Atoms on a Worldline: A Path-Integral Approach to Electromagnetic Casimir

Energies

by

He Zheng

A dissertation accepted and approved in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

in Physics

Dissertation Committee:

Steven van Enk , Chair

Daniel Steck , Advisor

Dietrich Belitz , Core Member

Michael Kellman , Institutional Representative

University of Oregon

Summer 2025



© 2025 He Zheng

2



DISSERTATION ABSTRACT

He Zheng

Doctor of Philosophy in Physics

Title: Atoms on a Worldline: A Path-Integral Approach to Electromagnetic Casimir
Energies

The Casimir effect arises from quantum fluctuations of the electromagnetic field

and leads to observable forces between material bodies even in vacuum. While the

Casimir force between simple geometries has been well studied, accurately calculating

Casimir interactions in general or many-body configurations remains a challenging

problem, especially when electromagnetic vector properties and material responses

are taken into account.

Among the primary advancements in computational tools for Casimir physics, the

worldline path-integral approach provides a powerful alternative to traditional mode

summation and scattering approaches by reformulating the quantum vacuum energy

in terms of an ensemble of fluctuating particle paths. The worldline method offers

strong potential for handling arbitrary geometries through intuitive Monte Carlo

sampling and parallelizable algorithms. However, most prior worldline formulations

are restricted to scalar fields or simplified boundary conditions.

This dissertation aims to extend and strengthen the worldline formalism

in both numerical and analytical directions. First, it develops a pathwise

3



differentiation technique that enables efficient computation of Casimir forces and

higher derivatives of the energy numerically. Second, the thesis contrasts with the

Green-tensor formalism and investigates the breakdown of scalar approximations in

electromagnetic Casimir worldlines between discrete polarizable atoms, highlighting

the necessity of vectorial field treatments in the worldline method. These findings

demonstrate the critical effects of electromagnetic polarization mixing in Casimir

energy computations and suggest new pathways for studying dispersion forces in

many-body systems.

This dissertation contains previously published as well as unpublished co-

authored materials.

4



CURRICULUM VITAE

NAME OF AUTHOR: He Zheng

GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:

University of Oregon, Eugene, OR, USA

University of California Los Angeles, Los Angeles, CA, USA

DEGREES AWARDED:

Doctor of Philosophy, Physics, 2025, University of Oregon

Bachelor of Science, Physics, 2017, UCLA

Bachelor of Science, Mathematics, 2017, UCLA

AREAS OF SPECIAL INTEREST:

Quantum Optics

Atomic Physics

PROFESSIONAL EXPERIENCE:

Graduate Teaching Assistant, University of Oregon, 2018 – 2025

GRANTS, AWARDS AND HONORS:

Departmental Highest Honors, UCLA, 2017

Weiser Physics Teaching Award, University of Oregon, 2023

Emanuel Optical, Molecular, and Quantum Sciences Fellowship, 2025

5



PUBLICATIONS:

H. Zheng and D. A. Steck. Preprint: arXiv:2502.15997 [quant-ph].

J. B. Mackrory, H. Zheng, and D. A. Steck. Phys. Rev. A 110, 042826 (2024).

T. Cohen, S. Majewski, B. Ostdiek and P. Zheng. JHEP 06 (2020) 019.

6



ACKNOWLEDGEMENTS

I would like to express deep gratitude to my advisor, Prof. Daniel Steck, for

the guidance and scientific insights throughout the doctoral journey. I am thankful

to the committee members for their constructive feedback and thought-provoking

questions, which have helped me refine and strengthen my research work.

I also wish to thank my collaborators and fellow graduate students for the many

valuable discussions and shared moments in physics.

Lastly, I am grateful to my parents for their support and encouragement along

the way.

7



TABLE OF CONTENTS

Chapter Page

I INTRODUCTION TO CASIMIR PHYSICS . . . . . . . . . . . . . . . 13

1.1 Historical Background and Physical Origin . . . . . . . . . . . . . 13

1.2 Casimir Pressure Between Parallel Plates . . . . . . . . . . . . . . 17

1.3 Quantum Electrodynamics with Magnetodielectric Media . . . . . 22

1.4 Atom Interacting with a Planar Surface . . . . . . . . . . . . . . . 24

1.5 An Overview of Experiments . . . . . . . . . . . . . . . . . . . . . 29

1.6 An Overview of Computational Methods . . . . . . . . . . . . . . 35

II ELECTROMAGNETIC WORLDLINE FORMALISM . . . . . . . . . 41

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Scalar Field Worldline Formulation . . . . . . . . . . . . . . . . . . 47

2.4 Worldline Form of Scalar Electromagnetism . . . . . . . . . . . . . 53

2.5 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . 65

III PATHWISE DIFFERENTIATIONOFWORLDLINE PATH INTEGRALS 76

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2 Derivatives of Worldline Casimir–Polder Energies . . . . . . . . . . 79

8



Chapter Page

3.3 Differentiation of Worldline Casimir Energies . . . . . . . . . . . . 94

3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Appendix to Chapter III . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.1 Boundary Crossing Statistics . . . . . . . . . . . . . . . . . . . . . 119

A.2 Derivation of the TE-Mode Casimir Energy Between Two Thin
Delta-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.3 Fitting Range and Parameters . . . . . . . . . . . . . . . . . . . . 133

IV THE VALIDITY OF SCALAR APPROXIMATION . . . . . . . . . . . 134

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.2 Green Tensor Calculation of the Casimir–Polder Potential Between
Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.3 Multi-Atom Generalization of Scalar Worldlines . . . . . . . . . . 144

4.4 The Breakdown of Scalar Approximation at N = 3 . . . . . . . . . 156

4.5 Physical Interpretation and Future Direction . . . . . . . . . . . . 160

Appendix to Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . 163

A.1 Derivation of N = 3 Co-linear Casimir–Polder Potential with
Green-Tensor Decomposition . . . . . . . . . . . . . . . . . . . 163

REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9



LIST OF FIGURES

Figure Page

1.1. A sketch of the parallel-plate setup and the constrained electromagnetic
modes. Only blue modes with half integer multiples of wavelength are
allowed between the plates. . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2. Feynman diagram representing a ground-state atom interacting with EM
field through emission and re-absorption of virtual photons. . . . . . . . 27

2.1. Decomposition of electromagnetic modes into TE and TM polarizations
for a planar stratified system. TE modes have the electric field
perpendicular to the plane of incidence, while TM modes have the
magnetic field perpendicular to it. . . . . . . . . . . . . . . . . . . . . . 55

2.2. Casimir–Polder worldlines pinned to the atomic position. (a) The
vacuum loop does not contribute to the potential. (b) Only the one
that sojourns the medium contributes. . . . . . . . . . . . . . . . . . . 64

2.3. Linear drift applied to Wiener processes to force the bridge closure. . . 67

3.1. Schematic path showing partial averaging up to the mth discrete path
increment up to and after the source point x0. The diagram shows the
example geometry of the Casimir–Polder potential for an atom in vacuum
near a planar interface with a dielectric material of susceptibility χ. . . 87

3.2. Schematic of an atom between two dielectric half-spaces with respective
susceptibilities χ1 and χ2, illustrating the prototype problem for the
partial-averaging technique of Section 3.2.3 close to one body. The dashed
spheres show the typical extent of the paths at various running times T
(i.e., times over which the paths can diffuse). At T1 the paths typically
touch only the nearer body, while at T2 they typically touch both bodies.
The time interval for partial averaging the paths is Tm. The method of
Sec. 3.2.2 requires the background dielectric to be approximately uniform
in the vicinity of the atom, and thus that Tm ≪ T1 so that the paths
are unlikely to touch either interface during the partial-averaging time.
However, because the path integral can be evaluated in the presence of a
single, planar interface, the partial averaging can be extended such that
Tm ∼ T1, provided Tm ≪ T2, so that most paths only hit the nearer
interface during the partial-averaging time. . . . . . . . . . . . . . . . . 90

10



Figure Page

3.3. Geometry for interacting dielectric bodies of susceptibility χj, centered
at Rj relative to the origin. The normal vectors n̂j to the surface of the
jth body are also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.4. Numerical evaluation of the path integral (3.4) for the (normalized,
dimensionless) Casimir–Polder potential of an atom near a dielectric
half-space, as a function of the (dimensionless) dielectric susceptibility
χ (data shown as circles). The remaining curves, each successively offset
by a factor of 10−2 for clarity, show spatial derivatives of the potential
according to the method of Eq. (3.11), for n = 1 (squares), 2 (diamonds),
4 (triangles), 7 (inverted triangles), and 10 (stars) derivatives. The solid
lines give the analytic result (3.45) for comparison in each case. Error
bars delimit one standard error. Details of the calculations are given in
the text, but the computational difficulty increases with increasing n,
and accuracy noticeably suffers by the n = 10 case. The same random
numbers are used for all the points in the same curve. . . . . . . . . . . 104

3.5. Magnitude of the relative error in the computation of (a) one spatial
derivative and (b) two derivatives of the atom–surface Casimir–Polder
potential, for a surface dielectric susceptibility χ = 100. The plots
compare the performance of the finite-difference (circles) and partial-
averaging (triangles) methods. The finite-difference data are plotted with
respect to the spatial displacement δ, normalized to the atom–surface
distance d, while the partial-averaging results are plotted as a function
of the averaging fraction m/N , where m appears in Eq. (3.16) and N
is the number of points per discrete path. The calculations employed
paths with N = 104 points per path, averaging over 1011 paths. Error
bars delimit one standard error. The lines are fits to the data (except for
the second derivative in the finite-difference case), and the intersection
(marked by a star) gives the estimated best performance of the method.
Data points within each method are statistically independent. . . . . . 105

3.6. Magnitude of the relative error in the computation of (a) one spatial
derivative and (b) two derivatives of the atom–surface Casimir–Polder
potential. The calculations here are similar to those in Fig. 3.5, except
here the surface is a perfect conductor (χ −→ ∞). Also, the calculations
here employed paths with N = 106 points per path (with subpaths of 105

points on either side of the point closest to the surface), averaging over
109 paths. Other details are as in Fig. 3.5. . . . . . . . . . . . . . . . . 107

3.7. Numerical evaluation of the derivatives of the path integral (3.2) for
the (normalized, dimensionless) Casimir potential of two dielectric thin
planes, as a function of the (dimensionless) susceptibility parameter χ̂

11



Figure Page

(data shown as circles). The first (n = 1) derivative is shown (circles)
along with the second (n = 2) derivative (squares, offset by a factor 10−2

for clarity). The solid lines give the derivatives of the analytic result
(3.50) for comparison in each case. The calculation used 109 paths of 105

points per path. Error bars delimit one standard error (on the order of
a few times 10−4 to a few times 10−3 in this plot). The same random
numbers are used for the points in each curve. . . . . . . . . . . . . . . 111

3.8. Magnitude of the relative error in the computation of (a) one spatial
derivative and (b) two derivatives of the Casimir potential for two
thin planes, for susceptibility parameter χ̂ = 1. The plots compare
the performance of the finite-difference (circles) and partial-averaging
(triangles) methods; the second derivative additionally has one finite
difference applied to each surface (squares). The finite-difference data
are plotted with respect to the spatial displacement δ, normalized to the
atom–surface distance d, while the partial-averaging results are plotted as
a function of the averaging fraction m/N , where m appears in Eq. (3.16)
and N is the number of points per discrete path. The calculations
employed paths with N = 105 points per path, averaging over 109

paths. Error bars delimit one standard error. The lines are fits to
the data, and the intersection (marked by a star) gives the estimated
best performance of the method. Data points within each method are
statistically independent. . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.9. Magnitude of the relative error in the computation of (a) one spatial
derivative and (b) two derivatives of the Casimir potential for 2 thin
planes, for susceptibility parameter χ̂ = 0.01. Details are as in Fig. 3.8.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.1. A sketch of two atoms interacting with each other in the worldline picture.
The path of total running time T starts from one atom and is set to hit
the second atom at an arbitrary intermediate time τ ′. . . . . . . . . . . 147

4.2. The Casimir–Polder coefficients for the three-body interaction obtained
from the worldline scalar sum and the Green tensor method, as a
function of the cosine of the angle between the displacement vectors. The
comparison for b/c = 0.5 is shown in (a), where the collinear configuration
corresponds to cos θ = 1. The case b/c = 1 is shown in (b), where the
equilateral-triangle configuration corresponds to cos θ = 0.5. In both
cases, the energy shift is plotted in units of ℏcα1α2α3/(4πε0)

3πR10. . . 159

12



CHAPTER I

INTRODUCTION TO CASIMIR PHYSICS

This thesis comprises research projects unified by the goal of extending the

worldline approach to Casimir energy calculations toward broader applicability.

