Stress-Energy Must be Singular on the Misner Space Horizon even for Automorphic Fields

Abstract

We use the image sum method to reproduce Sushkov's result that for a massless automorphic field on the initial globally hyperbolic region $IGH$ of Misner space, one can always find a special value of the automorphic parameter $\alpha$ such that the renormalized expectation value $\langle\alpha|T_{ab}|\alpha\rangle$ in the {\it Sushkov state} ``$\langle\alpha|\cdot|\alpha\rangle$'' (i.e. the automorphic generalization of the Hiscock-Konkowski state) vanishes. However, we shall prove by elementary methods that the conclusions of a recent general theorem of Kay-Radzikowski-Wald apply in this case. I.e. for any value of $\alpha$ and any neighbourhood $N$ of any point $b$ on the chronology horizon there exists at least one pair of non-null related points $(x,x') \in (N\cap IGH)\times (N\cap IGH)$ such that the renormalized two-point function of an automorphic field $G^\alpha_{\rm ren}(x,x')$ in the Sushkov state is singular. In consequence $\langle\alpha|T_{ab}|\alpha\rangle$ (as well as other renormalized expectation values such as $\langle\alpha|\phi^2|\alpha\rangle$) is necessarily singular {\it on} the chronology horizon. We point out that a similar situation (i.e. singularity {\it on} the chronology horizon) holds for states on Gott space and Grant space.