integrable on account of the action of
$K^{-\frac{1}{2}}_{\alpha}\cos\left[K^{\frac{1}{2}}_{\alpha}(t-t^{\prime})\right]$.
This potential singularity is of infrared type since it involves the low
energy behaviour of the operator $P$ as in Equation (1) and, in turn, it is
directly dependent on the spectrum of $K_{\alpha}$, whose form depends
explicitly on the underlying boundary condition as per Equation (4).

Having established at a generic and abstract level where is the source of a
possible infrared singularity, in the next section, we discuss an explicit
example that highlights it more concretely, making particularly evident the
interplay with the underlying boundary condition.

##  3 Infrared Divergences on Bertotti-Robinson Spacetime

In order to give an example of the feature outlined in the previous section,
we consider as background $(M,g)$ a Bertotti-Robinson spacetime, which is a
solution to the Einstein-Maxwell equation and it approximates the near horizon
geometry of an extremal black hole with unit charge, see e.g. [19, 20, 21,
22]. As a manifold, $M$ is globally diffeomorphic to
$\textrm{CAdS}_{2}\times\mathbb{S}^{2}$, where $\textrm{CAdS}_{2}$ stands for
the universal cover of the two-dimensional anti-de Sitter spacetime. In the
following we start by considering a distinguished patch $(\widetilde{M},g)$ of
$(M,g)$ which makes manifest the connection with a black hole spacetime, see
[21]. Herein the line-element reads

$ds^{2}=-(\rho^{2}-1)dt^{2}+(\rho^{2}-1)^{-1}d\rho^{2}+d\Omega^{2}(\theta,\varphi),$
(9)

where $t\in\mathbb{R}$, $\rho\in(1,\infty)$ while
$d\Omega^{2}(\theta,\varphi)=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the line
element of the unit two-sphere. On top of $(\widetilde{M},g)$ we consider a
massive, real scalar field $\Psi:\widetilde{M}\to\mathbb{R}$, whose dynamics
is ruled by

$P_{0}\Psi=\left(\Box_{g}-m^{2}\right)\Psi=0,$ (10)

where $\Box_{g}$ is the D’Alembert wave operator built out of $g$ and where we
are discarding any coupling to the underlying, scalar curvature since it is
vanishing. Since $(\widetilde{M},g)$ is a static spacetime with a (conformal)
timelike boundary at $\rho\to\infty$, c.f. Equation (9), it is legitimate to
look for a bi-distribution
$\omega_{2}\in\mathcal{D}^{\prime}(\widetilde{M}\times\widetilde{M})$ which
represents the two-point correlation function of a ground state. In the
following we follow a procedure based on a mode decomposition which yields
ultimately a counterpart both of Equation (5) and of Equation (7) in the case
in hand. Let us therefore consider a solution $\Psi$ of Equation (10) which
admits the following decomposition

$\Psi(t,\rho,\theta,\phi)=\sum\limits_{l=0}^{\infty}\sum\limits_{p=-l}^{l}\int\limits_{\mathbb{R}}e^{-i\omega
t}\psi_{\omega lp}(\rho)Y^{p}_{l}(\theta,\phi),$ (11)

where $Y^{p}_{l}(\theta,\varphi)$ is the standard spherical harmonic,
eigenfunction of the Laplacian on the unit $2$-sphere, namely
$\Delta_{\mathbb{S}^{2}}Y_{lp}(\theta,\varphi)=-l(l+1)Y_{lp}(\theta,\varphi)$.
On account of Equation (10), $\psi_{\omega lp}(\rho)$ satisfies the radial
equation

$(\mathbf{L}+\omega^{2})\psi(\rho):=\bigg{\\{}(\rho^{2}-1)\bigg{[}\frac{d}{d\rho}\bigg{(}(\rho^{2}-1)\frac{d}{d\rho}\bigg{)}-l(l+1)-m^{2}\bigg{]}+\omega^{2}\bigg{\\}}\psi(\rho)=0,$
(12)

which is a Sturm-Liouville problem with eigenvalue $-\omega^{2}$, see [27].
Observe that we have suppressed the subscripts highlighting the dependence of
$\mathbf{L}$ as well as of the solutions of the radial equation on the
spectral parameters $\omega,l,p$ to avoid an unnecessarily heavy notation. A
basis of solutions of Equation (12) is given by the associated Legendre
polynomials

$\displaystyle\psi_{1}(\rho)$ $\displaystyle=P^{i\omega}_{\nu}(\rho),$ (13a)
$\displaystyle\psi_{2}(\rho)$ $\displaystyle=Q^{i\omega}_{\nu}(\rho),$ (13b)

with $\nu=\frac{1}{2}(\sqrt{1+4l(l+1)+4m^{2}}-1)$, see [26, §14.2]. The next
step consists of constructing a fundamental solution associated to the
differential operator $\mathbf{L}$. Following the theory of Sturm-Liouville
operators, see [27], first of all we allow $\omega$ to take any complex value
and we look for solutions of Equation (10) that belong to
$\mathcal{H}_{1}:=L^{2}((1,c),d\mu)$ or to
$\mathcal{H}_{\infty}:=L^{2}((c,\infty),d\mu)$, with
$d\mu=(\rho^{2}-1)^{-1}d\rho$, while $c\in(1,\infty)$ is an arbitrary, but
fixed number.

