Vacuum polarization on the spinning circle

TL;DR

This paper studies vacuum polarization effects of a scalar field around a two-dimensional spinning cosmic string, highlighting issues with non-globally hyperbolic spacetimes and mode choices affecting physical acceptability.

Contribution

It demonstrates the importance of global hyperbolicity for well-behaved vacuum fluctuations and critiques previous mode choices that lead to unphysical two-point functions.

Findings

Vacuum fluctuations are well-behaved in globally hyperbolic spacetime.

Divergences occur at the chronology horizon in non-globally hyperbolic spacetime.

Previous mode choices yield non-Hadamard two-point functions, which are unphysical.

Abstract

Vacuum polarization of a massive scalar field in the background of a two-dimensional version of a spinning cosmic string is investigated. It is shown that when the `radius of the universe' is such that spacetime is globally hyperbolic the vacuum fluctuations are well behaved, diverging though on the `chronology horizon'. Naive use of the formulae when spacetime is nonglobally hyperbolic leads to unphysical results. It is also pointed out that the set of normal modes used previously in the literature to address the problem gives rise to two-point functions which do not have a Hadamard form, and therefore are not physically acceptable. Such normal modes correspond to a locally (but not globally) Minkowski time, which appears to be at first sight a natural choice of time to implement quantization.