Some Aspects of Quantum Field Theory in Non-Minkowskian Space-Times

This thesis is concerned with several aspects of quantum field theory in space-times which are different from Minkowski space-time. The difference may be due either to the presence of a non-zero curvature, or else be a consequence of the topology of the manifold. An example of quantum field theory in a space -time which is flat, but with a topology which is distinct from that of Minkowski space-time is the Casimir effect. A short review of some of the popular derivations are presented along with comments. The analogy with field theory at a finite temperature is discussed. We also consider why it is sometimes possible to obtain results which are finite by adopting a regularization technique only without the necessity of performing renormalization, and discuss some limitations of this. Renormalization of interacting scalar field theories in a flat space-time with a non-Minkowskian topology is considered. The Schwinger-Dyson equations for a general scalar field theory with a quartic interaction are derived using functional techniques. An introduction to twisted fields is presented. The renormalization of interacting twisted and untwisted scalar fields to second order in perturbation theory in a flat spacetime with the topology S('1) x R('3) is discussed. It is seen that the presence of a non-trivial topology can lead to additional non-local divergent terms, beyond those divergences which are present in Minkowski space-time; however, when all graphs of the same order are combined, all such non-local divergences cancel among each other, and the theory may be renormalized with the same choice of counterterms as in Minkowski space -time. A further feature which is discussed is that the existence of a non-trivial topology can cause the propagators to develop poles corresponding to the generation of a topological mass. The functional approach to the effective potential is reviewed, and it is shown how zeta-function regularization fits naturally into this scheme. This formalism is applied to calculate the effective potential for some scalar field theories in non-Minkowskian space-times. Topological mass generation is discussed, and it is also shown how radiative corrections can lead to spontaneous symmetry breaking in some cases. One- and two-loop contributions to the vacuum energy density are obtained for both massless and massive fields. In the massive case the role of renormalization in removing non-local divergences is discussed.