Since both $\psi_{1}$ and $\psi_{2}$ are smooth functions, it suffices to look
for their asymptotic expansion in a neighborhood of $\rho=1$. Using [26,
§14.8], it turns out that $Q^{\pm i\omega}_{\nu}$, $P^{\pm
i\omega}_{\nu}\in\mathcal{H}_{1}$ if $\pm\operatorname{Im}(\omega)>0$, while
they both lie in $\mathcal{H}_{\infty}$ if and only if $0\leq\nu<\frac{1}{2}$,
which is equivalent to $l=0$ and $m^{2}\in[-\frac{1}{4},\frac{3}{4})$.
Whenever $\nu>\frac{1}{2}$, only $Q^{\pm i\omega}_{\nu}$ belongs to
$\mathcal{H_{\infty}}$.

Hence, using the standard terminology proper of Sturm-Liouville problems
summarized in [5], at $\rho=\infty$, we say that
$\mathcal{Q}^{\omega}_{\nu}(\rho):=e^{\pi\omega}Q^{i\omega}_{\nu}(\rho)$ is
the principal solution, since, as $\rho\to\infty$, it falls off to $0$ faster
than any other solution of Equation (12). At the same time
$\mathcal{P}^{\omega}_{\nu}(\rho):=\frac{\pi}{2i}\sinh^{-1}(\pi\omega)P^{i\omega}_{\nu}(\rho)$
is referred to as being a choice for one secondary solution, see [28, Def.
4.2] as well as [24] for a discussion of the physical significance. We
highlight that the $\omega$-dependent coefficients have also been chosen for
later computational efficiency.

Using Weyl classification of the behaviour of a Sturm-Liouville problem at the
end points, we can infer that both $\rho=1$ and $\rho\to\infty$ are limit
circle if $0<\nu<\frac{1}{2}$, while only $\rho\to\infty$ falls in this class
if $\nu\geq\frac{1}{2}$. This entails that, in order to solve Equation (12),
in addition to initial data we need to assign boundary conditions at the
endpoints whenever they are limit circle. In the following, since we are more
concerned about the rôle of the time-like boundary in the analysis of the
underlying quantum theory, we consider only Dirichlet boundary conditions on
the horizon ($\rho=1$), while, at $\rho\to\infty$, we assign Robin boundary
conditions. Observe that, if we switch back to a fully-covariant description
of the underlying field, Equation (9) entails that the locus $\rho=1$ is an
event horizon, hence a light-like hypersurface. This

This translates in the choice of the following solutions of Equation (12)
whenever $\operatorname{Im}(\omega)\neq 0$:

$\displaystyle u_{1}(\rho)$
$\displaystyle:=\Theta(\operatorname{Im}(\omega))P^{i\omega}_{\nu}(\rho)+\Theta(-\operatorname{Im}(\omega))P^{-i\omega}_{\nu}(\rho),$
(14a) $\displaystyle u_{\infty}(\rho)$
$\displaystyle:=\cos\gamma\,e^{\pi\omega}Q^{i\omega}_{\nu}(\rho)+\sin\gamma\,\frac{\pi}{2i}\sinh^{-1}(\pi\omega)P^{i\omega}_{\nu}(\rho),\qquad\gamma\in[0,\pi/2],$
(14b)

where the Heaviside step function in $u_{1}$ helps us to take into account
both $\operatorname{Im}\omega$ positive and negative in a single expression,
while the exponential coefficients appearing in $u_{\infty}$ are chosen for
future computational convenience. As a consequence of the choices above, it
turns out that $u_{\infty}$ abides by Robin boundary conditions, namely

$\lim_{\rho\to\infty}(\cos\gamma\\{u_{\infty}(\rho),\mathcal{P}^{\omega}_{\nu}(\rho)\\}+\sin\gamma\\{u_{\infty}(\rho),\mathcal{Q}^{\omega}_{\nu}(\rho)\\})=0,$
(15)

where $\\{\cdot,\cdot\\}:=(\rho^{2}-1)W[\cdot,\cdot]$, $W$ being the
Wronskian.

