# Casimir Physics beyond the Proximity Force Approximation: The Derivative
Expansion

César D. Fosco  Centro Atómico Bariloche, Instituto Balseiro, Comisión
Nacional de Energía Atómica, 8400 Bariloche, Argentina  Fernando C. Lombardo
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad
de Buenos Aires, Buenos Aires, Argentina  Instituto de Física de Buenos Aires
(IFIBA), CONICET, Universidad de Buenos Aires, Argentina  Fransisc D.
Mazzitelli  Centro Atómico Bariloche, Instituto Balseiro, Comisión Nacional de
Energía Atómica, 8400 Bariloche, Argentina

###### Abstract

We review the derivative expansion (DE) method in Casimir physics, an approach
which extends the proximity force approximation (PFA). After introducing and
motivating the DE in contexts other than the Casimir effect, we present
different examples which correspond to that realm. We focus on different
particular geometries, boundary conditions, types of fields, and quantum and
thermal fluctuations. Besides providing various examples where the method can
be applied, we discuss a concrete example for which the DE cannot be applied;
namely, the case of perfect Neumann conditions in 2 + 1 dimensions. By the
same example, we show how a more realistic type of boundary condition
circumvents the problem. We also comment on the application of the DE to the
Casimir–Polder interaction which provides a broader perspective on
particle–surface interactions.

##  I Introduction

Casimir forces are one of the most intriguing macroscopic manifestations of
quantum fluctuations in Nature. Their existence, first realized in the
specific context of the interaction between the quantum electromagnetic (EM)
field and the boundaries of two neutral bodies, manifests itself as an
attractive force between them. That force depends, in an intricate manner, on
the shape and EM properties of the objects. Since the discovery of this effect
by Hendrik Casimir 75 years ago Casimir48  this, and closely related
phenomena, have been subjected to intense theoretical and experimental
research Milonnibook ; Miltonbook ; Bordagbook ; Dalvitbook . The outcome of
that work has not just revealed fundamental aspects of quantum field theory,
but also subtle aspects of the models used to describe the EM properties of
material bodies. Besides, it has become increasingly clear that this research
has potential applications to nanotechnology.

Theoretical and experimental reasons have called for the calculation of the
Casimir energies and forces for different geometries and materials
Mostepanenko2021 , and with an ever increasing accuracy. The simplicity of the
theoretical predictions when two parallel plates are involved, corresponds to
a difficult experimental setup, due to alignment problems (in spite of this,
the Casimir force for this geometry has been measured at the $10\%$ accuracy
level Carugno ). Equivalently, geometries which are more convenient from the
experimental point of view, and allow for higher precision measurements, lead
to more involved theoretical calculations. Such is the case of a cylinder
facing a plane onofrio , or a sphere facing a plate, which is free from the
alluded alignment problems precisionera1 ; precisionera2 ; precisionera3 ;
precisionera4 ; precisionera5 ; precisionera6 ; precisionera7 .

From a theoretical standpoint, finding the dependence of the Casimir energies
and forces on the geometry of the objects, poses an interesting challenge.
Indeed, even when evaluating the self-energies which result from the coupling
on an object to the vacuum field fluctuations, results may be rather non-
intuitive; as in the case of a single spherical surface Boyer68 .

For a long time, calculations attempting to find analytical results for the
Casimir and related interactions had been restricted to using the so called
proximity force approximation (PFA). In this approach, the interaction
energies and the resulting forces are computed approximating the geometry by a
collection of parallel plates and then adding up the contributions obtained
for this approximate geometry. This procedure was presumed to work well
enough, at least for smooth surfaces when they are sufficiently close to each
other; in more precise terms: when the curvature radii of the surfaces $R_{i}$
are much larger than the distance $d$ between them. Indeed, this is the main
content of the Derjaguin approximation (DA), developed by Boris Derjaguin in
the 1930s Derjaguin1 ; Derjaguin2 ; Derjaguin3  , which is pivotal in the
study of surface interactions, especially in the context of colloidal
particles and biological cells. This approach has significant implications in
understanding colloidal stability, adhesion, and thin film formation.

It is worth introducing some essentials of the DA, in particular, of the
geometrical aspects involved. Assuming the interaction energy per unit area
between two parallel planes at a distance $h$ is known, and given by
$E_{\shortparallel}(h)$, the DA yields an expression for the interaction
energy between two curved surfaces, $U_{\rm DA}$ Milonnibook ; Bordagbook ;
Derjaguin1 ; Derjaguin2 ; Derjaguin3 ; Intermolecular . Indeed,

$U_{\rm DA}(a)=2\pi R_{\rm eff}\int_{a}^{\infty}E_{\shortparallel}(h)dh\,,$
(1)

where $a$ denotes the distance between the surfaces, $R_{1}$ and $R_{2}$ are
their curvature radii (at the point of closest distance), while $R_{\rm
eff}=R_{1}R_{2}/(R_{1}+R_{2})$. It is rather straightforward to implement the
approximation at the level of the force $f_{\rm DA}$ between surfaces:

$f_{\rm DA}(a)=2\pi R_{\rm eff}E_{\shortparallel}(a)\,.$ (2)

This approximation is usually derived from a quite reasonable assumption,
namely, that the interaction energy can be approximated by means of the PFA
expression:

$U_{\rm PFA}=\int\,dS\,E_{\shortparallel}\;.$ (3)

Here, the surface integration may be performed over one of the participating
surfaces, but it could also be over an imaginary, “interpolating surface”,
which lies between them. The DA is obtained from the expression above, by
approximating the surfaces by (portions of) the osculating spheres (with radii
$R_{1}$ and $R_{2}$) at the point of closest approach.

Based on this hypothesis, on dimensional grounds one can expect the
corrections to the PFA to be of order ${\mathcal{O}}(a/R_{i})$. Note, however,
that since the PFA had not been obtained as the leading-order term in a well-
defined expansion, the approximation itself did not provide any quantitative
method to asses the validity of that assumption.

A need for reliable measure of the accuracy of the results obtained using
different methods became increasingly crucial, specially since the development
of the “precision era” in the measurement of the Casimir forces precisionera1
; precisionera2 ; precisionera3 ; precisionera4 ; precisionera5 ;
precisionera6 ; precisionera7 . It was in this context that the Derivative
Expansion (DE) approach, was first introduced by us in 2011 FLM2011 , as a
tool to asses the validity of the PFA, by putting it in the framework of an
expansion, and to calculate corrections to the PFA using that very same
expansion. When one realizes that the PFA had previously been proposed in
contexts which are rather different to Casimir physics, it becomes clear that
the improvement on the PFA which represents the DE may and does have relevance
on those realms, regardless of them having an origin in vacuum fluctuations or
not. Indeed, when one strips off the DE of the particularities of Casimir
physics, one can see the ingredients that allowed one to implement it are also
found, for example, in electrostatics, nuclear physics, and colloidal surface
interactions.

Here, we present the essential features of the DE, its derivation, and
consider some examples of its applications. The review is organized as
follows. In Section II , we recall some aspects of the DA which stem from its
application to nuclear and colloidal physics. We start with the DA not just
for historical reasons, but also because we believe that this sheds light on
some geometrical aspects of the approximation, in a rather direct way (like
the relevance of curvature radii and distances).

Then, in Section III, we introduce the DE in one of its simplest realizations,
namely, in the context of electrostatics, for a system consisting of two
conducting surfaces kept at different potentials FLMelectro . We first
evaluate the PFA in this example, and then introduce the DE as a method to
improve on that approximation. In Section IV, we introduce a more abstract,
and therefore more general, formulation of the DE FLMrev2014 . By putting
aside the particular features of an specific interaction, and keeping just the
ones that are common to all of them, we are lead to formulate the problem as
follows: the DE is a particular kind of expansion of a functional having as
argument a surface (or surfaces). We mean “functional” here in its
mathematical sense: a function that assigns a number to a function or
functions. We elucidate and demonstrate some of the aspects of the DE in this
general context; the purpose of presenting those aspects are not just a matter
of consistency or justification, but they also provide a concrete way of
applying and implementing the DE to any example where it is applicable.

Then, in Section V, we focus on the DE in the specific context of the Casimir
interaction between surfaces, for perfect boundary conditions at zero
temperature; i.e., vacuum fluctuations FLM2011 ; Bimonte1 . Then in Section VI
we review the extension of those results to the case of finite temperatures
and real materials Bimonte2 ; FLMTdim . As we shall see, the temperature
introduces another scale, which affects the form one must adopt for the
different terms in the DE. Then we comment on an aspect which first manifests
itself here: as it happens with any expansion, it is to be expected to break
down for some specific examples, when the hypothesis that justified it are not
satisfied. We show this for the case of the Casimir effect with Neumann
conditions at finite temperatures FLMTdim ; FLMNeu . We also show that the
application of the DE to the EM field is free of this problem, if dissipative
effects are included in the model describing the media FLMTem .

The application of the DE to Casimir-Polder forces for atoms near smooth
surfaces BimonteCP  is described in Section VII. Other alternatives to compute
Casimir energies beyond PFA paulo22  are described in Section VIII. Section IX
contains our conclusions.

##  II Proximity Approximations in Nuclear and Colloidal Physics

The introduction of the Derjaguin Approximation (DA) to nuclear physics dates
back to the seminal paper Blocki1 . In this paper, the DA was rediscovered and
applied to calculate nuclear interactions, starting with a Derjaguin-like
formula for the surface interaction energies. The approach was based on a
crucial ”universal function” - a term referring here to the interaction energy
between flat surfaces, calculated using a Thomas-Fermi approximation. In spite
of the rather different context, the analogy with the approach followed in the
DA becomes clear when one introduces three surfaces, the physical ones,
$\Sigma_{L}$ and $\Sigma_{R}$, and the intermediate one $\Sigma$ which one
uses to parametrize the interacting ones. Then, if the physical surfaces are
sufficiently smooth, the interaction energy should, to a reasonable
approximation, be described by the PFA, in a similar fashion as in Equation
(3). To render the assertion above more concrete, we yet again use the
function $h:\Sigma\to{\mathbb{R}}$, measuring the distance between
$\Sigma_{L}$ and $\Sigma_{R}$ at each point on $\Sigma$. Since $h$ will have
level sets which are, except for a zero measure set, one-dimensional (closed
curves), and the interaction depends just on $h$, the PFA expression for the
interaction energy $U$ may be rendered as a one-dimensional integral:

$U_{\rm PFA}=\int dh\,J(h)\,E_{\shortparallel}(h)\;,$ (4)

where $J(h)dh$ is the infinitesimal area between two level curves on $\Sigma$:
the ones between $h$ and $h+dh$, while $E_{\shortparallel}$ is the universal
function.

We now assume that $\Sigma$ is a plane, and that the physical surfaces may be
both described by means of just one Monge patch based on $\Sigma$. This
surface is then naturally thought of (in descriptive geometry terms) as the
projection plane. Using Cartesian coordinates
$(x_{1},x_{2})\equiv{\mathbf{x}_{\shortparallel}}$ on $\Sigma$, assuming (for
smooth enough surfaces) that $J$ may be regarded as constant, and using a
second-order Taylor expansion of $h$ around $a$ (the distance of closest
approach):

$h({\mathbf{x}_{\shortparallel}})\simeq
a+\frac{1}{2}\big{(}\frac{x_{1}^{2}}{R_{1}}+\frac{x_{2}^{2}}{R_{2}}\big{)}$
(5)

produces, when evaluating the PFA interaction energy (4), the DA energy (1).
Here, $R_{1}$ and $R_{2}$ are the radii of curvature of the surface by
$x_{3}=h({\mathbf{x}_{\shortparallel}})$ at $x_{3}=a$.

This result may be improved, even within the spirit of the PFA, by introducing
some refinements. Indeed, in Blocki2 , a generalization of the PFA has been
introduced such that the starting point was Equation (4), but now allowing for
the surfaces to have larger curvatures, as long as they remained almost
parallel locally. The main difference that follows from those weaker
assumptions is that, now, the Jacobian $J$ may become a non-trivial function
of $h$. For instance, introducing a linear expansion:

$J(h)\approx J_{0}+J_{1}h\;,$ (6)

a straightforward calculation shows that the force $f$ becomes:

$f_{\rm
PFA}(a)\,=\,J_{0}E_{\shortparallel}(a)-J_{1}(a)\int_{a}^{\infty}\,dh\,E_{\shortparallel}(h)\;.$
(7)

Note that the result is the sum of the DA term plus a second term proportional
to the derivative of the Jacobian with respect to $h$. This is a correction to
the DA obtained from the same starting point we used for the DA: $U_{\rm
PFA}$. In other words, Equation (7) is still determined by the energy density
for parallel plates. As we shall see, the DE will introduce corrections that
go beyond $E_{\shortparallel}(a)$. The correction will depend on both the
geometry and the nature of the interaction.