1.1 Historical Background and Physical Origin

The Casimir effect is one of the most remarkable and conceptually illuminating

phenomena arising from the interplay between quantum field theory and macroscopic

boundary conditions. First predicted by Hendrik Casimir in 1948 [1], the effect refers

to the emergence of a measurable force between two neutral, perfectly conducting

plates in vacuum due solely to the quantization of the electromagnetic field. The

original calculation, carried out in the context of quantum electrodynamics (QED),

showed that the modification of vacuum fluctuations between the plates results in

a net attractive force. This prediction challenged the classical view of the vacuum

as inert and featureless, instead portraying it as a dynamic entity teeming with

zero-point fluctuations.

In Casimir’s canonical setup, two parallel plates are placed at a fixed separation

a, and the electromagnetic field is required to vanish on their surfaces. A sketch

is shown in Fig. 1.1. The boundary conditions discretize the allowed modes of the

field, effectively excluding long-wavelength modes between the plates compared to

the unbounded vacuum outside. The vacuum energy, which is formally the sum of

the zero-point energies 1
2
ℏω of all allowed modes, becomes geometry-dependent due

to this restriction. Casimir computed the difference in vacuum energy between the

13



confined and free cases and found the resulting force per unit area to be

F = − π2ℏc
240a4

, (1.1)

a strikingly simple and universal result that depends only on fundamental constants

and the separation between the plates.

FIGURE 1.1. A sketch of the parallel-plate setup and the constrained
electromagnetic modes. Only blue modes with half integer multiples of wavelength
are allowed between the plates.

At the time of Casimir’s prediction, the physical reality of zero-point energy

itself was still a topic of debate. However, subsequent theoretical developments

and increasingly precise experiments have confirmed the existence of Casimir forces,

solidifying their role in modern physics. The first clear experimental observation was

achieved by Sparnaay in the 1950s, though with limited precision. More definitive

measurements came decades later, notably by Lamoreaux (1997) and Mohideen &

Roy (1998), who confirmed the Casimir force between metallic surfaces to within a

few percent of theoretical predictions. These experiments, together with advances

in nanotechnology, have propelled the Casimir effect from a theoretical curiosity

14



to a practical consideration in micro-electromechanical systems (MEMS), where

unwanted stiction due to Casimir forces can lead to device failure.

While the Casimir effect can be regarded as a general feature of quantum

field theory subject to boundary conditions, the electromagnetic Casimir effect is

particularly significant. Unlike scalar fields, the electromagnetic field possesses

massless quanta—photons—whose long-range interactions dominate over other forces

at macroscopic distances. Moreover, the electromagnetic coupling to matter is vastly

stronger than gravity, the only other long-ranged fundamental interaction, making

Casimir forces observable even at micrometer scales.

Physically, the Casimir effect can be interpreted in several complementary

ways. In the field-theoretic picture, it results from alterations in the vacuum zero-

point energy due to boundary-induced modifications of the electromagnetic mode

structure. Alternatively, at the microscopic level, it can be viewed as a manifestation

of the interaction between fluctuating electric dipoles in different bodies via the

exchange of virtual photons. This viewpoint connects the Casimir effect seamlessly

to van der Waals forces, first described by van der Waals in 1873, where quantum

fluctuations of instantaneous dipole moments lead to attractive forces between

molecules. In fact, the Casimir–Polder potential, derived by Casimir and Polder in

1948, explicitly describes the interaction between a neutral atom and a conducting

wall as a retarded extension of the van der Waals force, highlighting the unification

of these phenomena under the framework of quantum electrodynamics.

Beyond its foundational role as a signature of quantum vacuum fluctuations,

the Casimir effect has grown increasingly important in both experimental physics

and technological applications. Its first unambiguous measurement was achieved

by Lamoreaux in 1997 [2], who introduced a sphere-plane geometry to circumvent

15



the alignment challenges of parallel-plate setups. This configuration, combined

with precision torsion balance techniques, enabled reliable force measurements at

submicron separations. Since then, the Casimir force has been confirmed across a

wide range of materials and geometries. In micro- and nano-scale systems, vacuum-

induced forces can significantly affect device operation, making Casimir effects

an essential consideration in nanoscale engineering. Moreover, precision Casimir

experiments have been proposed as sensitive tests of physics beyond the Standard

Model, including searches for hypothetical forces or light scalar particles coupling to

matter at short range. A more detailed account of key experimental milestones is

presented in Sec. 1.5.

In addition to being a testbed for quantum field theory, the Casimir effect plays an

increasingly critical role in modern technologies and quantum information science. In

microelectromechanical systems (MEMS), Casimir forces can induce stiction, causing

mechanical components to adhere and potentially fail [3]. Similarly, Casimir forces

influence the behavior of ultracold atoms near dielectric surfaces, setting fundamental

limits on trapping distances and coherence times [4, 5, 6]. These technological

implications have driven the need for accurate and flexible computational methods

capable of predicting Casimir interactions in realistic geometries and materials.

The nontrivial dependence of the Casimir effect on geometry, material properties,

and temperature has spurred the development of a wide range of theoretical

approaches. While early calculations relied on mode summation and idealized

assumptions, more sophisticated methods have since emerged, including scattering

theory approaches, Green’s function techniques, and path integral formulations.

Of particular interest is the worldline method — the focus of this thesis —

which provides a geometrically intuitive and numerically powerful tool for tackling

16



general configurations. Initially developed for scalar fields interacting with idealized

backgrounds, extending the worldline approach to the full electromagnetic case

remains a challenging but promising direction for Casimir physics.

1.2 Casimir Pressure Between Parallel Plates

The Casimir effect provides one of the most tangible demonstrations of the

nontrivial structure of the quantum vacuum. Quantum field theory reveals that

the vacuum state is populated by fluctuating fields, a consequence of the Heisenberg

uncertainty principle. These fluctuations give rise to a nonzero ground state energy,

known as zero-point energy, for each mode of the quantized field.

For the electromagnetic field, each mode with wavevector k and polarization

λ behaves as an independent harmonic oscillator, contributing a zero-point energy

1
2
ℏωk, where ωk = c|k|. The formal expression for the vacuum energy is therefore

Evac =
1
2

∑
k,λ ℏωk.

Although the sum diverges, differences in vacuum energy between physical

configurations, such as the presence or absence of boundaries, yield finite and

measurable quantities. Introducing material boundaries modifies the allowed field

modes, resulting in a shift in the vacuum energy E0, which manifests as a force:

F = −∂∆E0

∂a
, (1.2)

where a denotes the separation between the bodies. For a simple, parallel

configuration as shown in Fig. 1.1, the quantum zero-point energy of the

electromagnetic field between two perfectly conducting plates of area A at zero

17



temperature can be formally expressed as

E0(a) =
1

2

∑
k,λ

ℏωk

=
1

2
ℏc
∑
k⊥

∞∑
n=−∞

√
k2
⊥ + (

πn

a
)2

=
1

2
ℏcA

∫ ∞

−∞

d2k⊥
(2π)2

∞∑
n=−∞

√
k2
⊥ + (

πn

a
)2. (1.3)

The direct mode-summation approach to the Casimir effect leads to formally

divergent expressions due to the contribution of infinitely many high-frequency

vacuum modes. To make these expressions mathematically well-defined, a high-

frequency cutoff function f(k), where k = |k|, is introduced to suppress contributions

from large wavevectors. Physically, this reflects the fact that real materials cannot

reflect arbitrarily high-frequency radiation: the conduction electrons responsible for

reflectivity have finite mass and response time, and cannot follow field oscillations

beyond a certain threshold. At asymptotically high frequencies (ωk → ∞), materials

become effectively transparent, and the electromagnetic modes behave as though no

boundary were present.

The cutoff function f(k) → 0 as k → ∞, and suppresses these unphysical

contributions. Thus, it serves as a proxy for the finite-frequency response of

real materials. In more complete treatments, this behavior is incorporated

explicitly through a complex, frequency-dependent dielectric function ε(ω). From

a computational perspective though, f(k) is often just a formal trick to regulate

divergent sums.

Importantly, the cutoff function does not appear in the final physical answer. It

cancels out when computing differences in vacuum energies — such as the energy

18



between configurations with and without boundaries — because high-frequency

modes are unaffected by the presence of the surfaces and contribute equally in both

cases. Thus, while f(k) is crucial to intermediate steps in the derivation, it ultimately

drops out, leaving a finite and physically meaningful Casimir energy.

After subtracting the offset corresponding to the zero-point energy in the free

space, the difference in energy becomes

∆E0(a) =
1

4π
ℏcA

∫ ∞

0

k⊥dk⊥

[ ∞∑
n=−∞

√
k2⊥ + (

πn

a
)2 f(k)

− a

π

∫ ∞

−∞
dkz

√
k2⊥ + k2z f(

√
k2⊥ + k2z)

]
=
π2ℏcA
4a3

[
h(0)/2 +

∞∑
n=1

h(n)−
∫ ∞

0

dz h(z)
]
, (1.4)

where

h(z) ≡
∫ ∞

0

dρ
√
ρ+ z2f(π

√
ρ+ z2/a) = 2

∫ ∞

|z|
dt t2f(πt/a), (1.5)

with the change of variables as ρ ≡ (ak⊥/π)
2 and z ≡ akz/π. To extract a meaningful

finite result, one standard technique is to apply the Euler–Maclaurin summation

formula, which relates sums to integrals through correction terms involving higher

derivatives of the summand. A truncated approximation up to the third derivative

is given by

h(0)/2 +
∞∑
n=1

h(n) ≈
∫ ∞

0

dz h(z)− h′(0)

12
+
h′′′(0)

720
. (1.6)

19



By Eq. (1.5), for z ≥ 0, g′(z) vanishes at z = 0. The third derivative therefore

dominates the energy difference and yields

∆E0(a) = −π
2ℏcA
720a3

. (1.7)

The Casimir pressure between two conducting plates is thus

P0 = ∂a∆E0/A = − π2ℏc
240a4

, (1.8)

which is the Casimir force per unit area from Eq. (1.1). The pressure is negative,

indicating that the force between the plates is attractive. Notably, the result is

universal — it does not depend on any material parameter such as charge, mass, or

coupling strength. This universality emerges from the fact that, in the idealized limit

of perfect conductors, the only relevant scale in the problem is the plate separation

itself. While this derivation is based on a highly symmetric geometry and idealized

boundary conditions, it sets the foundation for understanding Casimir forces in more

realistic configurations.

1.2.1 Lifshitz Theory for Dielectric Half-Spaces

The original Casimir effect based on idealized configurations revealed the physical

significance of zero-point energy, but real materials are not perfect reflectors, and

real interfaces are seldom idealized planes. To address this limitation, the theory

was extended by Lifshitz in 1956 [7] to account for frequency-dependent dielectric

response, finite temperature, and continuous media.

