Casimir effect in quantum field theory

Abstract

A new conceptual foundation for renormalizing ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ on locally flat space-time---to obtain the so-called Casimir effect---is presented. The Casimir ground state is viewed locally as a (nonvacuum) state on Minkowski space-time and the expectation value of the normal-ordered ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ is taken. The same ideas allow us to treat, for the first time, self-interacting fields for arbitrary mass in perturbation theory---using traditional flat-space-time renormalization theory. First-order results for zero-mass $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory agree with those recently announced by Ford. We point out the crucial role played by the simple renormalization condition that the vacuum expectation value of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ must vanish in Minkowski space-time, and in a critical discussion of other approaches, we clarify the question of renormalization ambiguities for ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ in curved space-times. In an Appendix, we show how the Casimir effect arises in the ${C}^{*}$-algebra approach to quantum field theory.