We wish to point out that the lack of knowledge of an exact expression for
$E_{\shortparallel}$ is not specific to nuclear physics, but of course it may
appear in other applications. The general PFA approach can nevertheless be
introduced; the accuracy of its predictions will then be limited not just by
the fulfillment or not of the geometrical assumptions, but also by the
reliability of the expression for $E_{\shortparallel}$. Using different
approximations for $E_{\shortparallel}$ gives as many results for the PFA. For
a recent review in the case of nuclear physics, see Ref. RevNuc1 ; RevNuc2 .

An apparently unrelated approximation, based on different physical
assumptions, was introduced in the context of colloidal physics. Let us now
see how it yields a result which agrees with the DA: it is the so called
Surface Element Integration (SEI) SEI , or Surface Integration Approach (SIA)
SIA . This approach may be introduced as follows: let us consider a compact
object facing the $x_{3}=0$ plane. $x_{3}$ is then the normal coordinate to
the plane, pointing towards the compact object. With this conventions, the SEI
approximation applied to the interaction energy amounts to the following:

$U_{\rm SEI}=-\int_{\rm
plane}dx_{1}dx_{2}\,\frac{\hat{n}\cdot\hat{e}_{3}}{|\hat{n}\cdot\hat{e}_{3}|}\,E_{\shortparallel}\;.$
(8)

Here, $\hat{n}$ denotes the outwards pointing unit normal to each surface
element of the object. We see that, when the compact object may be thought of
as delimited by just two surfaces, one of them facing the plane and the other
away from it, the SEI consists of the difference between the PFA energies of
those surfaces. This (possibly startling) fact is, as we shall see, related to
the fact that the SEI becomes exact for almost transparent bodies, a situation
characterized by the fact that the interaction is the result of adding all the
(volumetric ) pairwise contributions.

In the context of colloidal physics , the SEI method relies heavily upon the
existence of a pressure on the compact object. The effect of that pressure
should be integrated over the closed surface surrounding the compact object,
in order to find the total force SEI . An alternative route to understand the
SEI is to showthat Equation (8) becomes exact when the interaction between
macroscopic bodies is the superposition of the interactions for the pair
potentials of their constituents SIA . That may be interpreted by using a
simple example. Consider two media, one of them, the left medium $L$,
corresponding to the $x_{3}\leq 0$ half-space, while the right medium, $R$, is
defined as the region:

$R\;=\;\\{(x_{1},x_{2},x_{3}):\psi_{1}({\mathbf{x}_{\shortparallel}})\leq
x_{3}\leq\psi_{2}({\mathbf{x}_{\shortparallel}})\\}\,.$ (9)

The interaction energy $U$ is a functional of the two functions $\psi_{1,2}$.
When the media are diluted, we expect the interaction energy to have the form

$U[\psi_{1},\psi_{2}]\;=\;\int
d^{2}{\mathbf{x}_{\shortparallel}}\left(E_{\shortparallel}(\psi_{1})-E_{\shortparallel}(\psi_{2})\right)\,,$
(10)

where $E_{\shortparallel}(a)$ is the interaction energy per unit area, between
two half-spaces at a distance $a$. This formula can be interpreted as follows:
to obtain the interaction energy for the configuration described by $\psi_{1}$
and $\psi_{2}$, one must certainly subtract from
$E_{\shortparallel}(\psi_{1})$ the contributions from $x_{3}>\psi_{2}$. This
“linearity” is expected to be valid only for dilute media, and in that
situation it coincides with the result obtained using the SEI. One expects
then the SEI to give an exact result for almost-transparent media, for which
the superposition principle holds true, and the total interaction energy is
due to the sum of all the different pairwise potentials SIA . It is worth
noting, at this point, the important fact that the PFA also becomes exact in
Casimir physics when the media constituting the objects are dilute. Indeed,
this has been pointed out in Decca ; Miltondilute .

The examples just described illustrate the relevance of the DA, and of some of
its variants, to different areas of physics. At the same time, the main
drawback is made rather evident: in spite of being based on reasonable
physical assumptions, it is difficult to assess its validity. The reason for
this difficulty is that the approximation is uncontrolled, and therefore the
estimation of the error incurred is difficult, within a self-contained
approach.

The DE provides a systematic method to improve the PFA, and to compute its
next-to-leading-order (NTLO) correction in a consistent set up.

##  III Introducing the Derivative Expansion

###  III.1 The PFA in an Electrostatic Example

We introduce the PFA, and then the DE, in an example which neatly illustrates
the DE main aspects, in the context of electrostatics. Here, contrary to what
happens when dealing with more involved systems, like, say, Van der Waals,
nuclear or Casimir forces, the physical assumptions and their implementations
are more transparent. We follow closely Ref.FLMelectro

The set-up we want to describe consists of two perfectly-conducting surfaces,
one of them an infinite grounded plane and the other a smoothly curved surface
kept at an electrostatic potential $V_{0}$. We use coordinates such that the
plane corresponds to $z=0$ while the smooth surface is such that it can be
described by a single function, namely, by an equation of the form
$z=\psi(x_{\shortparallel})$. The electrostatic energy contained between
surfaces can then be written as follows:

$U=\frac{\epsilon_{0}}{2}\int
d^{2}{\mathbf{x}_{\shortparallel}}\int_{0}^{\psi({\mathbf{x}_{\shortparallel}})}dz\,|\mathbf{E}|^{2}\,,$
(11)

where $\epsilon_{0}$ denotes the permittivity of vacuum. In terms of $U$ and
$V_{0}$, the capacitance $C$ of the system is then given by $C=2U/V_{0}^{2}$.

Let us see how one implements the PFA in order to calculate $U$ (from which
one can extract, for instance, an approximate expression for $C$) expecting it
to be accurate when the distance between the two surfaces is shorter than the
curvature radius of the curved conductor. To that end, one first finds and
approximation to the electric field between the conductors, by proceeding as
follows: the smooth conductor is regarded as a set of parallel plates (Fig.1),
in the sense that the electric field ${\mathbf{E}}$ points along the $z$
direction and has a $z$-independent value. The electric field does, however,
depend on ${\mathbf{x}_{\shortparallel}}$ since it is assumed to have, for
every ${\mathbf{x}_{\shortparallel}}$, the same intensity as the electric
field due to two (infinite) conducting planes at a distance
$\psi({\mathbf{x}_{\shortparallel}})$. Namely,
${\mathbf{E}}({\mathbf{x}})=-V_{0}/\psi({\mathbf{x}_{\shortparallel}})\,\mathbf{\hat{z}}$.
Therefore, the approximated expression for the electrostatic energy becomes:

$U_{\rm PFA}=\frac{\epsilon_{0}V_{0}^{2}}{2}\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\frac{1}{\psi({\mathbf{x}_{\shortparallel}})}\,.$
(12)

It is implicitly understood in the equation above, that the region to
integrate is such that the assumption on the distance and curvature is
satisfied. On the contrary, regions such that the assumption is not satisfied
can be consistently ignored (see the example below).

It should be evident that Equation (12) provides a rather convenient tool to
obtain estimates for the electrostatic energy in many relevant situations.
Indeed, to illustrate this point we consider a cylinder of length $L$ and
radius $R$ in front of a plane, and denote by $a$ the minimum distance between
the two surfaces. The cylinder is not a surface that can be described by a
single patch; namely, one needs at least two functions. However, in the
context of the PFA, it is reasonably to assume that only the half that is
closer to the plane should be relevant. Assuming the axis of the cylinder to
be along $y$, the function $\psi$ reads:

$\psi(x)=a+R\left(1-\sqrt{1-\frac{x^{2}}{R^{2}}}\right)\,,$ (13)

with the variable $x$ assumed to be in the range $-x_{M}<x<x_{M}<R$. Note that
for $x_{M}/R=O(1)<1$ the assumption on the distance and the curvature is
satisfied. It is to be expected that, as long as $R\gg a$ (where the PFA gives
an accurate value of the electrostatic energy), the final result will not
depend on $x_{M}$. This can be readily checked by inserting Equation (13) into
Equation (12), computing the integral, and expanding that result for $a\ll R$.
Doing this we obtain:

$U_{\rm PFA}^{\rm
cp}\approx\frac{\epsilon_{0}V_{0}^{2}L\pi}{\sqrt{2}}\sqrt{\frac{R}{a}}\,,$
(14)

which is independent of $x_{M}$. An immediate consequence of this is that,
when the cylinder approaches the plane, the electrostatic force behaves as
$a^{-3/2}$.

Figure 1: In the PFA, the interaction between a smoothly curved surface and a
plane is approximated by that of a set of parallel plates. For each pair of
parallel plates, border effects are ignored.

Let us check now the accuracy of $U_{\rm PFA}^{\rm cp}$. We take advantage of
the knowledge of the exact expression for the electrostatic interaction
energy:

$U^{\rm cp}=\frac{\pi L\epsilon_{0}V_{0}^{2}}{{\rm
arccosh}\left(1+\frac{a}{R}\right)}\,.$ (15)

For $a/R\ll 1$, $U^{\rm cp}$ yields the PFA result $U^{\rm cp}_{\rm PFA}$
(14). The relevance of the corrections to the PFA can be estimated by
expanding the exact result, but keeping also the next-to-leading order (NTLO)
when $a<<R$:

$U^{\rm
cp}\approx\frac{\epsilon_{0}V_{0}^{2}L\pi}{\sqrt{2}}\sqrt{\frac{R}{a}}\left(1+\frac{1}{12}\frac{a}{R}\right)\;.$
(16)

We will now introduce the DE. By construction, it should produce the NTLO
result (for this an other surfaces), without resorting to the expansion of any
exact expression (the knowledge of which, needless to say, is usually
lacking).

###  III.2 Improvement of the PFA Using a Derivative Expansion

We begin by noting that the electrostatic energy is a functional of the
function which defines the shape of the surface. A second observation is that,
in principle, there is no reason to assume that the functional is local in
$\psi$. Here, “local” means that it contains just one integral over
${\mathbf{x}_{\shortparallel}}$ of a sum of terms involving powers of
$\psi({\mathbf{x}_{\shortparallel}})$ and derivatives at
$\psi({\mathbf{x}_{\shortparallel}})$. On the contrary, the exact functional
will generally involve terms where, for example, there are two or more
integrals over ${\mathbf{x}_{\shortparallel}}$, and kernels depending of those
variables, and products of $\psi$ with different arguments. However,
regardless of the non locality of the exact expression, it must become local
when the surfaces are sufficiently smooth and close to each other. Indeed, if
the PFA becomes valid asymptotically in that limit, then the energy must
approach a result which is a local function of $\psi$. Not whatever local
functional but just one without derivatives.

The way we found to depart slightly but significantly from the PFA, has been
to add terms involving derivatives of $\psi$. Namely, we shall assume that the
electrostatic energy can be expanded in local terms involving derivatives of
$\psi$. One can think of the condition $|\nabla\psi|\ll 1$, as introducing a
small, dimensionless expansion parameter. In physical terms, this means that
the curved surface is almost parallel to the plane on the points where it is
satisfied.

To introduce the first departure from the PFA, we include terms with up to two
derivatives. Then the electrostatic energy has to be (up to this order) of the
form:

$U_{\rm DE}\simeq\int
d^{2}{\mathbf{x}_{\shortparallel}}\left[V(\psi)+Z(\psi)|\nabla\psi|^{2}\right]\,,$
(17)

for some functions $V$ and $Z$. The gradient is the two-dimensional one, and
it can only appear in such a way that the energy is a scalar ($\psi$ is a
scalar under changes of coordinates on the plane). Besides, recalling the
equations of electrostatics, and on dimensional grounds, the result must be
proportional to $\epsilon_{0}V^{2}$. On top of that it must reproduce $U_{\rm
PFA}$ for constant $\psi$. Furthermore, as $\psi$ is the only other
dimensionful quantity, both functions $V$ and $Z$ have to be proportional to
$\psi^{-1}$. Thus, we have restricted even further the functional to:

$U_{\rm DE}\simeq\frac{\epsilon_{0}V_{0}^{2}}{2}\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\frac{1}{\psi}(1+\beta_{\rm
E}|\nabla\psi|^{2})\,,$ (18)

where $\beta_{\rm E}$ is a numerical coefficient to be determined (the
subindex $\rm E$ stands for electrostatics). It is worth stressing that it is
independent of the specific surface being considered, as long as it is smooth.
Therefore, it can be obtained once and for all just from its evaluation for a
particular case. A simple procedure to obtain the coefficient $\beta_{\rm E}$,
when an exact analytic solution to the problem is known, would be to retrieve
its value by expanding that solution. Let us do that for the configuration of
a cylinder in front of a plane. Inserting Equation (13) into Equation (18),
and performing the integrals, an expansion of the result in powers of $a/R$,
allow us to fix $\beta_{\rm E}$. Indeed, in order to agree with the expansion
of the exact result in Equation (16), this fixes its value to $\beta_{\rm
E}=1/3$. Of course, one will obtain the same value for any other particular
example for which the exact solution was known.

