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In this dissertation we study the Casimir effect, which is demonstrated to be a manifestation of the quantum vacuum. The boundary conditions are imposed by constructing δ-function potentials, so-called semitransparent boundaries. The coupling to the δ-function potential reduces to the Dirichlet boundary condition in the strong coupling limit. In the case of electrodynamics the strong coupling limit corresponds to metallic plates, and the weak coupling limit corresponds to tenuous dielectrics.
In chapter 1 we derive the expression for the vacuum energy in the presence of a background for the case of scalar fields. The vacuum energy, or the Casimir energy, is expressed in terms of the trace of the logarithm of Green's functions. Thus the problem is reduced to solving a second order differential equation for the Green's function in the presence of a background potential. The Casimir energy is compared to the alternate expression derived from the energy-momentum tensor associated with the field in the presence of a background. The Casimir force is interpreted as the force between two objects arising due to the change in energy when the distance between their center of masses is varied. The Casimir force is also identified in terms of the divergence of the energy-momentum tensor.
In Chapter 2 we study the differential equations satisfied by Green's function and present solutions to the Green's functions which will be used in this thesis.
Using the expressions derived in chapters 1 and 2 we derive the Casimir energy associated with a single plate, and two parallel plates, in chapter 3. The infinite energy associated with the vacuum in the absence of a potential, or the plates, which is a divergent quantity, is isolated. The Casimir energy associated with a single plate is shown to be divergent. The Casimir energy for two parallel plates is written as a sum of three terms. Two of these terms correspond to the energy associated with the two single plates constituting the parallel plates, each of which is divergent. The third term which is finite is interpreted to arise due to the interaction between the two plates. The Casimir force on a single plate is evaluated to be zero. This is expected because we do not change the energy by shifting a single plate due to translational symmetry. The same argument applies for the two divergent terms associated with the single plates forming the parallel plates. But the third finite term does contribute to a change in the energy when the distance between the plates is changed. We derive an explicit expression for the Casimir force between parallel plates for arbitrary couplings to the δ-function potentials. In the strong coupling limit we recover the Casimir force in the Dirichlet limit for the case of scalar fields.
Even though the divergent terms in the total energy associated with the single plates constituting two parallel plates do not contribute to the Casimir force, the situation still appears unsatisfactory. Any form of energy acts as a 'source' in the Einstein's equation and thus falls under gravity. The classic example is that of bending of light around a planet. How do the divergent terms associated with single plates interact with gravity? In chapter 4 we consider parallel plates falling in a weak gravitational field described by a space-time given in terms of Rindler coordinates. For the case of parallel plates we show that all the three terms contributing to the total energy interact with a weak gravitational field exactly like a conventional mass. We interpret this as meaning that the divergent terms associated with the single plates serve to renormalize the bare masses of the single plates.
In chapter 5 we generalize our results for other inertial forces. In particular we demonstrate that the results of chapter 4 hold for the case of two parallel plates rotating with a constant angular speed. We show that exactly as in the case of Rindler acceleration, the Casimir energy, including its divergent parts, experiences a centripetal force exactly like a conventional mass. In chapter 4 our analysis is restricted to the case when the direction of the Rindler acceleration is in a direction perpendicular to the plane containing the plates. In chapter 5 we generalize this result for the case when the parallel plates makes an arbitrary angle with respect to the tangent to the circle of rotation. We show that the centripetal force is independent of the orientation of the parallel plates in a frame in which the center of inertia is at the origin.
In chapter 6 we study parallel plates with sinusoidal corrugations. (Abstract shortened by UMI.)
Advisor: | Milton, Kimball A. |
Commitee: | Branch, David, Dickey, Leonid, Kantowski, Ronald, Kao, Chung |
School: | The University of Oklahoma |
Department: | Department of Physics and Astronomy |
School Location: | United States -- Oklahoma |
Source: | DAI-B 69/07, Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Physics, Particle physics |
Keywords: | |
Publication Number: | 3315897 |
ISBN: | 9780549675174 |