###  3.1 The radial Green function

As a next step, denoting by $I$ the interval $(1,\infty)$, we look for
$G_{\gamma}\in\mathcal{D}^{\prime}(I\times I)$, $\gamma\in[0,\frac{\pi}{2}]$,
a distribution whose integral kernel
$G_{\gamma}(\omega,\rho,\rho^{\prime})\equiv G_{\gamma}(\rho,\rho^{\prime})$
abides by

$(\mathbf{L}\otimes\mathbb{I})G_{\gamma}(\rho,\rho^{\prime})=(\mathbb{I}\otimes\mathbf{L})G_{\gamma}(\rho,\rho^{\prime})=\delta(\rho-\rho^{\prime}),$
(16)

where $\mathbf{L}$ is the operator in Equation (12) and where the subscript
$\gamma$ highlights that we are looking for a fundamental solution which
encodes Robin boundary conditions as per Equation (15). Observe that, once
more, we are suppressing for convenience of the notation any explicit
reference on the dependence of $G_{\gamma}$ on $\omega$ as well as on $l,p$.
Using Equation (14) and standard results of theory of second order ODEs, it
holds that

$G_{\gamma}(\rho,\rho^{\prime})=\frac{u_{1}(\rho_{<})u_{\infty}(\rho^{\prime}_{>})}{(\rho^{2}-1)W[u_{\infty},u_{1}](\rho)},$
(17)

where

$u_{1}(\rho_{<})u_{\infty}(\rho^{\prime}_{>})=\Theta(\rho-\rho^{\prime})u_{1}(\rho^{\prime})u_{\infty}(\rho)+\Theta(\rho^{\prime}-\rho)u_{1}(\rho)u_{\infty}(\rho^{\prime}),$

while $W[\cdot,\cdot]$ is the Wronskian between $u_{1}$ and $u_{\infty}$. As
an example, for Dirichlet ($\gamma=0$) and Neumann boundary conditions
($\gamma=\frac{\pi}{2}$), Equation (17) boils down to

$G_{0}(\rho,\rho^{\prime})=e^{\pi\omega}P^{-i\omega}_{\nu}(\rho_{<})Q^{i\omega}_{\nu}(\rho_{>}),$
(18a)
$G_{\frac{\pi}{2}}(\rho,\rho^{\prime})=\frac{-i\pi}{2\sinh\pi\omega}P^{-i\omega}_{\nu}(\rho_{<})P^{i\omega}_{\nu}(\rho_{>}),$
(18b)

while, for $\gamma\in(0,\pi/2)$, i.e. for Robin boundary conditions, it holds
that

$G_{\gamma}(\rho,\rho^{\prime})=\frac{\cos\gamma
G_{0}(\rho,\rho^{\prime})+\sin\gamma
G_{\frac{\pi}{2}}(\rho,\rho^{\prime})}{\cos\gamma+\sin\gamma}.$ (19)

Observe that if we reinstate the $\omega$-dependence in the radial Green
function, it holds that, for all $\gamma\in[0,\frac{\pi}{2}]$,
$G_{\gamma}(-\omega,\rho,\rho^{\prime})=\overline{G_{\gamma}(\omega,\rho,\rho^{\prime})}$
whenever $\omega\in\mathbb{R}$. We remark that given $G_{\gamma}$ there is
always the freedom to add a bi-solution of the radial equation
$\mathbf{L}G_{0}=0$ which abides by the Robin boundary condition parameterized
by $\gamma\in[0,\frac{\pi}{2}]$. In our construction this freedom amounts to
the possibility of making different choices of the secondary solution in
Equation (13b). The consequences of this leeway have been thoroughly
investigated in [28, 29] and we shall not delve further into them.

###  3.2 Resolution of the Identity

Having established an expression for the radial Green function, in order to
construct the two-point correlation function for the ground state of the
Klein-Gordon field as in Equation (10) endowed with Robin boundary conditions,
we need to individuate a resolution of the identity written in terms of the
Green function $G_{\gamma}$ as in Equation (19). Following [30, Ch. 7],
reinstating the explicit dependence on $\omega$ and recalling that we are
allowing $\omega\in\mathbb{C}$, it holds that

$-(\rho^{2}-1)\delta(\rho-\rho^{\prime})=\frac{1}{2\pi
i}\int_{C_{\infty}}G_{\gamma}(\lambda,\rho,\rho^{\prime})d\lambda,\quad\lambda=\omega^{2},$
(20)

where $C_{\infty}$ is a suitable “Pac-Man” contour in the complex
$\lambda$-plane. Using Jordan’s lemma, the right hand side of Equation (20)
can be shown to abide by the following chain of identities:

$\displaystyle\frac{1}{2\pi
i}\int_{C_{\infty}}G_{\gamma}(\lambda,\rho,\rho^{\prime})d\lambda\overset{\lambda=\omega^{2}}{=}\frac{1}{\pi
i}\int_{\mathbb{R}}d\omega\,\omega\,G_{\gamma}(\omega,\rho,\rho^{\prime})$
$\displaystyle=\frac{1}{\pi
i}\int_{0}^{\infty}d\omega\,\omega\,[G_{\gamma}(\omega,\rho,\rho^{\prime})-G_{\gamma}(-\omega,\rho,\rho^{\prime})]=$
(21)
$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}d\omega\,\omega\,\operatorname{Im}[G_{\gamma}(\omega,\rho,\rho^{\prime})].$
(22)