Lifshitz’s formalism marks a significant generalization of Casimir’s result,

replacing mode summation with a fluctuation-electrodynamic treatment grounded in

20



the principles of statistical physics and linear response theory. In contrast to idealized

boundary conditions, Lifshitz theory incorporates material-specific response through

their complex, frequency-dependent dielectric functions. It provides a framework

for calculating dispersion forces — commonly known as van der Waals and Casimir

forces—between planar dielectric half-spaces by evaluating the stress tensor in the

intervening vacuum, sourced by fluctuating polarization currents in the surrounding

media.

Subsequent developments by Dzyaloshinskii and Pitaevskii [8] refined the

formalism using Green’s function methods and fluctuation-dissipation theorems,

making the theory amenable to both analytic and numerical analysis. Beyond its

foundational importance, Lifshitz theory is indispensable in applications ranging

from nanomechanics to precision metrology. Experimental verifications of Casimir

forces between real materials invariably require this framework, especially when

dielectric contrast, temperature dependence, or material absorption are non-

negligible. As such, the Lifshitz formalism provides the essential starting point for

any rigorous study of Casimir forces in realistic settings.

We consider two semi-infinite, planar, non-magnetic dielectrics with permittivities

ε1(ω) and ε2(ω), separated by a vacuum gap of width d. At zero temperature,

the Casimir pressure arises from the modification of the electromagnetic zero-point

fluctuations due to reflection at the interfaces.

After Wick rotation to imaginary frequency ω → iξ, the field in the vacuum gap is

decomposed into transverse-electric (TE) and transverse-magnetic (TM) plane-wave

modes with in-plane wavevector k⊥. Define the decay constants

κ0 =
√
k2⊥ + ξ2/c2, κa =

√
k2⊥ + εa(iξ) ξ2/c2, (a = 1, 2). (1.9)

21



The Fresnel reflection coefficients for vacuum–medium a interfaces on the imaginary

axis are

rTM
a (iξ, k⊥) =

εa(iξ)κ0 − κa
εa(iξ)κ0 + κa

, rTE
a (iξ, k⊥) =

κ0 − κa
κ0 + κa

. (1.10)

The free energy per unit area is then

F(d) =
ℏ
2π

∫ ∞

0

dξ

∫
d2k⊥

(2π)2

∑
p=TE,TM

log
[
1− r

(p)
1 (iξ, k⊥) r

(p)
2 (iξ, k⊥) e

−2κ0d
]
. (1.11)

Differentiating with respect to d gives the pressure,

P (d) = −∂F
∂d

= −ℏ
π

∫ ∞

0

dξ

∫ ∞

0

k⊥ dk⊥
2π

∑
p=TE,TM

κ0 r
(p)
1 r

(p)
2 e−2κ0d

1− r
(p)
1 r

(p)
2 e−2κ0d

. (1.12)

This is the standard zero-temperature Lifshitz formula for the Casimir pressure

between two parallel dielectric half-spaces separated by vacuum — a significant step

forward in describing interactions between real materials.

1.3 Quantum Electrodynamics with Magnetodielectric Media

Quantum electrodynamics (QED) in linear media extends the formalism of free-

space QED to systems in which the electromagnetic field interacts with material

bodies characterized by spatially and frequency-dependent dielectric permittivity

ε(r, ω) and magnetic permeability µ(r, ω). These properties modify the quantization

of the field and determine the structure of vacuum fluctuations, which are central to

Casimir and Casimir–Polder interactions.

22



1.3.1 Green Tensors in Linear Media

The starting point is Maxwell’s equations in the presence of linear media. In

the frequency domain, assuming a time dependence ∼ e−iωt, the source-modified

Maxwell equations for time-harmonic fields E(r, ω) and B(r, ω) read:

∇ ·D(r, ω) = 0, (1.13)

∇ ·B(r, ω) = 0, (1.14)

∇× E(r, ω) = iωB(r, ω), (1.15)

∇×H(r, ω) = −iωD(r, ω) + Jext(r, ω), (1.16)

with constitutive relations for isotropic, homogeneous linear media:

D(r, ω) = ε0ε(r, ω)E(r, ω), B(r, ω) = µ0µ(r, ω)H(r, ω). (1.17)

Combining Eqs. (1.15) and (1.16) leads to the inhomogeneous vector Helmholtz

equation for the electric field:

∇× µ−1(r, ω)∇× E(r, ω)− ω2

c2
ε(r, ω)E(r, ω) = iωµ0Jext(r, ω). (1.18)

The solution to this equation is expressed in terms of the dyadic electromagnetic

Green tensor G(r, r′, ω), defined as the response function satisfying:

[
∇× µ−1(r, ω)∇×−ω

2

c2
ε(r, ω)

]
G(r, r′, ω) = δ(r− r′)I. (1.19)

23



This Green tensor encodes how a point dipole source at position r′ generates

an electric field at r, incorporating the full influence of material boundaries and

dispersion.

Given an external current distribution Jext(r, ω), the electric field is obtained by

convolution:

E(r, ω) = iωµ0

∫
d3r′ G(r, r′, ω) · Jext(r

′, ω). (1.20)

In quantum electrodynamics with media, the Green tensor also governs the spectral

and spatial structure of field fluctuations. In particular, linear response theory

relates the field commutator to the imaginary part of the Green tensor. At zero

temperature, the vacuum expectation value of the electric field commutator is given

by the fluctuation-dissipation theorem:

⟨[Êi(x, t), Êj(x
′, t′)]⟩ = ℏ

π

∫ ∞

−∞
dω Im[Gij(x,x

′, ω)] exp[−iω(t− t′)]. (1.21)

This relation will serve as the cornerstone for computing Casimir–Polder shifts in

the subsequent analysis.

1.4 Atom Interacting with a Planar Surface

The Casimir–Polder effect predicts that a ground-state atom near a conducting

surface is attracted to the surface due to the local modification of the field

modes in the vacuum state. The interaction between an atom and the quantized

electromagnetic field is central to understanding a wide range of physical phenomena,

including spontaneous emission, van der Waals and Casimir–Polder forces, and

radiative energy shifts. At the microscopic level, the interaction between an atom and

the quantized electromagnetic field originates from the minimal coupling prescription

24



in the Coulomb gauge, where the vector potential Â(r̂, t) satisfies the transversality

condition ∇ · Â = 0. For a single electron bound in an external potential V (r̂), the

Hamiltonian takes the form

Ĥ =
1

2m

[
p̂+ eÂ(r̂, t)

]2
+ V (r̂), (1.22)

where p̂ = −iℏ∇ is the canonical momentum operator and e is the electron charge.

Expanding the square yields three distinct contributions:

Ĥ =
p̂2

2m
+

e

2m

(
p̂ · Â+ Â · p̂

)
+

e2

2m
Â2 + V (r̂). (1.23)

The first term describes the kinetic energy of the bound electron. The second term

represents the interaction between the particle and the quantized electromagnetic

field and gives rise, via second-order perturbation theory, to spontaneous emission,

van der Waals and Casimir–Polder forces. The third term contributes to radiative

corrections such as the Lamb shift, though in simplified treatments it is often

absorbed into renormalized parameters or neglected in the dipole approximation.

In general, p̂ and Â(r̂, t) do not commute

[p̂i, Âj(r̂)] = −iℏ ∂iÂj(r̂) ̸= 0, (1.24)

so the linear coupling term must be symmetrized to ensure Hermiticity. However, in

the electric-dipole approximation, which assumes that the field varies negligibly over

the extent of the atom, one approximates Â(r̂, t) ≈ Â(rN , t), where rN is the nuclear

position. Under this assumption, Â becomes a constant operator with respect to r̂,

and thus commutes with p̂, allowing the minimal-coupling interaction to be written

25



simply as

Ĥint =
e

m
p̂ · Â. (1.25)

While Eq. (1.22) is exact, it is often more convenient to perform a unitary

transformation to the multipolar gauge, in which the field couples directly to the

atomic multipole moments. This transformation is achieved using the Power–Zienau–

Woolley (PZW) unitary operator:

ÛPZW = exp

[
− i

ℏ
∑
j

ej r̂j · Â(R̂)

]
, (1.26)

where R̂ is the center-of-mass coordinate of the atom, and the sum is over charged

constituents j (for simplicity, we focus on a single electron with charge −e and

position r̂).

To leading order in the multipole expansion of the atom–field coupling, the

interaction energy reduces to that between the atomic electric dipole moment and

the electric field evaluated at the position of the atom:

Ĥint = −d̂ · Ê(rA), (1.27)

where d̂ = −er̂ is the electric dipole moment operator of the atom, and Ê(rA) is the

quantized electric field operator at the atomic center-of-mass position rA.

This derivation rests on the dipole approximation, which assumes that the

wavelength of the electromagnetic field is much larger than the spatial extent of the

atom. As a result, the spatial dependence of Â(r̂) is neglected over the volume of the

atom and approximated by its value at the center-of-mass coordinate R̂. This form

26



is especially useful in time-dependent perturbation theory and is the starting point

for analyzing spontaneous emission, energy shifts, and fluctuation-induced forces.

Given the interaction Hamiltonian, the energy shift of the ground state can be

calculated with second-order perturbation theory

VCP = −
∑
j

∑
k,ζ

|⟨g|d|ej⟩ · ⟨0|E|1k,ζ⟩|2

ℏ(ωj0 + ωk)
, (1.28)

where ωj0 = (Ej −E0)/ℏ is the transition frequency between the excited and ground

states of the atom, and ωk is the energy of the mode (k, ζ) labelled by the wave vector

k and the polarization index ζ. The field matrix element ⟨0|Ê(rA)|1k,ζ⟩ describes

|"⟩

|"⟩

|$⟩

FIGURE 1.2. Feynman diagram representing a ground-state atom interacting with
EM field through emission and re-absorption of virtual photons.

the electric field of a single-photon excitation in the mode (k, ζ) evaluated at the

atomic position rA, while the atomic dipole matrix element ⟨g|d̂|ej⟩ characterizes

the transition strength between the ground state |g⟩ and excited state |ej⟩. This

second-order expression represents the sum over virtual photon exchange processes

where the atom briefly transitions to an excited state and the field to a one-photon

state, followed by a return to the ground state. This process is illustrated by the

diagram in Fig. 1.2.

27



The energy shift may be recast into a form that separates the atomic and field

degrees of freedom. Defining the atomic polarizability tensor via the Kramers–

Heisenberg relation,

αik(ω) =
2

ℏ
∑
j

ωj0⟨g|d̂i|ej⟩⟨ej|d̂k|g⟩
ω2
j0 − ω2

, (1.29)

we may write the energy shift as an integral over the vacuum field correlation function

evaluated at the position of the atom. The zero-point electric field fluctuations in

the presence of a dielectric medium are characterized by the symmetrized two-point

function as given by the fluctuation–dissipation theorem in Eq. (1.21).

Substituting the polarizability and the time-ordered field correlator into the

perturbative expression, the Casimir–Polder potential appears as an integral over

real frequencies. To render the expression suitable for both analytic continuation

and numerical evaluation, the frequency integral is rotated to the imaginary axis via

a Wick rotation ω → iξ. This contour deformation is justified by the analytic

properties of the Green tensor and polarizability as causal response functions.

Eventually, the Casimir–Polder potential takes the widely used form

VCP(rA) = − ℏ
2π

∫ ∞

0

dξ αij(iξ)Gij(rA, rA; iξ), (1.30)

where the integration is over imaginary frequencies and repeated indices are summed.

The Green tensor characterizes the medium-modified vacuum fluctuations of the

electric field, while the atomic polarizability encapsulates the atom’s internal

response to those fluctuations.

