GEOMETRICAL INVESTIGATIONS OF THE CASIMIR EFFECT: THICKNESS AND CORRUGATION DEPENDENCIES

Abstract

In the quantum theory the vacuum is not empty space. It is considered as a state

of infinite energy arising due to zero point fluctuations of the vacuum. Calculation

of any physically relevant process requires subtracting this infinite energy using a

procedure called normalization. As such the vacuum energy is treated as an infinite

constant. However, it has been established beyond doubt that mere subtraction

of this infinite constant does not remove the effect of vacuum fluctuations and it

cannot be treated just as a mathematical artifact. The presence of boundaries, which

restricts the vacuum field, causes vacuum polarization. Any non-trivial space-time

topology can cause similar effects. This is manifested as the Casimir effect, whereby

the boundaries experience a force due to a change in the energy of the vacuum. To

calculate the vacuum energy we treat the boundaries or other restrictive conditions

as classical backgrounds, which impose boundary conditions on the solution of the

vacuum field equations. Alternatively, we can incorporate the classical background in

the Lagrangian of the system as classical potentials, which automatically include the

boundary conditions in the field equations. Any change in the boundary conditions

changes the vacuum energy and consequently the Casimir force is experienced by the

boundaries.

In this dissertation we study the geometric aspect of the Casimir effect. We consider both the scalar field and the physically relevant electromagnetic field. After

a brief survey of the field in Chapter 1, we derive the energy expression using the

Schwinger\'s quantum action principle in Chapter 2. We present the multiple scattering

formalism for calculating the vacuum energy, which allows us to calculate the

interaction energy between disjoint bodies and subtract out the divergent terms from

the beginning. We then solve the Green\'s dyadic equation for the electromagnetic

field interacting with the planar background surfaces, where we can decompose the

problem into two transverse scalar modes. In Chapter 3 we collect all the solutions

for the scalar Green\'s functions for the planar and the cylindrical geometries, which

are relevant for this dissertation.

In Chapter 4 we derive the interaction energy between two dielectric slabs of finite

thickness. Taking the thickness of the slabs to infinity leads to the Lifshitz results

for the two infinite dielectric semi-spaces, while taking the dielectric permittivity to

infinity gives the well-known Casimir energy between two perfect conductors. We

then present a simple model to consider the thin-plate limit (taking the thickness

of the slabs to zero) based on Drude-Sommerfeld free electron gas model, which

modifies the plasma frequency of the material to include the finite size dependence.

We get a non-vanishing result for the Lifshitz energy in the slab thickness going to

zero limit. This is remarkable progress as it allows us to understand the infinitesimal

thickness limit and opens a possibility of extending this model to apply it to graphene

and other two dimensional surfaces. The Casimir and Casimir-Polder results in the

perfect conductor limit give us the expected results.

In Chapter 5 we study the lateral Casimir torque between two concentric corrugated

cylinders described by -potentials, which interact through a scalar field. We

derive analytic expressions for the Casimir torque for the case when the corrugation amplitudes are small in comparison to the corrugation wavelengths. We derive explicit

results for the Dirichlet case, and exact results for the weak coupling limit, in

the leading order. The results for the corrugated cylinders approach the corresponding

expressions for the case of corrugated parallel plates in the limit of large radii of

the cylinders (relative to the difference in their radii) while keeping the corrugation

wavelength fixed.

In Chapter 6 we calculate the lateral Casimir energy between corrugated parallel

dielectric slabs of finite thickness using the multiple scattering formalism in the perturbative

approximation and obtain a general expression, which is applicable to real

materials. Taking the thickness of the plates to infinity leads us to the lateral Lifshitz

formula for the force between corrugated dielectric surfaces of infinite thickness. Taking

the dielectric constant to infinity leads us to the conductor limit which has been

evaluated earlier in the literature. Taking the dilute dielectric limit gives the van der

Waals interaction energy for the corrugated slabs to the second order in corrugation

amplitude. The thin plate approximation proposed in Chapter 4 is used to derive the

Casimir energy between two corrugated thin plates. We note that the lateral force

between corrugated perfectly conducting thin plates is identical to the ones involving

perfectly conducting thick plates. We also evaluate an exact expression (in terms of

a single integral) for the lateral force between corrugated (dilute) dielectric slabs.