Introducing the function
$R_{\gamma}(\omega,\rho,\rho^{\prime}):=\frac{2}{\pi}\operatorname{Im}[G_{\gamma}(\omega,\rho,\rho^{\prime})]$,
by comparison with Equation (20), it holds that

$\int_{\mathbb{R}}d\omega\,\omega\,R_{\gamma}(\omega,\rho,\rho^{\prime}):=(\rho^{2}-1)\delta(\rho-\rho^{\prime}),$
(23)

which is a real-valued odd function in $\omega$, cf. Equation (21). Using
Equation (19), one can infer that

$\displaystyle
R_{0}(\omega,\rho,\rho^{\prime})=\frac{i}{\pi}\big{(}e^{\pi\omega}P^{-i\omega}_{\nu}(\rho)Q^{i\omega}_{\nu}(\rho^{\prime})-e^{-\pi\omega}P^{i\omega}_{\nu}(\rho)Q^{-i\omega}_{\nu}(\rho^{\prime})\big{)},$
(24a) $\displaystyle
R_{\frac{\pi}{2}}(\omega,\rho,\rho^{\prime})=\frac{1}{2\sinh\pi\omega}(P^{-i\omega}_{\nu}(\rho)P^{i\omega}_{\nu}(\rho^{\prime})+P^{i\omega}_{\nu}(\rho)P^{-i\omega}_{\nu}(\rho^{\prime})),$
(24b) $\displaystyle R_{\gamma}(\omega,\rho,\rho^{\prime})=\frac{\cos\gamma
R_{0}(\omega,\rho,\rho^{\prime})+\sin\gamma
R_{\frac{\pi}{2}}(\omega,\rho,\rho^{\prime})}{\cos\gamma+\sin\gamma},\qquad\gamma\in\big{(}0,\,\frac{\pi}{2}\big{)}.$
(24c)

###  3.3 Two-point correlation function of the ground state

Having individuated the necessary building blocks, we are in a position to
address the question of the construction of a two-point correlation function
for the ground state of a Klein-Gordon field as per Equation (10) with Robin
boundary conditions. In other words, denoting by $(M,g)$ the underlying
Bertotti-Robinson spacetime as per Equation (9), we are looking in the first
place for $\omega_{2,\gamma}\in\mathcal{D}^{\prime}(M\times M)$, the subscript
$\gamma$ highlighting the dependence on the Robin boundary condition, such
that

$(P\otimes\mathbb{I})\omega_{2,\gamma}=(\mathbb{I}\otimes
P)\omega_{2,\gamma}=0,$

while its antisymmetric part is such that, working at the level of integral
kernels

$\omega_{2,\gamma}(x,x^{\prime})-\omega_{2,\gamma}(x^{\prime},x)=i\mathcal{E}_{\gamma}(x,x^{\prime}),$
(25)

where
$\mathcal{E}_{\gamma}\in\mathcal{D}^{\prime}(\mathcal{M}\times\mathcal{M})$ is
the causal propagator, namely the difference between the retarded and the
advanced fundamental solutions of the operator $P$ as in Equation (10)
supplemented with Robin boundary conditions – see also Section 2. In turn
$\mathcal{E}_{\gamma}$ abides by the following initial value problem:

$\displaystyle(P\otimes\mathbb{I})\mathcal{E}_{\gamma}=(\mathbb{I}\otimes
P)\mathcal{E}_{\gamma}=0,$ (26)
$\displaystyle\mathcal{E}_{\gamma}|_{\Sigma_{t}\times\Sigma_{t}}=0,\qquad\partial_{t}\mathcal{E}_{\gamma}|_{\Sigma_{t}\times\Sigma_{t}}=\delta_{\Sigma_{t}},$
(27)

where $\Sigma_{t}$ is a generic Cauchy surface at constant time
$t\in\mathbb{R}$, while $\delta_{\Sigma_{t}}$ is the Dirac delta on
$\Sigma_{t}$. Following [4, Ch. 2.3], a solution to this initial value problem
can be written as

$\mathcal{E}_{\gamma}(x,x^{\prime})=\lim_{\varepsilon\to
0^{+}}\sum_{l=0}^{\infty}\sum_{p=-l}^{+l}\int_{\mathbb{R}}d\omega\sin{(\omega(t-t^{\prime}-i\varepsilon))}Y_{l}^{p}(\theta,\phi)\overline{Y_{l}^{p}(\theta^{\prime},\phi^{\prime})}R_{\gamma}(\omega,\rho,\rho^{\prime}),$
(28)

where $R_{\gamma}(\omega,\rho,\rho^{\prime})$ ought to satisfy the integral
identity