Although this compact result arises from a formulation in which both the atom

and the electromagnetic field are quantized, the final expression for the energy shift

28



does not depend explicitly on the details of how the field is quantized. This is

particularly important, as the quantization of electromagnetic fields in dissipative or

absorbing media is technically challenging. The simplification here lies in the fact

that the result only requires knowledge of the classical electromagnetic response

functions — namely, the dielectric and magnetic susceptibilities — through the

Green tensor. Equally noteworthy is that this expression emerges from a linear,

dipole-type interaction Hamiltonian, and thus only captures lowest-order (linear)

response between the atom and field. Nevertheless, it provides a powerful and general

framework for computing dispersion interactions in complicated environments.

In addition to calculating the interaction of atoms with macroscopic bodies, the

Green tensor formalism can also be used for calculating Casimir–Polder potential

between two polarizable atoms in vacuum. This case will be discussed in detail in

Chapter IV.

1.5 An Overview of Experiments

1.5.1 Experimental Confirmation and Technological Advancements

The experimental confirmation of the Casimir effect, originally predicted by

Hendrik Casimir in 1948, has evolved into a cornerstone of modern precision

measurement and quantum vacuum physics. Initial attempts to detect Casimir

forces faced formidable challenges due to the minuscule scale of the interaction, which

becomes significant only at submicron separations. Nonetheless, rapid advances in

experimental control, materials fabrication, and surface metrology by the late 20th

century paved the way for high-precision measurements, culminating in a series of

landmark experiments that established the Casimir effect as a real and measurable

consequence of quantum electrodynamics.

29



A pivotal breakthrough occurred in 1997, when Lamoreaux performed the first

quantitative measurement of the Casimir force between macroscopic bodies [2]. In

this experiment, a torsion pendulum apparatus was used to measure the attractive

force between a gold-coated spherical lens and a parallel gold-coated flat plate. The

sphere-plate configuration approximates the parallel-plate geometry while avoiding

the problem of maintaining strict parallelism. Electrostatic calibrations were carried

out by applying known voltages between the surfaces to isolate and subtract Coulomb

contributions. The measured force was found to be in agreement with theoretical

predictions, including corrections for finite conductivity and surface roughness,

within approximately 5%. This work marked the first unambiguous confirmation of

the Casimir effect in a macroscopic setup and validated the foundational predictions

of Casimir’s original theory in a real experimental context.

Building upon this success, Mohideen and Roy introduced a new methodology

in 1998 based on atomic force microscopy (AFM) [9]. Their experiment used a

much smaller polystyrene sphere (radius ∼ 100 µm) coated with a thin layer of gold,

attached to an AFM cantilever interacting with a planar gold-coated surface. This

setup allowed for direct force curve measurements with sub-pico-Newton sensitivity

and nanometer spatial resolution, particularly in the critical 100–500 nm separation

range. Interferometric techniques were employed to accurately determine the sphere-

plate separation, and electrostatic calibrations ensured that residual potentials did

not contaminate the measurements. Their results showed excellent agreement with

theoretical predictions incorporating finite conductivity corrections. Notably, this

experiment was the first to resolve the Casimir force in the deep submicron regime

with high precision, establishing AFM-based force measurements as the standard

tool for Casimir metrology.

30



A third major milestone came in 2001, when Chan, Aksyuk, Kleiman, Bishop,

and Capasso demonstrated the influence of Casimir forces in microelectromechanical

systems (MEMS) [10]. Their experiment featured a microfabricated gold-coated

cantilever suspended above a stationary gold-coated substrate, forming a nearly

parallel-plate geometry with gap distances ranging from hundreds of nanometers

to below 100 nanometers. As the separation decreased, the Casimir attraction

eventually overwhelmed the mechanical restoring force of the cantilever, inducing

a ”pull-in” instability where the beam snapped into contact with the substrate.

By comparing the critical pull-in distance with theoretical models incorporating

the measured optical response of the gold coatings, they verified the magnitude

and distance dependence of the Casimir force in a microstructured system. This

experiment not only provided another precise confirmation of Casimir theory but

also highlighted the importance of quantum vacuum forces in the engineering of

nanoscale devices, ushering in the field of Casimir-aware MEMS design.

Aside from a quantum vacuum phenomena, an intriguing classical analogue of the

Casimir effect emerges in the context of acoustic wave phenomena, where fluctuating

quantum fields are replaced by externally generated classical noise. In this so-called

acoustic Casimir effect, two bodies immersed in a fluid or gas experience an effective

force due to the suppression of acoustic modes between them relative to the ambient

field outside. Unlike conventional Casimir forces — which originate from quantum

vacuum or thermal fluctuations—the acoustic Casimir effect is driven by engineered

noise fields with tunable spectral properties. When the acoustic field exhibits

a flat (white) frequency spectrum, the induced force between plates is typically

attractive and monotonic with respect to separation, closely mirroring the behavior

of quantum Casimir forces. However, if the noise spectrum is frequency-dependent,

31



non-monotonic and even repulsive forces can arise. The effect was experimentally

demonstrated by Larraza and Denardo in 1998 [11], who observed tunable acoustic-

induced forces between parallel plates subjected to controlled band-limited noise,

providing a macroscopic, classical platform to explore Casimir-like physics in a regime

entirely decoupled from quantum field fluctuations.

These foundational experiments collectively transformed the Casimir effect from

a theoretical curiosity into an experimentally verified and technologically relevant

phenomenon. They established the feasibility of measuring vacuum-induced forces

with pico-Newton precision, and spurred further investigations into temperature

dependence, material-specific effects and nontrivial geometries. The legacy of

these experiments continues to influence modern research at the intersection of

quantum field theory, materials science, and precision measurement, solidifying the

Casimir effect as a crucial phenomenon in both fundamental physics and applied

nanotechnology.

1.5.2 Searches for BSM Physics and Modifications of Gravity

Beyond metrology, precision measurements of Casimir forces have gained

increasing prominence in the context of probing new physics beyond the Standard

Model (BSM) and testing possible deviations from Newtonian gravity at sub-micron

scales.

The principal reason why Casimir experiments are relevant for fundamental

physics searches lies in the strength and range of the Casimir force: at separations

below a few microns, it dominates over the gravitational interaction between

laboratory-scale test bodies by many orders of magnitude. Consequently, the Casimir

background must be carefully modeled and subtracted in any attempt to isolate

32



weaker hypothetical interactions, but also provides a well-understood quantum

phenomenon that can be leveraged to place constraints on new forces.

In many theoretical scenarios, such hypothetical interactions are mediated by

light scalar or vector bosons, leading to Yukawa-type modifications of the Newtonian

gravitational potential:

V (r) = −GNm1m2

r

(
1 + αe−r/λ

)
, (1.31)

where α represents the strength of the new interaction relative to gravity, and λ =

ℏ/(mc) sets the force range, determined by the inverse mass m of the hypothetical

mediator. Such interactions naturally arise in theories involving dilatons, moduli

fields from string compactifications, axion-like particles, chameleon fields, or Kaluza-

Klein gravitons in models with large extra dimensions.

Experimental efforts to probe these effects in the context of Casimir physics

generally follow two complementary approaches. The first involves precise

measurements of Casimir forces in well-defined geometries, typically between metallic

or dielectric plates or spheres, and comparison with theoretical predictions based

on Lifshitz theory or scattering formalisms. Any deviation between theory and

experiment may signal new physics, provided systematic uncertainties such as surface

roughness, finite conductivity, patch potentials, and thermal corrections are well

controlled.

A notable example is the series of torsion pendulum experiments conducted by

Decca et al., in which the Casimir force between a metallic plate and a gold-coated

sphere was measured with sub-pico-Newton sensitivity. By achieving agreement with

theoretical predictions at the level of 1% or better in the 0.2 ∼ 1 µm range, these

33



experiments were able to place stringent bounds on Yukawa-type forces with α ∼ 102

and λ ∼ 0.5 µm, ruling out wide classes of BSM scenarios [12, 13].

The second approach, often termed the isoelectronic or Casimir-less technique,

aims to null out the standard Casimir background by clever material design. In these

experiments, test masses are fabricated such that the Casimir forces from different

regions cancel out due to identical electronic properties (e.g., gold versus germanium

layers with similar optical response), while the hypothetical new interaction remains

sensitive to differences in mass density. The INFN Padova group pioneered such

experiments, achieving differential force sensitivity at the femto-Newton level. These

null experiments provide especially clean constraints on composition-dependent or

density-coupled exotic forces [14].

Theoretical modeling of new interactions in this context typically assumes that

the additional force is additive over the bulk volumes of the test bodies. Thus, the

total hypothetical signal is computed by integrating the Yukawa potential over the

geometry of interest, taking into account the densities and geometrical configuration.

The resulting constraints on α and λ are typically plotted in a two-dimensional

exclusion plot, with Casimir-based experiments providing some of the strongest

bounds in the range λ ∼ 10 nm to λ ∼ 10 µm.

Beyond Yukawa-type modifications, Casimir experiments have also been

employed to probe models involving environmentally sensitive scalar fields, non-

Newtonian power-law potentials, axion-mediated dipole forces and axion dark matter

[15, 16, 17, 18]. While Casimir experiments are not directly sensitive to spin-

dependent interactions, they remain among the most effective probes of scalar-

mediated, composition-dependent fifth forces in the mesoscopic regime.

34



In summary, the Casimir effect has transcended its role as a canonical prediction

of quantum electrodynamics to become a sensitive tool in the search for new forces

and modifications of gravity. Through a combination of theoretical precision and

experimental ingenuity, Casimir-based setups continue to refine the landscape of

allowable BSM interactions, providing a unique window into potential new physics

at the micron and submicron frontier.

1.6 An Overview of Computational Methods

Modern applications increasingly demand exact or systematically improvable

methods. We introduce here the proximity force approximation as a simple but

crude method and the scattering formalism as the general-purpose method so far.

These computational methods motivate the development of the path-integral-based

worldline formalism.

1.6.1 Proximity Force Approximation

The Proximity Force Approximation (PFA) is one of the most widely

used computational tools for estimating Casimir forces between curved surfaces.

Originally developed in the context of van der Waals forces and surface adhesion,

it has since become an essential approximation technique for Casimir interactions,

particularly in experimental configurations where non-planar geometries are involved.

The core idea behind PFA is to approximate the Casimir energy (or force) between

gently curved objects by summing over infinitesimal contributions from locally

parallel surface elements, assuming each local region behaves like a pair of parallel

plates separated by a local distance.

35



For a sphere of radius R near a flat plate at closest separation a ≪ R, the PFA

yields an approximate expression for the Casimir energy by integrating the energy

per unit area between parallel plates over the curved surface of the sphere:

EPFA(a) = 2πR

∫ ∞

a

dz E∥(z), (1.32)

where E∥(z) is the Casimir energy density per unit area between two parallel plates

separated by distance z. For perfect conductors at zero temperature,

E∥(z) = − π2ℏc
720z3

, (1.33)

leading to the standard PFA result:

EPFA(a) = −π
3ℏcR
360a2

, FPFA(a) = −π
3ℏcR
120a3

. (1.34)

This expression is widely used in interpreting experimental data, especially in sphere-

plane and cylinder-plane geometries, where exact solutions are not easily tractable.

Despite its practical utility, the PFA is not a systematically controlled

approximation. It lacks a small parameter expansion in a rigorous sense. Its validity

rests on the assumption that the radius of curvature R of the surfaces is much larger

than the minimal separation a:

a

R
≪ 1. (1.35)

Under this condition, the local curvature changes slowly compared to the decay length

of the Casimir interaction, justifying the assumption of near-planarity. However, the

PFA does not correctly capture diffraction, retardation, or multiple-scattering effects

that become important when a/R increases. Corrections beyond PFA are nontrivial

36



and have been actively studied through both analytical expansions and numerical

methods such as the scattering formalism.

