Smaller11  $\BODY$ Smaller08  $\BODY$ MnLargeSymbols’164 MnLargeSymbols’171

Scalar-Mediated Quantum Forces Between Macroscopic Bodies and Interferometry

Philippe Brax<EMAIL_ADDRESS>and Sylvain Fichet
<EMAIL_ADDRESS>

a Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS,

F-91191 Gif/Yvette Cedex, France

b ICTP South American Institute for Fundamental Research & IFT-UNESP,

R. Dr. Bento Teobaldo Ferraz 271, São Paulo, Brazil

c Centro de Ciencias Naturais e Humanas, Universidade Federal do ABC,

Santo Andre, 09210-580 SP, Brazil

Abstract

We study the quantum force between classical objects mediated by massive
scalar fields bilinearly coupled to matter. The existence of such fields is
motivated by dark matter, dark energy, and by the possibility of a hidden
sector beyond the Standard Model. We introduce the quantum work felt by an
arbitrary (either rigid or deformable) classical body in the presence of the
scalar and show that it is finite upon requiring conservation of matter. As an
example, we explicitly show that the quantum pressure inside a Dirichlet
sphere is finite — up to renormalizable divergences. With this method we
compute the scalar-induced quantum force in simple planar geometries. In
plane-point geometry we show how to compute the contribution of the quantum
force to the phase shift observable in atom interferometers. We show that atom
interferometry is likely to become a competitive search method for light
particles bilinearly coupled to matter, provided that the interferometer arms
have lengths below $\sim 10$ cm.

###### Contents

  1. 1 Introduction
  2. 2 The Quantum Work
    1. 2.1 Action and Quantum Vacuum Energy
    2. 2.2 The Source and its Deformation
    3. 2.3 Quantum Work and Force
    4. 2.4  $W_{\lambda}$ is Finite (Renormalizable Case)
  3. 3 Weak Coupling: Finiteness Properties and Thin Shell Limit
    1. 3.1 $W_{\lambda}$ is Finite (EFT Case)
    2. 3.2 The Thin Shell Limit
    3. 3.3 The Dirichlet Limit
  4. 4 Scalar Quantum Forces between Rigid Bodies
    1. 4.1 Rigid Bodies
    2. 4.2 Vanishing of Tadpoles
    3. 4.3 The Scalar Casimir-Polder Limit
    4. 4.4 The Scalar Casimir Limit
  5. 5 The Dirichlet Sphere
    1. 5.1 Is the quantum pressure on the sphere finite in QFT?
    2. 5.2 Review and Discussion
    3. 5.3 The Quantum Work on the Dirichlet Sphere
  6. 6 Planar Geometry
    1. 6.1 Force Between two Plates
    2. 6.2 Force Between a Plate and a Point Source
  7. 7 Bounding Quantum Forces with Atom Interferometry
    1. 7.1 The setting
    2. 7.2 Computing the Phase Shift
    3. 7.3 Limits
    4. 7.4 Sensitivity to New Particles
  8. 8 Conclusion
  9. A Derivation of the Plate-Point Casimir-Polder Potential
  10. B On Divergences from the Heat Kernel Expansion
  11. C The Loop Divergence from Momentum Space

##  Introduction

The exchange of virtual particles induces macroscopic forces between bodies.
Beyond the tree level, such forces are relativistic and quantum in essence,
they are properly described in the framework of quantum field theory (QFT).
Here we refer to such forces simply as “quantum forces”. Even though the
seminal works on Casimir forces are from the nineteen forties CP_original ;
Casimir_original , the topic of quantum forces is still very much active (see
e.g. Milton:2004ya ; Klimchitskaya:2009cw ; 2015AnP…527…45R ; Woods:2015pla ;
Bimonte:2017bir ; Bimonte:2021maf ; Bimonte:2022een  for recent papers and
reviews). In the case of electromagnetic interactions, refined calculations
take into account medium properties such as electromagnetic permittivities as
well as effects from finite temperature, see bordag2009advances  for a
comprehensive review. In the present paper we work in the simple framework of
scalar-mediated quantum forces at zero temperature, with the key assumption
that the scalar couples bilinearly to the sources. 222See e.g. Milton:2002vm ;
Graham:2002fw ; Jaffe:2005vp ; Mobassem:2014jma  for a selection of related
works involving scalar quantum forces. In this scalar setting, we use a simple
variational method to derive quantum forces between bodies of various shapes
and positions.

The scalar case is of prime relevance for cosmology Hui:2016ltb ;
Joyce:2014kja  and deriving quantum forces mediated by massive scalars could
lead to new laboratory tests of important cosmological models, from scalar
dark matter to dark energy. In this paper, we argue that atom interferometry
Hamilton:2015zga  could be such a promising technique. Our primary motivation
comes from the pervasiveness of scalar fields bilinearly coupled to matter in
cosmology Joyce:2014kja  and in extensions of the Standard Model of particle
physics Allanach:2016yth . For instance, the dark matter in our Universe can
be modelled as a scalar particle with $Z_{2}$ symmetry, which couples thus
bilinearly to matter. A wealth of dark energy models and related modified
gravity models Brax:2021wcv  also involve a light scalar field. For these,
when the classical force induced by the linear coupling to sources is screened
(see e.g. Damour:1994zq ; Khoury:2003aq ; Khoury:2003rn ), the quantum force
arising from the bilinear coupling to sources can become dominant Brax:2018grq
, hence motivating our study. Finally, irrespective of observational
motivations, the existence of a sector hidden beyond the Standard Model and
featuring a light scalar with a $Z_{2}$ symmetry (e.g. a pseudo-Nambu
Goldstone boson of a symmetry of the hidden sector) is a logical possibility
that requires investigation. There is thus, in short, a cornucopia of reasons
to study scalar fields bilinearly coupled to matter.

A secondary motivation is that some of the technical and conceptual results
that we present — such as the proof of finiteness of the quantum work — are
best exposed with a scalar field. We use an approach based on the variation of
the quantum vacuum energy which is similar in spirit to the one found in the
seminal work by Schwinger Schwinger:1977pa . The developments presented here
can be taken as a streamlined presentation of this variational method based in
the context of scalar QFT.

The vacuum energy of QFT features divergences which are treated via standard
renormalization methods, e.g. using dimensional regularization and introducing
local counterterms Peskin:257493 , which maps these divergences onto the RG
flow of the Lagrangian parameters. The fact that such divergences do not
affect the quantum force can be shown in complete generality at one-loop using
e.g. the heat kernel expansion bordag2009advances . Apart from these well-
behaved divergences, other “spurious” divergences which are non-removable by
renormalization can appear in certain Casimir calculations. This happens for
example for a calculation of the pressure in the Dirichlet sphere
Milton_Sphere  (see detailed discussion in Sec. 5), and even for a version of
the calculation of the pressure between plates (see detailed discussion in
Sec. 6.1). While such spurious divergences might be easily identified and
removed in an ad hoc way, in this work we will show that they are
systematically removed by the requirement of conservation of matter in the
sources. Matter conservation has so far not been taken into account in
calculations of quantum forces, to the best of our knowledge. At the
conceptual level it is the new ingredient of our calculations.

Hopefully our presentation will be useful to cosmologists who would like to
see their models tested in the laboratory. In particular we transparently
explain in terms of Feynman diagrams how the screening of the scalar inside
the sources gives rise to two distinct regimes for the quantum force. This
situation is similar to the Casimir and Casimir-Polder forces arising in
electromagnetism. The common nature and unified description of these
electromagnetic quantum forces has been long known, see e.g. Refs.
Dzyaloshinskii_1961 ; Schwinger:1977pa . Here we will give a formula for the
quantum force in the massive scalar case and relate the “scalar Casimir” and
“scalar Casimir-Polder” forces. 333See e.g. Feinberg:1968zz ;
FeinbergSucherNeutrinos ; Grifols:1996fk ; Fichet:2017bng ; Costantino:2019ixl
for Casimir-Polder-like forces in the context of particle physics.  The use of
this general formula is mandatory for computing certain observables such as
phase shifts in interferometry.

Finally, another purpose of this work is to lay out the foundations for more
phenomenological studies, for which the effect of the quantum force from a
particle beyond the Standard Model needs to be predicted in realistic
experiments. As such, we will present a calculation of the phase shift induced
by a quantum force and measurable in atom interferometers. We will then study
to which extent atom interferometry is a competitive method to search for
light dark particles.

### Technical Review

The present study bears some technical similarities with other works from the
Casimir literature. Here we briefly list a few of these references. Our
definition of the quantum work involves the variation of the quantum vacuum
energy with respect to a deformation of the source. A similar approach relying
on energy variation has been used in Schwinger:1977pa , expressed in the
language of Schwinger’s source theory. The source we consider has finite
density, and the Dirichlet limit is obtained by sending the density to
infinity. A similar framework was considered in Graham:2002fw ; Graham:2002xq
; Graham:2003ib  (the conservation equation is not exploited in these
references). In our formalism we consider arbitrary deformations of an
arbitrary body. A somewhat similar approach was taken in Schwinger:1977pa  in
the case of deformation of a dielectric fluid. We sometimes use regularisation
by point-splitting, in the context of Casimir calculations this method has for
example been used in Milton:2004ya . Finally, under certain conditions
including that the bodies are incompressible, finiteness of the quantum work
at one-loop can be shown using the heat kernel formalism. This is reviewed and
discussed in App. B.
Comparison to Franchino-Vinas:2021lbl  : After pre-publication we became aware
of the recent article Franchino-Vinas:2021lbl  whose scope and results have
partial overlap with the present study — upon appropriate matching of terms
and concepts. Both works introduce the concept of variable/effective mass. The
principle of virtual work whose proof for arbitrary geometry in a specific
model is presented in Franchino-Vinas:2021lbl  is fully compatible with the
general formula of the quantum work presented here. The translation of the key
quantities between the model of Franchino-Vinas:2021lbl  and the present work
is $\phi|_{[\color[rgb]{0,0,1}{35}]}\equiv\Phi$,
$\frac{\lambda_{1}}{4}\sigma^{2}\phi^{2}|_{[\color[rgb]{0,0,1}{35}]}\equiv{\cal
B}_{m}J$. An important difference of scope is that in Franchino-Vinas:2021lbl
the deformation flow of the body is rigid, which implies that no discussion of
matter conservation is needed. In contrast, the compressible deformation flow
and its relation to matter conservation are a key aspect of our study.
Finally, the formula for the quantum work resulting from the rigid deformation
of an interface is obtained in Franchino-Vinas:2021lbl  in terms of a surface
integral of the stress tensor. This result turns out to precisely match the
one we obtain in our Eq. (45) upon setting the variation of density to zero,
even though the intermediate steps of the derivations are rather different.
This is a nontrivial verification of the consistency of both works.

### Outline

The paper is arranged as follows. Section 2 presents the framework for scalar
quantum forces, giving a formula for the quantum work valid at the non-
perturbative level for arbitrary deformable sources. Section 3 specializes to
the weak coupling case (which includes the case of effective field theories).
The quantum work in the limiting case of thin-shell geometries is further
evaluated. Section 4 specializes to two rigid bodies, showing that the Casimir
and Casimir-Polder forces for massive scalar fields are asymptotically
recovered as limits of our unifying formula for quantum forces. The Casimir
pressure on the Dirichlet sphere is revisited and shown to be finite in
section 5. The generalized Casimir forces for finite density objects and a
finite vacuum mass for the scalar field in plane-plane and point-plane
geometry are respectively computed in sections 6. In section 7 we then
calculate the phase shift in atom interferometers. In App. A we compare a
computation of the Casimir-Polder interaction from a scattering amplitude and
our derivation, showing explicitly that they coincide. In App. B we review a
proof of the finiteness of the quantum work at the one-loop level using the
heat kernel expansion following the steps of bordag2009advances , which
contributes to motivating our approach. App. C contains details on a loop
calculation in momentum space .

### Definitions and Conventions

We assume $d+1$-dimensional Minkowski spacetime ${\cal M}_{d+1}$ with mostly-
minus signature $(+,-,\ldots,-)$. The $d+1$ Cartesian coordinates are denoted
by $x_{\mu}$, spatial coordinates are denoted by $x_{i}\equiv{\bm{x}}$. We
will be considering a source $J({\bm{x}})$ with arbitrary shape and dimension.
The support of the source is described by the indicator function
${\bm{1}}_{J}({\bm{x}})$ or equivalently using a continuous support function
$l({\bm{x}})$ which is positive where $J$ is supported and negative where it
is not, with ${\bm{1}}_{J}(x)\equiv\Theta(l({\bm{x}}))$ where $\Theta$ is the
Heaviside distribution. The boundary of the source is denoted by $\partial
J=\left\\{{\bm{x}}\in{\cal M}_{d+1}|l({\bm{x}})=0\right\\}$. Integration over
the support of the source $J$ is denoted $\int_{J}d^{d}{\bm{x}}$. Integration
over the support of the boundary $\partial J$ is denoted $\int_{\partial
J}d\sigma({\bm{x}})$. Although our main focus is ultimately the $d=3$ case, we
keep $d$ general when possible and specialize to $d=3$ in specific settings.

##  The Quantum Work

In this section we compute the quantum work felt by a source bilinearly
coupled to a quantum field under an arbitrary deformation of the source. We
focus on a scalar field $\Phi$ for simplicity — generalization to spinning
fields is identical although more technical. The bodies subject to the Casimir
forces are assumed to be classical and static. The set of bodies is
collectively represented in the partition function by a static source term
$J({\bm{x}})$. More precisely, the $J({\bm{x}})$ distribution corresponds to
the vacuum expectation of the density operator $\hat{n}({\bm{x}})$ in the
presence of matter,
$J({\bm{x}})=\langle\Omega|\hat{n}({\bm{x}})|\Omega\rangle$.

###  Action and Quantum Vacuum Energy

We consider the fundamental Lagrangian

${\cal
L}[\Phi]=\frac{1}{2}(\partial_{M}\Phi)^{2}-\frac{1}{2}m^{2}\Phi^{2}+\ldots\,.$
(1)

The ellipses include possible interactions of $\Phi$, which do not need to be
specified. The interacting theory for $\Phi$ can either be renormalizable —
with either weak or strong coupling, or may also be an effective field theory
(EFT) involving a series of operators of arbitrary dimensions. In this latter
case the theory is weakly coupled below the EFT cutoff scale on distances
larger than $\Delta x\sim\frac{1}{\Lambda}$. The scale $\Lambda$ is the energy
cut-off of the theory.

We consider the partition function in Minkowski spacetime 444We call the
generating functional $Z[J]$ the partition function in analogy to the
Euclidean case.

$Z[J]=\int{\cal D}\Phi e^{i\left(S[\Phi]-\int d^{d+1}x\,{\cal
B}[\Phi]J({\bm{x}})\right)}$ (2)

where ${\cal B}[\Phi]$ is a bilinear operator in $\Phi$. This operator can
encode an arbitrary number of field derivatives. We distinguish two cases. If
the bilinear operator has no derivative, then the scalar theory can be
renormalizable. In this case we write the operator as

${\cal B}_{m}[\Phi]=\frac{1}{2\Lambda}\Phi^{2}$ (3)

If the bilinear operator has derivatives, then the scalar theory is an EFT and
in general contains a whole series of higher dimensional operators. In this
case, including an arbitrary number of such terms, we can write the operator
as

${\cal B}_{\rm EFT}[\Phi]=\sum_{n>1,i}\frac{c_{n,i}}{2\Lambda^{2n+1}}\Phi{\cal
O}_{n,i}\Phi\,$ (4)

where ${\cal O}_{n,i}$ is a scalar derivative operator with $2n$ derivatives
which act either to the left or to the right, 555When writing the bilinear
interaction in the form Eq. (4), consistently with the source term defined in
(2), we take into account that any derivative acting on $J$ has been removed
using integration by part, producing derivatives that act on the fields. With
this constraint, the most general structure is the one given in Eq. (5).