It is worthy of noting that, since the DE is a perturbative approach, it
should be desirable to have a perturbative method to calculate the coefficient
$\beta_{\rm EM}$. In other words, to compute it from first principles, using
the appropriate expansion. One can do that, for instance, by solving
perturbatively the Laplace equation and then resorting to the method described
in Section V. We have performed that calculation in Ref.FLMelectro , and refer
the reader to that work for details, and also for the application of the DE to
other electrostatic examples.

###  III.3 Two Smooth Surfaces

As a natural generalization of the previously discussed situation, let us now
consider two surfaces described by the two functions
$\psi_{1}({\mathbf{x}_{\shortparallel}})$ and
$\psi_{2}({\mathbf{x}_{\shortparallel}})$, each one of them measuring the
respective height of a surface with respect to a reference plane $\Sigma$.
This geometry was first considered in the context of the DE for the Casimir
effect in Ref.Bimonte1 .

To construct the DE for the electrostatic energy in this case, we keep up to
two derivatives of the functions. This allows we to write the general
expression:

$\displaystyle U_{\rm DE}[\psi_{1},\psi_{2}]$ $\displaystyle=$
$\displaystyle\int_{\Sigma}d^{2}{\mathbf{x}_{\shortparallel}}\,U_{\shortparallel}({\psi)}\,\left[1+\beta_{1}|\nabla\psi_{1}|^{2}\right.$
(19) $\displaystyle+$
$\displaystyle\beta_{2}|\nabla\psi_{2}|^{2}+\beta_{\times}\nabla\psi_{1}\cdot\nabla\psi_{2}$
$\displaystyle+$
$\displaystyle\left.\beta_{\rm-}\,{\hat{\mathbf{z}}}\cdot\nabla\psi_{1}\times\nabla\psi_{2})+\cdots\right],$

where $\psi=|\psi_{2}-\psi_{1}|$ is the height difference,
$U_{\shortparallel}(\psi)=\epsilon_{0}V_{0}^{2}/(2\psi)$ is the electrostatic
energy between parallel plates, and the dots denote higher derivative terms.
Equation (19) actually contains four numerical constants: $\beta_{1}$,
$\beta_{2}$, $\beta_{\times}$, and $\beta_{-}$. However, symmetry
considerations imply some constraints on them: the energy must be invariant
under the interchange of $\psi_{1}$ and $\psi_{2}$, since that is just a
relabeling: $\beta_{1}=\beta_{2}$ and $\beta_{-}=0$. Furthermore, in order to
reproduce the result for a single smooth surface in front of a plane we must
have $\beta_{1}=\beta_{2}=1/3$. The coefficient $\beta_{\times}$ can be
determined taking into account that the energy should be invariant under a
simultaneous rotation of both surfaces Bimonte1 . Indeed, for an infinitesimal
rotation of each surface by an angle $\epsilon$ in the plane $(x,z)$, the
changes induced on the functions $\psi_{i}$ are
$\delta\psi_{i}=\epsilon(x+\psi_{i}\partial_{x}\psi_{i})$, for $i=1,2$. To
simplify the determination of $\beta_{\times}$ we can assume that, initially,
$\psi_{1}=0$ and that $\psi_{2}$ is only a function of $x$. Computing
explicitly the variation of $U_{\rm DE}$ to linear order in $\epsilon$ one can
show that

$\delta U_{\rm DE}=0\Rightarrow\beta_{\times}={1}/{3}\,,$ (20)

and therefore

$U_{\rm
DE}[\psi_{1},\psi_{2}]=\frac{\epsilon_{0}V_{0}^{2}}{2}\int_{\Sigma}d^{2}{\mathbf{x}_{\shortparallel}}\,\frac{1}{\psi}\,\left[1+\frac{1}{3}\left(|\nabla\psi_{1}|^{2}+|\nabla\psi_{2}|^{2}+\nabla\psi_{1}\cdot\nabla\psi_{2}\right)\right].$
(21)

Note that, by taking the variation of the electrostatic energy Equation (19)
with respect to translations or rotations of one of the surfaces, one can
obtain the vertical and lateral components of the force, as well as the
torque, due to the remaining surface.

The identities $\beta_{1}=\beta_{2}$ and $\beta_{-}=0$ are universally valid,
regardless of the interaction (as long as the surfaces are of an identical
nature), but $\beta_{\times}=\beta_{1}$ holds true for the electrostatic
interaction. This depends upon the fact that the leading term is proportional
to $\psi^{-1}$ (i.e. it is then not valid for the Casimir energy).

For later use, let us recall that, for a general function
$E_{\shortparallel}(\psi)$, the relation between the different coefficients
becomes Bimonte1 :

$2(\beta_{1}+\beta_{2})+2\beta_{\times}+\psi\frac{d\log
E_{\shortparallel}}{d\psi}=1\,.$ (22)

The relation (22) shows that, for any interaction, the DE for the interaction
energy between two curved surfaces can be reduced to the problem of a single
surface in front of a plane. Indeed, in the later case one can determine
$\beta_{1}$ and $\beta_{2}$, while Equation (22) determines the remaining
coefficient $\beta_{\times}$.

To summarize: when computing the electrostatic energy associated with a
configuration of two conductors at different potentials, with smoothly curved
surfaces, one can go beyond the PFA by simply assuming that the energy admits
an expansion in derivatives of the functions that define the shapes of the
conductors. If the exact electrostatic energy for a single non trivial curved
configuration is known, one can determine all the free parameters in the
expansion.

Finally, the NTLO correction produces an appreciable improvement in the DA
and, by the same token, also provides an assessment for its validity. An
interesting alternative approach to compute electrostatic forces beyond the
PFA can be found in Ref. Hudlet .

##  IV Obtaining the DE from a Perturbative Expansion

Regardless of the interaction considered, the DA and its improvement, the DE,
can be obtained by performing the proper resummation of a perturbative
expansion FLMrev2014 . The required expansion is in powers of the departure of
the surfaces, about a two flat parallel planes configuration. This connection
yields a systematic and quite general approach to obtain the DE, even when an
exact solution is not available.

To keep things general, we work with a general functional of the surface; that
functional may correspond to an energy, free energy, force, etc. Besides, we
do not make any assumption about the kind of interaction involved, not even
about whether it satisfies a superposition principle or not.

To begin, let us we assume a geometry where there are two surfaces, one of
which, $L$, is a plane, which with a proper choice of Cartesian coordinates
($x_{1}$, $x_{2}$, $x_{3}$), is described by $x_{3}=0$. The other one, $R$, is
assumed to be describable by $x_{3}=\psi({\mathbf{x}_{\shortparallel}})$.

The object for which we implement the approximation is denoted by $F[\psi]$, a
functional of $\psi$. Then we note that the PFA for $F$, to be denoted here by
$F_{0}$, is obtained as follows: add, for each
${\mathbf{x}_{\shortparallel}}$, the product of a local surface density
${\mathcal{F}}_{0}(\psi({\mathbf{x}_{\shortparallel}}))$ depending only on the
value of $\psi$ at the point ${\mathbf{x}_{\shortparallel}}$, times the
surface element area; namely,

$F_{0}[\psi]\;=\;\int
d^{2}{\mathbf{x}_{\shortparallel}}\,{\mathcal{F}}_{0}(\psi({\mathbf{x}_{\shortparallel}}))\;.$
(23)

The surface density is, in turn, determined by the (assumed) knowledge of the
exact form of $F$ for the case of two parallel surfaces, as follows:

${\mathcal{F}}_{0}(a)\;=\;\lim_{{\mathcal{S}}\to\infty}\big{[}\frac{F[a]}{\mathcal{S}}\big{]}\;,$
(24)

where ${\mathcal{S}}$ denotes the area of the $L$ plate and $a$ is a constant.
Namely, to determine the density one needs to know the functional $F$ just for
constant functions $\psi\equiv a$. Note that, if the functional $F$ is the
interaction energy between the surfaces, ${\mathcal{F}}_{0}$ becomes the
interaction energy per unit area $E_{\shortparallel}$, and $F_{0}$ becomes
$U_{\rm PFA}$ (see Equation (3)).

Let us now show how to derive the PFA (and its corrections) by the resummation
of a perturbative expansion. To that end, we evaluate $F$ for a $\psi$ having
the form:

$\psi({\mathbf{x}_{\shortparallel}})=a+\eta({\mathbf{x}_{\shortparallel}})\;.$
(25)

and write the resulting perturbative expansion in powers of $\eta$, which has
the general form:

$F[\psi]={\mathcal{S}\mathcal{F}_{0}}(a)+\sum_{n\geq
1}\int\frac{d^{2}k_{\shortparallel}^{(1)}}{(2\pi)^{2}}...\frac{d^{2}k_{\shortparallel}^{(n)}}{(2\pi)^{2}}\,\delta(k_{\shortparallel}^{(1)}+...+k_{\shortparallel}^{(n)})\,h^{(n)}(k_{\shortparallel}^{(1)},...,k_{\shortparallel}^{(n)})\,{\widetilde{\eta}}(k^{(1)}_{\shortparallel})...{\widetilde{\eta}}(k^{(n)}_{\shortparallel})\;,$
(26)

where $\delta(\cdot)$ is the Dirac delta function, and the form factors
$h^{(n)}$ can be computed by using perturbative techniques. For the Dirichlet-
Casimir effect, this can be done in a rather systematic way 4thorder .
Although the approach to follow in order to obtain those form factors may
depend strongly on the kind of system considered, the form of the expansion
shall be the same. Note that the form factors may depend on $a$, although, in
order to simplify the notation, we will not make that dependence explicit.

Up to now, we have not used the hypothesis of smoothness of the $R$ surface.
We do that now by assuming that the Fourier transform $\widetilde{\eta}$ is
peaked at the zero momentum. What follows is to make use of this assumption
for all terms in the expansion. In Equation (26), we set then:
$h^{(n)}(k_{\shortparallel}^{(1)},...,k_{\shortparallel}^{(n)})\simeq
h^{(n)}(0,...,0)$, and, as a consequence:

$F(\psi)\simeq{\mathcal{S}\mathcal{F}_{0}}(a)+\sum_{n\geq
1}h^{(n)}(0,..,0)\int
d^{2}{\mathbf{x}_{\shortparallel}}[\eta({\mathbf{x}_{\shortparallel}})]^{n}\;.$
(27)

One could evaluate the form factors at the zero momentum straighforwardly.
However, there is a shortcut here that allows one to obtain all of them
immediately: consider a constant
$\eta({\mathbf{x}_{\shortparallel}})=\eta_{0}$, so that the interaction energy
is given by Equation (27) with the replacement $\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\eta({\mathbf{x}_{\shortparallel}})^{n}\rightarrow{\mathcal{S}}\eta_{0}^{n}$.
For this particular case, $F$ becomes just the functional corresponding to
parallel plates, which are separated by a distance $a+\eta_{0}$:

${\mathcal{F}_{0}}(a+\eta_{0})={\mathcal{F}_{0}}(a)+\sum_{n\geq
1}h^{(n)}(0,..,0)\eta_{0}^{n}\,.$ (28)

We then conclude that, in this low-momentum approximation, the series can be
summed up with the result:

$F_{0}[\psi]\simeq\int
d^{2}{\mathbf{x}_{\shortparallel}}{\mathcal{F}_{0}}\left(a+\eta({\mathbf{x}_{\shortparallel}})\right)=\int
d^{2}{\mathbf{x}_{\shortparallel}}{\mathcal{F}_{0}}(\psi)\,,$ (29)

which is just the PFA.

The calculation just above shows that, for the class of geometries considered
in this paper, the PFA can be justified from first principles as the result of
a resummation of a perturbative calculation corresponding to almost flat
surfaces. In order to be well defined, the PFA requires that the form factors
$h^{(n)}(k_{\shortparallel}^{(1)},...,k_{\shortparallel}^{(n)})$ have a finite
limit as $k_{\shortparallel}^{(i)}\to 0$.