$\int_{\mathbb{R}}d\omega\,\omega\,R_{\gamma}(\omega,\rho,\rho^{\prime})=(\rho^{2}-1)\delta(\rho-\rho^{\prime}).$
(29)

This is nothing but Equation (23) which justifies why we have used the symbol
$R_{\gamma}$. Keeping in mind Equation (25) as well as Equation (28), we have
all ingredients to write the formal expression of the two-point correlation
function of the ground state associated to a Klein-Gordon field with Robin
boundary conditions on a Bertotti-Robinson spacetime, namely, using [4, Th.
2.22]:

$w_{2,\gamma}(x,x^{\prime})=\lim_{\varepsilon\to
0^{+}}\sum_{l=0}^{\infty}\sum_{p=-l}^{+l}\int_{\mathbb{R}}d\omega\,\Theta(\omega)e^{-i\omega(t-t^{\prime}-i\varepsilon)}Y_{l}^{p}(\theta,\phi)\overline{Y_{l}^{p}(\theta^{\prime},\phi^{\prime})}R_{\gamma}(\omega,\rho,\rho^{\prime}),$
(30)

which can be rewritten in a more compact form, summing over $m$ and using [26,
§14.30.9], as

$w_{2,\gamma}(x,x^{\prime})=\lim_{\varepsilon\to
0^{+}}\sum_{l=0}^{\infty}\int_{0}^{\infty}d\omega
e^{-i\omega(t-t^{\prime}-i\varepsilon)}\frac{2l+1}{4\pi}P_{l}(\cos\Gamma(\theta,\theta^{\prime},\phi,\phi^{\prime}))R_{\gamma}(\omega,\rho,\rho^{\prime}),$
(31)

where $P_{l}$ is the Ferrers function of the first kind while
$\Gamma:\mathcal{S}^{2}\times\mathcal{S}^{2}\to\mathbb{R}$ is the geodesic
distance on the unit 2-sphere. We highlight that we have called $w_{2,\gamma}$
a formal expression since if we expand $R_{\gamma}$ near $\omega=0$, for both
$\gamma=0$ and $\gamma\in(0,\frac{\pi}{2}]$, the asymptotic behaviour of
$R_{\gamma}$ reads – see Figure 1

$|R_{0}(\omega,\rho,\rho^{\prime})|\overset{\omega\to
0}{\sim}\omega,\qquad|R_{\gamma}(\omega,\rho,\rho^{\prime})|\overset{\omega\to
0}{\sim}\omega^{-1}.$ (32)

Hence the expression in Equation (31) identifies a well defined distribution
only for $\gamma=0$, namely Dirichlet boundary conditions, while for
$\gamma\in(0,\frac{\pi}{2}]$ an infrared divergence occurs, therefore Equation
(31) cannot identify the two-point correlation function of a ground state.

Figure 1: The radial function $R_{\gamma}$ as in Equation (24a) where we set
$l=0$. To highlight the behaviour near $\omega=0$, we have considered the
product $\omega R_{\gamma}$ as a function of $\omega$, the other parameters
being fixed. The plot shows that only for $\gamma=0$ $\omega R_{\gamma}$ tends
to $0$ as $\omega\to 0$, all other cases displaying therefore a singular
behaviour.

###  3.4 A different scenario: the Poincaré patch

In the previous section we have shown that infrared divergences can occur in a
Bertotti-Robinson spacetime when the underlying Klein-Gordon field is not
endowed with Dirichlet boundary conditions. In the following, we highlight
that the correspondence between the existence of a singular behaviour at large
distances and the choice of a boundary condition is highly dependent on the
underlying geometry. To achieve this goal, we consider a different patch of
the Bertotti-Robinson spacetime, denoted by $(\widetilde{M}_{1},g_{1})$. It
has been already analyzed in [29] and it can be realized starting from
Equation (9) and switching from the coordinates $(t,\rho)$ to $(\tau,r)$ by
means of the transformation [22]

$\tau=\frac{(\rho^{2}-1)^{1/2}\sinh t}{\rho+(\rho^{2}-1)^{1/2}\cosh t},\qquad
r=\frac{1}{\rho+(\rho^{2}-1)^{1/2}\cosh t}.$ (33)

The line element in Equation (9) becomes

$ds^{2}=\frac{1}{r^{2}}(-d\tau^{2}+dr^{2}+r^{2}d\Omega^{2}),$ (34)

where the new coordinates can be assigned the following domain:
$\tau\in\mathbb{R}$ while $r\in(0,\infty)$. Subsequently, on top of
$\widetilde{M}_{1}$, we consider a massive, real, Klein-Gordon field
$\widetilde{\Psi}:\widetilde{M}_{1}\to\mathbb{R}$ whose dynamics is ruled by
Equation (10) where the D’Alembert wave operator is built out of the metric in
Equation (34). In particular we are interested in constructing the two-point
correlation function of the ground state associated to $\widetilde{\Psi}$ and
we can follow the same step-by-step construction outlined in the previous
section. Here we will not dwell into the details which have been already
accounted for in [29] and we report the main ingredients and results. We
observe that in the last cited paper the main interest lies in the analysis of
the response function of a Unruh-de Witt detector rather than in the
identification of infrared singularities. As in the previous section we
consider a mode decomposition