${\cal
O}_{n,i}=(\overleftarrow{\partial}^{2})^{p}\overleftarrow{\partial}_{\mu_{1}}\ldots\overleftarrow{\partial}_{\mu_{a}}(\overrightarrow{\partial}^{2})^{n-p-a}\overrightarrow{\partial}_{\mu_{1}}\ldots\overrightarrow{\partial}_{\mu_{a}}$
(5)

The $c_{n,i}$ coefficients are dimensionless.

Let us comment further about some aspects of EFT. Here we distinguished
between the ${\cal B}_{m}$ and ${\cal B}_{\rm EFT}$ operators, essentially
because our subsequent analysis of finiteness of the quantum work slightly
differs between both cases. In general, a given EFT can feature both the
${\cal B}_{m}$ and ${\cal B}_{\rm EFT}$ terms. In such case the ${\cal B}_{m}$
contribution would tend to dominate, unless it is suppressed by a symmetry
(e.g. a shift symmetry). Still in the EFT framework, we may also notice that
non-derivative higher order operators involving powers of $\Phi$ and $J$ are
in general present. While the higher order operators which are bilinear in
$\Phi$ could be simply accounted in the generic source $J$, the broader view
is that the EFT validity domain prevents these operators from becoming
important, i.e. schematically $\partial/\Lambda\ll 1$. Finally we emphasize
that our subsequent analyses involving UV divergences can be done at the level
of the EFT, without having to specify the underlying completion, see e.g.
Manohar:1996cq ; Manohar:2018aog  for more details on divergences and
renormalization in EFT.

Since the source is static, the partition function takes the form

$Z[J]=e^{-iE[J]T}$ (6)

where $E[J]$ is referred to as the quantum vacuum energy and $T$ is an
arbitrary time interval specified in evaluating the time integrals. This time
scale will drop from the subsequent calculations. 666 In Minkowski space we
have in general $Z=\langle 0|e^{-i{\cal H}T}|0\rangle$, with ${\cal H}$ the
Hamiltonian of the system. This is why the eigenvalue $E[J]$ is identified as
the vacuum energy. We also mention that upon Wick rotation to Euclidean space,
$E[J]$ corresponds to the free energy of the system.  In general, we can set
$J$ as an abstract quantity that can be used to generate the correlators of
the theory. In our case, since the source couples to ${\cal B}$ (see Eq. (2)),
taking functional derivatives of $E[J]$ in $J$ generates the connected
correlators of the composite operator $\cal B$, i.e. $\langle{\cal
B}(x_{1}){\cal B}(x_{2})\ldots\rangle$. In this work, we consider that $J$
represents a physical distribution of matter, i.e. $J({\bm{x}})$ is taken to
be the expectation value of the density operator $\hat{n}({\bm{x}})$ in the
presence of matter,
$J({\bm{x}})=\langle\Omega|\hat{n}({\bm{x}})|\Omega\rangle$. For concreteness,
one can for instance think of a nonrelativistic fermion density, appearing for
example via $\bar{\psi}\psi=n({\bm{x}})$ or
$\bar{\psi}\gamma_{\mu}\psi=\delta_{\mu 0}n({\bm{x}})$ in the relativistic
formulation. 777 Higher monomials contributions such as
$\frac{(\bar{\psi}\psi)^{2}}{\Lambda^{3}}$ are neglected in this example. We
mention that in the presence of screening, i.e. in the Dirichlet or Casimir
regime defined in next sections, the validity of the EFT tends to be improved
because the coupling to the source tends to be suppressed. Validity of the EFT
in the presence of screening has been discussed in Brax:2018grq .

###  The Source and its Deformation

The source is parametrized by

$J({\bm{x}})=n({\bm{x}}){\bm{1}}_{J}({\bm{x}})$ (7)

and corresponds to a particle number distribution of mass dimension $d$. The
support of this distribution is encoded in
${\bm{1}}_{J}({\bm{x}})=\Theta(l({\bm{x}}))$ where the continuous function
$l({\bm{x}})$ is positive where $J$ is supported and negative where it is not.
The number density $n({\bm{x}})$ is in general an arbitrary distribution over
the support. The integral $N_{J}=\int d\mu_{i}J(\mu_{i})$ amounts to the total
particle number of the source.

We then introduce a deformation of the source. We assume that matter is
deformable i.e. both the support and the number density can vary under the
deformation. We will see that such a generalization from rigid to deformable
matter is necessary in order to ensure that the calculation is well-defined
and that no infinities show up.

The infinitesimal deformation of the source is parametrized by a scalar
parameter $\lambda$. Under our assumptions the source depends on this
deformation parameter as
$J_{\lambda}({\bm{x}})=n_{\lambda}({\bm{x}})\Theta[l_{\lambda}({\bm{x}})]$.
The deformed source takes the form
$J_{\lambda+d\lambda}({\bm{x}})=n_{\lambda+d\lambda}({\bm{x}})\Theta[l_{\lambda+d\lambda}({\bm{x}})]$.
The deformation of the support of the source is parametrized by

$l_{\lambda+d\lambda}({\bm{x}})=l_{\lambda}({\bm{x}}-{\bf
L}({\bm{x}})d\lambda)$ (8)

where the ${\bf L}$ vector is the deformation flow. Defining
$\frac{\partial}{\partial\lambda}\equiv\partial_{\lambda}$ the variation of
the source under the $\lambda$ deformation is then given by
$\partial_{\lambda}J_{\lambda}({\bm{x}})=\partial_{\lambda}n_{\lambda}\Theta[l_{\lambda}({\bm{x}})]-n_{\lambda}{\bf
L}\cdot{\bm{\partial}}l_{\lambda}\,\delta[l_{\lambda}({\bm{x}})]$. An
arbitrary deformation of a generic source is pictured in Fig. 1.

We assume that $J({\bm{x}})$ is made out of classical matter and is not a
completely abstract distribution. As the source is made of classical matter,
then its local number density must be conserved. Any deformation of the source
must be therefore subject to the conservation of the number density. The local
conservation equation under the deformation parametrized by $\lambda$ is

$\partial_{\lambda}n_{\lambda}+{\bm{\partial}}\cdot\left(n_{\lambda}{\bf
L}\right)=0\,.$ (9)

It implies the integral form

$\partial_{\lambda}\int_{J_{\lambda}}d^{d}{\bm{x}}\,n_{\lambda}({\bm{x}})=\int
d^{d}{\bm{x}}\,\partial_{\lambda}J_{\lambda}({\bm{x}})=0$ (10)

where the second integral is over all space.

In the special case where the density $n$ is constant in $\lambda$ and $x$,
Eq. (9) reduces to the condition of an incompressible deformation flow,
${\bm{\partial}}\cdot{\bf L}=0$. In this case the source describes
incompressible matter (e.g. a fluid). If ${\bm{\partial}}\cdot{\bf L}=0$ and
${\bf L}$ is piecewise constant in $x$, then the source describes rigid
matter. Section. 4 specializes to such rigid sources.

###  Quantum Work and Force

Figure 1:  An arbitrary infinitesimal deformation of a generic source. The
dotted and plain contours respectively correspond to the boundary of
$J_{\lambda}$ and $J_{\lambda+d\lambda}$. The arrows denote the deformation
flow ${\bf L}$.

We now study how the quantum system evolves upon a general, infinitesimal
deformation of the source, $\lambda\to\lambda+d\lambda$. To proceed we
introduce the quantum work under a variation in $\lambda$,

$W_{\lambda}=-\partial_{\lambda}E[J_{\lambda}]\,.$ (11)

In the particular case of a rigid body and if the deformation field can be
factored out, then a quantum force can be defined as

$W_{\lambda}={\bf L}\cdot{\bm{F}}.$ (12)

This defines the quantum force between the objects. Such a simplification is
however generally not possible, the most fundamental quantity to consider is
the quantum work. Our definition of quantum work is closely related to
variational approaches such as the principle of virtual work applied to
quantum physics, see e.g. Schwinger:1977pa  and Li:2019ohr ; Franchino-
Vinas:2020okl  for related recent studies.

The quantum vacuum energy $E[J]$ is a formally divergent quantity. However, if
one varies it with respect to a physical parameter, the resulting variation is
a physical observable and thus should be finite. 888Throughout this work, the
physical parameter is typically a geometric variable such as the distance
between two bodies.  Hence even though $W_{\lambda}$ encodes all the quantum
effects felt by the source(s), the only divergences remaining in this quantity
are those with a physical meaning, i.e. the ones which must be treated in the
framework of renormalization. One origin for such divergences is the field
interactions, another can be the curvature of spacetime, as pointed out in
Fichet:2021xfn  in AdS. Even for a free theory in flat space, there can be
renormalization of surface tension terms, as pointed out in Bordag:2004rx . We
refer to such divergences as “physical” ones — these are the familiar
divergences from QFT, that only appear for specific integer values of
spacetime dimension. One of our goals in the following is to show that there
are no other divergences than the physical ones in the QFT calculation of the
quantum force.

Using the definition of Eqs. (2), (6), the quantum work is given by

$W_{\lambda}=-\int d^{d}{\bm{x}}\left\langle{\cal
B}\right\rangle_{J_{\lambda}}(x,x)\partial_{\lambda}J_{\lambda}({\bm{x}})\,$
(13)

where $\left\langle{\cal B}\right\rangle_{J_{\lambda}}$ is the time-ordered
quantum average of ${\cal B}$ in the presence of the source $J_{\lambda}$.
Here and throughout we use the shortcut notation
$W_{J_{\lambda}}=W_{\lambda}$, $\left\langle{\cal
B}\right\rangle_{J_{\lambda}}=\left\langle{\cal B}\right\rangle_{\lambda}$.
While all quantities depend on $\lambda$, the $\lambda$ dependence is relevant
only under the $\partial_{\lambda}$ variation and is often dropped elsewhere.
The two-point function in presence of the source at coinciding points will be
sometimes denoted by $\left\langle{\cal B}\right\rangle_{J}$ in the rest of
the paper.

Using the general form of the bilinear coupling, ${\cal B}$ is expressed in
terms of the two point function of $\Phi$ in the presence of $J$, $\langle
T\Phi(x)\Phi(y)\rangle_{J}$, evaluated at coinciding points. We assume that
the classical value of $\Phi$ is zero. 999This can be the consequence of a
$Z_{2}$ symmetry, enforcing that $\Phi$ only appears bilinearly in the action
such that $\langle\Phi\rangle=0$. In the presence of nonzero
$\langle\Phi\rangle$, the possible $Z_{2}$ symmetry is broken, implying that
at weak coupling the fluctuation of $\Phi$ over the $\langle\Phi\rangle$
background has a linear coupling to the source. As a result a classical force
is also present in addition to the quantum force. This case does not need to
be investigated in the present paper. This extra classical force appearing in
the presence of $\langle\Phi\rangle\neq 0$ does not automatically dominate
over the quantum force — instead, various regimes arise as a function of the
geometry. These aspects have been partly investigated in Brax:2018grq .  Hence
the disconnected part of the two-point function vanishes and the two-point
function reduces to $\langle T\Phi(x)\Phi(y)\rangle_{J}=\Delta_{J}(x,y)$,
where $\Delta_{J}(x,y)$ is the Feynman propagator in the presence of the
source $J$. The quantum average of ${\cal B}$ is then expressed in terms of
the Feynmann propagator $\Delta_{J}$ as

$\langle{\cal B}_{m}\rangle_{J}=\frac{1}{2\Lambda}\Delta_{J}(x_{1},x_{2})$
(14) $\langle{\cal B}_{\rm
EFT}\rangle_{J}=\sum_{n>1,i}\frac{c_{n,i}}{2\Lambda^{2n+1}}{\cal
O}_{n,i}\Delta_{J}(x_{1},x_{2})\,.$ (15)

Here in the EFT case we introduce the shortcut $\langle T\Phi(x){\cal
O}_{n,i}\Phi(y)\rangle_{J}\equiv{\cal O}_{n,i}\Delta_{J}(x,y)$ where the
${\cal O}_{n,i}$ operator as defined in Eq. (5) has derivatives acting to the
left and to the right. The general formula for the quantum work presented in
Eq. (13) is valid at the non-perturbative level.

###  $W_{\lambda}$ is Finite (Renormalizable Case)

The quantity $\left\langle{\cal B}\right\rangle_{J}(x,x)$ is formally
divergent since it contains the propagator at coinciding points. It is thus
not obvious why the quantum work $W_{\lambda}$ should be finite. Throughout
this subsection, we regulate the divergence in $\left\langle{\cal
B}\right\rangle_{J}$ by introducing a small splitting of the endpoints,
$\left\langle{\cal B}\right\rangle_{J}(x,x)\equiv\left\langle{\cal
B}\right\rangle_{J}(x,x_{\epsilon})|_{\epsilon\to 0}$ where we defined
$x_{\epsilon}=x+\epsilon$. 101010 The analogous regularization in Fourier
space is momentum cutoff, $p<\frac{1}{\epsilon}$. These regularizations admit
a physical meaning. $\epsilon$ can be thought as the distance scale below
which the description of the classical matter as a continuous distribution
breaks down. In this view, the cutoff length $\epsilon$ has a physically
meaningful value, and the statement of existence of a divergence turns into a
statement on $\epsilon$-dependence of the result.

We consider the renormalizable case i.e. the $\langle{\cal B}\rangle_{m}$
operator (the EFT case is addressed in section 3.1). We assume that, as a
preliminary step, all the divergences which can be removed by the
renormalization of the coupling constants of local operators have been
performed, e.g. the fundamental mass $m$ is the renormalized mass. We assume
that $n({\bm{x}})$ is finite for any ${\bm{x}}$ over the support of $J$ (the
$n\to\infty$ limit is discussed in section 3.2). No assumption on the
interactions of $\Phi$ is necessary thus $\Phi$ can be strongly coupled. Under
these conditions the finiteness of the quantum work can be shown as follows.

