Vacuum polarization on the spinning circle

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 formulas 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