$\widetilde{\Psi}(\tau,r,\theta,\phi)=\sum\limits_{l=0}^{\infty}\sum\limits_{p=-l}^{l}\int\limits_{\mathbb{R}}e^{-i\omega\tau}\widetilde{\psi}_{\omega
lp}(r)Y^{p}_{l}(\theta,\phi),$ (35)

where $\widetilde{\psi}_{\omega lp}$ satisfies the radial equation

$(\mathbf{\tilde{L}}+\omega^{2})\widetilde{\psi}_{\omega
lp}(r):=\bigg{[}\frac{d^{2}}{dr^{2}}-\frac{l(l+1)+m^{2}}{r^{2}}+\omega^{2}\bigg{]}\widetilde{\psi}_{\omega
lp}(r).$ (36)

A basis of solutions of this equation can be written in terms of Bessel
functions of the first kind and, using once more the language of Sturm-
Liouville problems, the primary and secondary solutions read

$\tilde{\mathcal{P}}(r)=\sqrt{r}J_{\eta}(\omega
r),\qquad\tilde{\mathcal{S}}(r)=-\omega^{2\eta}\sqrt{r}J_{-\eta}(\omega r),$
(37)

where $\eta=\frac{1}{2}\sqrt{1+4l(l+1)+4m^{2}}$ and where, for simplicity of
the notation, we have dropped the subscript highlighting the dependence on
$\omega,l,p$. As discussed in [29, Sec. 2], Robin boundary conditions,
parameterized by $\gamma\in[0,\frac{\pi}{2}]$, can be imposed whenever $l=0$
and $m^{2}\in[-\frac{1}{4},\frac{3}{4})$ exactly as in the previous section.
Using Equation (37), it turns out that the two-point correlation function for
the ground state in this specific patch reads formally

$\widetilde{w}_{2,\gamma}(x,x^{\prime})=\lim\limits_{\varepsilon\to
0^{+}}\sum_{l=0}^{\infty}\int_{0}^{\infty}d\omega
e^{-i\omega(\tau-\tau^{\prime}-i\varepsilon)}\frac{2l+1}{4\pi}P_{l}(\cos\Gamma(\theta,\theta^{\prime},\phi,\phi^{\prime}))\tilde{R}_{\gamma}(\omega,r,r^{\prime}),$
(38)

where, similarly to Equation (7) we have taken the sum over all admissible
values of $p$ and where

$\tilde{R}_{\gamma}(\omega,r,r^{\prime})=\frac{\sqrt{rr^{\prime}}(\cos(\gamma)J_{\eta}(\omega
r)+\sin(\gamma)\omega^{2\eta}J_{-\eta}(\omega r))(\cos(\gamma)J_{\eta}(\omega
r^{\prime})+\sin(\gamma)\omega^{2\eta}J_{-\eta}(\omega
r^{\prime}))}{2(\sin^{2}(\gamma)\omega^{4\eta}+\sin(2\gamma)\cos(\pi\eta)\omega^{2\eta}+\cos^{2}(\gamma))}.$
(39)

If we expand $\tilde{R}_{\gamma}$ near $\omega=0$, for both
$\gamma\in[0,\frac{\pi}{2})$ and $\gamma=\frac{\pi}{2}$, the asymptotic
behaviour of $\tilde{R}_{\gamma}$ reads

$|\tilde{R}_{\gamma}(\omega,r,r^{\prime})|\overset{\omega\to
0}{\sim}\omega^{2\eta},\qquad|\tilde{R}_{\frac{\pi}{2}}(\omega,r,r^{\prime})|\overset{\omega\to
0}{\sim}\omega^{-2\eta}.$ (40)

Since we are considering mass values in the range $m^{2}\in[0,\frac{3}{4})$,
then the parameter $\eta\in[\frac{1}{2},1)$ which entails that the right hand
side of Equation (38) identifies a well-defined distribution whenever
$\gamma\in[0,\frac{\pi}{2})$ while for $\gamma=\frac{\pi}{2}$, namely only for
Neumann boundary conditions, an infrared divergence occurs in the integrand,
see Figure 2. Therefore one cannot conclude that Equation (38) identifies the
two-point correlation function of a ground state when $\gamma=\frac{\pi}{2}$.

Figure 2: The radial function $\tilde{R}_{\gamma}$ as in Equation (39) where
we set $l=0$. To highlight the behaviour near $\omega=0$, we have considered
the product $\omega\tilde{R}_{\gamma}$ as a function of $\omega$, the other
parameters being fixed. The plot shows that the function tends to $0$ as
$\omega\to 0$ for all boundary conditions except for $\gamma=\frac{\pi}{2}$.