In the $\epsilon\to 0$ limit, we can decompose the expectation value of $\cal
B$ as the sum of i) the $\epsilon$-dependent, would-be divergent term, and ii)
a finite term in which the $\epsilon$ dependence amounts to $O(\epsilon)$
corrections that can be neglected for $\epsilon\to 0$. This gives

$\left\langle{\cal B}\right\rangle_{J}(x,x_{\epsilon})=\left\langle{\cal
B}\right\rangle_{J}^{\rm div}(x,x_{\epsilon})+\left\langle{\cal
B}\right\rangle^{\rm fin}_{J}(x,x)+O(\epsilon)\,.$ (16)

We then use the assumption that the number density is finite. It implies that
the effective squared mass of $\Phi$ inside the source,
$m^{2}+\frac{n({\bm{x}})}{\Lambda}$, is finite. We remind that $m$ has been
renormalized already. A divergence in the short distance behaviour of
$\langle{\cal B}\rangle(x,x_{\epsilon})$ can arise from a propagator going
from $x$ to $x_{\epsilon}$. As the effective mass term amounts to a relevant
operator with finite value, it is negligible in the short distance limit of
the propagator, i.e. in the large momentum limit. As the source $J$ appears in
the propagator only via the effective mass term, we conclude that the
divergent piece $\left\langle{\cal B}\right\rangle_{J}^{\rm
div}(x,x_{\epsilon})$ is independent of $J$. Furthermore, in that short
distance limit, the propagator must be Lorentz invariant, and we conclude that
the divergent piece in Eq. (16) is independent of $x$,

$\left\langle{\cal B}\right\rangle_{J}^{\rm div}(x,x_{\epsilon})|_{{\rm
small}\,\epsilon}=\left\langle{\cal B}\right\rangle^{\rm
div}(x,x_{\epsilon})=\left\langle{\cal B}\right\rangle^{\rm
div}(0,\epsilon)\equiv\left\langle{\cal B}\right\rangle^{\rm
div}_{\epsilon}\,.$ (17)

We thus see that the divergent piece depends only on $\epsilon$ and diverges
in the $\epsilon\to 0$ limit. Using the decomposition Eq. (16) and the
definition of the quantum work, we obtain the decomposition

$W_{\lambda}=W^{\rm fin}_{\lambda}+W^{\rm div}_{\lambda}$ (18)

with

$W^{\rm fin,div}_{\lambda}=-\int d^{d}{\bm{x}}\left\langle{\cal
B}\right\rangle_{J}^{\rm
fin,div}(x,x)\partial_{\lambda}J_{\lambda}({\bm{x}})\,.$ (19)

In the divergent piece, $\left\langle{\cal B}\right\rangle^{\rm
div}_{\epsilon}$ factors out of the integral because it is independent of $x$.
This gives

$W^{\rm div}_{\lambda}=-\left\langle{\cal B}\right\rangle_{\epsilon}^{\rm
div}\int d^{d}{\bm{x}}\,\partial_{\lambda}J_{\lambda}({\bm{x}})\,.$ (20)

The remaining integral corresponds exactly to the variation of the total
density of the source under the deformation, appearing in the integral form of
the conservation equation Eq. (10). Thus if the equation of conservation Eq.
(10) is satisfied, then Eq. (20) vanishes. We conclude that, upon conservation
of matter in the source, for any deformation and finite $n$ the quantum work
is finite:

$W^{\rm div}_{\lambda}=0\,.$ (21)

This is true at the nonperturbative level. In the particular case of
incompressible matter, Eq. (20) reduces to

$W^{\rm div}_{\lambda}|_{\rm incompressible}=-n\left\langle{\cal
B}\right\rangle^{\rm div}\int_{J}d^{d}{\bm{x}}\,{\bm{\partial}}\cdot
L({\bm{x}})\,.$ (22)

In that case we can say that, upon conservation of matter in the source, for
any divergent-free deformation flow and finite $n$, the quantum work is
finite:

$W^{\rm div}_{\lambda}|_{\rm incompressible}=0$ (23)

The finiteness of the quantum work will be exemplified in the upcoming
sections.

Finally we comment on the finite part of the quantum work. The finite part can
be put in the useful alternative form by evaluating the integrand, using the
divergence theorem and using the conservation equation,

$W^{\rm fin}_{\lambda}=-\int d^{d}{\bm{x}}\,n_{\lambda}({\bm{x}}){\bf
L}\cdot{\bm{\partial}}\left[\left\langle{\cal
B}\right\rangle_{\lambda}(x,x)\right]\,.$ (24)

This is another way to verify that any constant piece in $\left\langle{\cal
B}\right\rangle(x,x)$ does not contribute to the quantum work as it appears
under a gradient in the integrand.

##  Weak Coupling: Finiteness Properties and Thin Shell Limit

At weak coupling the $\Phi$ field has an equation of motion (EOM) that we can
use to evaluate the quantum work. We introduce the bilinear operator ${\cal
B}^{\prime\prime}$, defined by

${\cal B}=\frac{1}{2}\Phi(x){\cal B}^{\prime\prime}\Phi(x)\,.$ (25)

This is the operator that appears in the EOM. For example, when applied to
${\cal B}_{m}$ this is ${\cal B}^{\prime\prime}_{m}=\frac{1}{\Lambda}$. In the
EFT case, ${\cal B}^{\prime\prime}$ is the differential operator appearing in
Eq. (4),

${\cal B}_{\rm
EFT}^{\prime\prime}=\sum_{n>1,i}\frac{c_{n,i}}{\Lambda^{2n+1}}{\cal
O}_{n,i}\,.$ (26)

The left and right derivatives in ${\cal O}_{n,i}$ act on the propagators
attached respectively to the left and right of the vertex.

At leading order in the perturbative expansion, the $\Delta_{J}(x,x^{\prime})$
propagator satisfies the equation of motion

${\cal D}_{x}\Delta_{J}(x,x^{\prime})+{\cal
B}^{\prime\prime}J({\bm{x}})\Delta_{J}(x,x^{\prime})=-i\delta^{d+1}(x-x^{\prime})$
(27)

where ${\cal D}=\square+m^{2}$ is the wave operator and $\square$ is the
scalar d’Alembertian. The solution to Eq. (27) is a Born series that describes
the bare propagator $\Delta_{0}$ (i.e. $\Delta_{J}|_{J\to 0}$) dressed by
insertions of ${\cal B}^{\prime\prime}J$. For convenience we define the
insertion

$\Sigma(x,y)=-i{\cal B}^{\prime\prime}J({\bm{x}})\delta^{d+1}(x-y)$ (28)

and we use the inner product $f\star g=\int d^{d+1}u\,f(u)g(u)$. With these
definitions the dressed propagator is given by

$\displaystyle\Delta_{J}(x,x^{\prime})$
$\displaystyle=\sum^{\infty}_{q=0}\Delta_{0}\left[\star\Sigma\star\Delta_{0}\right]^{q}(x,x^{\prime})$
(29) $\displaystyle=\Delta_{0}(x,x^{\prime})-\int
d^{d+1}u\,\Delta_{0}(x,u)i{\cal
B}^{\prime\prime}J({\bm{u}})\Delta_{0}(u,x^{\prime})+\ldots$ (30)

Putting this result back into Eq. (13) provides the leading, one-loop
contribution to the quantum work, 111111 Another, conceptually similar way to
derive this formula is via the heat kernel formalism, see e.g. Bordag:2004rx ;
Franchino-Vinas:2020okl

$W_{\lambda}^{\rm 1-loop}=-\frac{1}{2\Lambda}\int d^{d}{{\bm{x}}}{\cal
B}^{\prime\prime}\sum^{\infty}_{q=0}\Delta_{0}\left[\star\Sigma\star\Delta_{0}\right]^{q}(x,x)\partial_{\lambda}J({\bm{x}})$
(31)

This is valid for both ${\cal B}_{m}$ and ${\cal B}_{\rm EFT}$ insertions. In
terms of Feynman diagrams Eq. (31) is simply a loop with an arbitrary number
of insertions of ${\cal B}^{\prime\prime}J$ and one insertion of
$\partial_{\lambda}J$. A term of the series is represented (without the
$\partial_{\lambda}$ variation) in Fig. 2, where each insertion is represented
by a black dot.

In the EFT case, the validity domain of the EFT implies that higher
derivatives terms in ${\cal B}_{\rm EFT}$ must remain small. Still, in the
series of Eq. (31), the effect of lower derivative terms may become important
in some regime. When the effect of these derivatives is important, it may
happen that convergences issues in Eq. (31) appear, that need careful
consideration taking into account EFT validity. A related example is discussed
in details in Bordag:2001ta . In the following, we will show the finiteness of
every term in $W_{\lambda}^{\rm 1-loop}$ and simply assume that the overall
series is convergent.

Figure 2:  A sample one-loop diagram in the presence of an arbitrary source.
Black dots represent insertions of $-i{\cal B}^{\prime\prime}J({\bm{x}})$.
Each black dot is integrated over the support of the source. Under an
infinitesimal deformation of the source ($\lambda\to\lambda+d\lambda$), the
corresponding variation (i.e. $\partial_{\lambda}$) of this diagram
contributes to the one-loop quantum work Eq. (31).

###  $W_{\lambda}$ is Finite (EFT Case)

Our proof of Eq. (21) uses that the effective mass is a relevant operator that
becomes negligible at short distances. In contrast, the insertions from ${\cal
B}^{\rm EFT}$ correspond to irrelevant operators hence the same reasoning
cannot apply — the operators become more important at short distance. The
solution to this apparent puzzle is that the EFT in its domain of validity is
necessarily weakly coupled, hence instead of using a non-perturbative argument
one can use the series representation Eq. (31) to prove finiteness.

Let us verify finiteness term-by-term. We single out a term from Eq. (31) and
introduce point-splitting, ${\cal
B}^{\prime\prime}\Delta_{0}\left[\star\Sigma\star\Delta_{0}\right]^{q}(x,x_{\epsilon})$,
with $x_{\epsilon}=x+\epsilon$. Our goal is to show that the divergent piece
in this quantity is independent of $x$. The term is

${\cal
B}^{\prime\prime}\Delta_{0}\left[\star\Sigma\star\Delta_{0}\right]^{q}(x,x_{\epsilon})=(-i)^{q}\left(\prod^{q}_{i=1}\int
d^{d}{\bm{\mu}}_{i}J({\bm{\mu}}_{i})\int dt_{i}\right)\prod^{q}_{i=0}{\cal
B}^{\prime\prime}\Delta_{0}(\mu_{i},\mu_{i+1})\bigg{|}_{\mu_{0}=x,\mu_{q+1}=x_{\epsilon}}$
(32)

It is understood that one of the block of derivatives in ${\cal
B}^{\prime\prime}$ acts to the left and the other acts to the right.

The divergence in the diagram defined by Eq. (32) occurs when all the
positions coincide. This is more easily verified in momentum space, in which
case there is a single loop integral over the internal momentum flowing around
the loop. The divergence is tied to the large momentum for all the
propagators, which in position space corresponds to the limit of coincident
endpoints. We present the explicit momentum space calculation in App. C. The
divergent piece of Eq. (32) is $(-i)^{q}c_{J}^{q}L^{\rm div}_{q,\epsilon}$
where $c_{J}$ is finite and $L^{\rm div}_{q,\epsilon}$ is the divergent part.
The key point is that $L^{\rm div}_{q,\epsilon}$ is position-independent.

Putting this piece into the definition of the quantum work Eq. (31) gives the
divergent piece of the quantum work,

$W_{\lambda}^{\rm
1-loop,div}=-\frac{1}{2\Lambda}\sum_{q=0}^{\infty}(-ic_{J})^{q}L^{\rm
div}_{q,\epsilon}\int d^{d}{\bm{x}}\partial_{\lambda}J\,.$ (33)

The remaining integral corresponds exactly to the variation of the total
density of the source under the deformation, appearing in the integral form of
the conservation equation Eq. (10). Thus if the equation of conservation Eq.
(10) is satisfied, then Eq. (33) vanishes.

It follows that, upon conservation of matter in the source, for any
deformation and finite $n$ the quantum work is finite:

$W_{\lambda}^{\rm 1-loop,div}=0\,.$ (34)

The incompressible version of this finiteness property trivially follows, like
for Eq. (23).

###  The Thin Shell Limit

So far we have considered a generic source as an arbitrary volume in
$d$-dimensional space. Here we investigate a subset of sources for which the
support is a thin shell approaching a codimension-one hypersurface.

We denote the source by
$J_{\eta}=n(x){\bm{1}}_{\mathcal{S}_{\eta},\lambda}(x)$ where $\eta$
parametrizes the small width of the shell. For $\eta\to 0$ the support of the
shell tends to a hypersurface denoted by $\mathcal{S}$. To avoid any ambiguity
we always keep $\eta$ small but nonzero in the following calculations. The
volume element can be split as

$\int_{\mathcal{S}_{\eta}}d^{d}{\bm{x}}\overset{{\rm
small}\,\eta}{=}\int_{\mathcal{S}}d\sigma({\bm{x}})\int_{\rm width}dx_{\perp}$
(35)

where the $x_{\perp}$ coordinate parametrizes the direction normal to
$\mathcal{S}$. The boundary of ${\cal S}_{\eta}$ can also be decomposed as

$\partial{\cal S}_{\eta\to 0}={\cal S}_{\rm in}\cup{\cal S}_{\rm out}$ (36)

where ${\cal S}_{\rm in,out}$ are the two hypersurfaces bounding the volume
enclosed by $\cal S_{\eta}$ in the limit $\eta\to 0$. The propagator in the
presence of the thin shell is denoted by $\Delta_{\cal S}(x,x^{\prime})$. The
density can be chosen to scale with $\eta$ such that it remains finite for
$\eta\to 0$. This happens if the density scales as $\eta n=$cst. In the
following, the deformation of the source is kept arbitrary. All quantities
depend on the deformation parameter $\lambda$. We will drop the $\lambda$
index when appropriate.

We evaluate the quantum work for this specific class of sources, taking $\eta$
small but finite. For simplicity we consider the coupling to the source
induced via the ${\cal B}_{m}$ operator. Our evaluation involves various
manipulations of the EOMs and of the divergence theorem. We emphasize that we
do not use the conservation equation. That way we can explicitly demonstrate
later on, at the level of applications, that the conservation equation is
required to obtain finiteness of the quantum work. For clarity we split the
calculation in various steps.

##### Step 1:

Starting from the general expression of the quantum work Eq. (13), we evaluate
the $\partial_{\lambda}J_{\lambda}$ variation. We use
$\frac{{\bm{\partial}}l}{\lVert{\bm{\partial}}l\lVert}=\mathbf{n}_{\rm in}$
with $\mathbf{n}_{\rm in}$ the inward-pointing normal vector, then use the
divergence theorem $\int_{\partial{\cal S}}d\sigma({\bm{x}})\mathbf{n}_{\rm
out}\cdot{\bm{f}}({\bm{x}})=\int_{{\cal
S}}d^{d}{\bm{x}}{\bm{\partial}}\left({\bm{f}}({\bm{x}})\right)$ with
$\mathbf{n}_{\rm out}=-\mathbf{n}_{\rm in}$. We obtain

$W_{{\cal
S}_{\eta},\lambda}=-\frac{1}{2\Lambda}\int_{\mathcal{S}_{\eta}}d^{d}{\bm{x}}\,\Delta_{\cal
S}({\bm{x}},{\bm{x}})\partial_{\lambda}n_{\lambda}(x)-\frac{1}{2\Lambda}\int_{\mathcal{S}_{\eta}}d^{d}{\bm{x}}\,{\bm{\partial}}\left[{\bf
L}n_{\lambda}(x)\Delta_{\cal S}(x,x)\right]\,.$ (37)

##### Step 2: (Using the equations of motion)

We will further simplify the second term of Eq.(37) by observing that it can
be related to discontinuities determined from the equation of motion. In order
to proceed we introduce the notation for derivatives acting on either the
first or second argument of the propagator,
${\bm{\partial}}_{1}\Delta(x,x^{\prime})\equiv{\bm{\partial}}_{x}\Delta(x,x^{\prime})$,
${\bm{\partial}}_{2}\Delta(x,x^{\prime})\equiv{\bm{\partial}}_{x^{\prime}}\Delta(x,x^{\prime})$.
The coincident-point propagator is regularized via point-splitting using
$x_{\epsilon}=x+\epsilon$ where the shifted point is taken to belong to ${\cal
S}$ when $x\in{\cal S}$.