This procedure also suggests how the PFA could be improved; one can include
the NTLO terms in the low-momentum expansions of the form factors. We assume
that they can be expanded in powers of the momenta up to the second order. We
stress that this is by no means a trivial assumption. Indeed, depending on the
the interaction considered, the form factors could include nonanalyticities
(we will discuss some explicit examples below). In case of no
nonanalyticities, one can introduce the expansions:

$h^{(n)}(k_{\shortparallel}^{(1)},...,k_{\shortparallel}^{(n)})=h^{(n)}(0,...,0)+\sum_{i,\alpha}A^{(n)}_{i\alpha}k_{\shortparallel\,\alpha}^{(i)}+\sum_{i,j,\alpha,\beta}B^{(n)}_{ij\alpha\beta}k_{\shortparallel\,\alpha}^{(i)}k_{\shortparallel\,\beta}^{(j)}\,\ldots,$
(30)

for some $a-$dependent coefficients $A^{(n)}_{i\alpha}$ and
$B^{(n)}_{ij\alpha\beta}$. Here $i,j=1,...,n$ label arguments while
$\alpha,\beta=1,2$ label their components. Symmetry considerations are
crucial, since they allow us to simplify the above expression (30), as
follows: rotational invariance implies that the form factors depend only on
the scalar products $k_{\shortparallel}^{(i)}\cdot k_{\shortparallel}^{(j)}$.
Additionally, they have to be symmetric under the interchange of any two
momenta. This thus leads to

$h^{(n)}(k_{\shortparallel}^{(1)},...,k_{\shortparallel}^{(n)})=h^{(n)}(0,...,0)+B^{(n)}\sum_{i}k_{\shortparallel}^{(i)\,2}+C^{(n)}\sum_{i\neq
j}k_{\shortparallel}^{(i)}\cdot k_{\shortparallel}^{(j)}\,,$ (31)

for some coefficients $B^{(n)}$ and $C^{(n)}$.

Inserting Equation (31) into Equation (26) and taking integrations by parts,
one then finds the form of the first correction to the PFA:

$F_{2}[\psi]=\int d^{2}{\mathbf{x}_{\shortparallel}}\,\left[\sum_{n\geq
2}D^{(n)}\,\eta^{n-2}\right]|\nabla\eta|^{2}\;,$ (32)

where the coefficients $D^{(n)}$ are linear combinations of $B^{(n)}$ and
$C^{(n)}$. The subindex $2$ in $F$ indicates that this is the part of the
functional containing two derivatives.

We complete the calculation by calculating the sum in Equation (32). To that
end, we evaluate the correction $F_{2}$ for a particular case:
$\eta({\mathbf{x}_{\shortparallel}})=\eta_{0}+\epsilon({\mathbf{x}_{\shortparallel}})$,
with $\epsilon\ll\eta_{0}$, and expand up to the second order in $\epsilon$.
Thus,

$F_{2}[a+\eta_{0}+\epsilon]=\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\left[\sum_{n\geq
2}D^{(n)}\,\eta_{0}^{n-2}\right]|\nabla\epsilon|^{2}\,.$ (33)

The resummation can be obtained in this case, by considering the usual
perturbative evaluation of the interaction energy up to second order in
$\epsilon$. This evaluation does, naturally, depend on the interaction
considered, but, once one has that result one can obtain the sum of the series
above. We we will denote by $Z$ that sum, namely:

$Z(a+\eta_{0})\equiv\sum_{n\geq 2}D^{(n)}\,\eta_{0}^{n-2}\,.$ (34)

Upon replacement $\eta_{0}\to\eta$ in Equation (34), one obtains

$F_{2}[\psi]=\int
d^{2}{\mathbf{x}_{\shortparallel}}\,Z(\psi)|\nabla\psi|^{2}\,.$ (35)

This is the NTLO correction to the PFA. This concludes our systematic
derivation of the PFA, including its first correction, a result which may be
put as follows:

$F_{\rm DE}[\psi]\;=\;\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\Big{[}V(\psi)\,+\,Z(\psi)|\nabla\psi|^{2}\Big{]}\;,$
(36)

where $V(\psi)={\mathcal{F}}_{0}(\psi)$ is determined from the (known)
expression for the interaction energy between parallel surfaces, while
$Z(\psi)$ can be computed using a perturbative technique. In practice,
$Z(\psi)$ can be evaluated setting $\eta_{0}=0$ in Equation (34).

The higher orders may be derived by an extension of the procedure described
just above. It should be evident that, for the expansion to be well-defined,
the analytic structure of the form factors is quite relevant. Indeed, the
existence of nonanalytic zero-momentum contributions can render the DE non
applicable. This should be expected on physical grounds, since the presence of
nonanalytic terms implies that the functional cannot be approximated, in
coordinate space, by the single integral of a local density. Physically, it is
a signal that the nonlocal aspects of the interaction cannot be ignored. That
should not come up as a surprise, when one recalls that the same kind of
phenomenon does happen when evaluating the effective action in quantum field
theory, and the quantum effects contain contributions due to virtual massless
particles. In this case, the effective action may develop nonanalyticities at
zero momentum.

The main messages of this Section are the following: irrespective of the
nature of the interaction, the energy and forces between objects are
functionals of their shapes. The PFA is recovered when the form factors of the
functionals are evaluated at zero momentum. Enhancements to this approximation
are achievable by expanding these form factors at low momenta. If the
expansion is analytic, a resummation of the form factors produces the DE.

##  V DE for the Zero-Temperature Casimir Effect

The application of the DE to the Casimir interaction energy between two
objects was, actually, our original motivation to introduce the approximation,
and it is useful briefly review some aspects of this application here. We
consider first a real vacuum scalar field satisfying Dirichlet boundary
conditions (Section V.1) and then we move to the EM field with perfect-
conductor boundary conditions (Section V.2). We follow Ref. FLM2011  for the
derivation of the DE in the Dirichlet case.

###  V.1 Scalar Field with Dirichlet Boundary Conditions

We consider here a massless real scalar field $\varphi$ in $3+1$ dimensions,
coupled to two mirrors which impose Dirichlet boundary conditions. In our
Euclidean conventions, we use $x_{0},x_{1},x_{2},x_{3}$ to denote the
spacetime coordinates, $x_{0}$ being the imaginary time. As before, the
mirrors occupy two surfaces, denoted by $L$ and $R$, defined by the equations
$x_{3}=0$ and $x_{3}=\psi({\mathbf{x}_{\shortparallel}})$, respectively.

On only dimensional grounds, and using natural units ($\hbar\equiv c\equiv
1$), the DE approximation to the interaction energy to be of the form

$E_{\rm DE}=-\frac{\pi^{2}}{1440}\int
d^{2}{\mathbf{x}_{\shortparallel}}\;\frac{1}{\psi^{3}}\left[\alpha_{\rm
D}+\beta_{\rm D}(\partial_{\alpha}\psi)^{2}\right],$ (37)

where $\alpha_{\rm D}$ and $\beta_{\rm D}$ are dimensionless coefficients that
do not depend on the geometry. The subindex $D$ stands for Dirichlet. An
evaluation of the above expression for parallel plates fixes $\alpha_{D}\equiv
1$. As in the electrostatic case, the coefficient $\beta_{\rm D}$ could be
computed from explicit examples where the interaction energy is known exactly.

Let us recall, from Section IV, that the interaction energy can also be
computed from an expansion of the Casimir energy in powers of $\eta$ for

$\psi({\mathbf{x}_{\shortparallel}})\;=\;a\,+\,\eta({\mathbf{x}_{\shortparallel}})\;,$
(38)

where $a$ (assumed to be greater than zero) is the spatial average of $\psi$
whereas $\eta$ contains its varying piece. The expansion needed is of the
second order in $\eta$, and with up to two spatial derivatives.

To obtain such an expansion, we start from a rather general yet formal
expression for the energy (for earlier perturbative computations of the
Casimir force see, for example, Ref.earlier-pert1 ; earlier-pert2 ). That
formal expression follows from the functional approach to the Casimir effect,
where we deal with ${\mathcal{Z}}$, the zero-temperature limit of a partition
function. That partition function, for a scalar field in the presence of two
Dirichlet mirrors is given by

${\mathcal{Z}}\;=\;\int{\mathcal{D}}\varphi\;\delta_{L}(\varphi)\,\delta_{R}(\varphi)\;e^{-S_{0}(\varphi)}\;,$
(39)

with $S$ denoting the real scalar field free (Euclidean) action

$S_{0}(\varphi)\;=\;\frac{1}{2}\,\int d^{4}x\,(\partial\varphi)^{2}\;,$ (40)

while the $\delta_{L}$ and $\delta_{R}$ impose Dirichlet boundary conditions
on the $L$ and $R$ surface, respectively.

The vacuum energy, $E$, is then obtained as follows:

$E\;=\;-\,\lim_{T\to\infty}\log{\mathcal{Z}}/T\;,$ (41)

where $T$ is the extent of the time dimension (or $\beta^{-1}$, in a thermal
partition function setting). We discard from $E$ the terms that do not
contribute to the Casimir interaction energy between the two surfaces. These
terms will appear as factors in ${\mathcal{Z}}$; among them the one describing
the zero point energy of the field in the absence of the plates, and also the
‘self-energy’ contributions, due to the vacuum distortion produced by each
mirror, even when the other is infinitely far apart.

Exponentiating the two Dirac delta functions by introducing two auxiliary
fields, $\lambda_{L}$ and $\lambda_{R}$, we obtain for ${\mathcal{Z}}$ an
equivalent expression:

${\mathcal{Z}}\;=\;\int{\mathcal{D}}\varphi{\mathcal{D}}\lambda_{L}\,{\mathcal{D}}\lambda_{R}\;e^{-S(\varphi;\lambda_{L},\lambda_{R})}\;,$
(42)

with

$S(\varphi;\lambda_{L},\lambda_{R})=S_{0}(\varphi)-i\int
d^{4}x\varphi(x)\left[\lambda_{L}(x_{\shortparallel})\delta(x_{3})+\lambda_{R}(x_{\shortparallel})\sqrt{g_{R}({\mathbf{x}_{\shortparallel}})}\,\delta(x_{3}-\psi({\mathbf{x}_{\shortparallel}}))\right]$

where we have introduced
$x_{\shortparallel}\equiv(x_{0},x_{1},x_{2})=(x_{0},{\mathbf{x}_{\shortparallel}})$.
The factor depending on the determinant of the induced metric on the $R$,
$g_{R}({\mathbf{x}_{\shortparallel}})\equiv
1+|\nabla\psi({\mathbf{x}_{\shortparallel}})|^{2}$ makes the expression above
reparametrization invariant. However, by a redefinition of the auxiliary field
$\lambda_{R}$ one gets rid of that factor, at the expense of generating a
Jacobian. That Jacobian does not depend on the distance between the two
surfaces, since only derivatives of $\psi$ are involved. Therefore it will not
contribute the the Casimir interaction energy and thus we shall subsequently
ignore such factor, as well as others that will appear in the course of the
calculations.

Integrating out $\varphi$, we see that ${\mathcal{Z}}_{0}$, corresponding to
the field $\varphi$ in the absence of boundary conditions factors out, while
the rest becomes an integral over the auxiliary fields:

${\mathcal{Z}}={\mathcal{Z}}_{0}\int{\mathcal{D}}\lambda_{L}{\mathcal{D}}\lambda_{R}e^{-\frac{1}{2}\int
d^{3}x_{\shortparallel}\int
d^{3}y_{\shortparallel}\sum_{\alpha,\beta}\lambda_{\alpha}(x_{\shortparallel}){\mathbb{T}}_{\alpha\beta}\lambda_{\beta}(y_{\shortparallel})},$
(43)

with:

${\mathcal{Z}}_{0}\;=\;\int{\mathcal{D}}\varphi\;e^{-S_{0}(\varphi)}\;,$ (44)

and $\alpha,\,\beta=L,R$ . We have introduced the objects:

$\displaystyle{\mathbb{T}}_{LL}(x_{\shortparallel},y_{\shortparallel})$
$\displaystyle=\langle
x_{\shortparallel},0|(-\partial^{2})^{-1}|y_{\shortparallel},0\rangle$ (45)
$\displaystyle{\mathbb{T}}_{LR}(x_{\shortparallel},y_{\shortparallel})$
$\displaystyle=\langle
x_{\shortparallel},0|(-\partial^{2})^{-1}|y_{\shortparallel},\psi({\mathbf{y}_{\shortparallel}})\rangle$
(46) $\displaystyle{\mathbb{T}}_{RL}(x_{\shortparallel},y_{\shortparallel})$
$\displaystyle=\langle
x_{\shortparallel},\psi({\mathbf{x}_{\shortparallel}})|(-\partial^{2})^{-1}|y_{\shortparallel},0\rangle$
(47) $\displaystyle{\mathbb{T}}_{RR}(x_{\shortparallel},y_{\shortparallel})$
$\displaystyle=\langle
x_{\shortparallel},\psi({\mathbf{x}_{\shortparallel}})|(-\partial^{2})^{-1}|y_{\shortparallel},\psi({\mathbf{y}_{\shortparallel}})\rangle$
(48)

where we use a “bra-ket” notation to denote matrix elements of operators, and
$\partial^{2}$ is the four-dimensional Laplacian. Thus, for example,

$\langle
x|(-\partial^{2})^{-1}|y\rangle\,=\,\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{e^{ik\cdot(x-y)}}{k^{2}}\;.$
(49)

A subtraction of the zero point contribution contained in ${\mathcal{Z}}_{0}$
leads to:

$E\;=\;\lim_{T\to\infty}\big{(}\frac{1}{2T}{\rm Tr}\log{\mathbb{T}}\big{)}\;,$
(50)

which still contains self-energies. Up to now, we have obtained a formal
expression for the vacuum energy; let us now proceed to evaluate its DE.

