Quantum field theory on Clifford-Klein space-times. The effective Lagrangian and vacuum stress-energy tensor

Abstract

The vacuum-averaged stress-energy tensor is calculated for a scalar field on a variety of static space-times, T(X)M₃, where the spatial section, M₃, is a Clifford-Klein space form of the flat or spherical type, R³/ Gamma , or S³/ Gamma . The particular examples when M₃ is a Klein-bottle waveguide, R(X)K₂, or a lens-space, S³/Z_m, are treated in most detail. It is found that the vacuum stress on quotient spaces is not of the same tensorial structure as R_{mu nu} -¹/₂g_{mu nu} R. This leads to difficulties with the back-reaction problem. It is further found that 'twisting' the field alters the vacuum stress compared to the untwisted theory. Values for the total energies in the three types of polyhedral cellular decompositions of S³ are given. Dirichlet boundary conditions for rectangular cavities are also considered.