In the thin shell limit, the second term in Eq. (37) takes the form

$-\frac{1}{2\Lambda}\eta\,\int_{\cal
S}d\sigma({\bm{x}})\left({\bm{\partial}}_{1}[{\bf L}(x)n(x)\Delta_{\cal
S}(x,x_{\epsilon})])+{\bf L}(x)n(x){\bm{\partial}}_{2}[\Delta_{\cal
S}(x,x_{\epsilon})]\right)(1+O(\eta))$ (38)

We used the volume element Eq. (35) and that the integrand is continuous over
$\mathcal{S}_{\eta}$. In the first term of Eq. (38) the vector ${\bf L}$ is
kept inside the derivative for further convenience.

The EOM is

${\cal D}_{x}\Delta_{\cal
S}(x,x^{\prime})+\frac{1}{\Lambda}J_{\eta}({\bm{x}})\Delta_{\cal
S}(x,x^{\prime})=-i\delta^{d+1}(x-x^{\prime})$ (39)

where ${\cal D}_{x}=\square_{x}+m^{2}$. Below the $\delta^{d+1}$ is always
zero due to point splitting. Each of the terms in Eq. (38) can be expressed
using an appropriate derivative of the EOM, with the remaining endpoint set to
an appropriate value.

The first term in Eq. (38) is obtained by multiplying the EOM with ${\bf
L}({\bm{x}})$ then applying ${\bm{\partial}}_{x}$. We then integrate across
the normal coordinate and set the remaining endpoint of the propagator
$x^{\prime}$ to coincide with $x$ in the transverse coordinates. This gives
the identity

$-\frac{\eta}{\Lambda}{\bm{\partial}}_{1}\left[{\bf L}n\Delta_{\cal
S}(x,x_{\epsilon})\right](1+O(\eta))\overset{{\rm
small}\,\eta}{=}\left[\int_{\rm width}dx_{\perp}{\bm{\partial}}_{x}\left[{\bf
L}({\bm{x}})\square_{x}\Delta_{\cal
S}(x,x^{\prime})\right]\right]_{x^{\prime}\to x_{\epsilon}}$ (40)

The fundamental mass contributes as $O(\eta)$, it is thus neglected . After
integrating over the transverse coordinates (i.e. applying $\int_{\cal
S}d\sigma({\bm{x}})$), the l.h.s of Eq. (40) coincides with the first term of
Eq. (38).

The second term in Eq. (38) is obtained by applying
${\bm{\partial}}_{x^{\prime}}$ to the EOM then contracting with ${\bf
L}({\bm{x}}^{\prime})$. The subsequent steps are the same as above and the
result is

$-\frac{\eta}{\Lambda}{\bf L}n{\bm{\partial}}_{2}\left[\Delta_{\cal
S}(x,x_{\epsilon})\right](1+O(\eta))\overset{{\rm
small}\,\eta}{=}\left[\int_{\rm width}dx_{\perp}{\bf
L}({\bm{x}}^{\prime}){\bm{\partial}}_{x^{\prime}}\left[\square_{x}\Delta_{\cal
S}(x,x^{\prime})\right]\right]_{x^{\prime}\to x_{\epsilon}}\,.$ (41)

Using the identities (40) and (41) in Eq. (38) one can eliminate the presence
of the density in favor of d’Alembertians.

##### Step 3:

Finally we use the divergence theorem on the right hand side of both Eqs. (40)
and (41). In Eq.(41) the divergence theorem turns the ${\square}_{x}$ into
$\mathbf{n}\cdot{\bm{\partial}}$. 121212Notice that $x^{\prime}$ is set to
$x_{\epsilon}$ only outside of the integral. Hence the $x^{\prime}$ dependence
of the integrand is irrelevant when applying the divergence theorem. Replacing
these results in Eq. (37) we obtain the final form for the quantum work on a
thin shell,

$\displaystyle W_{\lambda}^{\cal S}=$
$\displaystyle-\frac{1}{2}\int_{\mathcal{S}_{\rm in}\cup\mathcal{S}_{\rm
out}}d\sigma({\bm{x}})\left(\,n_{i}L_{j}\,\partial^{i}_{1}\partial^{j}_{2}\Delta_{\cal
S}(x,x_{\epsilon})+\mathbf{n}.{\bf
L}\,\square_{1}\Delta_{\mathcal{S}}(x,x_{\epsilon})\right)$
$\displaystyle-\frac{1}{2\Lambda}\int_{\mathcal{S}_{\eta}}d^{d}{\bm{x}}\,\Delta_{\cal
S}(x,x_{\epsilon})\partial_{\lambda}n_{\lambda}({\bm{x}})+O(\eta)$ (42)

The volume term in the second line encodes the variation of the number density
of the source under the deformation. Notice that we have not used the
conservation equation yet in deriving Eq. (42). The conservation equation must
ensure that the quantum work is finite, along the line of (21). This will be
exemplified in Sec. 5.

###  The Dirichlet Limit

We have so far assumed that $\eta n\equiv n_{\cal S}$ is finite. $n_{\cal S}$
can be understood as the number density on the hypersurface $\cal S$. In this
subsection we take the limit $n_{\cal S}\to\infty$. In this limit we obtain
that the propagator vanishes anywhere inside ${\cal S}_{\eta}$, including on
the boundaries ${\cal S}_{\rm in}$, ${\cal S}_{\rm out}$, i.e. we have
$\Delta_{\mathcal{S}}(x_{,}x^{\prime})=0$ for any $x^{\prime}$ or $x\in{\cal
S}$. On the other hand the derivatives normal to the boundary do not vanish on
the boundary, i.e. $\partial_{1}^{\perp}\Delta_{\cal
S}(x,x^{\prime})|_{x\in{\cal S}_{\rm in,out}}\neq 0$. 131313 These properties
are shown at the level of the EOM using field continuity. We integrate the EOM
on any domain crossing ${\cal S_{\eta}}$ and use the divergence theorem. This
makes appear the well-known ”jump” in the normal derivatives
$\partial_{1}^{\perp}\Delta_{\cal S}(x,x^{\prime})|_{x\in{\cal S}_{\rm
out}}-\partial_{1}^{\perp}\Delta_{\cal S}(x,x^{\prime})|_{x\in{\cal S}_{\rm
in}}=\frac{n_{\cal S}}{\Lambda}\Delta_{\cal S}(x,x^{\prime})|_{x\in{\cal
S}}\times(1+O(\eta))\,.$ (43) The propagator is continuous everywhere, hence
on the rhs is it enough to simply write $x\in{\cal S}$ without further detail.
Due to the requirement of continuity of the propagator, the discontinuity of
derivatives must remain finite. As a result, taking the limit $n_{\cal
S}\to\infty$ implies that $\Delta_{\cal S}(x,x^{\prime})|_{x\in{\cal S}}\to 0$
for any $x^{\prime}$.  In summary, the limit $n_{\cal S}\to\infty$ amounts to
a Dirichlet boundary condition for the propagator.

We now apply this Dirichlet limit to the quantum work Eq. (42). The second
surface term in the first line of Eq. (42) vanishes in the Dirichlet limit
since the second endpoint belongs to the boundary and has no derivative acting
on it. In contrast the first surface term involves derivatives on both
endpoints and thus does not vanish in the Dirichlet limit. This term further
simplifies: the first derivatives of $\Delta(x,x^{\prime})$ across ${\cal S}$
are discontinuous only in the normal coordinate, while they are continuous in
the other directions, therefore
$n_{i}L_{j}\,\partial^{i}_{1}\partial^{j}_{2}\Delta_{\cal
S}(x,x_{\epsilon})=L_{\perp}\,\partial^{\perp}_{1}\partial^{\perp}_{2}\Delta_{\cal
S}(x,x_{\epsilon})$. We see that only the normal component of the deformation
flow $L_{\perp}=\mathbf{n}\cdot{\bf L}$ contributes in the Dirichlet limit.

We can then make contact with the scalar stress-energy tensor
$T^{\mu\nu}=-\partial^{\mu}\phi\partial^{\nu}\phi+\frac{1}{2}\eta^{\mu\nu}(\partial_{\rho}\phi\partial^{\rho}\phi-m^{2}\phi^{2})$.
Namely, when considering the normal component $T^{\perp\perp}$, we recognize
that the difference of the time-ordered expectation value $\langle
T^{\perp\perp}\rangle$ between ${\cal S}_{\rm out}$ and ${\cal S}_{\rm in}$ is

$\left[\langle T^{\perp\perp}\rangle\right]^{{\cal S}_{\rm out}}_{{\cal
S}_{\rm
in}}=\frac{1}{2}\left[\langle\partial^{\perp}\phi\partial^{\perp}\phi\rangle\right]^{{\cal
S}_{\rm out}}_{{\cal S}_{\rm
in}}=\frac{1}{2}\left[\partial_{1}^{\perp}\partial_{2}^{\perp}\Delta_{\cal
S}(x,x)\right]^{{\cal S}_{\rm out}}_{{\cal S}_{\rm in}}$ (44)

Notice that there is a contribution from the normal derivatives inside the
isotropic $\eta_{\mu\nu}$ term. The derivatives in the transverse directions
do not contribute due to continuity. It follows that the quantum work on a
thin shell in the Dirichlet limit can be expressed using the stress-energy
tensor as

$W_{\lambda}^{\cal S}=-\int_{\mathcal{S}_{\rm in}\cup\mathcal{S}_{\rm
out}}d\sigma({\bm{x}})\,L_{\perp}\,\langle\Omega|T^{\perp\perp}|\Omega\rangle-\frac{1}{2\Lambda}\int_{\mathcal{S}_{\eta}}d^{d}{\bm{x}}\,\Delta_{\cal
S}(x,x_{\epsilon})\partial_{\lambda}n_{\lambda}({\bm{x}})+O(\eta)$ (45)

This is, of course, not a coincidence. This contribution to the quantum work
reproduces exactly the difference of stress-energy tensors that is used to
compute the Casimir forces or pressures on thin shells, see e.g. Milton_Sphere
. The second term, which is the new term arising in our calculation, must
ensure that any spurious divergence arising from the first term cancels out
upon requiring matter conservation of the source, as dictated by the
finiteness property Eq. (21). We will exemplify this in the case of the
Dirichlet sphere in Sec. 5.

##  Scalar Quantum Forces between Rigid Bodies

Figure 3:  Sample Feynman diagrams for a scalar field in the presence of two
extended sources. (i): A generic one-loop contribution. (ii): “Tadpole” loops
vanishing under $\partial_{\lambda}$. (iii): Casimir limit (Strong coupling to
sources) (iv): Casimir-Polder limit (Weak coupling to sources)

###  Rigid Bodies

In this section we choose a specific shape for the source and the deformation.
The source is assumed to be the compound of two rigid bodies $J=J_{1}+J_{2}$
with number densities $n_{1,2}$. We assume that the $J_{2}$ source moves
rigidly with respect to $J_{1}$. The deformation flow ${\bf L}$ thus reduces
to a constant vector over $J_{2}$ and vanishes elsewhere. In this particular
case the ${\bf L}$ factors out in $W_{\lambda}$ and we can talk about the
quantum force ${\bf F}_{1\to 2}$ between $J_{1}$ and $J_{2}$. Using Eq. (31)
the general expression for the quantum work is expressed as

$W_{\lambda}^{\rm 1-loop}={\bf L}\cdot{\bf F}_{1\to 2}=-\frac{1}{2\Lambda}\int
d^{d}{\bm{x}}\sum^{\infty}_{q=0}\Delta_{0}\left[\star\Sigma\star\Delta_{0}\right]^{q}(x,x){\bf
L}\cdot{\bm{\partial}}J_{2}(x)$ (46)

where $\Sigma=-\frac{i}{\Lambda}(J_{1}+J_{2})\delta^{d+1}(x-y)$. We will then
evaluate the general formula Eq. (46) in specific limits. An arbitrary term of
the series is represented in Fig. 3i.

###  Vanishing of Tadpoles

In Eq.(46), each insertion of $\Sigma$ contains both $J_{1}$ and $J_{2}$. Let
us focus on subterms involving only $J_{2}$. Such terms amount to the
generalization of “tadpole” diagrams for an extended source, here $J_{2}$.
Using integrations by parts and the fact that the propagators $\Delta_{0}$ in
empty spacetime are Lorentz invariant, i.e are functions of $u-v$ only, one
can check that any such tadpole term is equal to minus itself and thus
vanishes. This makes sense since such terms do not involve the $J_{1}$ source
at all and should not contribute to the force between the two sources. A
tadpole diagram is represented in Fig. 3ii. These diagrams contain divergent
contributions to the quantum work. Therefore the vanishing of the tadpole
diagrams ensures that the perturbative finiteness Eq. (34) is satisfied.

###  The Scalar Casimir-Polder Limit

Let us assume that the values of $\frac{n_{1,2}}{\Lambda}$ are small enough
such that the leading contributions come from the first terms in the series.
The first term of the series has $q=0$. This term amounts to a tadpole
diagram, hence it vanishes as shown in Sec. 4.2. We thus turn to the $q=1$
term. This term is

$W^{\rm 1-loop}_{q=1}=\frac{i}{2\Lambda^{2}}\int
d^{d}{\bm{u}}\,d^{d+1}v\,\Delta_{0}(u,v)J(v)\Delta_{0}(v,u){\bf
L}\cdot{\bm{\partial}}J_{2}(u)\,.$ (47)

We then decompose $J(u)=J_{1}(u)+J_{2}(u)$. The $\int
J_{2}\Delta_{0}^{2}{\bm{\partial}}J_{2}$ piece is again a tadpole and thus
vanishes as shown in Sec. 4.2 — using integration by parts, on can check that
it is equal to minus itself. The remaining term is

$W^{\rm 1-loop}_{q=1}=\frac{i}{2\Lambda^{2}}\int
d^{d}{\bm{u}}d^{d+1}v\,\Delta_{0}(u,v)J_{1}(v)\Delta_{0}(v,u){\bf
L}\cdot{\bm{\partial}}J_{2}(u)\,.$ (48)

Upon integrating by parts (or evaluating ${\bm{\partial}}J_{2}$ and using the
divergence theorem), we recognize the variation of a bubble diagram that
corresponds precisely to the definition of the Casimir-Polder potential
$V_{\rm CP}(R)$ between two point sources. Namely,

$W^{\rm 1-loop}_{q=1}=-n_{1}n_{2}\int d^{d}{\bm{u}}d^{d}{\bm{v}}\,{\bf
L}\cdot{\bm{\partial}}V_{\rm CP}(u-v)\,$ (49)

where we have defined the potential

$V_{\rm CP}(r)=-\frac{i}{2}\int
dt\left(\Delta_{0}(0;r,t)\right)^{2}=-\frac{1}{32\pi^{3}\Lambda^{2}}\frac{m}{r^{2}}K_{1}(2mr)\,.$
(50)

We can see the explicit dependence on the fundamental mass $m$ in this result.
Details of the explicit evaluation in the last step can be found in e.g. Ref.
Costantino:2019ixl .

We can notice that Eq. (49) amounts to the integral over the $J_{1,2}$
supports of the quantum work between two point sources generated by the
directional derivative of the potential $W_{\rm CP}$, i.e. $W^{\rm
1-loop}_{n=1}=n_{1}n_{2}\int d^{d}{\bm{u}}d^{d}{\bm{v}}\,W_{\rm CP}$ with
$W_{\rm CP}=-{\bf L}\cdot{\bm{\partial}}V_{\rm CP}$.