Moreover, the PFA does not account for non-additivity of Casimir interactions —

the fact that the total Casimir energy of a system is not simply the sum of pairwise

interactions. This becomes especially important in many-body configurations or in

the presence of structured materials such as photonic crystals or metasurfaces.

The PFA is widely used in analyzing Casimir force measurements involving

sphere-plate geometries, such as in the high-precision experiments of Lamoreaux [2]

and Mohideen and Roy [9]. In these setups, a metallic sphere is brought close

to a flat metallic surface, and the resulting force is measured using torsional

balances or atomic force microscopy. The PFA provides a practical means of

comparing experimental data with theoretical predictions, especially when the

surface separation remains small.

However, to achieve quantitative agreement at the few-percent level or better,

experiments must incorporate corrections beyond PFA, including those due to surface

roughness, finite conductivity, temperature, and geometry-induced deviations.

Theoretical work has provided gradient expansions and exact scattering-theoretic

corrections that extend PFA systematically [19, 20].

The proximity force approximation remains a central computational tool in

Casimir physics, especially in interpreting experimental data in curved geometries.

While limited in accuracy beyond the short-distance, large-radius regime, its

simplicity and physical transparency continue to make it an essential part of the

Casimir toolkit. Its limitations have also motivated the development of more robust

numerical techniques, which aim to extend precision Casimir force predictions to

arbitrary geometries and material properties.

37



1.6.2 Surface Current Scattering Formalism

As Casimir experiments and applications have progressed toward increasingly

intricate geometries and material models, the demand for general, accurate, and

systematically extensible computational frameworks has intensified. Among the most

powerful and rigorously developed approaches is the scattering matrix formalism [21],

which reformulates Casimir interactions in terms of the electromagnetic response of

objects via their individual scattering matrices. This method is both conceptually

clear and computationally robust, enabling the calculation of Casimir forces between

bodies of arbitrary shape, material composition, and spatial arrangement.

The central idea of the scattering formalism is that the Casimir energy arises

from multiple reflections of virtual photons between distinct bodies. Each object

is characterized by a T -matrix (or reflection matrix), which encodes how incident

electromagnetic multipole fields are scattered into outgoing ones. The interaction

between objects is governed by translation matrices, which relate the multipole

expansions centered on different bodies.

The finite-temperature Casimir free energy between two compact bodies, labeled

1 and 2, is given by a Matsubara frequency sum:

F(a) = kBT

∞∑
n=0

′ log det [I− T1(iξn)U12(iξn)T2(iξn)U21(iξn)] , (1.36)

where ξn = 2πnkBT/ℏ are the Matsubara frequencies, Ti(iξn) are the T -matrices

of the objects evaluated at imaginary frequencies, and Uij(iξn) are the translation

operators mapping fields from one object to the coordinate frame of the other. The

38



prime on the sum indicates that the n = 0 term carries half-weight:

∞∑
n=0

′f(ξn) =
1
2
f(ξ0) +

∞∑
n=1

f(ξn). (1.37)

This formulation can be rigorously derived from a functional integral over the

electromagnetic field, constrained by boundary conditions enforced through auxiliary

surface currents J(r) on each object. Integrating out the field yields an effective

action coupling the induced currents via the vacuum Green tensor G(0)(r, r′; iξ):

Seff [J] =
1

2

∫
d3r

∫
d3r′ J(r) ·G(0)(r, r′; iξ) · J(r′). (1.38)

This makes manifest that the Casimir interaction results from quantum correlations

between fluctuating surface currents.

For numerical evaluation, the surface currents are expanded in a suitable vector

basis, such as vector spherical harmonics (for spheres), cylindrical modes (for rods),

or plane-wave modes (for planar objects). The T -matrix encapsulates the object’s

response in this basis, while the translation matrices handle the coupling between

distinct coordinate systems. The determinant in Eq. (1.36) compactly encodes the

entire multiple-scattering series, and generalizes naturally to more than two bodies

through an expansion in irreducible many-body scattering amplitudes.

This scattering formalism has enabled exact numerical calculations in a wide

variety of settings: sphere-plane, sphere-sphere, cylinder-cylinder, sphere-grating,

and more. Its strength lies in its ability to incorporate material dispersion,

anisotropy, and temperature, all within a unified matrix-based framework. It

continues to serve as a cornerstone of modern theoretical and computational Casimir

physics.

39



While the scattering matrix formalism is highly successful, the computational cost

can grow rapidly with the number of scattering channels required for convergence,

especially in many-body or non-spherical systems. In contrast, the worldline

path-integral method offers a complementary approach that is basis-independent,

parallelizable, more intuitive, and can efficiently capture Casimir interactions

via stochastic sampling of fluctuating paths, naturally accommodating arbitrary

geometries. Thus, the worldline formalism fills a crucial gap in Casimir computations

where traditional operator-based methods become intractable.

40



CHAPTER II

ELECTROMAGNETIC WORLDLINE FORMALISM

2.1 Introduction

The previous chapter presents two primary approaches to calculating

atom–surface Casimir interactions: explicit mode summation — typically assuming

perfectly conducting boundaries — and the more general QED Green tensor

formalism, which accommodates dielectric response. While both methods are

analytically powerful in highly symmetric geometries, they become intractable when

the surfaces involved lack symmetry or involve arbitrary spatial profiles. This

motivates the development of alternative techniques that are more amenable to

numerical implementation in general geometries.

In this chapter, we introduce the worldline path integral method — a powerful

approach for computing Casimir energies and forces that extends naturally to a

wide range of geometries, including macroscopic bodies. Originally developed for

scalar fields in quantum field theory [22, 23], the method provides both a conceptual

and computational foundation for more complete electromagnetic treatments. While

scalar models omit the full vector structure of electromagnetism, they remain

physically meaningful in many contexts, such as scalar decompositions in symmetric

settings, and serve as a practical starting point for worldline-based techniques.

The worldline approach reformulates the Casimir energy as a path integral over

ensembles of fluctuating field configurations, which can be interpreted as closed

particle worldlines in Euclidean spacetime. This path integral is evaluated using

stochastic sampling — effectively a Monte Carlo method over Brownian trajectories.

41



Unlike mode summation or Green-function-based methods, which require solving

field equations for each geometry, the worldline method encodes geometry implicitly

via the interaction of these worldlines with the background bodies. In the sections

that follow, we develop this framework in electromagnetism, with particular attention

to its numerical realization and to the connection between worldline averages and

physically observable quantities such as Casimir forces.

2.2 Path Integrals

The path integral approach to quantum mechanics [24, 25] provides an alternative

to the operator-based formulation by expressing quantum amplitudes as a sum over

all possible trajectories weighted by an action-dependent phase. This formulation

offers deep insight into quantum fluctuations and serves as the foundation for modern

approaches to quantum field theory and semiclassical approximations.

2.2.1 Propagators in Quantum Mechanics

Consider a non-relativistic particle of mass m moving in one dimension under a

potential V (x). The transition amplitude for the particle to move from position xi

at time ti to xf at time tf is given by the kernel:

K(xf , tf ;xi, ti) = ⟨xf |e−iĤ(tf−ti)/ℏ|xi⟩, (2.1)

where the Hamiltonian operator is Ĥ = T̂ + V̂ = p̂2

2m
+ V (x̂).

To evaluate this, we partition the time interval into N equal slices of step size

ϵ = (tf − ti)/N , and insert complete sets of position eigenstates at each intermediate

42



time:

K(xf , tf ;xi, ti) = lim
N→∞

( m

2πiℏϵ

)N/2
∫ N−1∏

j=1

dxj

N−1∏
j=0

⟨xj+1|e−iϵĤ/ℏ|xj⟩ (2.2)

≡ lim
N→∞

∫
Dx

N−1∏
j=0

Kϵ(xj+1, xj), (2.3)

with x0 = xi, xN = xf , and the path integral measure defined by

Dx ≡
( m

2πiℏϵ

)N/2
N−1∏
j=1

dxj. (2.4)

The short-time propagator Kϵ is evaluated using the Trotter formula:

e−iϵĤ/ℏ = e−iϵ(T̂+V̂ )/ℏ ≈ e−iϵT̂ /ℏe−iϵV̂ /ℏ +O(ϵ2). (2.5)

The kinetic term is most naturally evaluated in the momentum basis. Using

⟨x|p⟩ = (2πℏ)−1/2eipx/ℏ, we have:

⟨xj+1|e−iϵT̂ /ℏ|xj⟩ =
∫

dp

2πℏ
eip(xj+1−xj)/ℏe−iϵp2/2mℏ

=

√
m

2πiℏϵ
exp

[
im

2ℏϵ
(xj+1 − xj)

2

]
. (2.6)

The potential term is local in position space:

⟨xj+1|e−iϵV̂ /ℏ|xj⟩ ≈ δ(xj+1 − xj) exp

[
−iϵ
ℏ
V (xj)

]
. (2.7)

43



Combining both terms yields the short-time propagator:

Kϵ(xj+1, xj) =

√
m

2πiℏϵ
exp

[
iϵ

ℏ

(
m

2

(
xj+1 − xj

ϵ

)2

− V (xj)

)]
, (2.8)

and the full transition amplitude becomes:

K(xf , tf ;xi, ti) = lim
N→∞

∫ N−1∏
j=1

dxj

( m

2πiℏϵ

)N/2

× exp

[
i

ℏ

N−1∑
j=0

ϵ

(
m

2

(
xj+1 − xj

ϵ

)2

− V (xj)

)]
. (2.9)

In the continuum limit ϵ→ 0, this becomes a path integral over all continuous paths

x(t) with endpoints x(ti) = xi, x(tf ) = xf :

K(xf , tf ;xi, ti) =

∫ x(tf )=xf

x(ti)=xi

Dx(t) exp
[
i

ℏ
S[x(t)]

]
, (2.10)

where the classical action is

S[x(t)] =

∫ tf

ti

dt

[
1

2
mẋ2(t)− V (x(t))

]
. (2.11)

To render the integral convergent and connect to statistical mechanics, one often

performs a Wick rotation t→ −iτ to make the action become Euclidean:

SE[x(τ)] =

∫ τf

τi

dτ

[
1

2
m

(
dx

dτ

)2

+ V (x(τ))

]
, (2.12)

and the propagator becomes

KE(xf , τf ;xi, τi) =

∫
Dx(τ) exp

[
−1

ℏ
SE[x(τ)]

]
. (2.13)

44



In this form, the path integral has the structure of a partition function over Brownian

motion trajectories, and is foundational in the worldline representation of quantum

field theory.

This formulation provides an intuitive and powerful method for analyzing

quantum systems via classical paths weighted by the action. Its extension to multiple

dimensions and field configurations underpins path integral approaches to quantum

field theory, statistical field theory, and the numerical worldline methods used in this

thesis to evaluate Casimir energies.

2.2.2 The Feynman–Kac Formula

The Feynman–Kac formula provides a fundamental bridge between stochastic

processes and the solutions to a class of parabolic partial differential equations

(PDEs), and plays a central role in the probabilistic formulation of quantum

mechanics, statistical physics, and finance.

Consider a non-relativistic quantum particle of mass m in one dimension,

governed by the Hamiltonian

Ĥ = − ℏ2

2m

d2

dx2
+ V (x). (2.14)

The propagator in imaginary time, which is the Euclidean analog of the quantum

mechanical kernel, reads:

KE(xf , τf ;xi, τi) = ⟨xf |e−(τf−τi)Ĥ/ℏ|xi⟩. (2.15)

In the context of statistical mechanics, this object is interpreted as the amplitude for

a particle to diffuse from xi to xf in time τf − τi, under the influence of a potential

45



V (x). The goal is to show that this propagator can be expressed as an ensemble

average over Brownian paths weighted by the Euclidean action.

