Field Quantization and Vacuum Energy in the Presence of Boundaries
From the standpoint of quantum field theory, the Casimir effect is related to the vacuum polarization that arises in quantization volumes restricted by boundaries or in spaces with nontrivial topology. Both boundaries and the nontrivial topology of space-time can be considered as classical external conditions, on which background the field quantization should be performed. This chapter presents the basic facts related to the quantization procedure for fields of various spins obeying boundary (or identification) conditions. It starts with the classical wave equations and then considers various boundary conditions. The rest of the chapter is devoted to both the canonical and path-integral field quantization procedures in the presence of boundaries and to different representations for the vacuum energy. Propagators with boundary conditions are also introduced. Although fields of different spin are touched upon, the presentation is primarily devoted to the case of the electromagnetic field in the presence of material boundaries.
Keywords: Noether theorem, boundary conditions, identification conditions, curved space-time, metrical energy-momentum tensor, field tensor, Maxwell equations, gauge fixing term, Lorentz gauge, Coulomb gauge
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