A diagram in the Casimir-Polder limit is shown in Fig. 3iv. In this limit the
quantum loop penetrates the whole bodies.

###  The Scalar Casimir Limit

A different limit is obtained when the effective mass inside the sources,
$m^{2}+\frac{n_{1,2}({\bm{x}})}{\Lambda}$, is large enough for the dressed
propagator to be repelled from the sources. This occurs whenever ${\cal
D}_{x}\Delta(x,x^{\prime})\ll J({\bm{x}})\Delta_{J}(x,x^{\prime})$ for any
$x,x^{\prime}$. In this Dirichlet limit, the EOM Eq. (27) gives then
$J({\bm{x}})\Delta_{J}(x,x^{\prime})\approx 0$, which enforces
$\Delta_{J}(x,x^{\prime})\approx 0$ for any $x$ in the source and any
$x^{\prime}$ in the whole space. The propagator vanishes on the boundary,
$\Delta_{J}(x\in\partial J,x^{\prime})\approx 0$, by continuity. Therefore the
propagator has Dirichlet boundary conditions in this regime. We refer to this
limit as the scalar Casimir limit since it reproduces a Dirichlet problem for
which the quantum force is usually referred to as ”Casimir” even when the
underlying theory is not electrodynamics, e.g. here a massive scalar field
theory.

A sample diagram from the Casimir limit is shown in Fig. 3iii. In this limit
the quantum loop does not penetrate inside the bodies.

Summarizing, we have shown that our formula for the one-loop quantum work
Eq.(46) interpolates between the scalar Casimir-Polder force and the scalar
Casimir force. The two limits are realised in different physical regimes,
which essentially depend on the competition between the magnitudes of the
effective mass and of the d’Alembertian in the EOM. Qualitatively, we can say
that for fixed fundamental parameters and densities, the Casimir-Polder limit
emerges in the short separation regime while the Casimir limit emerges in the
large separation regime.

##  The Dirichlet Sphere

In this section we consider the Casimir pressure on a spherical shell in the
presence of a scalar field with Dirichlet boundary condition on the shell,
i.e. a “Dirichlet sphere”. This is a standard problem, that we wish here to
revisit. An early calculation can be found in Boyer:1968uf , and an expression
describing the Casimir pressure on a $d-1$-sphere has been derived in
Milton_Sphere , which will be our main reference.

###  Is the quantum pressure on the sphere finite in QFT?

The QFT prediction obtained in Milton_Sphere  features a “spurious”
divergence. Here we discuss why such a divergence is not expected in light of
the finiteness properties derived in Sec. 2. We have emphasised in Sec. 2 the
role of matter conservation to obtain a finite quantum work. In the case of
the sphere, the deformation flow describing the radial deformation of the
sphere is not divergence-free, ${\bm{\partial}}\cdot L_{\rm Sphere}\neq 0$ for
arbitrary spacetime dimension $d$ except $d=1$. If the sphere density were to
be assumed to be constant, then the matter of the sphere would not conserved
under the deformation and it would then follow that neither of the finiteness
properties (21) and (23) would apply. Hence a divergent piece would show up in
the expression of the quantum pressure as an artefact. This is exactly what we
find below. We also show how this spurious divergence exactly cancels when
matter conservation is used.

###  Review and Discussion

We first review the result from Milton_Sphere . The radius of the sphere is
denoted by $a$. The pressure on the sphere obtained in this reference can be
put in the form

$\frac{F_{S_{d-1},[\color[rgb]{0,0,1}{26}]}}{A_{d-1}}=\frac{F^{\rm
fin}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}}{A_{d-1}}+\frac{F^{\rm
div}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}}{A_{d-1}}$ (51)

where

$\displaystyle\frac{F^{\rm
fin}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}}{A_{d-1}}=i\frac{1}{a^{d}}\sum^{\infty}_{h=0}c_{h}\int^{\infty}_{-\infty}d\omega\omega\frac{d}{d\omega}\log\left(\omega
aJ_{h-1+\frac{d}{2}}(|\omega|a)H^{(1)}_{h-1+\frac{d}{2}}(|\omega|a)\right)$
$\displaystyle\frac{F^{\rm
div}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}}{A_{d-1}}=i\frac{1-d}{a^{d}}\sum^{\infty}_{h=0}c_{h}\int^{\infty}_{-\infty}d\omega.$

and the coefficients are given by

$c_{h}=\frac{(h-1+\frac{d}{2})\Gamma(h+d-2))}{2^{d}\pi^{\frac{d+1}{2}}h!\Gamma(\frac{d-1}{2})}.$
(53)

$A_{d-1}$ is the area of the $d-1$-sphere with radius $a$. All the terms are
real upon rotation to Euclidean time, here for convenience we keep the
Lorentzian integrals. The result Eq. (51) is obtained based on the difference
between the radial component of the stress tensor on each side of the sphere.
In our language, this is equivalently obtained by considering the sphere as a
rigid source with infinite density and deforming it along the radial flow
${\bf L}={\bm{e}}_{r}$.

The $F_{S}^{\rm fin}$ term is finite for any $d$ different from positive even
integers. The divergence showing up when $d=2k$, $k=1,2\ldots$ is a familiar
feature in QFT that can be treated in the framework of renormalization. This
physical divergence is not our focus here. In contrast, the $F_{S}^{\rm div}$
is infinite for any $d\neq 1$. This behavior is not the one of a meromorphic
function in $d$, and should thus draw our attention.

In Milton_Sphere  it was nicely observed that for $d<1$ the
$\sum^{\infty}_{h=0}c_{h}$ series vanishes identically. A proposal was then
made to remove the $F^{\rm div}$ term using an analytical continuation of $d$
to the $d<1$ region. In summary the argument amounts to stating that since in
this region one has $\sum^{\infty}_{h=0}c_{h}=0$ identically, any term
constant in $h$ arising from the integral is irrelevant since it is multiplied
by zero and can thus be subtracted. Notice that the proposed argument is
different from usual dimensional regularization which simply turns the
quantity of interest into a meromorphic function of $d$.

An issue remains, however, as $\sum^{\infty}_{h=0}c_{h}=0$ in the $d<1$ region
only guarantees that the divergent term $F^{\rm div}$ takes the indefinite
form “$0\times\infty$”. As a result, even if one requires the
$\sum^{\infty}_{h=0}c_{h}$ sum to vanish for any $d>1$ by analytical
continuation of zero, the proposed method remains inconclusive in the sense
that $0\times\infty$ is undefined. For this reason, we will see that the use
of our formalism (namely the quantum work on a Dirichlet shell Eq. (42)
together with matter conservation) allows one to bypass these ambiguities and
make the result finite.

###  The Quantum Work on the Dirichlet Sphere

We now proceed with our calculation of the quantum work on the Dirichlet
sphere with radial deformation. We first define the source and the
deformation. We consider a source with the geometry of a spherical shell of
width $\eta$ and with a finite number density $n$ in the $\eta\to 0$ limit,

$J_{\eta}({\bm{x}})=\frac{n_{\lambda}}{\eta}{\bm{1}}_{a-\frac{\eta}{2}<r<a+\frac{\eta}{2}}({\bm{x}})\,$
(54)

The propagator in the presence of this source is denoted by
$\Delta_{S}(x,x^{\prime})$. The boundary of the shell for small $\eta$ is
identified with $\partial S_{\eta\to 0}=S_{\rm in}\cup S_{\rm out}$ where
$S_{\rm in,out}$ are the $d-1$-spheres with respective radii $r=a_{-},a_{+}$.
The deformation of the source is parametrized by $\lambda$ and changes the
sphere radius such that $a_{\lambda+d\lambda}=a_{\lambda}+Ld\lambda$.
Equivalently, in terms of the support function, the deformation is
$l_{\lambda+d\lambda}(r)=l_{\lambda}(r-Ld\lambda)$. The deformation flow
vector is thus ${\bf L}=L{\bm{e}}_{r}$. Using the conservation equation Eq.
(9) we can easily derive the variation of density corresponding to such a
deformation. We find

$\partial_{\lambda}n=-\frac{d-1}{a}nL\,.$ (55)

We can now compute the quantum work. Since we are interested in a sphere we
can readily use the general formula for the quantum work on a thin shell Eq.
(42). In the Dirichlet limit the quantum work reads

$\displaystyle W_{S_{d-1}}=$
$\displaystyle-\frac{1}{2}A_{d-1}L\left[\partial_{r^{\prime}}\partial_{r^{\prime\prime}}\Delta_{S}(r^{\prime},t;r^{\prime\prime},t)|_{r^{\prime}=r^{\prime\prime}\in
S_{\rm
out}}-\partial_{r^{\prime}}\partial_{r^{\prime\prime}}\Delta_{S}(r^{\prime},t;r^{\prime\prime},t)|_{r^{\prime}=r^{\prime\prime}\in
S_{\rm in}}\right]$
$\displaystyle-\frac{1}{2\Lambda}\int_{S}d^{d}{\bm{x}}\Delta_{S}(x,x)\partial_{\lambda}n_{\lambda}\Big{|}_{n\to\infty}\,.$
(56)

The first term in Eq. (56) matches precisely the quantity computed in
Milton_Sphere . Namely we find

$\displaystyle W_{S_{d-1}}=L\left(F^{\rm
fin}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}+F^{\rm
div}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}\right)-\frac{1}{2\Lambda}\int_{S}d\sigma({\bm{x}})\Delta_{S}(x,x)\partial_{\lambda}n_{\lambda}\Big{|}_{n\to\infty}\,$
(57)

where the components of the force are defined in Eqs. (5.2).

The remaining task is to evaluate the last term in Eq. (57), which encodes the
variation of density. To this end we first have to evaluate the propagator in
the presence of the sphere with finite density. This is done by recomputing
the propagator in Milton_Sphere , replacing the two Dirichlet boundary
conditions on $S$ by two boundary conditions obtained from integrating the EOM
on the shell enclosing $r=a$ and using the divergence theorem. Introducing the
Fourier transform in time
$\Delta(x,x)=\int\frac{d\omega}{2\pi}\Delta_{\omega}({\bm{x}},{\bm{x}})$, we
find for the propagator at coinciding points

$\Delta_{\omega}(a,a)\overset{{\rm
large}\,n}{\to}i\frac{\Lambda}{na^{d-1}}\sum^{\infty}_{h=0}\frac{(h-1+\frac{d}{2})\Gamma(h+d-2))}{2^{d-2}\pi^{\frac{d-1}{2}}h!\Gamma(\frac{d-1}{2})}=i\frac{\Lambda}{na^{d-1}}\sum^{\infty}_{h=0}4\pi
c_{h}$ (58)

with $c_{h}$ defined in Eq. (53). Using the variation of density dictated by
matter conservation Eq. (55) we then obtain

$\frac{1}{2\Lambda}\int_{S}d\sigma({\bm{x}})\Delta_{S}(x,x)\partial_{\lambda}n_{\lambda}\Big{|}_{n\to\infty}=i\frac{1-d}{a^{d}}L\sum^{\infty}_{h=0}c_{h}\int^{\infty}_{-\infty}d\omega=LF^{\rm
div}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}\,.$ (59)

We see that this contribution from the variation of density exactly cancels
the divergent piece in Eq. (57). It follows that our final result for the
quantum work on the Dirichlet sphere amounts to the finite part of the result
from Milton_Sphere , namely

$W_{S_{d-1}}=LF^{\rm fin}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}\,.$ (60)

The fact that the term from the variation of density cancels the $F^{\rm
div}_{S_{d-1},[\color[rgb]{0,0,1}{26}]}$ divergence upon requirement of matter
conservation is non trivial. This cancellation provides a check of our
expression for the quantum work on a thin shell Eq. (42) and illustrates how
the finiteness of the quantum work can manifest itself concretely.

##  Planar Geometry

In this section we evaluate the quantum force in simple planar geometries.
While the geometric setup in itself is very well-known, the exact results for
the specific quantum force considered here in the case of a massive scalar
have not been presented elsewhere. This detailed calculation serves to
illustrate how the quantum work remains finite as a result of matter
conservation. It also exhibits the transition between the scalar Casimir and
Casimir-Polder regimes. The plate-point result obtained here is also key for
the search for new particles via atom interferometry that is presented in
section 7.

###  Force Between two Plates

We focus on the classic Casimir setup with two plates facing each other and
separated by a distance $\ell$ along the $z$ axis. The deformation we consider
amounts to a variation of $\ell$. We compute the force induced by a massive
scalar field with bilinear coupling to the constituents of the plates.

The quantum field $\Phi$ is described by the Lagrangian

${\cal
L}=\frac{1}{2}(\partial_{\mu}\Phi)^{2}-\frac{m^{2}}{2}\Phi^{2}-\frac{1}{2\Lambda^{2}}\Phi^{2}J({\bf
x})\,.$ (61)

In this application, it is convenient to define $J$ to be the mass density
distribution of the sources. In this setting there are five regions along $z$,
with the plates supported on region $1$ and $3$. The source $J$ is defined as

$J(z)=\rho_{1}\Theta(-\bar{z}_{\infty}<z<0)+\rho_{3}\Theta(\ell<z<z_{\infty})$
(62)

The width of the plates is taken to be much larger than the separation,
$z_{\infty},\bar{z}_{\infty}\gg\ell$. The fact that the plates actually end
instead of continuing to infinity i.e.
$|z_{\infty}|,|\bar{z}_{\infty}|<\infty$ is crucial in order to ensure matter
conservation, and therefore that the quantum work is finite as dictated by Eq.
(23). The effective mass can be written as

$\displaystyle
m^{2}(z)=m_{\infty}^{2}\Theta(z<-\bar{z}_{\infty})+m_{1}^{2}\Theta(-\bar{z}_{\infty}<z<0)+m_{2}^{2}\Theta(0<z<\ell)$
$\displaystyle+m_{3}^{2}\Theta(\ell<z<z_{\infty})+m_{\infty}^{2}\Theta(z>z_{\infty})\,,$
(63)

where

$m_{\infty,2}^{2}=m^{2}$ (64)

and

$m_{1,3}^{2}=\frac{\rho_{1,3}}{\Lambda^{2}}+m^{2}\,.$ (65)

In Eq. (61) the boundary operator amounts to

${\cal B}_{m}(\Phi)=\frac{\Phi^{2}}{2\Lambda^{2}}\,.$ (66)

Derivative operators such as ${\cal
B}(\Phi)=\frac{(\partial\Phi)^{2}}{2\Lambda^{4}}$ could be treated along the
same lines.

The deformation of the source that we consider amounts to shifting the right
plate (i.e. region 3, the second term in Eq. (62). This corresponds to an
infinitesimal shift of $\ell$ and $z_{\infty}$,
$\ell_{\lambda+d\lambda}=\ell_{\lambda}+Ld\lambda$,
$z_{\infty,\lambda+d\lambda}=z_{\infty,\lambda}+Ld\lambda$. Notice that the
left plate is left untouched, hence $\bar{z}_{\infty}$ does not vary.
Equivalently, in terms of the support function of the right plate, this is
described by $l_{\lambda+d\lambda}(z)=l_{\lambda}(z-Ld\lambda)$. The geometry
is summarized as

Since the source moves rigidly, the formula for the quantum work Eq. (46)
applies and a quantum force can be defined out from the quantum work, $W^{\rm
1-loop}=LF_{\rm quant}$. In the following we determine $F_{\rm quant}$.