From the Feynman path integral in imaginary time (2.13), the Euclidean

propagator is:

KE(xf , τf ;xi, τi) =

∫ x(τf )=xf

x(τi)=xi

Dx(τ) exp
[
−1

ℏ

∫ τf

τi

dτ

(
1

2
mẋ2(τ) + V (x(τ))

)]
.

(2.16)

This path integral resembles the probability distribution for Brownian motion with

a weight factor from the potential.

To make this precise, we write the kinetic term as a Wiener measure:

P [x(τ)] ∼ exp

(
−1

ℏ

∫ τf

τi

1

2
mẋ2(τ)dτ

)
, (2.17)

which is just the Gaussian weight for Brownian motion (after rescaling time). Thus,

we interpret the measure Dx(τ) as an average over Brownian paths:

KE(xf , τf ;xi, τi) = EBM
xi→xf

[
exp

(
−1

ℏ

∫ τf

τi

V (x(τ)) dτ

)]
, (2.18)

where the expectation EBM denotes averaging over all Brownian paths connecting xi

to xf in time τf − τi.

Let us now define:

u(x, τ) := ⟨x|e−(T−τ)Ĥ/ℏ|ψ0⟩, (2.19)

where |ψ0⟩ is some initial state and T is a fixed Euclidean time. Then u(x, τ) satisfies

the Euclidean Schrödinger (or heat) equation:

∂u

∂τ
=

(
ℏ2

2m

∂2

∂x2
− V (x)

)
u(x, τ), (2.20)

46



with final condition u(x, T ) = ψ0(x). The Feynman–Kac formula now asserts:

u(x, τ) = Ex

[
exp

(
−1

ℏ

∫ T

τ

V (x(s)) ds

)
ψ0(x(T ))

]
, (2.21)

where x(s) is a Brownian path starting at x(τ) = x and evolving under free diffusion

until time T .

The Feynman–Kac formula thus provides a probabilistic representation of the

solution to the heat equation with a potential. The term exp(−
∫
V ) acts like a

weight that penalizes paths passing through regions of large potential energy. This

notable result offers an elegant route from quantum mechanics to stochastic processes

and forms the basis of practical numerical methods for Casimir energies and effective

actions in curved or bounded geometries.

2.3 Scalar Field Worldline Formulation

In the following, we develop the path-integral, or worldline, approach to

computing Casimir energies in general geometries, including interactions between

extended macroscopic bodies. The original formulation of this method was

introduced in the context of scalar fields, which, while not representing the full

electromagnetic field, provide a valuable conceptual and computational framework

for introducing the technique. The method recasts the effective action of a quantum

field theory in terms of a functional integral over ensembles of closed particle

trajectories, enabling a numerical Monte Carlo evaluation of Casimir energies. This

stochastic formulation contrasts sharply with the traditional mode summation and

Green tensor approaches, and is particularly well suited for problems with nontrivial

geometries where analytical methods become intractable.

47



2.3.1 Schwinger Proper-Time Formalism

The Schwinger proper-time formalism [26] is a powerful technique in quantum

field theory that rewrites operator inverses — such as propagators or determinants

arising in one-loop effective actions — into integrals over an auxiliary parameter

known as the proper time. This method provides a unifying framework that is not

only elegant but also computationally advantageous, especially in curved or bounded

geometries where mode-summation techniques become intractable.

In quantum field theory, loop diagrams often lead to divergent integrals.

Schwinger introduced the proper-time method in 1951 to systematically treat such

divergences and calculate vacuum polarization effects in quantum electrodynamics

(QED). His formulation recasts loop amplitudes as proper-time integrals, thereby

exposing their mathematical structure and allowing for efficient regularization. This

method also forms the basis for worldline path-integral techniques used in modern

computations of Casimir energies.

Let us consider a free scalar field of mass m. The Feynman propagator in position

space is the Green’s function of the Klein–Gordon operator:

G(x, x′) = ⟨x| 1

−□+m2 − iϵ
|x′⟩. (2.22)

Using the identity

1

A
=

∫ ∞

0

ds e−sA, (2.23)

which holds for a positive-definite operator A, we may write

G(x, x′) =

∫ ∞

0

ds ⟨x|e−s(−□+m2)|x′⟩. (2.24)

48



The quantity inside the integrand is the heat kernel:

K(x, x′; s) ≡ ⟨x|e−s(−□+m2)|x′⟩, (2.25)

which solves the heat equation

(
∂

∂s
+ (−□+m2)

)
K(x, x′; s) = 0, K(x, x′; 0) = δ(x− x′). (2.26)

Thus, the scalar propagator becomes an integral over the proper time s:

G(x, x′) =

∫ ∞

0

dsK(x, x′; s). (2.27)

For a scalar field interacting with a background potential V (x), the one-loop

effective action is formally:

Γ[V ] =
1

2
Tr log

(
−□+m2 + V (x)

)
. (2.28)

Using the identity

logA = −
∫ ∞

0

ds

s
e−sA, (2.29)

the effective action becomes:

Γ[V ] = −1

2

∫ ∞

0

ds

s
Tr
(
e−s(−□+m2+V (x))

)
. (2.30)

In position space, this trace reads:

Γ[V ] = −1

2

∫ ∞

0

ds

s

∫
ddxK(x, x; s), (2.31)

49



where K(x, x; s) is the diagonal part of the heat kernel.

This representation makes divergences manifest as small-s behavior in the

integrand. Various regularization schemes — such as dimensional regularization,

zeta function regularization, or cutoff regularization — can be introduced at this

stage.

The heat kernelK(x, x′; s) admits a representation as a quantum mechanical path

integral over all trajectories of fixed proper-time s:

K(x, x′; s) =

∫ x(s)=x

x(0)=x′
Dx(t) exp

[
−
∫ s

0

dt

(
ẋ2(t)

4
+ V (x(t))

)]
. (2.32)

This expression treats the field-theoretic propagator as an average over first-

quantized particle trajectories, with each path weighted by a Euclidean action. When

x = x′, the path is closed and periodic, forming the basis of the worldline formalism.

The one-loop effective action is then:

Γ[V ] = −1

2

∫ ∞

0

ds

s

∫
ddx0

∫
x(0)=x(s)=x0

Dx(t) exp
[
−
∫ s

0

dt

(
ẋ2(t)

4
+ V (x(t))

)]
.

(2.33)

This functional arises from the Schwinger proper-time representation of the operator

trace Tr log(−∂2 + V ), which encodes the quantum corrections to the classical

action at one-loop order. The integration over closed paths x(t) with coincident

endpoints, x(0) = x(s) = x0, reflects the contribution of virtual particle loops in

configuration space. The proper-time parameter s plays the role of an internal

Schwinger parameter, controlling the scale of fluctuation modes. This formulation

effectively rewrites the one-loop effective action as a path integral over particle

trajectories, and is therefore known as the worldline formalism.

50



2.3.2 Partition Function, Ground-State Energy, and Functional

Determinants

Having established the one-loop effective action in the Schwinger proper-time

representation, we now interpret this formalism through the lens of statistical field

theory, where thermodynamic quantities are naturally expressed via the partition

function. The Euclidean path integral serves as a partition function encoding vacuum

fluctuations and energy levels of the system.

In statistical mechanics, the partition function Z of a quantum system at

temperature T is defined by tracing over the Boltzmann factor:

Z(β) = Tr
(
e−βĤ

)
, (2.34)

where Ĥ is the Hamiltonian operator, and β = 1/(kBT ) is the inverse temperature.

This quantity encodes the complete thermodynamic information of the system. In

the low-temperature limit β → ∞, the partition function is dominated by the ground

state energy E0, such that

Z(β) ∼ e−βE0 , ⇒ E0 = − lim
β→∞

1

β
logZ(β). (2.35)

Let us consider a free real scalar field ϕ(x) in d-dimensional spacetime with a

quadratic Hamiltonian. After performing a Wick rotation t 7→ −iτ , the partition

function becomes a Euclidean path integral over field configurations that are periodic

in imaginary time with period β:

Z(β) =

∫
ϕ(0,x)=ϕ(β,x)

Dϕ exp

[
−
∫ β

0

dτ

∫
dd−1xLE(ϕ)

]
, (2.36)

51



where the Euclidean Lagrangian density is given by

LE(ϕ) =
1

2

(
∂ϕ

∂τ

)2

+
1

2
(∇ϕ)2 + 1

2
m2ϕ2 +

1

2
V (x)ϕ2. (2.37)

This is a Gaussian path integral over a quadratic action. To evaluate it, we identify

the quadratic form operator acting on ϕ:

A = −∂2τ −∇2 +m2 + V (x), (2.38)

so the functional integral takes the standard form

Z(β) ∝ [detA]−1/2 , (2.39)

up to normalization factors irrelevant for thermodynamics. This is the infinite-

dimensional analog of the standard Gaussian integral in finite dimensions.

The free energy is defined thermodynamically as

F (β) = − 1

β
logZ(β), (2.40)

and substituting Eq. (2.39) gives

F (β) =
1

2β
log detA. (2.41)

Using the identity log detA = Tr logA, valid for trace-class operators, we write

F (β) =
1

2β
Tr logA. (2.42)

52



Therefore, the ground-state energy of the system can be expressed as

E0 = − lim
β→∞

1

β
logZ(β) = lim

β→∞
F (β). (2.43)

Equivalently, using the product rule,

∂

∂β
logZ(β) = −F (β)− β

∂F

∂β
, (2.44)

so

E0 = − lim
β→∞

∂

∂β
logZ(β), (2.45)

which is often taken as the operational definition of vacuum energy in terms of the

partition function.

This result establishes a rigorous connection between the spectral properties

of the differential operator A, the Euclidean functional integral, and the vacuum

energy of the system. In worldline approaches to quantum field theory, this trace-log

formula becomes the foundation for computing Casimir energies via a proper-time

representation and Monte Carlo sampling of closed paths.

2.4 Worldline Form of Scalar Electromagnetism

2.4.1 Scalar Decomposition of Electromagnetism

The scalar field considered in the previous section is artificial and does not relate

to realistic physical settings, since electromagnetism is a massless spin-1 gauge theory

with two physical polarization degrees of freedom. In many applications of quantum

electrodynamics, particularly those involving fluctuations or Casimir phenomena,

it is advantageous to exploit symmetries in the system to simplify the vectorial

53



structure of the electromagnetic field. One widely used simplification is the scalar

decomposition of the electromagnetic field, where the full vector field is separated

into independent scalar components. This decomposition is especially useful in

geometries with high symmetry, such as planar or cylindrical stratified media, where

translational invariance in transverse directions enables a reduction of Maxwell’s

equations to scalar wave equations. These scalar reductions provide the foundation

for scalar approximations in worldline path integral methods.

We begin with the action for a source-free electromagnetic field in a linear,

isotropic, and inhomogeneous medium characterized by spatially varying permittivity

ε(z) and permeability µ(z). Working in the temporal gauge ϕ = 0, the action

becomes:

S[A] =
1

2

∫
d4x

[
ε(z)

(
∂A

∂t

)2

− 1

µ(z)
(∇×A)2

]
, (2.46)

where A(r, t) is the vector potential. Varying this action with respect to A gives the

Euler–Lagrange equations:

−ε(z)
c2

∂2A

∂t2
+∇×

[
1

µ(z)
∇×A

]
= 0. (2.47)

We now specialize to a planar stratified geometry, in which ε(z) and µ(z) depend

only on the coordinate z. The system is translationally invariant in the transverse

x- and y-directions. Therefore, we Fourier expand the fields in those directions:

A(r, t) = A(z), ei(k⊥·r⊥−ωt), (2.48)

with transverse momentum k⊥ = (kx, ky). The residual freedom in gauge fixing is

removed by imposing the Coulomb gauge ∇ · A = 0, which, for the above ansatz,

54



reduces to:

ik⊥ ·A⊥(z) + ∂zAz(z) = 0. (2.49)

This constraint implies that only two independent field components remain —

corresponding to the physical polarizations of the photon. These naturally split

into TE and TM sectors.

