| Summary: | The conventional Casimir effect manifests itself as a quantum mechanical force between two plates, that arises from the quantization of the electromagnetic field in the enclosed vacuum. In this thesis the existence is discussed of an extra, topological term in the Casimir energy at finite temperatures. This topological Casimir effect emerges due to the nontrivial topological features of the gauge theory: the extra energy is the result of tunneling transitions between states that are physically the same but topologically distinct. It becomes apparent when examining, for instance, periodic boundary conditions. I explicitly calculate the new term for the simplest example of such a system, a Euclidean 4-torus. By dimensional reduction, this system is closely related to two dimensional electromagnetism on a torus, which is well understood. It turns out that the topological term is extremely small compared to the conventional Casimir energy, but that the effect is very sensitive to an external magnetic field. The external field plays the role of a topological theta parameter, analogous to the theta vacuum in Yang-Mills theory. The topological Casimir pressure and the induced magnetic field show a distinctive oscillation as a function of the external field strength, something that can hopefully be observed experimentally. === Science, Faculty of === Physics and Astronomy, Department of === Graduate
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