#### Propagator

The Feynman propagator in position-momentum space, defined by

$\Delta(x,x^{\prime})=\int\frac{d^{3}p}{(2\pi)^{3}}e^{ip^{\alpha}(x-x^{\prime})_{\alpha}}\Delta_{p}(z,z^{\prime})$
(67)

with $(p^{\alpha},z)$, $\alpha=(0,1,2)$, has been calculated in the presence
of a piece-wise constant mass in Brax:2018grq . Defining the $z$-dependent
momentum with the Feynman $\epsilon$-prescription

$\omega(z)=\sqrt{(p_{0})^{2}-(p_{1})^{2}-(p_{2})^{2}+i\epsilon-m^{2}(z)},$
(68)

the homogeneous equation of motion becomes

$(\partial_{z}^{2}+\omega^{2}(z))\Phi(z)=0$ (69)

whose solutions in a given region $i$ are simply $e^{\pm i\omega_{i}z}$. The
solution everywhere can be found by continuity of the solution and its
derivative at each of the interfaces. The propagator is obtained by solving
the equations of motion in the five regions and matching them at the boundary.
Details can be found in the appendix of Brax:2018grq . For the calculation of
the quantum force, we will only need to evaluate the propagator at coinciding
points on the two boundaries of the right-hand plate, $z=\ell$ and
$z=z_{\infty}$.

#### Quantum Force

The deformation of the source term is found to be

$\partial_{\lambda}J=-(\rho_{3}-\rho_{2})L\left(\delta(z-\ell)-\delta(z-z_{\infty})\right)\,.$
(70)

Putting this variation back into the definition of the quantum work, we can
factor out the $L$ term and obtain the quantum force as defined in Eq. (46).
As a result the quantum force is given by

$\displaystyle{F}_{\rm quant}$
$\displaystyle=\frac{1}{2}(m_{3}^{2}-m_{2}^{2})\int
d^{2}{\bm{x}}_{\parallel}(\Delta(x^{\alpha},\ell;x^{\alpha},\ell)-\Delta(x^{\alpha},z_{\infty};x^{\alpha},z_{\infty}))$
$\displaystyle=\frac{1}{2}(m_{3}^{2}-m_{2}^{2})\int
d^{2}{\bm{x}}_{\parallel}\int\frac{d^{3}p}{(2\pi)^{3}}(\Delta_{p}(\ell,\ell)-\Delta_{p}(z_{\infty},z_{\infty}))$
(71)

with ${\bm{x}}_{\parallel}=(x_{1},x_{2})$. The cancellation between the
divergent parts of the two propagators at coinciding points is evident in
Eq.(71), using the fact that the divergence is location-independent. In the
second line we have introduced the propagator in position-momentum space. Here
we have explicitly

$\Delta_{p}(\ell,\ell)-\Delta_{p}(z_{\infty},z_{\infty})=\frac{(\omega_{1}+\omega_{2})+e^{2i\ell\omega_{2}}(\omega_{2}-\omega_{1})}{(\omega_{1}+\omega_{2})(\omega_{2}+\omega_{3})-e^{2i\ell\omega_{2}}(\omega_{2}-\omega_{1})(\omega_{2}-\omega_{3})}-\frac{1}{\omega_{2}+\omega_{3}}\,.$
(72)

with $\omega_{i}=\sqrt{(p_{\alpha})^{2}+i\epsilon-m^{2}_{i}}$.

The surface integral $\int d^{2}{\bm{x}}_{\parallel}=S$ is factored out hence
defining a pressure. The final expression for the quantum pressure between the
two plates (i.e. regions $1$ and $3$) is then

$\frac{F_{\rm
quant}}{S}=\int_{0}^{\infty}\frac{dkk^{2}}{2\pi^{2}}\frac{\gamma_{2}(\gamma_{2}-\gamma_{1})(\gamma_{2}-\gamma_{3})}{(\gamma_{2}-\gamma_{1})(\gamma_{2}-\gamma_{3})-e^{2\ell\gamma_{2}}(\gamma_{1}+\gamma_{2})(\gamma_{2}+\gamma_{3})}\,\,$
(73)

after Wick’s rotation with $\omega_{i}=i\gamma_{i}=i\sqrt{k^{2}+m^{2}_{i}}$.
This is the general expression of the plate-plate quantum pressure in the
presence of a scalar coupled quadratically to matter.

In the limit of large density-induced effective mass
$m_{1,3}\rightarrow\infty$, the general expression Eq. (73) becomes

$\frac{F_{\rm
quant}}{S}=\int_{0}^{\infty}\frac{dkk^{2}}{2\pi^{2}}\frac{\gamma_{2}}{1-e^{2\ell\gamma_{2}}}\,.$
(74)

In this limit the effective mass in the plates become so large that the field
obeys Dirichlet boundary conditions. Accordingly, Eq. (74) matches exactly the
Casimir pressure from a massive scalar with Dirichlet boundary conditions. For
a massless scalar the integral can be explicitly performed and we retrieve

$\frac{F_{\rm quant}}{S}=-\frac{\pi^{2}}{480\ell^{4}}\,.$ (75)

This is the classic Casimir pressure for a massless scalar field.

In the limit of small density-induced effective mass defined as
$(m^{2}_{1,3}-m_{2}^{2})/m^{2}_{2}\ll 1$, i.e. when the contribution of the
density to the effective mass is small with respect to the fundamental mass,
the pressure becomes

$\frac{F_{\rm
quant}}{S}=-(m^{2}_{1}-m_{2}^{2})(m^{2}_{3}-m_{2}^{2})\int_{0}^{\infty}\frac{dkk^{2}}{2\pi^{2}}\frac{e^{-2\ell\gamma_{2}}}{16(\gamma_{2})^{3}}\,.$
(76)

We checked that this corresponds exactly to the Casimir-Polder force
integrated over regions 1 and 3. We will see how this calculation can be
cross-checked in the next example.

In summary, we have verified that both the scalar Casimir and scalar Casimir-
Polder pressures are recovered as limits of the more general expression of the
plate-plate quantum pressure Eq. (73). Qualitative considerations are given in
the next example.

#### On Finiteness

In Eq. (71) we have observed the cancellation between the divergent piece of
$\Delta_{p}(\ell,\ell)$ and $\Delta_{p}(z_{\infty},z_{\infty})$ in the
integrand. This cancellation makes the expression for the force finite. To
illustrate how the presence of the $\Delta_{p}(z_{\infty},z_{\infty})$ term is
tied to matter conservation, we consider the following counterexample.

Let us imagine that we ignored the displacement of the outer edge
($z=z_{\infty}$) of the plate. This would imply that the matter of the plate
is not conserved since the plate’s width would change while the density
remains constant. In such a setup, the result would be the same as Eq. (71)
but without the $\Delta_{p}(z_{\infty},z_{\infty})$ contribution. Therefore
the expression would be infinite. This simple counterexample illustrates that,
when dropping the requirement of matter conservation, the prediction of the
force becomes infinite i.e. property (23) does not hold.

###  Force Between a Plate and a Point Source

As a third application of our formalism, we focus on the interaction between a
point particle and a plate. As before we assume the Lagrangian

${\cal
L}=\frac{1}{2}(\partial_{\mu}\Phi)^{2}-\frac{m^{2}}{2}\Phi^{2}-\frac{1}{2\Lambda^{2}}\Phi^{2}\,\,J({\bf
x})\,.$ (77)

The plate is supported on $z<0$ and has mass density $\rho_{1}$. The source is
taken to be

$J({\bf x})=\rho_{1}\Theta(-z_{\infty}<z<0)+m_{N}{\delta^{2}}({\bf
x_{\parallel}})\delta(z-\ell)\,.$ (78)

The mass of the point particle is $m_{N}$. We define the effective mass of
$\Phi$ in the plate as

$m^{2}_{1}=\frac{\rho_{1}}{\Lambda^{2}}+m^{2}\,$ (79)

which depends on the coupling to matter, $\frac{1}{\Lambda^{2}}$. The
effective mass of $\Phi$ is then piecewise constant,

$m^{2}(z)=m_{\infty}^{2}\Theta(z<-z_{\infty})+m_{1}^{2}\Theta(-z_{\infty}<z<0)+m_{2}^{2}\Theta(z>0)$
(80)

with $m_{\infty,2}=m$.

The deformation we consider is an infinitesimal shift of the point particle
position $\ell$, $\ell_{\lambda+d\lambda}=\ell_{\lambda}+Ld\lambda$. The
geometry is summarized as

#### Propagator

The Feynman propagator in position-momentum space $(p^{\alpha},z)$,
$\alpha=(0,1,2)$ in the presence of a piecewise constant mass has been
calculated in Brax:2018grq . The effect of the point source on the propagation
is negligible. 141414This can be checked by evaluating the dressed propagator
in energy-position space $(p_{0},{\bm{x}})$. In the resummed propagator, the
effect of the insertion is small within the EFT validity range, leaving the
term with one point source insertion as the main non-vanishing contribution to
the quantum work.  For the present calculation, if the frame is chosen such
that the deformation changes the position of the point source and not of the
plate, one can safely ignore the region at $-z_{\infty}$ and thus consider the
propagator over two regions, with
$m^{2}(z)=m_{1}^{2}\Theta(z<z_{12})+m_{2}^{2}\Theta(z>z_{12})$. The propagator
is found to be

$\Delta_{p}(z,z^{\prime})=\begin{cases}\frac{e^{i\omega_{2}(z_{>}-z_{<})}}{2\omega_{2}}E_{2}(z_{<})\quad\quad
z_{12}<z_{<}\\\
\frac{e^{i(\omega_{2}(z_{>}-z_{12})-\omega_{1}(z_{<}-z_{12})}}{\omega_{1}+\omega_{2}}\quad
z_{<}<z_{12}<z_{>}\\\
\frac{e^{i\omega_{1}(z_{>}-z_{<})}}{2\omega_{1}}E_{1}(z_{>})\quad\quad
z_{>}<z_{12}\end{cases}\,$ (81)

where

$\displaystyle
E_{1}(z)=1+e^{i2(z_{12}-z)\omega_{1}}\frac{\omega_{1}-\omega_{2}}{\omega_{1}+\omega_{2}}$
$\displaystyle
E_{2}(z)=1+e^{i2(z-z_{12})\omega_{2}}\frac{\omega_{2}-\omega_{1}}{\omega_{1}+\omega_{2}}\,.$

We have defined $z_{<}=\min(z,z^{\prime})$ and $z_{>}=\max(z,z^{\prime})$. We
have introduced $\omega_{i}=\sqrt{(p_{\alpha})^{2}-m^{2}_{i}+i\epsilon}$. The
$\epsilon$ prescription guarantees that the propagators decay at infinity. The
$E_{1}$, $E_{2}$ functions essentially describe how the presence of the
boundary affects the propagator with both endpoints in the same region. When
the boundary $z_{12}$ is rejected to infinity, one recovers the usual
expression for a fully homogeneous space.

#### Quantum Force

The deformation of the source is

$\partial_{\lambda}J=-m_{N}L{\delta^{2}}({\bf
x_{\parallel}})\partial_{z}\delta(z-\ell)\,.$ (83)

Using this expression into the quantum work, one obtains after one integration
by parts the quantum force

$\displaystyle{F}_{\rm quant}$
$\displaystyle=-\frac{1}{2}\frac{m_{N}}{\Lambda^{2}}\partial_{z}\Delta_{J}(x^{\alpha},z;x^{\alpha},z)|_{z\to\ell}$
(84)
$\displaystyle=-\frac{1}{2}\frac{m_{N}}{\Lambda^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\partial_{z}\Delta_{J}({\bm{k}};z,z)|_{z\to\ell}$

In the last line we have introduced the position-momentum space propagator.

Using Eq. (81) with $z_{12}=0$ since the plate is placed at the origin, we
have

$\displaystyle F_{\rm quant}=$
$\displaystyle-\frac{m_{N}}{\Lambda^{2}}\frac{1}{2}\int\frac{d^{3}{p}}{(2\pi)^{3}}\frac{1}{2\omega_{2}}\partial_{z}E_{2}(z)|_{z=\ell}$
$\displaystyle=$
$\displaystyle-(m_{1}^{2}-m_{2}^{2})\frac{m_{N}}{\Lambda^{2}}\frac{1}{4\pi^{2}}\int
dk\,k^{2}\frac{e^{-2\ell\gamma_{2}}}{(\gamma_{1}+\gamma_{2})^{2}}\,$ (85)

where we have performed a Wick rotation and introduced
$\gamma_{i}=\sqrt{k^{2}+m^{2}_{i}}$. This is the general expression of the
plate-point quantum force in the presence of a scalar coupled quadratically to
matter. Diagrammatically, the particle will interact with the plate via loops
starting at the point particle, going into the plate and coming back to the
point particle.

In the limit of large density-induced effective mass $m_{1}\to\infty$, the
force takes the form

$F_{\rm quant}=-\frac{m_{N}}{\Lambda^{2}}\frac{1}{4\pi^{2}}\int
dk\,k^{2}e^{-2\ell\gamma_{2}}\,.$ (86)

This is the limit for which the density is so large that the field is repelled
by the plate and the field obeys Dirichlet boundary conditions at the boundary
of the plate. We refer to this case as the Casimir limit. In the massless case
we obtain

$F_{\rm quant}=-\frac{m_{N}}{16\pi^{2}\Lambda^{2}}\frac{1}{\ell^{3}}\,.$ (87)

In the limit of small density-induced effective mass we expand in
$(m^{2}_{1}-m_{2}^{2})\ll m_{i}^{2}$ and obtain

$F_{\rm
quant}=-(m_{1}^{2}-m_{2}^{2})\frac{m_{N}}{\Lambda^{2}}\frac{1}{4\pi^{2}}\int
dk\,k^{2}e^{-2\ell\gamma_{2}}\frac{1}{4\gamma_{2}^{2}}$ (88)

In the massless case we obtain

$F_{\rm
quant}=-\frac{m_{1}^{2}\,m_{N}}{16\pi^{2}\Lambda^{2}}\frac{1}{\ell}\,.$ (89)

This is the Casimir-Polder limit. As a cross check, in the Appendix we show
that this limit is exactly recovered by integrating the point-point Casimir-
Polder potential over the extended source.

In summary, we have verified that both the scalar Casimir and scalar Casimir-
Polder pressures are recovered as limits of the more general expression of the
plate-point quantum pressure Eq. (73). The two limits of the massless formula
Eqs. (87) and (89) make transparent that there is a transition between the two
regimes as a function of the separation $\ell$. Namely, the Casimir regime
occurs for $\ell\gg m^{-1}_{1}$ while the Casimir-Polder regime occurs for
$\ell\ll m^{-1}_{1}$, with $m^{2}_{1}=\frac{\rho_{1}}{\Lambda^{2}}$. One way
to think about this phenomenon is that, while at large distance the plate
behaves as a mirror, leading to a Casimir force, at short distance the quantum
fluctuations start penetrating the mirror. As a result the behaviour of the
Casimir force gets softened into the the Casimir-Polder one at short distance.
This behaviour is confirmed numerically.

##  Bounding Quantum Forces with Atom Interferometry

###  The setting

In this section we calculate a somewhat more evolved observable in the context
of atom (or neutron) interferometry experiments. Namely we compute the matter-
wave phase shift measurable in atom interferometers, which is generated in the
presence of a quantum force between the atom and a neighbouring plate. The
calculation uses the results from section 6.2.