##  4 Conclusions

In this short paper, we have highlighted that, in the construction of the two-
point correlation functions for a free, Klein-Gordon field on globally
hyperbolic manifolds with a timelike boundary, one might face the occurrence
of infrared singularities in the construction of the two-point correlation
function of the ground state even when considering simple boundary conditions
such as those of Robin type. Although this pathology is a well-known feature
especially in two-dimensional globally hyperbolic spacetimes with an empty
boundary, we have shown that it can occur even in higher dimensions, e.g. in a
Bertotti-Robinson spacetime, and that it is strictly linked to the choice of
boundary conditions for the underlying field theory. It is interesting to
remark that, since this phenomenon occurs for ground states, then it is also
present for thermal/KMS states whose infrared behaviour is more singular due
to the contribution of the Bose factor in the mode decomposition. Hence, one
might envisage scenarios, behaving similarly to free Bosonic quantum field
theories in three-dimensional globally hyperbolic spacetimes with an empty
boundary, for which the ground state exists while the thermal counterpart
displays infrared singularities. Finally, our work sets the ground for
multiple future analyses among which we reckon that the following are
especially worth mentioning:

  * 1.

investigating the occurrence of infrared singularities when we consider a more
general class of boundary conditions such as those of Wentzell type [31, 32]
and those associated to a dynamic wall [33, 34],

  * 2.

a comparison of the phenomenon highlighted in this work with the freedom of
choosing the secondary solution in a Sturm-Liouville problem as highlighted in
[28].

##  5 Acknowledgements

The work of L.C. is supported by a postdoctoral fellowship of the Department
of Physics of the University of Pavia, while that of L.S. by a PhD fellowship
of the University of Pavia.

