The importance of quantum effects in Kaluza–Klein theory

Abstract

This paper presents a discussion of the role of quantum effects in Kaluza–Klein theories. It is demonstrated why it is not possible to examine the existence of self-consistent solutions induced by quantum corrections to the classical theory if only the vacuum energy is used. The importance of the induced gravity and induced Yang–Mills terms in the effective action are emphasized. General criteria are given for the existence of self-consistent solutions in certain cases, and an expression is given for the gauge-coupling constant. Quantization of five-dimensional gravity with a cosmological constant is considered. Expressions are given for the constants that multiply the induced gravity and Yang–Mills terms in the one-loop effective action for this theory. Although the theory is one-loop finite, the necessity for performing finite renormalizations—a fact that has hitherto been overlooked—is discussed. Results of an analysis of the stability of self-consistent solutions are given, where it is shown why many of the solutions are unstable to small perturbations. A number of prospects for future work are given.