We need to expand $E$ to the second order in $\eta$, keeping up to the second
order term in an expansion in derivatives. It is convenient to do so first for
$\Gamma\;\equiv\;\frac{1}{2}{\rm Tr}\log{\mathbb{T}}$ . Namely,

$\Gamma(a,\eta)\;=\;\Gamma^{(0)}(a)\;+\;\Gamma^{(1)}(a,\eta)\;+\;\Gamma^{(2)}(a,\eta)\;+\;\ldots$
(51)

where the upper index denotes the order in derivatives. Each term will be a
certain coefficient times the spatial integral over
${\mathbf{x}_{\shortparallel}}$ of a local term, depending on $a$ and on
$\eta$-derivatives. Additionally, because the configuration is time-
independent, they should be proportional to $T$ (a factor that will cancel
out). Expanding first the matrix ${\mathbb{T}}$ in powers of $\eta$

${\mathbb{T}}={\mathbb{T}}^{(0)}+{\mathbb{T}}^{(1)}+{\mathbb{T}}^{(2)}+\ldots\,,$
(52)

we obtain: $\Gamma\,=\,\Gamma^{(0)}+\Gamma^{(1)}+\Gamma^{(2)}\,+\,\ldots$,

$\displaystyle\Gamma^{(0)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}\log{\mathbb{T}}^{(0)}$ $\displaystyle\Gamma^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm
Tr}\log\left[({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}^{(1)}}\right]$
$\displaystyle\Gamma^{(2)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
Tr}\log\left[({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}}^{(2)}\right]$ (53)
$\displaystyle-$ $\displaystyle\frac{1}{4}{\rm
Tr}\log\left[({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}}^{(1)}({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}}^{(1)}\right]\;,$

where, in $\Gamma^{(l)}$, we need to keep up to $l$ derivatives of $\eta$.

Then, the zeroth-order term is obtained as follows: replace $\psi$ by a
constant, $a$, and then subtract from the result its $a\to\infty$ limit (this
gets rid of self-energies). This leads to:

$\Gamma^{(0)}(a)\;=\;\frac{1}{2}\,{\rm
Tr}\log[1-(T_{LL}^{(0)})^{-1}T_{LR}^{(0)}(T_{RR}^{(0)})^{-1}T_{RL}^{(0)}\big{]}\;.$
(54)

Here, the $T_{\alpha\beta}^{(0)}$ are identical to the ones for two flat
parallel mirrors separated by a distance $a$ .

Taking the trace, leads to:

$\Gamma^{(0)}\,=\,\frac{T}{2}\,\int
d^{2}{\mathbf{x}_{\shortparallel}}\int\frac{d^{3}k_{\shortparallel}}{(2\pi)^{3}}\log[1-e^{-2k_{\shortparallel}a}]\,.$
(55)

Then, we recall the general derivation to note that the replacement $a\to\psi$
leads to:

$\displaystyle E^{(0)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\,\int
d^{2}{\mathbf{x}_{\shortparallel}}\int\frac{d^{3}k_{\shortparallel}}{(2\pi)^{3}}\log[1-e^{-2k_{\shortparallel}\psi({\mathbf{x}_{\shortparallel}})}]$
(56) $\displaystyle=$ $\displaystyle-\frac{\pi^{2}}{1440}\int
d^{2}{\mathbf{x}_{\shortparallel}}\;\frac{1}{\psi({\mathbf{x}_{\shortparallel}})^{3}},$

which is the PFA expression for the vacuum energy.

To improve on the previous result, we consider its first non trivial
correction. There can be no first order term because of symmetry
considerations. while to terms contribute to the second order

$\Gamma^{(2)}\;=\;\Gamma^{(2,1)}\,+\,\Gamma^{(2,2)}$ (57)

where,

$\Gamma^{(2,1)}\,=\,\frac{1}{2}{\rm
Tr}\log\left[({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}}^{(2)}\right]$ (58)

and

$\Gamma^{(2,2)}\,=\,-\frac{1}{4}{\rm
Tr}\log\left[({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}}^{(1)}({\mathbb{T}}^{(0)})^{-1}{\mathbb{T}}^{(1)}\right]\;.$
(59)

In the terms above, we have to keep just up to two derivatives of $\eta$. We
see that, in Fourier space, and before implementing any expansion in momentum
(derivatives), they have the structure:

$\Gamma^{(2,j)}\;=\;\frac{T}{2}\,\int\frac{d^{2}{\mathbf{k}_{\shortparallel}}}{(2\pi)^{2}}f^{(2,j)}({\mathbf{k}_{\shortparallel}})\,|\tilde{\eta}({\mathbf{k}_{\shortparallel}})|^{2}$
(60)

($j=1,2$), with $\tilde{\eta}$ denoting the Fourier transform of $\eta$, and
with the $f^{(2,j)}$ kernels denoting the $k_{0}\to 0$ (i.e., static) limits
of the more general expressions:

$\displaystyle f^{(2,1)}(k_{\shortparallel})$ $\displaystyle=$
$\displaystyle-\int\frac{d^{3}p_{\shortparallel}}{(2\pi)^{3}}\frac{|p_{\shortparallel}||p_{\shortparallel}+k_{\shortparallel}|}{1-e^{-2|p_{\shortparallel}+k_{\shortparallel}|a}}$
$\displaystyle f^{(2,2)}(k_{\shortparallel})$ $\displaystyle=$
$\displaystyle-\int\frac{d^{3}p_{\shortparallel}}{(2\pi)^{3}}\frac{|p_{\shortparallel}||p_{\shortparallel}+k_{\shortparallel}|e^{-2|p_{\shortparallel}+k_{\shortparallel}|a}(1+e^{-2|p_{\shortparallel}|a})}{(1-e^{-2|p_{\shortparallel}|a})(1-e^{-2|p_{\shortparallel}+k_{\shortparallel}|a})}.$

By subtracting all the $a$-independent contributions, one finds:

$\Gamma^{(2)}\;=\;\frac{T}{2}\,\int\frac{d^{2}{\mathbf{k}_{\shortparallel}}}{(2\pi)^{2}}f^{(2)}({\mathbf{k}_{\shortparallel}})\,|\tilde{\eta}({\mathbf{k}_{\shortparallel}})|^{2}$
(61)

with:

$f^{(2)}(k_{\shortparallel})\;=\;-2\int\frac{d^{3}p_{\shortparallel}}{(2\pi)^{3}}\frac{|p_{\shortparallel}|\,|p_{\shortparallel}+k_{\shortparallel}|}{(1-e^{-2|p_{\shortparallel}|a})(e^{2|p_{\shortparallel}+k_{\shortparallel}|a}-1)}\,.$
(62)

The low-momentum behaviour of $f^{(2)}$ determines whether the DE can be
applied or not. In this case, the function is analytic and therefore a local
expansion of the vacuum energy exists. We need to extract its
${\mathbf{k}}^{2}$ order term in a Taylor expansion at zero momentum, namely
$f^{(2)}({\mathbf{k}_{\shortparallel}})\;\simeq\;\chi\,{\mathbf{k}_{\shortparallel}}^{2}$.
We find:

$\chi\,=\,-\frac{\pi^{2}}{1080\,a^{3}}\;.$ (63)

Thus,

$\displaystyle\Gamma^{(2)}(a,\eta)$ $\displaystyle=$
$\displaystyle-\frac{T}{2}\frac{\pi^{2}}{1080}\int\frac{d^{2}{\mathbf{k}_{\shortparallel}}}{(2\pi)^{2}}\frac{{\mathbf{k}_{\shortparallel}}^{2}}{a^{3}}\,|\tilde{\eta}({\mathbf{k}_{\shortparallel}})|^{2}$
(64) $\displaystyle=$ $\displaystyle-\frac{T}{2}\frac{\pi^{2}}{1080}\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\frac{1}{a^{3}}\,(\partial_{\alpha}\eta)^{2}\;.$

Therefore, the NTLO term in the DE becomes:

$E^{(2)}=\frac{\Gamma^{(2)}(\psi)}{T}\,=\,-\frac{1}{2}\frac{\pi^{2}}{1080}\int
d^{2}{\mathbf{x}_{\shortparallel}}\,\frac{(\partial_{\alpha}\psi)^{2}}{\psi^{3}}\;,$
(65)

where the index $\alpha$ runs from $1$ to $2$.

Putting together the terms up to second order,

$E_{\rm DE}\equiv E^{(0)}+E^{(2)}=-\frac{\pi^{2}}{1440}\int
d^{2}{\mathbf{x}_{\shortparallel}}\;\frac{1}{\psi^{3}}\left[1+\frac{2}{3}(\partial_{\alpha}\psi)^{2}\right].$
(66)

The leading-order term above is the Casimir energy according to the PFA ,
while the second order one represents the first significant deviation from it.
We note that the structure of both terms had been anticipated by dimensional
analysis and symmetry considerations. The overall normalization, on the other
hand, had been fixed by our previous knowledge of the (well-established)
result for parallel plates.

We would like to insist on the fact that the relative weight between the PFA
and its correction term–the factor $\beta_{\rm D}=2/3$–is independent of the
surface geometry. This value of $\beta_{\rm D}$ has been independently
corroborated in concrete examples by expanding the exact Casimir energy
expressions. Interesting cases among them are, for example, either a sphere or
a cylinder positioned in front of a plane.

We conclude this Section with an application of the DE to the particular
geometry of a sphere in front of a plane. Let us express the function $\psi$
of Equation (13) in polar coordinates $\rho$, with $R$ the radius of the
sphere and $d$ the distance to the plane. The function $\psi(\rho)$ describes
an hemisphere when $0\leq\rho\leq R$. By inserting the expression of $\psi$
into the DE for the Casimir energy, it becomes possible to explicitly
calculate the integrals, to get a rather compact analytical expression:

$E_{\rm DE}=E_{\rm PFA}\left(1+\frac{1}{3}\frac{a}{R}\right),$ (67)

where $E_{\rm PFA}=-\hbar c\pi^{3}R/(1440a^{2})$.

###  V.2 The EM Case

The results for the scalar field satisfying Dirichlet boundary conditions,
described in Section (V.1) above, have been generalized to different boundary
conditions and fields. Results for the EM field case and two curved surfaces
have been presented in Ref.Bimonte1 . Note that, as pointed out at the end of
Section III.3, symmetry considerations allow for the two-surface problem to be
reduced to the one of a curved surface facing a plane, namely, the geometry we
have just dealt with in the Dirichlet case above. Indeed, as shown in Ref.
Bimonte1 , the extension of Ref.FLM2011  to two curved surfaces is restricted
among other things by the tilt invariance of the reference plane, to which the
two surfaces can be projected. This served as a rigorous test for the self-
consistency of perturbative results.

Venturing beyond the scalar Dirichlet (D) case of Ref.FLM2011 , they
calculated the DE for Neumann (N), mixed D/N, and electromagnetic (EM)
(perfect metal) surfaces. Interestingly, they observed that the EM correction
must align with the sum of D and N corrections. They also replicated previous
findings for cylinders under D, N, and mixed D/N conditions, as well as for
the sphere with D boundary conditions. However, their calculations did not
confirm previous results for the sphere/plane geometry, either with N or EM
boundary conditions. Indeed, the results for $\beta$ were found to disagree
with those obtained from Refs. bordag08 ; bordag101 ; bordag102 . This
discrepancy was later resolved in Ref.bordag-teo  in favour of the results in
Bimonte1 .

Another interesting concrete example presented in Bimonte1  is the DE for two
spheres of radii $R_{1}$ and $R_{2}$, both imposing the same boundary
conditions. It was found there that

$E=E_{\rm
PFA}\left[1-\frac{a}{R_{1}+R_{2}}+(2\beta-1)\left(\frac{a}{R_{1}}+\frac{a}{R_{2}}\right)\right],$
(68)

where $E_{\rm PFA}=-(\alpha\pi^{3}R_{1}R_{2}))/[1440a^{2}(R_{1}+R_{2})]$; $a$
is chosen to be the distance of closest separation, and $\beta$ is a number
that depends on the type of boundary condition, as can be seen from Table 1.
$\alpha=\alpha_{\rm EM}=2$ in the EM boundary conditions case. The
corresponding formula for the sphere/plane case can be obtained by taking one
of the two radii to infinity (in fact it coincides with the D case in Equation
(67) when $\alpha=\alpha_{\rm D}=1$ and $\beta=\beta_{\rm D}=2/3$).