!!, "! !, " !!, "! !, "

$"
$"

%"
%"

%#
%#

$#$#

$$
$$

&$&$

&" &"

&#&#

%$
%$

(TE polarization) (TM polarization)
FIGURE 2.1. Decomposition of electromagnetic modes into TE and TM
polarizations for a planar stratified system. TE modes have the electric field
perpendicular to the plane of incidence, while TM modes have the magnetic field
perpendicular to it.

Transverse Electric (TE) Modes. Choose the vector potential to be polarized along

the y-direction: A(z) = Ay(z) ŷ. The electric and magnetic fields are then:

E = −∂tA = iωAy(z)ŷ, B = ∇ × A = (∂zAy x̂− ikxAy ẑ). The electric field

is orthogonal to the x–z plane (plane of incidence), identifying this as the TE

polarization. From Maxwell’s equations, one obtains a scalar wave equation for

Ey(z, t):

∇ ·
(

1

µ(z)
∇Ey

)
− ε(z)

c2
∂2Ey

∂t2
= 0. (2.50)

55



Transverse Magnetic (TM) Modes. Now let the vector potential lie in the x–z plane:

A(z) = Ax(z) x̂ + Az(z) ẑ. The Coulomb gauge implies a relation between Ax

and Az. The resulting magnetic field is: B = ∇ × A = (0, ikxAz − ∂zAx, 0).

This configuration yields a magnetic field along ŷ, defining the TM mode. The

corresponding electric field lies in the x–z plane. The y-component By(z, t) obeys a

scalar wave equation:

∇ ·
(

1

ε(z)
∇By

)
− µ(z)

c2
∂2By

∂t2
= 0. (2.51)

Equations (2.50) and (2.51) describe the propagation of each polarization mode in

an inhomogeneous medium. These equations form the basis of scalar approximations

in electromagnetic Casimir computations. In the worldline formalism, each scalar

equation admits a path-integral representation of its Green function, enabling

efficient numerical evaluation of Casimir energies. As discussed in the next section,

the spatial variation of ε(z) and µ(z) enters as a position-dependent “mass” term, and

the curvature-induced corrections are treated using the variable-mass path integral

framework.

2.4.2 Electromagnetic Partition Function

As discussed in Sec. 2.3, the partition function of the physical system plays the

pivotal role in calculating the ground-state energies. With the scalar decomposition

of electromagnetic fields in Sec. 2.4.1, the equations of motion for TE (2.50) and TM

56



(2.51) fields respectively lead to the resulting actions:

S
TE

=
1

2µ0

∫
dx4

[ε(z)
c2
(∂ϕ
∂t

)2 − 1

µ(z)
|∇ϕ|2

]
, (2.52)

S
TM

=
1

2ε0

∫
dx4

[µ(z)
c2
(∂φ
∂t

)2 − 1

ε(z)
|∇φ|2

]
(2.53)

The partition function for either scalar field (TE or TM) is given by the Euclidean

path integral over periodic field configurations, with the action determined by the

corresponding scalar wave operator. Explicitly,

Z
TE

=

∫
Dϕ exp

[
− 1

2ℏcµ0

∫ βℏc

0

dτ

∫
dx3
(
ε(x)

(∂ϕ
∂τ

)2
+

1

µ(x)
|∇ϕ|2

)]
, (2.54)

where τ ≡ βℏc. The TM partition function can be obtained by the swapping

symmetry E ↔ H,B ↔ −D, ε↔ µ.

According to the development in Sec. 2.3.2, the partition function is totally

determined by the imaginary-time Green operator corresponding to each scalar mode

logZi =
1

2
Tr logGi

∣∣
t→iℏβ (2.55)

where i ∈ {TE,TM}.

In the TE sector, for instance, the Green operator for an inhomogeneous, linear,

isotropic medium with spatially varying permittivity and permeability takes the form

G
TE

=

[
−∇ ·

(
1

µ(r)
∇
)
+
ε(r)

c2
( ∂
∂t

)2]−1

. (2.56)

In the worldline representation, the term 1/µ(r) in the Laplacian acts as a position-

dependent mass, which leads to a nontrivial curved-space kinetic term in the

57



worldline action:

Skin[x(τ)] ∝
∫ T

0

dτ µ(x(τ)) ẋ2(τ). (2.57)

To simplify the structure of the kinetic term, we define a rescaled operator via a

transformation using µ1/2(r):

G̃−1
TE

= µ1/2(r)G−1
TE
µ1/2(r). (2.58)

The goal here is to use the operator commutation relation to remove space-dependent

masses on the ∇2 terms, effectively recasting the Green operator in flat space. The

consequence of this process is to introduce extra effective potentials. Specifically,

these matter-induced potentials are given by

VTE =
1

2
[(∇ log

√
µ)2 −∇2 log

√
µ]

VTM =
1

2
[(∇ log

√
ε)2 −∇2 log

√
ε]. (2.59)

Finally, the electromagnetic partition functions in scalar decomposition (2.54) are

evaluated to be

ZTE = det

[
− 1

2
εµ∂2τ −

1

2
∇2 − VTE

]−1/2

ZTM = det

[
− 1

2
εµ∂2τ −

1

2
∇2 − VTM

]−1/2

, (2.60)

2.4.3 Worldline Casimir Energies and Renormalization

To extract zero-temperature Casimir energies from the field-theoretic partition

function, one considers the ground-state energy encoded in the logarithm of the

58



partition function. Specifically, the vacuum energy is given formally by

E = − ∂

∂β
logZ(β)

∣∣∣∣
β→∞

, (2.61)

which captures the dominant contribution from the lowest energy eigenstate in the

zero-temperature limit. However, this energy includes divergent contributions from

the self-energies of individual objects and the infinite vacuum energy of the free field.

Since only energy differences are physically meaningful, a renormalization procedure

must be employed. This is typically achieved by subtracting the energy of a reference

configuration—usually the same system with objects placed at infinite separation—

which yields a finite, well-defined Casimir interaction energy:

ECasimir = − ∂

∂β
[logZ(β)− logZ0(β)]β→∞ . (2.62)

This subtraction removes geometry-independent vacuum divergences and isolates

the finite interaction energy due to boundary conditions or material interfaces. The

choice of reference configuration, while arbitrary in principle, is commonly taken

to be a configuration in which the objects are sufficiently far apart such that their

mutual interaction vanishes.

To evaluate Casimir energies, the key is to represent logZ in a convenient form.

Here we choose to utilize log detA = Tr logA to recast it into trace-log form,

and apply the integral representation of logarithm (2.29). After these transforms,

the development of worldline expressions then follows directly from standard path-

integral treatments. The unrenormalized Casimir energy in terms of worldlines is

then

Ei = − ℏc
2(2π)D/2

∫ ∞

0

dT
T 1+D/2

∫
dx0

〈〈
e−T ⟨Vi⟩

⟨εµ⟩1/2

〉〉
x(t)

, (2.63)

59



where i ∈ {TE,TM} and the double angled brackets denote an ensemble average

over Brownian bridges that start at x0, with running time T . The single brackets

represent path average of a functional

⟨f [x]⟩ = 1

T

∫ T

0

dτ f [x(τ)]. (2.64)

The variable T here is the “proper time” by analogy with the Schwinger formalism

in Sec. 2.3.1. In Casimir energy computations, however, it is important to emphasize

that T in this context is not the actual proper time of a physical relativistic

particle. Rather, it is a dimensionful auxiliary variable arising from the Schwinger

representation of the functional determinant, and it governs the scale of the virtual

loop fluctuations contributing to the Casimir energy. The corresponding proper

time variable T in fact has a dimension of length squared. This is consistent with

interpreting T as controlling the typical extent of the fluctuating worldline paths,

which scale as
√
T . In Casimir settings, T acts as a resolution parameter: short T

probes ultraviolet (small-scale) contributions near surfaces, while large T captures

long-range effects.

Details of the derivation of worldline path integrals can be found in Ref. [27].

2.4.4 Worldline Casimir–Polder Potentials

The Casimir energy of scalar EM in the worldline form is

E
EM

= − ℏc
2(2π)D/2

∫ ∞

0

dT
T 1+D/2

∫
dD−1x0

〈〈
⟨ϵrµr⟩−1/2

x(τ) e
−T ⟨V ⟩x(τ)

〉〉
x(τ)

, (2.65)

where V in the exponential is either V
TE

or V
TM

depending on the polarization mode

considered.

60



In principle, Casimir–Polder interactions between an atom and a macroscopic

object can be computed by modeling the atom as a small piece of magnetodielectric

material embedded within the environment. However, in the worldline numerical

framework, the naive generation of paths suffers from poor efficiency since the atom

occupies an extremely small region of space, and the vast majority of randomly

sampled closed loops will fail to intersect it, yielding negligible contribution to the

renormalized energy. In fact, only those loops that simultaneously intersect both

the atomic location and the macroscopic medium can contribute meaningfully to the

two-body Casimir interaction after renormalization.

To address this, it is advantageous to employ a tailored path-integral formulation

adapted specifically to the atom-surface configuration. The introduction of the atom

is modeled by a sharply localized perturbation to the medium’s permittivity and

permeability:

δε(r) =
α0

ε0
δ(D−1)(r− r′), δµ(r) = µ0β0 δ

(D−1)(r− r′), (2.66)

where α0 and β0 represent the static polarizability and magnetizability of the atom

located at r′. The relevant interaction energy is given by the difference between the

Casimir energies in the presence and absence of the atomic perturbation. Assuming

the perturbations are small, this energy shift can be computed via first-order

functional derivatives:

δE = E[ε+ δε, µ+ δµ]− E[ε, µ]

=

∫
dr

[
δE

δε(r)
δε(r) +

δE

δµ(r)
δµ(r) +O[(δε)2, (δµ)2]

]
, (2.67)

61



Substituting Eq. (2.66), we obtain:

δE =
α0

ε0

δE

δε(r)

∣∣∣∣
r=r′

+ µ0β0
δE

δµ(r)

∣∣∣∣
r=r′

. (2.68)

Although the perturbation δε is singular, the Taylor expansion is not performed with

respect to the delta function itself, but rather in terms of a small physical parameter

— the atomic polarizability α0. We model the permittivity profile as a background

plus a localized perturbation due to the atom:

ε(x) = εbg(x) + δε(x), with δε(x) =
α0

ϵ0
fη(x− r), (2.69)

where fη is a smooth, normalized bump function localized near the atomic position

r with characteristic width η ≪ λ (the field variation scale). The energy functional

is then expanded in α0:

E[ε+ δε] = E[ε] +
α0

ϵ0

∫
dDx

δE

δε(x)
fη(x− r) +O(α2

0). (2.70)

In the point-dipole limit η → 0, we identify fη(x− r) → δ(x− r), and the first-order

correction becomes

∆E =
α0

ϵ0

δE

δε(x)

∣∣∣∣
x=r

. (2.71)

Thus, the delta function is not introduced directly in the expansion, but rather

as a notational shorthand for a localized finite distribution, always accompanied by

the small factor α0. The limit is taken after the functional derivative is evaluated.

This procedure is consistent with the physical assumption that the atom is much

smaller than the wavelength of the field, justifying its treatment as a point dipole.

62



The delta function is a convenient and well-defined distributional representation of

this limit.

This expression (2.68) represents the Casimir–Polder potential experienced by

the atom due to its magnetodielectric environment. Physically, it corresponds to

the change in ground-state energy arising from the interaction of the atom’s dipole

moments with vacuum fluctuations modulated by the macroscopic body.