Atom interferometry has been used to test gravity (see e.g. KasChu ; KasChu2 ;
1999Natur.400..849P ; 2006PhyS…74C..15M ) and to search for classical fifth
forces, often in the context of dark energy-motivated models Burrage:2014oza ;
Hamilton:2015zga ; Jaffe:2016fsh ; Sabulsky:2018jma . Interferometry has never
been used to search for quantum forces such as the one modelled by the scalar
field used throughout this paper. In this section we thus i) compute the phase
shift induced by a quantum force and ii) demonstrate that interferometry is a
competitive method to search for a dark field bilinearly coupled to matter.

Figure 4:  Spacetime paths followed by the atoms in the interferometer. The
path in the presence of the force ($\Gamma$, plain lines) is deformed with
respect to the path in the absence of the force ($\Gamma_{0}$, dotted lines).
The pictures indicate the asymptotic behaviors (Casimir-Polder and Casimir) of
the force between the atom and the plate.

Our focus is on a setup which is essentially an adaptation of the simplest
“Kasevich-Chu” experiment (see KasChu ; KasChu2 ; 1999Natur.400..849P , and
e.g. 2006PhyS…74C..15M  for a review). We describe briefly the setup and refer
to the above references and to e.g. Storey:1994oka  for further details. Atom
or neutron interferometry uses the difference of phase of two coherent
wavepackets following two different spacetime paths as shown in Fig. 4. The
two paths amount to two broken worldlines, $ACB$ and $ADB$, with a change of
direction at $C$ and $D$ respectively, and with same endpoint $B$. The changes
of velocity of the wavepackets are induced by laser pulses.

We assume that the interferometry experiment is carried out along the $z$ axis
over a plate located at $z=0$. We assume that the setup is oriented
horizontally at the surface of the Earth since we are not interested in
measuring the strength of the gravity field. The phase shift is caused by the
force between the atom and the plate. Experimentally, the plate might be a
large ball whose radius is assumed to be much larger than the length of
$\Gamma$. In this setup the potential $V(z)$ is the one for the plane-point
quantum force derived in section 6.2.

Using the WKB approximation, the leading phase shift between the two paths
induced by a potential $V(z)$ is given by (see Storey:1994oka )

$\delta\phi=-\int_{\rm\Gamma_{0}}dt\ V(z(t))\ $ (90)

where $\Gamma_{0}=A_{0}C_{0}B_{0}D_{0}$ denotes the unperturbed closed path
and $z(t)$ the corresponding classical trajectory of the wavepackets. Along
each segment of $\Gamma_{0}$, the trajectories are straight,

$z(t)=z_{i}+v_{i}(t-t_{i})\,.$ (91)

The velocities along each segment of the path can in general differ. In the
present setup the time spacing between each pulse is $T$ and the velocities on
the segments of $\Gamma_{0}$ are $v_{A_{0}C_{0}}=v_{D_{0}B_{0}}=v$,
$v_{C_{0}B_{0}}=v_{A_{0}D_{0}}=v^{\prime}$, as shown in Fig. 4.

###  Computing the Phase Shift

We consider the scalar model defined in Eq. (6.2), where the fundamental mass
of the field is denoted by $m$ and the mass density of the plate is denoted
$\rho$. The effective mass of the $\Phi$ field for $z<0$ and $z>0$ is given by
$m_{1}^{2}=\frac{\rho}{\Lambda^{2}}+m^{2}$ and $m_{2}^{2}=m^{2}$. The plane-
point force (85) derives from a potential $F(\ell)=-\frac{\partial
V(\ell)}{\partial\ell}$ given by

$V(\ell)=-\frac{m_{N}\rho}{\Lambda^{4}}\frac{1}{4\pi^{2}}\int
dk\,k^{2}\frac{e^{-2\ell\,\gamma_{2}}}{2\gamma_{2}(\gamma_{1}+\gamma_{2})^{2}}\,$
(92)

where $\ell$ is the distance from the particle to the plate, here taken to be
along the $z$ direction. Let us consider one segment of the path $\Gamma_{0}$
where the particle evolves between times $t_{i}$ and $t_{j}$, where $i,j$
denote the endpoints of the segment. The associated phase shift is

$\displaystyle\delta\phi_{ij}$
$\displaystyle=\int_{t_{i}}^{t_{j}}dt\,\,\frac{m_{N}\rho}{\Lambda^{4}}\frac{1}{4\pi^{2}}\int
dk\,k^{2}\frac{e^{-2(z_{i}+v(t-t_{i}))\,\gamma_{2}}}{2\gamma_{2}(\gamma_{1}+\gamma_{2})^{2}}\,$
(93)
$\displaystyle=\frac{1}{v}\frac{m_{N}\rho}{\Lambda^{4}}\frac{1}{4\pi^{2}}\int
dk\,k^{2}\frac{e^{-2z_{i}\,\gamma_{2}}-e^{-2z_{j}\,\gamma_{2}}}{4\gamma^{2}_{2}(\gamma_{1}+\gamma_{2})^{2}}\,$
(94)

This is an exact result following from the exact expression Eq. (92). If one
instead used the approximate expressions for either the Casimir or the
Casimir-Polder regime to compute the phase shift, one would need to ensure
that all distances involved are respectively much bigger or much smaller than
the Compton wavelength in the plate, $m_{1}^{-1}$ (see section 6). Since in
the interferometry experiment the separation between the point source and the
plane varies, using either the Casimir or the Casimir-Polder approximation may
potentially give an erroneous result. The phase shift calculation provides a
concrete example of prediction for which the use of the exact result Eq. (92)
is in general mandatory.

###  Limits

The phase shift given by Eq. (94) can be further evaluated in some limits when
taking $m=0$. All approximations below have been checked numerically.

####  $m_{1}z_{i}\gg 1$, $m_{1}z_{j}\gg 1$

This case amounts to computing the phase shift in the Casimir regime. It is
obtained by approximating $\gamma_{1}+\gamma_{2}\approx m_{1}$ in the
denominator of Eq. (94). We obtain

$\displaystyle\delta\phi_{ij}$
$\displaystyle=\int^{t_{j}}_{t_{i}}dt\frac{m_{N}}{32\pi^{2}\Lambda^{2}z^{2}}$
(95)
$\displaystyle=\frac{m_{N}(z_{j}-z_{i})}{32\pi^{2}\Lambda^{2}\,v\,z_{i}z_{j}}\,=\frac{m_{N}(t_{j}-t_{i})}{32\pi^{2}\Lambda^{2}z_{i}z_{j}}\,.$
(96)

The overall factor $\Lambda^{-2}$ is characteristic of the Casimir regime.

####  $m_{1}z_{i}\ll 1$, $m_{1}z_{j}\ll 1$

This case would amount to computing the phase shift in the Casimir-Polder
regime. However, approximating $\gamma_{1}+\gamma_{2}\approx 2k$ in the
denominator gives a divergent result, therefore we have to go beyond the
Casimir-Polder approximation to obtain a finite expression. This is possible
only in our formalism: the small but nonzero effective mass in the plate
regularizes the divergence. It is obtained by taking
$\gamma_{1}+\gamma_{2}\approx 2k+\frac{m_{1}^{2}}{2k}$ in the denominator. We
obtain

$\delta\phi_{ij}=\frac{m_{N}(t_{j}-t_{i})\rho}{128\pi^{2}\Lambda^{4}}\left(\frac{1}{2}-\gamma+\frac{z_{j}\log\left(\frac{\Lambda}{z_{j}\sqrt{\rho}}\right)-z_{i}\log\left(\frac{\Lambda}{z_{i}\sqrt{\rho}}\right)}{z_{j}-z_{i}}\right)$
(97)

The overall $\Lambda^{-4}$ is characteristic of the Casimir-Polder regime.

####  $m_{1}z_{i}\ll 1$, $m_{1}z_{j}\gg 1$

In this nontrivial case we do not find an accurate approximation, only
expressions valid up to $O(1)$ uncertainty, which are nevertheless very
useful. Taking $\gamma_{1}+\gamma_{2}\approx m_{1}$ in the denominator gives

$\delta\phi_{ij}=\frac{m_{N}(t_{j}-t_{i})\rho}{16\pi^{2}z_{j}\Lambda^{3}}\,.$
(98)

Taking $\gamma_{1}+\gamma_{2}\approx 2k+m_{1}^{2}/2k$ in the denominator gives

$\delta\phi_{ij}=\frac{m_{N}(t_{j}-t_{i})\rho}{128\pi z_{j}\Lambda^{3}}\,.$
(99)

The exact result lies in between these two expressions. We can see that the
overall scaling for this case is $\Lambda^{-3}$.

###  Sensitivity to New Particles

Figure 5:  Bounds on the quantum force induced by a scalar field bilinearly
coupled to nucleons. Red lines correspond to the sensitivity from atom
interferometry using $a_{\rm ex}=10^{-8}$m2s-1. Yellow regions correspond to
bounds from other experiments and match Ref. Brax:2017xho . Those were
computed in the Casimir-Polder approximation, except the bound from bouncing
neutrons.

The phase shift over the closed path $\Gamma_{0}$ can be written as
$\Delta\Phi=a\kappa T^{2}$, where $\kappa=M(v-v^{\prime})$ is the transferred
momentum from the laser pulses and $T$ is the period of pulses (i.e. $2T$ is
the total time between splitting ($A$) and recombination ($B$) of the
wavepackets, see Fig. 4). The coefficient $a$ has dimension of an acceleration
and can be taken as the figure of merit for the precision of the atom
interferometer. The sensitivity of existing experiments can typically reach

$a_{\rm ex}\sim 10^{-9}g=10^{-8}\,{\rm m}^{2}{\rm s}^{-1}$ (100)

(see e.g. 1999Natur.400..849P ), that we use as our reference value.

The predicted value of $a$ given by the quantum force in our model is easily
obtained in the different regimes discussed above. Comparing it to $a_{\rm
ex}$ we obtain an experimental bound on the parameters of the $\Phi$ field,
i.e. an exclusion region in the $(m,\Lambda)$ plane. For better comparison to
other experimental constraints we introduce
$\bar{\Lambda}=\frac{\Lambda^{2}}{m_{N}}$, which then matches the convention
in Brax:2017xho .

The result is shown in Fig. 5. In the regime relevant for the presented
sensitivities, the phase shift is dominated by the contributions from the
segments near the plate, $\delta\phi\approx\delta\phi_{AC}-\delta\phi_{AD}$.
Moreover these contributions are typically in the nontrivial regime of section
7.3.3, which depends only on $z_{C}\approx z_{D}\equiv\bar{z}$, which we refer
to as the length of the arm of the interfometer. The sensitivity greatly
increases when $\bar{z}$ decreases, reflecting the fact that the quantum force
quickly increases at short distance. The universal diagonal line correspond to
the suppression of the quantum force due to the short Compton wavelength of
the $\Phi$ field. A different sensitivity in $a$ would amount to a change in
$\bar{z}$. All the lines are in the nontrivial regime of section 7.3.3,
because for the relevant values of $\Lambda$ the effective Compton wavelength
in the plate $m^{-1}_{1}$ is much smaller than $\bar{z}$.

Other experimental bounds are shown in Fig. 5 for comparison. 151515 In the
case of dark matter, searches via quantum forces are typically complementary
from direct detection searches. The former probe low DM masses while the
latter probe high DM masses. Fig. 2 in Fichet:2017bng  illustrates this
complementarity.  161616 Besides interferometry, another kind of experiment
carried out between a particle and a plane is the neutron bouncer, see e.g.
Nesvizhevsky:2007by ; Brax:2011hb ; Brax:2013cfa ; Jenke:2014yel ;
Cronenberg:2018qxf ; Brax:2017hna . The quantum levels of the neutrons in the
gravitational field of the Earth are probed, via e.g. Rabi oscillation
techniques, that constrain the difference between energy levels. The quantum
levels of the neutrons are classified by an integer $n$ and have the energies
$E_{n}=m_{N}gz_{0}\epsilon_{n}$ where $m_{N}$ is the neutron’s mass, $g$ the
acceleration of gravity on Earth and $z_{0}=(2m_{N}^{2}g)^{-1/3}$ is a
characteristic scale of the order of $6\,\mu$m. The number $-\epsilon_{n}$ are
the zeros of the Airy function ${\rm Ai}$ where the wave functions are
$\psi_{n}(z)\propto{\rm Ai}(\frac{z}{z_{0}}-\epsilon_{k})$ and $z$ is the
distance to the plate. The perturbation to the energy levels due to the
anomalous plate-neutron interaction is given by $\delta
E_{n}=\langle\psi_{n}|V(z)|\psi_{n}\rangle$ which is constrained by $|\delta
E_{3}-\delta E_{1}|\leq 10^{-14}{\rm eV}$ Cronenberg:2018qxf . The subsequent
bound obtained using the particle-plane force Eq. (85) is shown in Fig. 5.
This bound is subdominant with respect to the other bounds — also when using
the Casimir-Polder approximation Brax:2017hna .  These other bounds have been
so far computed only in the Casimir-Polder regime. In light of the present
work, we can see that, while this is exact for molecular bounds and neutron
scattering, the Casimir-Polder regime is in general only an approximation in
the presence of macroscopic bodies. Using the exact prediction would tend to
weaken the bound (see section 6).

We can thus conclude that atom interferometry turns out to be rather
competitive method to search for quantum dark forces, provided the
interferometer arms (or at least those near the plate, i.e. $AC$ and $AD$)
have length below $\bar{z}\sim 10$ cm. This is a reasonable length scale from
the experimental viewpoint, which already appears in recent experiments such
as the one of Hamilton:2015zga .

##  Conclusion

How does a classical body respond to an arbitrary deformation in the quantum
vacuum? This question can be tackled by introducing the notion of quantum work
$W$, an observable quantity which goes beyond the quantum force ${\bm{F}}$ and
reduces to $W={\bm{F}}\cdot{\bf L}$ in specific cases when the deformation
flow ${\bf L}$ is simple enough, e.g. when rigid bodies are displaced with
respect to each other. In this paper we have studied the quantum work induced
by a massive scalar field bilinearly coupled to macroscopic bodies made of
classical matter. Unlike for abstract sources, the number densities of such
bodies must satisfy the local conservation of matter. We have shown that the
prediction of the quantum work turns out to be finite — up to physical,
renormalizable divergences — upon requesting conservation of matter. This
result applies to any shape and geometry, either rigid or deformable. This is
shown both for a renormalizable — possibly strongly-coupled — theory, and in a
more general effective field theory setup allowing for higher derivative
interactions between the scalar and matter.

Our result about finiteness of the quantum work readily explains why the QFT
prediction of quantum forces sometimes feature seemingly “unremovable”
divergences for certain geometries. A key example is the quantum work felt by
a Dirichlet sphere under a radial deformation. The radial deformation flow is
not divergence-free and thus the sphere density must vary to ensure matter
conservation. Not taking this into account implies that matter of the sphere
is not conserved, thus the finiteness property is not ensured, and therefore
the expression of the quantum force can have a spurious divergence. We have
explicitly verified that taking into account the variation of the density
removes the spurious divergence in the case of the Dirichlet sphere.