A rather different example corresponds to two circular cylinders (with
identical boundary conditions) whose axes are inclined at a relative angle
$\theta$. Using the DE, the interaction Casimir energy reads:

$E=-\frac{\alpha\pi^{3}\sqrt{R_{1}R_{2}}}{1440a^{2}\sin\theta}\left[1+\left(\beta-\frac{3}{8}\right)\frac{a}{R_{1}+R_{2}}\right].$
(69)

For this particular geometry, the interaction energy has been computed
numerically in Rodriguez . The numerical results reproduce Equation (69) at
short distances.

The results obtained for the $\beta$-coefficients in each case are summarized
in Table 1.

$\beta_{\rm D}$ | $\beta_{\rm N}$ | $\beta_{\rm DN}$ | $\beta_{\rm ND}$ | $\beta_{\rm EM}$
---|---|---|---|---
2/3 | $2/3(1-30/\pi^{2})$ | 2/3 | $2/3-80/7\pi^{2}$ | $2/3(1-15/\pi^{2})$
Table 1: $\beta$ coefficient from (68) for the following five cases: a scalar
field obeying Dirichlet (${\rm D}$) or Neumann ($\rm N$) boundary conditions
on both surfaces, or ${\rm D}$ boundary condition on one surface and ${\rm N}$
boundary condition on the other, or vice versa, and for the electromagnetic
(${\rm EM}$) field with ideal metal boundary conditions Bimonte1

Having presented in this Section a derivation and some interesting results
obtained by applying the DE to the Casimir effect at zero temperature and for
perfect boundary conditions, we present in the rest of the review some
generalizations and applications.

##  VI Finite Temperature, Nonanalyticities, and DE

The DE can be extended to the finite temperature case Bimonte2 ; FLMTdim ;
FLMTem , the free energy being the relevant functional to approximate. There
are at least two reasons why this extension is not trivial: firstly, the
temperature introduces a dimensionful magnitude, and this will reflect itself
in the form of the DE (part of it was fixed by dimensional analysis). Second,
a known phenomenon in quantum field theory at finite temperature is the so-
called ”dimensional reduction”, by which a bosonic model which is defined in
$d+1$ dimensions at zero temperature, becomes effectively $d$-dimensional at
high temperatures. The DE should therefore manifest (and interpolate between)
those two cases.

We first describe, in Section VI.1, the results for a scalar field satisfying
Dirichlet conditions FLMTdim  in $d+1$ dimensions. Then, Section VI.2
discusses the appearance of nonanaliticities for Neumann boundary conditions
FLMTdim ; FLMNeu . Finally, we comment on the results for the EM field with
imperfect boundary conditions FLMTem ; Bimonte2  (Section VI.3) and on
semianalytic formula for plane–sphere geometry (Section VI.4).

###  VI.1 Dirichlet Boundary Conditions

In the finite-temperature case, and for the same geometry that we have
considered in the zero temperature case, the functions $V(\psi)$ and $Z(\psi)$
cannot be completely determined from dimensional analysis alone. Indeed, on
general grounds, we can assert that the Casimir free energy in $d+1$
dimensions, if the DE is applicable, must have the form:

$F_{\rm DE}[\psi]\;=\;\int
d^{d-1}{\mathbf{x}}_{\shortparallel}\,\Big{\\{}b_{0}(\frac{\psi}{\beta},d)\frac{1}{[\psi({\mathbf{x}}_{\shortparallel})]^{d}}\,+\,b_{2}(\frac{\psi}{\beta},d)\,\frac{(\nabla\psi)^{2}}{[\psi({\mathbf{x}}_{\shortparallel})]^{d}}\Big{\\}}$
(70)

where $b_{0}$ and $b_{2}$ are dimensionless and depends on the ratio of the
local distance between surfaces $\psi$ and the inverse temperature $\beta$.
They can be obtained from the knowledge of the Casimir free energy for small
departures around $\psi({\mathbf{x}_{\shortparallel}})=a={\rm constant}$. They
are given by FLMTdim

$\displaystyle b_{0}(\xi)$ $\displaystyle=$
$\displaystyle\frac{\xi}{2}\sum_{n=-\infty}^{\infty}\int\frac{d^{d-1}\mathbf{p}_{\shortparallel}}{(2\pi)^{d-1}}\log\left(1-e^{-2\sqrt{(2\pi
n\xi)^{2}+(\mathbf{p}_{\shortparallel})^{2}}}\right)$ $\displaystyle
b_{2}(\xi)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[\frac{\partial
F^{(2)}(\xi,n,\mathbf{l}_{\shortparallel})}{\partial|\mathbf{l}_{\shortparallel}|^{2}}\right]_{n\to
0,|\mathbf{l}_{\shortparallel}|\to 0}\,$ (71)

where

$F^{(2)}(\xi;n,|{\mathbf{l}}_{\shortparallel}|)=-2\xi\,\sum_{m=-\infty}^{+\infty}\int\frac{d^{d-1}{\mathbf{p}}_{\shortparallel}}{(2\pi)^{d-1}}\Big{\\{}\frac{\sqrt{(2\pi
m\,\xi)^{2}+{\mathbf{p}}^{2}_{\shortparallel}}}{1-\exp[-2\sqrt{(2\pi
m\xi)^{2}+{\mathbf{p}}^{2}_{\shortparallel}}\big{]}}\frac{\sqrt{(2\pi(m+n)\,\xi)^{2}+({\mathbf{p}}_{\shortparallel}+{\mathbf{l}}_{\shortparallel})^{2}}}{\exp\\{2\sqrt{[2\pi(m+n)\xi]^{2}+({\mathbf{p}}_{\shortparallel}+{\mathbf{l}}_{\shortparallel})^{2}}\big{\\}}-1}\Big{\\}}\,.$
(72)

In the zero temperature limit, the Matsubara sum becomes an integral that can
be analytically computed. The results are described in Table 2. The ratio
$b_{2}/b_{0}$ tends to $1$ for large values of $d$.

Dimension | ${b_{2}(d)}/{b_{0}(d)}$ | Approximate Value
---|---|---
$d=1$ | $\frac{1}{\pi^{2}}\left(1+\frac{\pi^{2}}{3}\right)$ | $0.435$
$d=2$ | $\frac{1+6\zeta(3)}{12\zeta(3)}$ | $0.569$
$d=3$ | $2/3$ | $0.667$
$d=4$ | $\frac{-\zeta(3)+10\zeta(5)}{12\zeta(5)}$ | $0.737$
$d=5$ | $\frac{10\pi^{2}-21}{10\pi^{2}}$ | $0.787$
$d=6$ | $\frac{-2\zeta(5)+7\zeta(7)}{6\zeta(7)}$ | $0.824$
Table 2: Values of the ratio ${b_{2}(d)}/{b_{0}(d)}$ for different dimensions.
The ratio tends to $1$ for $d\to\infty$. See text for details.

In the high temperature limit, we find

$\displaystyle\big{[}b_{0}(\xi,d)]_{\xi>>1}\;\simeq\;\xi\,\big{[}b_{0}(\xi,d-1)]_{\xi\to
0}\equiv\xi\,b_{0}(d-1)\;,$ (73)
$\displaystyle\big{[}b_{2}(\xi,d)\big{]}_{\xi>>1}\simeq\xi\big{[}b_{2}(\xi,d-1)\big{]}_{\xi\to
0}\equiv\xi b_{2}(d-1)$

where $\xi=\psi/\beta$. The coefficients $b_{0}(d-1)$ and $b_{2}(d-1)$ agree
with those for perfect mirrors at zero temperature, but in $d-1$ dimensions,
i.e., the ‘ dimensional reduction” effect.

An interesting result is found when this is applied to the (Dirichlet) Casimir
interaction for a system consisting of a sphere in front of an infinite plane.
Denoting by $a$ the distance between the surfaces, and by $R$ the radius of
the sphere, we get for the free energy at high temperatures:

$F_{\rm DE}[\psi]|_{\psi/\beta>>1,d=3}\,\sim\,-\frac{\zeta(3)R}{8\beta
a}\left(1-\frac{1}{6\zeta(3)}\frac{a}{R}\log\left(\frac{a}{R}\right)\right)\,.$
(74)

We see that the $R/a^{2}$-behavior corresponding to the dominant contribution
at zero temperature changes to $R/a\beta$ in the high temperature case. This
could be expected on dimensional grounds, if one assumes that the free energy
is linear in the temperature in this limit. Note that the same problem has
been exactly solved in Ref.bimonteprl , and one can show that Equation (74)
does agree with the small-distance expansion obtained from the exact solution.

It is worth to remark that the NTLO correction from the DE becomes
nonanalytic, because of the integration, in the ratio $a/R$. This behavior has
been observed in numerical calculations of the Casimir interaction energy for
this geometry, in the infinite temperature limit, for the electromagnetic case
(see Refs.bimonteprl ; Neto2012 ). It is important to recognize that this
nonanalyticity has nothing to do with the nonanalyticity in momenta of the
form factors described in Section 4, and is a non trivial prediction of the
DE.

###  VI.2 Neumann Boundary Conditions

This case, discussed in Ref.FLMNeu , highlights a potential warning to the
applicability of the DE, already mentioned previously: the appearance of
nonanalyticities in the form factors. To begin with, we deal with the zero
temperature case in $2+1$ dimensions, since the nonanalyticity appears because
of the existence of a Matsubara mode which behaves as a massless field in
$2+1$ dimensions, with Neumann boundary conditions.

The free Euclidean action for the vacuum (i.e., $T=0$) field $\varphi$ is
given by

${\mathcal{S}}_{0}\;=\;\frac{1}{2}\,\int d^{3}x\,(\partial\varphi)^{2}\;,$
(75)

and, instead of imposing perfect Neumann boundary conditions on the surfaces,
we add the following action to describe the interaction between the vacuum
field and the mirrors:

${\mathcal{S}}_{\rm I}[\varphi]=\frac{1}{2\bar{\mu}}\int
d^{3}x\left[\delta(x_{2})(\partial_{2}\varphi(x))^{2}+\sqrt{g(x_{\parallel})}\delta(x_{2}-\psi(x_{\parallel}))(\partial_{n}\varphi(x))^{2}\right]\;.$
(76)

The constant $\bar{\mu}$, which has the dimensions of a mass, is used to
impose Neumann boundary conditions in the $\bar{\mu}\to 0$ limit. We use the
same $\bar{\mu}$ on both L and R mirrors, since we will assume them to have
identical properties, differing just in their position and geometry.

The DE approximation to the Casimir energy can be computed following standard
steps. The result reads, in the limit $\bar{\mu}\psi\to 0$ FLMNeu ,

$E_{\rm
DE}[\psi]\;=\;-\frac{1}{16\pi}\int_{-\infty}^{\infty}dx_{1}\,\frac{1}{\psi(x_{1})^{2}}\left[\zeta(3)+\log[\bar{\mu}\psi(x_{1})]\left(\frac{d\psi(x_{1})}{dx_{1}}\right)^{2}\right]\;.$
(77)

In the expression above, the first term is the PFA contribution while the
second one is a non trivial correction to it, and depends on the shape of the
boundary (defined by $\psi$). It is then clear that, as this equation shows,
the DE is well posed when imposing imperfect Neumann boundary conditions in
$2+1$ dimensions. On the contrary, it cannot be applied when the boundary
conditions become perfect ($\bar{\mu}=0$). The reason is that the hypothesis
of analyticity in momentum, used to derive the DE, is clearly violated. The
non-existence of a local expansion is due to the existence of massless modes,
allowed by Neumann boundary conditions.

Since, at finite temperatures, a $3+1$ dimensional theory may be decomposed
into the sum of an infinite tower of decoupled $2+1$ dimensional Matsubara
modes, each one satisfying N boundary conditions, and with a mass
$\frac{2n\pi}{\beta}$, $n=0,1,2,\ldots$ The existence of the massless $n=0$
mode (the only one surviving in the high temperature limit) means that
analyticity will be lost in $3+1$ dimensions, for any non zero temperature.
That is indeed the case FLMTdim . We summarize here some of the main features
of that example: the free energy in the $d+1$ dimensional Neumann case can
again be written as before (see Eq.(70), but with coefficients $c_{0}$ and
$c_{2}$ instead of $b_{0}$ and $b_{2}$. The zero order term coincides with the
one for the Dirichlet case; namely: $c_{0}=b_{0}$.