Given the worldline expression of the energy Eq. (2.65), we explicitly calculate

the variations in VTE and the path-averaged material properties, which leads to the

TE-mode Casimir–Polder expression:

V TE
CP (r) =

ℏc
4(2π)D/2

∫ ∞

0

dT
T 1+D/2

〈〈(
α0ε

−1
0 µ(r) + β0µ0ε(r)

)
⟨εµ⟩−3/2e−T ⟨VTE⟩

− β0µ0T
2µ(r)

[
∇2 log µ+∇ log µ · ∇+∇2

]
⟨εµ⟩−1/2e−T ⟨VTE⟩

〉〉
x(t)|x(0)=r

. (2.72)

The TM expression can be obtained via the duality symmetry β0µ0 ↔ α0/ε0, µ ↔

ε, VTE ↔ VEM. The path is pinned to start and end at the atomic location. This is

the consequence of the delta functions present in the atomic perturbations. Detailed

calculations of inter-atomic Casimir–Polder potentials will be shown in Chapter IV.

The renormalization of Casimir–Polder worldlines is addressed as follows. For an

atom located at position r in otherwise empty space, the divergent contribution to

the Casimir–Polder potential arises from the field modes interacting with the point-

like atomic polarizability. This divergence is canceled by subtracting the reference

configuration in which all material boundaries are moved to infinity:

Vren(r) = V (r)− V∞(r), (2.73)

63



(")

($)

FIGURE 2.2. Casimir–Polder worldlines pinned to the atomic position. (a) The
vacuum loop does not contribute to the potential. (b) Only the one that sojourns
the medium contributes.

where V∞(r) denotes the potential the atom would experience if the material bodies

were infinitely far removed. This subtraction removes the T → 0 divergence in the

worldline path integral representation, yielding a finite, renormalized Casimir–Polder

energy.

On the other hand, consider an atom embedded within the dielectric half-space of

a vacuum-dielectric planar interface. Even though the atom is inside a homogeneous

material, the presence of the interface modifies the vacuum fluctuations and leads

to a nontrivial potential. However, the total energy again contains an unphysical

divergent part corresponding to the bulk dielectric environment. To isolate the

interaction with the interface, we subtract the contribution from a configuration in

which the interface is moved infinitely far away, while the atom remains embedded

in the same dielectric medium:

V (interface)
ren (r) = V (r)− Vbulk(r), (2.74)

64



where Vbulk(r) is the Casimir–Polder energy of the atom in an infinite homogeneous

dielectric with no boundaries. This ensures that the renormalized potential captures

only the influence of the interface, removing the background self-energy contribution.

In the worldline path-integral formalism, this subtraction is implemented by only

counting particle paths that intersect or are reflected by material boundaries. Paths

entirely confined to homogeneous regions contribute identically in both configurations

and cancel under subtraction. As such, the renormalized energy corresponds to

the difference in path averages between the actual geometry and its reference

configuration, ensuring a finite result. See Fig. 2.2 for an illustration.

2.5 Monte Carlo Simulation

Monte Carlo simulation is a class of numerical techniques that estimate quantities

of interest by averaging over random samples. The method is particularly powerful

for evaluating high-dimensional integrals, solving stochastic differential equations, or

sampling from probability distributions that are analytically intractable.

In a typical setting, Monte Carlo methods are used to compute an expectation

value of the form

E[f(X)] =

∫
f(x)p(x) dx, (2.75)

where p(x) is a probability density function and f(x) is an integrable function.

This expectation can be approximated by sampling independent random variables

x1, . . . , xN from p(x) and computing the sample mean:

E[f(X)] ≈ 1

N

N∑
i=1

f(xi). (2.76)

65



The central limit theorem ensures that, as N → ∞, the error in this

approximation scales as 1/
√
N , independent of the dimensionality of x. This

dimension-independence makes Monte Carlo methods especially attractive for

problems involving many degrees of freedom, such as statistical mechanics, quantum

field theory, and Bayesian inference.

Modern variants, such as importance sampling and Markov chain Monte Carlo

(MCMC), are designed to improve efficiency and reduce variance by tailoring the

sampling strategy to the specific structure of the problem. For the worldline

approach, Monte Carlo simulation provides a natural numerical method to estimate

functional integrals over stochastic paths, by generating ensembles of random

trajectories and computing observables as statistical averages over them.

In the context of Casimir calculations, a typical worldline expression of an

observable takes the form of

2.5.1 Path Generation

The numerical evaluation of worldline path integrals is central to the Casimir

calculations presented in this thesis. Essentially, it relies on sampling an ensemble

of stochastic trajectories. These trajectories are realizations of Brownian paths,

discretized in time and conditioned to satisfy certain boundary conditions. The

foundational stochastic process for this purpose is the Brownian bridge, which

represents a Brownian (Wiener) process constrained to begin and end at prescribed

values.

In its simplest form, a Brownian bridge B(t) over the interval t ∈ [0, 1] is a

one-dimensional process satisfying B(0) = B(1) = 0. For intermediate times, it

66



FIGURE 2.3. Linear drift applied to Wiener processes to force the bridge closure.

undergoes Gaussian diffusion according to standard Wiener statistics:

⟨⟨dB(t)⟩⟩ = 0, (2.77)

⟨⟨dB(t) dB(t′)⟩⟩ = δ(t− t′) dt. (2.78)

A discrete representation of such a path can be generated numerically by sampling

N steps {Bj}Nj=0 with B0 = BN = 0, and evolving the intermediate points via a

recurrence relation.

The algorithm used in this thesis to construct discrete Brownian bridges is

adapted from the so-called “v-loop” method introduced by Gies et al. [22] and

modified in [27]. It generates successive points Bj by

Bj =

√
cj
N
zj + cjBj−1, j = 1, . . . , N − 1, (2.79)

where zj are independent standard normal random variables, B0 = 0, BN = 0, and

the recursion coefficients are defined by

cj =
N − j

N − j + 1
. (2.80)

67



This procedure yields realizations of paths consistent with the Brownian bridge

measure corresponding to fixed endpoints at the origin.

In more sophisticated numerical procedures — such as the partial-averaging

method introduced in Chapter III — one requires Brownian bridges with arbitrary

endpoints, B(0) = a and B(1) = b. The recurrence (2.79) generalizes in this case to

Bj =
√
cj∆t zj + cj(Bj−1 − b) + b, j = 1, . . . , N − 1, (2.81)

where ∆t = 1/N is the time step, and the endpoint conditions B0 = a and BN = b

are enforced.

In several applications, such as the Hermite–Gaussian partial averaging method

in Sec. 3.2.2, only a segment of the path is constructed a priori (e.g., the first m

and last m time steps), while the intermediate section must be filled in using a

conditioned Brownian bridge. Suppose we have determined the value of the path at

some intermediate index k, i.e., Bk is known, and we wish to sample the remaining

Nk = N − k steps from t = k/N to t = 1 subject to the final endpoint BN = 0.

The remaining portion is then generated by shifting the generalized bridge

formula (2.81):

Bj =

√
cj
N
zj + cj(Bj−1 − b) + b, j = k + 1, . . . , N − 1, (2.82)

with Bk as the starting point, b = BN = 0 as the final endpoint, and recursion

coefficients

cj =
N − j

N − j + 1
. (2.83)

68



Note that the indices in Eq. (2.82) have been shifted to match the segment j ∈

{k + 1, . . . , N − 1}, and thus this process can be interpreted as the final Nk steps of

a standard bridge over [0, 1] translated to begin at t = k/N .

This completion method plays an essential role in ensuring the consistency of path

segments and facilitates recursive evaluation of quantities such as path derivatives,

curvature estimators, or reweighted observables. It also ensures compatibility with

boundary or symmetry constraints imposed by the underlying physics of the Casimir

interaction being modeled.

2.5.2 Proper Time Sampling

In the worldline representation of Casimir energies, the quantum effective action

is expressed as a path integral over an ensemble of closed trajectories, each scaled by

a proper time T that parametrizes the physical size of the loop. The total energy or

force is computed by averaging the contribution of such loops over a distribution of

T , weighted by a characteristic power-law kernel.

However, for a given Brownian path, its contribution to the renormalized

integrand is zero unless the loop intersects all of the relevant bodies in the geometry.

This condition is met only when T exceeds a minimal value T0, which is the first

proper time at which the path intersects the geometry as required. For example:

– In the worldline approach to Casimir energies, the renormalized interaction

energy arises from paths that intersect all relevant bodies. For instance,

consider a one-dimensional Brownian bridge starting at a point x0 located

between two parallel planes at positions L1 < x0 < L2. A given path only

contributes to the Casimir interaction energy if it touches both boundaries

during its evolution.

69



To determine the minimal proper time T required for such a path to reach both

boundaries, we must consider the maximal spatial excursion of the Brownian

path relative to its total proper time. In one dimension, the spatial extent of a

standard Brownian bridge scales as ∼
√
T . Therefore, the condition for a path

to reach the lower or upper plane is: |x0 − L1,2| ≲ xmax ∼
√
T . Inverting this

gives the minimal proper time needed to reach each boundary: T1,2 =
(x0−L1,2)2

x2
±

where x± is the normalized extremum (typically the maximum displacement

from the bridge trajectory, which is sampled over the ensemble). The smallest

such proper time T0 that ensures intersection with both objects is then:

T0 = max

[
(L1 − x0)

2

x2−
,
(L2 − x0)

2

x2+

]
. (2.84)

This cutoff time determines the lower limit of integration in the worldline

representation of the Casimir energy. Paths with T < T0 are too short to

reach both boundaries and hence do not contribute to the interaction part of

the energy.

– In Casimir–Polder energies, T0 is the time beyond which the path intersects

the extended body (excluding the atom itself).

This motivates the use of an importance sampling distribution for T , tailored

to each path, that captures both the lower bound T0 and the asymptotic power-law

behavior of the kernel. A particularly effective choice is the truncated power-law:

P (T ; T0,m) =
(m− 1)T m−1

0

T m
Θ(T − T0), (2.85)

70



where m > 1 and Θ is the Heaviside step function. This choice ensures that P (T )

is properly normalized over T ∈ [T0,∞), and mimics the power-law decay of the

integrand while avoiding unnecessary sampling below the physical cutoff T0.

To generate random samples of T ∼ P (T ; T0,m), one uses the inverse transform

sampling method. The cumulative distribution function (CDF) is

F (T ) =

∫ T

T0
P (T ′; T0,m) dT ′ = 1−

(
T0

T

)m−1

, (2.86)

whose inverse is

T (u) = T0 (1− u)−1/(m−1), (2.87)

where u ∈ (0, 1) is a uniformly distributed random number. Thus, the sampling

algorithm is as follows:

1. For each loop, estimate the minimal contributing time T0 based on the rescaled

geometry.

2. Generate a uniform random variable u ∈ (0, 1).

3. Compute T = T0(1− u)−1/(m−1).

In practice, T0 is estimated per path by scaling the Brownian loop and checking

whether it intersects the geometry. This may involve a search over T for which

the loop first touches all required surfaces or in symmetric geometries, an analytic

expression for the minimal spanning time.

Once T0 is determined, one draws T ∼ P (T ; T0,m) and rescales the path

accordingly. The integrand for the Casimir energy is then evaluated and reweighted

by T −1−D/2/P (T ), preserving the correctness of the total integral.

71



This strategy is essential in reducing the cost of computing Casimir interactions

across a wide range of length scales. By focusing sampling effort on physically

relevant regions of proper time, the truncated power-law distribution ensures

accurate estimates with fewer Monte Carlo samples.

2.5.3 Path Source Point Reweighting

In the worldline path integral formulation of Casimir energies, one often

encounters an integral over the spatial base point x0 of a closed Brownian path.

Physically, x0 represents the center of mass or anchor point of the loop, and

its integration accounts for all possible global translations of the loop in space.

Mathematically, this appears in expressions such as

E ∝
∫ ∞

0