When specializing to rigid bodies, the quantum work leads to a quantum force
that reduces to the scalar Casimir and Casimir-Polder forces as special
limits. There is a clear diagrammatic understanding of this interpolation. In
the short distance regime, the main contribution comes from the loop with only
one coupling to each body, which corresponds to Casimir-Polder. In the long
distance regime, loops with arbitrary number of insertions contribute, but
their resummation amounts to having a Dirichlet condition on the boundary of
the bodies, which corresponds to the traditional Casimir setup.

We have computed the quantum forces in plate-plate and plate-point geometries.
If, for example, the scalar has zero fundamental mass, in the plate-plate
geometry the force behaves as $\frac{1}{\ell^{4}}$ at long distance (Casimir)
and as $\frac{1}{\ell^{2}}$ at short distance (Casimir-Polder). For plate-
point geometry the force behaves as $\frac{1}{\ell^{3}}$ at long distance
(Casimir) and as $\frac{1}{\ell}$ at short distance (Casimir-Polder). For
nonzero fundamental mass the two limits can still exist, but the Casimir force
drops exponentially at separation $\ell\gg\frac{1}{2m}$.

These results have concrete applications for physical setups aimed at
searching for light dark particles. Such particles are ubiquitous in dark
matter models, dark energy models, and in extensions of the Standard Model.
Here we have briefly illustrated an application of the point-plane quantum
force for atom interferometers. In this type of experiment, a non-relativistic
atom with trajectory in the vicinity of a large sphere or a plane is sensitive
to the existence of a new force, which induces a phase shift that can be
computed in the WKB approximation. We have computed the atomic phase shift
induced by the scalar quantum force, and point out that the full result (as
opposed to a simple Casimir or Casimir-Polder approximation) is required to
obtain a sensible prediction. Using inputs from existing experiments, we show
that atom interferometry is likely to become a competitive method to search
for light dark particles bilinearly coupled to matter, provided that the
interferometer arms are shorter than $\sim 10$ cm, as summarized in Fig. 5.
This is a reasonable length scale from the experimental viewpoint, which
already appears in setups such as the one of Hamilton:2015zga .

In future work we will further investigate/revisit the constraints and
sensitivities of future experiments to macroscopic dark quantum forces with
Casimir-like behaviors.

## Acknowledgments

SF thanks Daniel Davies for a useful discussion. This work has been supported
by the São Paulo Research Foundation (FAPESP) under grants #2011/11973,
#2014/21477-2 and #2018/11721-4, by CAPES under grant #88887.194785, and by
the University of California, Riverside.

##  Appendix A Derivation of the Plate-Point Casimir-Polder Potential

The Casimir-Polder potential between a particle and a plate can be obtained by
direct calculation in the weak coupling regime (i.e. the limit of small
density in the plate). We start by computing the potential between two point
sources. The corresponding source term is

${\cal
L}\supset-\frac{1}{2}\Phi^{2}\left(\frac{m_{a}}{\Lambda^{2}}\delta^{3}({\bm{x}}-{\bm{x}}_{a})+\frac{m_{N}}{\Lambda^{2}}\delta^{3}({\bm{x}}-{\bm{x}}_{b})\right)\,.$
(101)

We remind that this is the non-relativistic approximation of the 4-point
interaction

${\cal
L}\supset-\frac{1}{2}\Phi^{2}\left(\frac{m_{a}}{\Lambda^{2}}\bar{\psi}_{a}\psi_{a}+\frac{m_{N}}{\Lambda^{2}}\bar{\psi}_{b}\psi_{b}\right)\,$
(102)

between the scalar and two fermion species. The scattering amplitude with two
insertions of two particle-anti particles pairs leads to a bubble diagram
which reads

$i{\cal
M}=-\frac{m_{N}m_{a}}{\Lambda^{4}}\,4m_{N}m_{a}\,\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{e^{i\omega_{2}|z_{1}-z_{2}|)}}{2\omega_{2}}\frac{e^{i\omega^{\prime}_{2}|z_{1}-z_{2}|)}}{2\omega^{\prime}_{2}}$
(103)

where $\omega_{2}=\sqrt{k^{2}-m_{2}^{2}}$,
$\omega^{\prime}_{2}=\sqrt{(k+p)^{2}-m_{2}^{2}}$. The factor of $\frac{1}{2}$
is a symmetry factor and the external fermions are such that their
nonrelativistic wavefunctions are normalised as $\bar{u}_{a}u_{a}=2m_{a},\
\bar{u}_{b}u_{b}=2m_{N}$. The non-relativistic scattering potential is given
by

$\displaystyle\tilde{V}(p,z_{1}-z_{2})=-\frac{\cal{M}}{4m_{a}m_{N}}$
$\displaystyle=-i\frac{m_{N}m_{a}}{\Lambda^{4}}\frac{1}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{e^{i\omega_{2}|z_{1}-z_{2}|)}}{2\omega_{2}}\frac{e^{i\omega_{2}^{\prime}|z_{1}-z_{2}|)}}{2\omega_{2}^{\prime}}\,.$

The spatial potential is then obtained from the Fourier transform of
$\tilde{V}$,

$V\left(\sqrt{(z_{1}-z_{2})^{2}+{\bm{x}}_{\parallel}^{2}}\right)=\int\frac{d^{2}{\bm{p}}_{\parallel}}{(2\pi)^{2}}\tilde{V}({\bm{p}}_{\parallel},z_{1},z_{2})e^{i{\bm{p}}_{\parallel}\cdot{\bm{x}}_{\parallel}}$
(105)

where ${\bm{x}}_{\parallel}=(x_{1},x_{2})$.

We now consider an ensemble of $N_{1}$ particle of the species $a$ in a volume
$V_{1}$ with a number density $n_{1}=\frac{N_{1}}{V_{1}}$. We average the
potential over the plate with a separation $\ell$ to the point particle as

$V(\ell)=n_{1}\int
d^{2}{\bm{x}}_{\parallel}\int_{-\infty}^{0}dz_{1}\int\frac{d^{2}{\bm{p}}_{\parallel}}{(2\pi)^{2}}\tilde{V}({\bm{p}}_{\parallel},z_{1},\ell)e^{i{\bm{p}}_{\parallel}\cdot{\bm{x}}_{\parallel}}.$
(106)

The transverse integrals simplify and the potential becomes simply

$V(\ell)=n_{1}\int_{-\infty}^{0}dz_{1}\tilde{V}(0,z_{1},\ell)=-n_{1}\frac{m_{a}m_{N}}{\Lambda^{4}}\,\int\frac{d^{3}k_{E}}{(2\pi)^{3}}\frac{e^{-2\gamma_{2}\ell}}{16(\gamma_{2})^{3}}$
(107)

after Wick’s rotation. Using $m_{1}^{2}-m_{2}^{2}=n_{1}m_{a}/\Lambda^{2}$, we
find

$V(\ell)=-(m_{1}^{2}-m_{2}^{2})\frac{m_{N}}{\Lambda^{2}}\,\int\frac{d^{3}k_{E}}{(2\pi)^{3}}\frac{e^{-2\gamma_{2}\ell}}{16(\gamma_{2})^{3}}\,.$
(108)

Finally the force is obtained by taking the derivative with respect to $\ell$

$F=-\partial_{\ell}V=-(m_{1}^{2}-m_{2}^{2})\frac{m_{N}}{\Lambda^{2}}\,\int\frac{d^{3}k_{E}}{(2\pi)^{3}}\frac{e^{-2\gamma_{2}\ell}}{8(\gamma_{2})^{2}}\,.$
(109)

This reproduces Eq. (88).

##  Appendix B On Divergences from the Heat Kernel Expansion

In this appendix we review how the heat kernel expansion allows for a proof of
the finiteness of the quantum work at one-loop level under the assumptions
that (i) the bodies are incompressible and (ii) the fluctuation is repelled
from the sources so that we can set boundary conditions at the source
surfaces, analogously to the Dirichlet limit described in section 4.4. We will
then see that the argument fails when condition (i) is dropped i.e.
considering a compressible body, which signals the need for an approach that
includes matter conservation.

We essentially review the exposition from bordag2009advances . We consider two
rigid bodies such that $J=J_{1}+J_{2}$ as in section 4. First of all, let us
rewrite the scalar action as

$S(\Phi)=S_{2}(\Phi)+S_{\rm int}(\Phi)$ (110)

where we have included the $J$-dependent contribution to the effective mass in
the quadratic part of the action. This action reads

$S_{2}(\Phi)=-\frac{1}{2}\int d^{4}x\Phi\Box_{J}\Phi$ (111)

where $\Box_{J}$ is the operator which includes the effective mass term. In
the renormalizable case, this is simply

$\Box_{J}=\Box+m^{2}+\frac{J}{\Lambda}\,.$ (112)

In the EFT case, higher order derivatives are present.

We assume weak coupling. When the sources are static, the quantum vacuum
energy reads

$E(J)=\frac{i}{T}\ln
Z(J)=-\frac{i}{2T}{\rm{Tr}}\ln\Box_{J}+\frac{i}{T}W_{2}(J)$ (113)

where the fist contribution is the one-loop contribution to the vacuum
diagrams, and $W_{2}(J)$ is the sum over all connected vacuum diagrams at two
loops and higher. Here we focus on the one-loop contribution, which is
independent of other possible interactions ( e.g. polynomial self-interactions
of $\Phi$). The one-loop piece can be evaluated using the heat kernel
$K_{J}=e^{-t\Box_{J}}$ and its trace as

${\rm Tr}\ln\Delta_{J}=-\int_{0}^{\infty}\frac{dt}{t}{\rm Tr}\,K_{J}(t)\,.$
(114)

See Vassilevich:2003xt  for a conceptual review on heat kernel methods. The
trace of the heat kernel has a $t\to 0$ expansion

${\rm Tr}\,K_{J}(t)=\frac{1}{(2\pi
t)^{3/2}}(a_{0}+a_{1/2}t^{1/2}+a_{1}t+a_{3/2}t^{3/2}+\dots)$ (115)

We can see that the heat kernel coefficients $a_{n/2},\ n=0,\dots 3$ are
responsible for the divergences of $E(J)$. These coefficients are universal
local quantities Vassilevich:2003xt ; bordag2009advances  and can be expressed
as volume integrals over the support of the $J_{i}$ sources and surface
integrals over their boundaries, $\partial J_{i}$. The first coefficient
simply amounts to the volume of the support of the fluctuation, which is the
space outside of the sources,

$a_{0}=V-V_{1}-V_{2}$ (116)

with $V_{1,2}={\rm Vol}(J_{1,2})$ and $V$ the volume of the total space.
Depending on the geometric setup, these various volumes may be finite or
infinite, this is a minor detail in the argument.

We can now vary the quantum vacuum energy under some deformation flow. We use
the notation of section 2. In the case of a rigid deformation flow (i.e.
satisfying ${\bm{\partial}}\cdot{\bf L}=0$), e.g. two bodies moving apart, the
coefficients are independent of the relative position of the bodies
Fulling:2003zx ; bordag2009advances . As a result all the divergences cancel
when computing the quantum work by varying the relative position of the
bodies. This is easily illustrated at the level of the $a_{0}$ coefficient,
for which we simply have

$\partial_{\lambda}a_{0}=-\partial_{\lambda}\int_{J_{\lambda}}d^{d}{{\bm{x}}}=-\int_{J_{\lambda}}d^{d}{{\bm{x}}}\,{\bm{\partial}}\cdot{\bf
L}=0\,.$ (117)

where we have used that $\partial_{\lambda}V=0$. Therefore under the
assumption of incompressibility the quantum work is finite. This is a version
of the classic argument given in bordag2009advances .

In contrast, in the case of a deformation flow for which
${\bm{\partial}}\cdot{\bf L}\neq 0$, i.e. dropping the assumption of
incompressibility, we can see that $\partial_{\lambda}a_{0}\neq 0$, which then
results in a divergence in the quantum work. There is no parameter in the
Lagrangian into which this divergence could be absorbed, therefore there is no
hope to renormalize this divergence away. The only way out we found is to let
the number density of the sources vary in such way that matter conservation in
the source is satisfied. As shown in section 2, 3.1 and exemplified in section
5, this is the condition that ensures that all possible divergences in the
quantum work vanish.

##  Appendix C The Loop Divergence from Momentum Space

We are interested in the loop integral given in Eq. (32),

$I=\left(\prod^{q}_{i=1}\int d^{d}{\bm{\mu}}_{i}J({\bm{\mu}}_{i})\int
dt_{i}\right)\prod^{q}_{i=0}{\cal
B}^{\prime\prime}\Delta_{0}(\mu_{i},\mu_{i+1})\bigg{|}_{\mu_{0}=x,\mu_{q+1}=x_{\epsilon}}.$
(118)

We introduce the $(d+1)$-dimensional Fourier transform of the propagators
defined as

$\Delta_{0}(\mu_{i},\mu_{i+1})=\int\not{d}^{d+1}{k}e^{i{k.(\mu_{i+1}-\mu_{i}})}\tilde{\Delta}_{0}(k)\,.$
(119)

We introduce the $d$-dimensional Fourier transform of the sources,

$J({\bm{\mu}}_{i})=\int\not{d}^{d}{\bf k}e^{i{\bf
k}\cdot{\bm{\mu}}_{i}}\tilde{J}({\bf k})\,.$ (120)

We have defined $\not{d}k=\frac{dk}{2\pi}$. The contraction between the
$(d+1)$-vectors is done using the Minkowski metric $k\cdot
y=\eta_{\mu\nu}k^{\mu}y^{\nu}$ with $k^{\mu}=(\omega,\bf k)$. The loop
integral becomes

$I=\int\not{d}^{d+1}{k}_{0}\left(\prod_{i=1}^{q}\int\not{d}^{d}{\bf
p}_{i}\tilde{J}({\bf
p}_{i})\right)\left(\prod_{i=0}^{q}\tilde{B}^{\prime\prime}(k_{i-1},k_{i})\tilde{\Delta}_{0}(k_{i})\right)e^{ik_{q}\cdot
x_{\epsilon}-ik_{0}x}$ (121)

with ${\bf p}_{i}={\bf k}_{i}-{\bf k}_{i-1}$ the momentum transferred to the
source.

Performing the time and position integrals we obtain an integral over the
$(d+1)$-momentum $k_{0}$ running through the loop. The derivative operators in
real space become simply multiplying operators depending on the two momenta
entering each vertex $\tilde{B}^{\prime\prime}(k_{i-1},k_{i})$. In the EFT
case it contains at least two powers of the momenta.

Since the source $J$ has compact support, the Riemann-Lebesgue lemma
guarantees that $\tilde{J}({\bf p})$ vanishes for large momenta and in fact
falls off for $|{\bf p}|\gtrsim R_{J}^{-1}$. This implies a smooth cutoff on
the value of ${\bf p}$. On the other hand the divergence in $I$ occurs for
arbitrarily large loop momentum $k_{0}$ . Since the $\bf p$ are bounded, at
large loop momentum we have $k_{0}\simeq k_{1}\simeq\ldots\simeq k_{q}$, thus
the divergent part of the diagram is $\bf p$-independent and is only a
function of $({x}_{\epsilon}-{x})=\epsilon$. Writing
$c_{J}=\int\not{d}^{d}{\bf p}_{i}\tilde{J}({\bf p}_{i})$, we have the
structure

$I\approx c_{J}^{q}L^{\rm div}_{\epsilon}\,$ (122)

where $L^{\rm div}_{\epsilon}$ is the position-independent divergent quantity.

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