When $d=3$, the NTLO term contains, besides a local term, a nonlocal
contribution which is linear in $T$, and thus present for any $T>0$. Hence,
there is no local DE for perfect Neumann boundary conditions at $d=2$ at zero
temperature and for $d=3$ at any finite temperature. Indeed, an expansion for
small values of $|k_{\shortparallel}|$ of the form factor contains, in
addition to a term proportional to $k_{\shortparallel}^{2}$, one proportional
to $(Ta)k_{\shortparallel}^{2}\log(k_{\shortparallel}^{2}a^{2})$.

###  VI.3 The Electromagnetic Case for Imperfect Boundary Conditions

We have seen that, for a real scalar field in the presence of Neumann boundary
conditions, the DE cannot be applied when in $2+1$ dimensions at zero
temperature, or in $3+1$ dimensions at a non-zero temperature FLMTdim . The
reason is that, as we have shown, nonanalyticities in the form factors appear.
We have shown that the nonanalyticity could be cured by introducing a small
departure from perfect Neumann conditions FLMNeu . It is natural to wonder
whether the nonanalyticities could also be cured by a similar approach for the
EM field in $3+1$ dimensions at finite temperatures. We know, based on the
insight obtained from Ref.FLMNeu , that nonanalyticities are originated in
contributions due to dimensionally reduced massless modes: zero Matsubara
frequency terms. To obtain an answer to this question, in Ref.FLMTem  we
singled out in detail the zero-mode contributions to the free energy, for a
media described by non trivial permittivity $\epsilon(\omega)$ and
permeability $\mu(\omega)$ functions.

We start from the free energy $F$ for the EM field, which can be written in
terms of the partition function ${\mathcal{Z}}(\psi)$, as follows:

$F(\psi)\,=\,-\frac{1}{\beta}\,\log[\frac{{\mathcal{Z}}(\psi)}{{\mathcal{Z}}_{0}}\big{]}\;,$
(78)

where the denominator, ${\mathcal{Z}}_{0}$, denotes the partition function for
the EM field in the absence of media and

${\mathcal{Z}}(\psi)\;=\;\int\big{[}{\mathcal{D}}A\big{]}\;e^{-{\mathcal{S}}_{\rm
inv}(A)}\,.$ (79)

The gauge invariant action ${\mathcal{S}}_{\rm inv}(A)$ reads

${\mathcal{S}}_{\rm
inv}(A)\;=\;\int_{0}^{\beta}d\tau\int_{0}^{\beta}d\tau^{\prime}\int
d^{3}{\mathbf{x}}\,\big{[}\frac{1}{2}\,F_{0j}(\tau,{\mathbf{x}})\epsilon(\tau-\tau^{\prime},{\mathbf{x}})F_{0j}(\tau^{\prime},{\mathbf{x}})+\frac{1}{4}F_{ij}(\tau,{\mathbf{x}})\mu^{-1}(\tau-\tau^{\prime},{\mathbf{x}})F_{ij}(\tau,{\mathbf{x}})\big{]}\;.$
(80)

Here, indices like $i,\,j\,\ldots$ run over spatial indices, Einstein
summation convention is assumed, and
$\epsilon(\tau-\tau^{\prime},{\mathbf{x}})$ and
$\mu(\tau-\tau^{\prime},{\mathbf{x}})$ denote the imaginary time versions of
the permittivity and permeability, respectively ($\mu^{-1}$ is the inverse
integral kernel of $\mu$).

The geometry of the system is determined by same two surfaces $L$ and $R$ we
have considered before, and defined by $x_{3}=0$ and
$x_{3}=\psi(\mathbf{x}_{\shortparallel})$, but now they correspond to the
boundaries of the media, i.e.,

$\displaystyle\epsilon(\tau-\tau^{\prime},{\mathbf{x}})$ $\displaystyle=$
$\displaystyle\theta(-x_{3})\epsilon_{L}(\tau-\tau^{\prime})+\theta(x_{3})\theta(\psi({\mathbf{x}}_{\shortparallel})-x_{3})+\theta(x_{3}-\psi({\mathbf{x}}_{\shortparallel}))\epsilon_{R}(\tau-\tau^{\prime})$
$\displaystyle\mu(\tau-\tau^{\prime},{\mathbf{x}})$ $\displaystyle=$
$\displaystyle\theta(-x_{3})\mu_{L}(\tau-\tau^{\prime})+\theta(x_{3})\theta(\psi({\mathbf{x}}_{\shortparallel})-x_{3})+\theta(x_{3}-\psi({\mathbf{x}}_{\shortparallel}))\,\mu_{R}(\tau-\tau^{\prime}),$

where $\epsilon_{L,R}(\tau-\tau^{\prime})$ and $\mu_{L,R}(\tau-\tau^{\prime})$
characterize the permittivity and permeability of the respective mirror.

We can expand the fields and the electromagnetic properties as

$\displaystyle A_{\mu}(\tau,{\mathbf{x}})$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\,\sum_{n=-\infty}^{+\infty}\widetilde{A}_{\mu}^{(n)}({\mathbf{x}})\,e^{i\omega_{n}\tau}$
$\displaystyle\epsilon(\tau-\tau^{\prime},{\mathbf{x}})$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\,\sum_{n=-\infty}^{+\infty}\widetilde{\epsilon}^{(n)}({\mathbf{x}})\,e^{i\omega_{n}(\tau-\tau^{\prime})}$
$\displaystyle\mu(\tau-\tau^{\prime},{\mathbf{x}})$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\,\sum_{n=-\infty}^{+\infty}\widetilde{\mu}^{(n)}({\mathbf{x}})\,e^{i\omega_{n}(\tau-\tau^{\prime})}$
(81)

where $\omega_{n}\equiv 2\pi n/\beta$ ($n\in{\mathbb{Z}}$) are the Matsubara
frequencies.

Inserting these expansions into the partition function one can readily check
the factorization

${\mathcal{Z}}(\psi)\;=\;\prod_{n=0}^{\infty}{\mathcal{Z}}^{(n)}(\psi)\;,$
(82)

and therefore

$F(\psi)\;=\;\sum_{n=0}^{\infty}\,F^{(n)}(\psi)\;.$ (83)

As mentioned, we are particularly interested in the $n=0$ contribution,

${\mathcal{Z}}^{(0)}(\psi)\;=\;\int[{\mathcal{D}}\widetilde{A}_{0}^{(0)}{\mathcal{D}}\widetilde{A}_{j}^{(0)}]\;e^{-{\mathcal{S}}^{(0)}(\widetilde{A}_{0}^{(0)},\widetilde{A}_{j}^{(0)})}\,,$
(84)

where

${\mathcal{S}}^{(0)}(\widetilde{A}_{0}^{(0)},\widetilde{A}_{j}^{(0)})\,=\,\frac{1}{\beta}\int
d^{3}{\mathbf{x}}\,\big{[}\frac{1}{2}\,\widetilde{\epsilon}^{(0)}({\mathbf{x}})(\partial_{j}\widetilde{A}_{0}^{(0)})^{2}\,+\,\frac{1}{4\,\widetilde{\mu}^{(0)}({\mathbf{x}})}(\widetilde{F}_{jk}^{(0)})^{2}\,+\,\frac{1}{2}\,\Omega_{0}^{2}({\mathbf{x}})(\widetilde{A}_{j}^{(0)})^{2}\big{]}\,,$
(85)

and

$\Omega_{0}^{2}({\mathbf{x}})\,\equiv\,\lim_{n\to
0}\,\big{[}\omega_{n}^{2}\,\widetilde{\epsilon}^{(n)}({\mathbf{x}})\big{]}\;.$
(86)

Note that $\Omega_{0}$ vanishes for a dielectric and also for a metal
described by the Drude model. On the other hand, it equals the plasma
frequency for a metal described by the plasma model.

The zero mode contribution to the free energy therefore splits into a scalar
($s$) and a vector ($v$) contribution, the former associated to the field
$\widetilde{A}_{0}^{(0)}$ and the later to $\widetilde{A}_{j}^{(0)}$

$F^{(0)}\;=\;F_{s}(\psi)\,+\,F_{v}(\psi)\;.$ (87)

To discuss the emergence of non-analyticities in the derivative expansion we
computed $F_{s}$ and $F_{v}$ assuming
$\psi(\mathbf{x}_{\shortparallel})=a+\eta(\mathbf{x}_{\shortparallel})$ up to
second order in $\eta$. The quadratic contributions can be written as

$F_{s,v}^{(2)}\,=\,\frac{1}{2}\int\frac{d^{2}k_{\parallel}}{(2\pi)^{2}}\,f_{s,v}^{(2)}(k_{\shortparallel},a)\,|\tilde{\eta}(k_{\shortparallel})|^{2}\;,$
(88)

and the crucial point is whether the functions $f_{s,v}^{(2)}$are analytic or
not in $k_{\parallel}$.

Omitting the details, we summarize the main results FLMTem : for finite values
of $\mu$ and $\epsilon$, the scalar contribution $f_{s}^{(2)}$ analytic,
including the limit $\epsilon\to\infty$, in which it tends to the $2+1$
dimensional Dirichlet value. It develops a nonanalytic (logarithmic)
contribution for $\mu=\infty$, since the kernel corresponds in this case to
that of a scalar field in $2+1$ dimensions satisfying Neumann boundary
conditions. In other words, magnetic materials regulate the non-analyticity of
the TE zero mode.

On the other hand, the TM zero mode is nonanalytic whenever
$\omega^{2}\epsilon(\omega)\to\Omega^{2}\neq 0$ as $\omega\to 0$ for both
mirrors. In terms of the models usually considered in the Casimir literature
to describe real materials, this condition corresponds to the plasma model.

In summary, the nonanalyticities we observed for perfect conductors in our
previous work FLMTdim , survive only under the assumption of perfectly
lossless materials. The NTLO corrections to PFA for metals (gold) at room
temperature have been computed in Ref. Bimonte2 .

###  VI.4 A Semianalytic Formula for Plane-Sphere Geometry

As a final application of the DE to compute the Casimir free energy we mention
the results of Ref.Bimonte3 , where the author combined exact calculations for
the zero mode and the DE to obtain a precise formula for the interaction
between a sphere and a plane at a finite temperature which is valid at all
separations. We briefly describe here these findings.

Formally, the free energy for this geometry can be written as

$F=k_{B}T\sum_{n\geq 0}^{\prime}Tr\log[1-\hat{M}(i\xi_{n})]\,,$ (89)

where the sum is over the Matsubara frequencies $\xi_{n}=2\pi nk_{B}T/\hbar$
and $\hat{M}$ denotes scattering matrix elements for this geometry. The prime
on the sum indicates that the $n=0$ term has an additional $1/2$ factor.

The $n=0$ contribution can be computed exactly using the Drude model to
describe the materials of the plane and the sphere, and plays a crucial role.
Indeed, the proposed approximation for the Casimir force on the sphere of
radius $R$ at a distance $a$ from the plane is

$F_{\rm approx}=F_{n=0}^{(\rm exact)}+F_{n>0}^{(\rm
PFA)}(1-\theta\frac{a}{R})\,,$ (90)

where $\theta$ can be computed using the DE. Notably, $F_{\rm approx}$
describes with high precision the Casimir force at all separations, as can be
checked by comparison with high precision numerical simulations of the exact
scattering formula.

These results have been generalized in subsequent studies to the case of the
two spheres- geometry Bimonte4 , also considering the differences that come
from the use of the Drude vs plasma models, as well as for grounded vs
isolated spheres Bimonte5 . The relevance of the use of grounded conductors in
Casimir experiments has also been discussed in Ref. FLMgrounded .

##  VII Casimir-Polder Forces

The DE approach has also been applied to the calculation of the Casimir-Polder
interaction between a polarizable particle and a gently curved surface
BimonteCP . We present in this Section a simplified version of the results
contained in that reference.