## References

  * [1] W. G. Brenna, R. B. Mann and E. Martin-Martinez, “Anti-Unruh Phenomena,” Phys. Lett. B 757 (2016), 307-311 [arXiv:1504.02468 [quant-ph]].
  * [2] L. J. Henderson, R. A. Hennigar, R. B. Mann, A. R. H. Smith and J. Zhang, “Anti-Hawking phenomena,” Phys. Lett. B 809 (2020), 135732 [arXiv:1911.02977 [gr-qc]].
  * [3] L. De Souza Campos and C. Dappiaggi, “The anti-Hawking effect on a BTZ black hole with Robin boundary conditions,” Phys. Lett. B 816 (2021), 136198 [arXiv:2009.07201 [hep-th]].
  * [4] L. De Souza Campos, “Probing thermal effects on static spacetimes with Unruh-DeWitt detectors,” (2022), Ph.D. Thesis at the University of Pavia, [arXiv:2203.09976 [gr-qc]].
  * [5] C. Dappiaggi and H. R. C. Ferreira, “Hadamard states for a scalar field in anti–de Sitter spacetime with arbitrary boundary conditions,” Phys. Rev. D 94 (2016) no.12, 125016 [arXiv:1610.01049 [gr-qc]].
  * [6] C. Dappiaggi, N. Drago and H. Ferreira, “Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary,” Lett. Math. Phys. 109 (2019) no.10, 2157-2186 [arXiv:1804.03434 [math-ph]].
  * [7] C. C. de Oliveira, R. A. Mosna and J. P. M. Pitelli, “Boundary conditions for isolated asymptotically anti-de Sitter spacetimes,” J. Math. Phys. 63 (2022) no.4, 042501 [arXiv:2203.16644 [gr-qc]].
  * [8] C. C. de Oliveira, R. A. Mosna and J. P. M. Pitelli, “Robin boundary conditions in acoustic BTZ black holes,” Phys. Rev. D 107 (2023) no.6, 064018 [arXiv:2303.05612 [gr-qc]].
  * [9] S. Namasivayam and E. Winstanley, “Vacuum polarization on three-dimensional anti-de Sitter space-time with Robin boundary conditions,” Gen. Rel. Grav. 55 (2023) no.1, 13 [arXiv:2209.01133 [hep-th]].
  * [10] T. Morley, P. Taylor and E. Winstanley, “Vacuum polarization on topological black holes with Robin boundary conditions,” Phys. Rev. D 103 (2021) no.4, 045007 [arXiv:2010.01562 [hep-th]].
  * [11] T. Morley, P. Taylor and E. Winstanley, “Quantum field theory on global anti-de Sitter space-time with Robin boundary conditions,” Class. Quant. Grav. 38 (2021) no.3, 035009 [arXiv:2004.02704 [gr-qc]].
  * [12] A. Garbarz, J. La Madrid and M. Leston, “Scalar field dynamics in a BTZ background with generic boundary conditions,” Eur. Phys. J. C 77 (2017) no.11, 807 [arXiv:1708.04905 [hep-th]].
  * [13] L. de Souza Campos and J. P. M. Pitelli, “Thermal effects on a global monopole with Robin boundary conditions,” Phys. Rev. D 104 (2021) no.8, 085020 [arXiv:2108.12236 [hep-th]].
  * [14] I. Khavkine and V. Moretti, “Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction,” Published in: Chapter 5, Advances in Algebraic Quantum Field Theory, R. Brunetti et al. (eds.), Springer, 2015 [arXiv:1412.5945 [math-ph]].
  * [15] C. J. Fewster and R. Verch, “The Necessity of the Hadamard Condition,” Class. Quant. Grav. 30 (2013), 235027 [arXiv:1307.5242 [gr-qc]].
  * [16] C. Dappiaggi, H. Ferreira and A. Marta, “Ground states of a Klein-Gordon field with Robin boundary conditions in global anti–de Sitter spacetime,” Phys. Rev. D 98 (2018) no.2, 025005 [arXiv:1805.03135 [hep-th]].
  * [17] F. Bussola, C. Dappiaggi, H. R. C. Ferreira and I. Khavkine, “Ground state for a massive scalar field in the BTZ spacetime with Robin boundary conditions,” Phys. Rev. D 96 (2017) no.10, 105016 [arXiv:1708.00271 [gr-qc]].
  * [18] L. de Souza Campos and C. Dappiaggi, “Ground and thermal states for the Klein-Gordon field on a massless hyperbolic black hole with applications to the anti-Hawking effect,” Phys. Rev. D 103 (2021) no.2, 025021 [arXiv:2011.03812 [hep-th]].
  * [19] B. Bertotti, “Uniform Electromagnetic Field in the Theory of General Relativity,” Phys. Rev. 116, 1331 (1959).
  * [20] I. Robinson, “A Solution of the Maxwell-Einstein Equations,” Bull. Acad. Pol. Sci. 7, 351 (1959).
  * [21] A. Conroy and P. Taylor, “Response of an Unruh-DeWitt detector near an extremal black hole,” Phys. Rev. D 105, no.8, 085001 (2022) [arXiv:2109.04486 [gr-qc]].
  * [22] A. C. Ottewill and P. Taylor, “Quantum field theory on the Bertotti-Robinson space-time,” Phys. Rev. D 86 (2012), 104067 [arXiv:1209.6080 [gr-qc]].
  * [23] L. Aké, J. L. Flores, M. Sánchez, “Structure of globally hyperbolic spacetimes with timelike boundary”, Rev. Matem. Iberoamericana, 37, Issue 1 (2021), 45, arXiv:1808.04412 [gr-qc].
  * [24] C. Dappiaggi and A. Marta, “Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes,” Math. Phys. Anal. Geom. 24 (2021) no.3, 28 [arXiv:2101.10290 [math-ph]].
  * [25] O. Gannot and M. Wrochna, “Propagation of Singularities on AdS Spacetimes for General Boundary Conditions and the Holographic Hadamard Condition,” J. Inst. Math. Jussieu 21 (2022) no.1, 67, [arXiv:1812.06564 [math.AP]].
  * [26] NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov,
  * [27] A. Zettl, Sturm-Liouville Theory, American Mathematical Society, (2005), 328p.
  * [28] L. d. Campos, C. Dappiaggi and L. Sinibaldi, “Hidden freedom in the mode expansion on static spacetimes,” Gen. Rel. Grav. 55 (2023) no.3, 50 [arXiv:2207.08662 [gr-qc]].
  * [29] L. d. Campos, C. Dappiaggi and L. Sinibaldi, “Physical significance of generalized boundary conditions: An Unruh-DeWitt detector viewpoint on PAdS2×S2,” Phys. Lett. B 836 (2023), 137597 [arXiv:2210.02395 [hep-th]].
  * [30] I. Stakgold and M.J. Holst, “Green’s functions and boundary value problems”, John Wiley & Sons, 99 (2011).
  * [31] C. Dappiaggi, H. R. C. Ferreira and B. A. Juárez-Aubry, “Mode solutions for a Klein-Gordon field in anti–de Sitter spacetime with dynamical boundary conditions of Wentzell type,” Phys. Rev. D 97 (2018) no.8, 085022 [arXiv:1802.00283 [hep-th]].
  * [32] C. Dappiaggi, B. A. Juárez-Aubry and A. Marta, “Ground state for the Klein-Gordon field in anti–de Sitter spacetime with dynamical Wentzell boundary conditions,” Phys. Rev. D 105 (2022) no.10, 105017 [arXiv:2203.04811 [hep-th]].
  * [33] N. Obadia and R. Parentani, “Notes on moving mirrors,” Phys. Rev. D 64 (2001), 044019 [arXiv:gr-qc/0103061 [gr-qc]].
  * [34] W. M. H. Wan Mokhtar and J. Louko, “The cost of building a wall for a fermion,” JHEP 08 (2022), 069 [arXiv:2205.10853 [gr-qc]].