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.