When a small polarizable particle is at a distance $a$ of a planar surface,
the Casimir-Polder potential reads Bordagbook

$U(a)=-\frac{1}{a^{4}}\int\frac{d\xi}{2\pi}\alpha(i\omega_{c}\xi)\beta^{(0)}(\xi)\,,$
(91)

where $\alpha(\omega)$ is the frequency dependent polarizability (which is
assumed isotropic), $\omega_{c}=c/a$, and

$\beta^{(0)}=\frac{e^{-2\xi}}{2}(1+2\xi+2\xi^{2})\,.$ (92)

For moderate distances such that $\alpha(\omega)\approx\alpha(0)$ one obtains
the usual Casimir-Polder potential Casimir-Polder

$U(a)=-\frac{3}{8\pi}\frac{\alpha(0)}{a^{4}}\,.$ (93)

Assume now that the particle is in front of a slightly curved surface. The
particle is at the origin of coordinates, and the surface is described, as
usual, by the height function $z=\psi(\mathbf{x}_{\shortparallel})$. The DE
for the Casimir-Polder interaction $U_{\rm DE}$ assumes that the interaction
depends on the derivatives of the height function $\psi$ evaluated at
$\mathbf{x}_{\shortparallel}=0$, the point on the surface closest to the
particle (a local minimum for $\psi$). If the surface is homogeneous and
isotropic, then the interaction energy must be invariant under rotations of
the $\mathbf{x}_{\shortparallel}$ coordinates. The more general expression
compatible with this properties describes the Casimir -Polder interaction
energy at $T=0$ reads BimonteCP :

$U_{\rm
DE}=-\frac{1}{\psi^{4}}\int\frac{d\xi}{2\pi}\alpha(i\omega_{c}\xi)\left(\beta^{(0)}(\xi)+\beta^{(1)}(\xi)\psi\nabla^{2}\psi\right)\,.$
(94)

The dimensionless function $\beta^{(1)}$ can be read from the perturbative
expansion of the potential $U$, carried to second order in the deformation,
that is, for
$\psi(\mathbf{x}_{\shortparallel})=a+\eta({\mathbf{x}_{\shortparallel}})$ with
$\eta({\mathbf{x}_{\shortparallel}})\ll a$. We stress that here the Casimir-
Polder energy is not a functional but a function of $\psi$ and its derivatives
evaluated at the origin of coordinates (recall that
$\nabla\psi(\mathbf{0})=0$). The DE is expected to be valid when $a\ll
R_{1},R_{2}$, the radii of curvature of the surface at
$\mathbf{x}_{\shortparallel}=0$. Note that $\psi(\mathbf{0})=d$ and

$\nabla^{2}\psi(\mathbf{0})=\frac{1}{R_{1}}+\frac{1}{R_{2}}\,.$ (95)

Using again the static polarizability approximation,
$\alpha(\omega)\approx\alpha(0)$, one obtains

$U_{\rm DE}=-\frac{1}{\pi
a^{4}}\alpha(0)\left[\frac{3}{8}-\frac{13}{60}a(\frac{1}{R_{1}}+\frac{1}{R_{2}})\right]\,.$
(96)

The results presented in Ref. BimonteCP  are much more general than those
described here: they include the Casimir-Polder potential for a general
polarization tensor $\alpha_{\mu\nu}(\omega)$ and higher order corrections
proportional to $(a/R_{i})^{2}$, as well as the details of the computation of
the corresponding functions $\beta^{(p)}$. Additional applications can be
found in moreCP1 ; moreCP2 .

##  VIII Other Techniques Beyond PFA

In Ref. pauloEPL  a detailed analysis of the Casimir effect’s roughness
correction in a setting involving parallel metallic plates is presented. The
plates were defined through the plasma model. The approach used is
perturbative, factoring in the roughness amplitude and allowing for the
consideration of diverse values of the plasma wavelength, plate separation,
and roughness correlation length. A notable finding was that the roughness
correction exceed the predictions of the PFA. The authors have calculated the
second-order response function, $G(k)$, across a spectrum of values
encompassing the plasma wavelength ($\lambda_{P}$), distance ($a$), and
roughness wave vector ($k$):

$G(k)=-\frac{A}{8\pi^{2}}\frac{1}{a^{5}q}\int_{0}^{\infty}\frac{dKe^{-2K}}{1-e^{-2K}}\int_{|K-q|}^{K+q}dK^{\prime}\frac{(KK^{\prime})^{2}+\frac{1}{4}(K^{2}+K^{\prime
2}-q^{2})^{2}}{1-e^{-2K^{\prime}}}\,,$ (97)

applicable when $\lambda_{p}\rightarrow 0$. Here, $A$ represents the plate
surface area, $K$ the dimensionless integration variable denoting the
imaginary wave vector’s $z$-component scaled by plate separation $d$,
$K^{\prime}$ the longitudinal component of the imaginary wave vector for the
diffracted wave, and $q=ka$.

The calculation in Ref. pauloEPL  helps to compute the second-order roughness
correction as a function of the surface profiles, $h_{1}$ and $h_{2}$.
Analytical solutions were determined for specific limiting cases, revealing a
more complex relationship with the perfect reflectors model than previously
recognized pauloEPL03 ; emigPRL01 , particularly in scenarios involving
extended distances and small roughness wavelengths. While the asymptotic case
of long roughness wavelengths aligns with PFA predictions, it was established
that PFA generally underestimates the roughness correction, a critical aspect
for exploring constraints on potentially new weak forces at sub-millimeter
ranges.

As a further expansion to pauloEPL , in Ref. pauloUniverse21 , the authors
explored the Casimir interaction between a plane and a sphere of radius $R$ at
a finite temperature $T$, in terms of the distance of closest approach, $a$.
Noting that, under the usual experimental conditions, the thermal wavelength
$\lambda_{T}$ satisfies $a\ll\lambda_{T}\ll R$, they evaluated the leading
correction to the PFA, applicable to such intermediate temperatures. They
resorted to developing the scattering formula in the plane-wave basis. The
result captures the combined effect of spherical geometry and temperature, and
is expressed as a sum of temperature-dependent logarithmic terms. Remarkably,
two of these logarithmic terms originated from the Matsubara zero-frequency
contribution.

Defining the variables $x=a/R$ and $\tau=a/\lambda_{T}$, and the deviation
$\delta{F}(T)={F}(T)-{F}(0)$, in the intermediate temperature regime
$x\ll\tau\ll 1$, it is found in Ref. pauloUniverse21  that

$\Delta=\frac{\delta{F}(T)-\delta{F}_{\rm PFA}(T)}{{F}_{\rm
PFA}(T)}\approx\frac{45}{\pi^{3}}x\tau\left[-\log^{2}(x)+2[1-\log(2)]\log(x)+2\log^{2}(\tau)+{\cal
O}(\log(\tau))\right].$ (98)

The leading neglected terms stem from non-zero Matsubara frequencies.

In Ref. pauloOSA , the leading-order correction to PFA in a plane-sphere
geometry was derived. The momentum representation connected this with
geometrical optics and semiclassical Mie scattering. The primary contributions
are shown to come from diffraction, with TE polarization becoming more
relevant than TM polarization. The diffraction contribution is calculated at
leading order, using the saddle-point approximation, considering leading order
curvature effects at the sphere tangent plane.

Additionally, the next-to-leading order (NTLO) term in the saddle-point
expansion contributed to the PFA correction. This involved computing the
round-trip operator within the WKB approximation, representing sequences of
reflections between the plane and the sphere. A key aspect was the tilt in the
scattering planes, allowing TE and TM polarizations to mix.

Comprehending the implications of polarization mixing channels on the
geometric optical correction applied to PFA holds considerable importance.
Indeed, these channels are recognized for inducing negative Casimir entropies
with a geometric foundation. 24–26OSA1; 24-26OSA2 ; 24-26OSA3 ; 49-51OSA1 ;
49-51OSA2 ; 49-51OSA3 . In spite of the non-vanishing contribution of the
polarization mixing matrix elements, the total correction associated with the
tilt between the scattering and Fresnel planes is zero at NTLO. This implies
that the primary correction to the PFA would remain unchanged even if the
complexities arising from the differences between the Fresnel and scattering
polarization bases were initially ignored. The latter points to the fact that
a different approach, one that completely omits the effect of polarization
mixing, could directly produce the leading order correction to PFA. Plane
waves proved to be a well-suited basis for studying the Casimir effect, as has
been evidenced in the more recent study paulo22 . The utility of that basis
ranges from analytical to numerical applications, particularly when dealing
with objects in close proximity, the most relevant situation in experiments.
It has been also shown that the use of plane waves was notably effective in
improving the interpretation of results in the realms of geometrical optics
and diffractive corrections.

In the context of a setup involving two spheres with arbitrary radii in
vacuum, it was shown in paulo22  that the PFA emerged as the leading term in
an asymptotic expansion for large radii. Extending a prior calculation based
on the saddle-point approximation, involving a trace over multiple round-trips
of electromagnetic waves between the spheres, the study encompassed spheres
made of bi-isotropic material, requiring the consideration of polarization
mixing during reflection processes. The result was naturally elucidated within
the framework of geometrical optics.

Then, by relying on a saddle-point approximation framework, the authors
derived leading-order corrections, of geometrical and diffractive origins.
Explicit results, at first obtained for perfect electromagnetic conductors
(PEMC) spheres at zero temperature, indicated that for certain material
parameters, the PFA contribution vanishes; should that be the case, the
leading-order correction would be the dominant term in the Casimir energy.

In the lowest-order saddle-point approximation, but including diffractive
corrections, one can show that the expression for the Casimir energy becomes:

$E_{\rm LO-SPA}=-\frac{\pi^{3}R_{\rm
eff}}{720a^{2}}\left[1-\frac{15}{\pi^{2}}x+\frac{15(10+3\pi)}{4\pi^{3}}x^{3/2}+..\right],$
(99)

where $x=a/R_{\rm eff}$. As expected, this result reproduces the PFA result
and its leading-order diffractive correction. The NTLO correction behaves as
$x^{3/2}$. However, the prefactor obtained accounts for about $90\%$ of the
one coming from numerical results pauloOSA . This discrepancy may be traced
back to having neglected the NTLO-SPA and NNTLO-SPA contributions.

##  IX Conclusions

In this review, we have discussed several properties and applications of the
DE approach, mostly as a method to improve the predictions of the Proximity
Force Approximation, of long standing use in many different fields.

We started the review by briefly discussing the precursor of the PFA: the
Derjaguin (and related) approximations, since we have found them rather
appropriate in order to display the essentially geometric nature of the kind
of problem we discuss: two quite close smooth surfaces, and an interaction
energy between them. Depending on the kind of system being considered, that
interaction between the two surfaces may or not be the result of the
superposition of the interactions between pairs. An example of an interaction
which is not the result of such a superposition is the Casimir effect. Note,
however, that even when the fundamental interaction satisfies a superposition
principle, like in electrostatics, the actual evaluation of the Coulomb
integral to calculate the total interaction energy could be a rather involved
problem because the actual charge density may not be known a priori. That is
indeed the case when the surfaces involved are conductors, since that usually
requires finding the electrostatic potential. We have used precisely this
problem in order to present the idea of the DE in a concrete example: to
calculate the electrostatic energy between two conducting surfaces held at
different potentials.

After introducing and applying the DE in that example, we have discussed its
more general proof of that expansion, by first putting the problem in a more
general and abstract way: how to approximate, under certain smoothness
assumptions, a functional of a pair of surfaces. At the same time, the proof
provides a concrete way to determine the PFA and its NTLO correction, the DE:
one just needs to perform an expansion in powers of the deformation of the
surfaces about the situation of two flat and parallel surfaces.

The derivations and examples here have been presented for a geometrical
setting were one surface is a plane, while the other may be described by a
single Monge patch based on that plane. However, as shown by other authors,
under quite reasonable and general assumptions, the results obtained for that
situation may be generalized to the case of two curved surfaces parametrized
by their respective patches, based on a common plane (which now does not
coincide with one of the physical surfaces).

Then we reviewed different applications of the DE to the zero temperature
Casimir effect, considering different fields and boundary conditions, staring
from the cases of the scalar field with Dirichlet boundary conditions, then
the EM field in the presence of perfectly conducting surfaces, and commented
on the scalar field with Neumann conditions.

We afterwards presented a description and brief review of the extension of DE
to finite temperature cases, and different numbers of spatial dimensions. The
temperature is a dimensionful magnitude and the phenomenon of dimensional
reduction presents a problem when there are Neumann boundary conditions or
when an EM field is involved. Indeed, dimensional reduction implies the
existence of a massless $2+1$ dimensional field (with Neumann conditions), and
this mode introduces a nonanalyticity in momentum space, which violated one of
the hypothesis of the DE, and therefore it cannot be applied. Nevertheless, we
have shown that the introduction of a small departure from ideal Neumann
conditions solves this issue, namely, analyticity is recovered and the DE may
be applied.

We also mentioned the application of DE to the Casimir-Polder interaction,
particularly between a polarizable particle and a gently curved surface. This
example highlights the broader implications of DE in understanding particle-
surface interactions beyond the Casimir force itself.

To conclude, we have presented in this review the main features of the DE
approach, with a focus in the Casimir effect, but pointing at the fact that
its applicability can certainly go beyond that realm. We have shown that
explicitly for electrostatics, but we expect it to be applicable to, for
example, the same kind of systems where the DA, SEI and SIA were introduced.

This research was funded by Agencia Nacional de Promoción Científica y
Tecnológica (ANPCyT), Consejo Nacional de Investigaciones Científicas y
Técnicas (CONICET), Universidad de Buenos Aires (UBA), and Universidad
Nacional de Cuyo (UNCuyo), Argentina.

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