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

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 $⟨\alpha \mathrm{\mid }{T}_{ab}\mathrm{\mid }\alpha ⟩$ in the {\it Sushkov state} ``$⟨\alpha \mathrm{\mid }\cdot \mathrm{\mid }\alpha ⟩$’’ (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 $(x,x’) \in (N\cap IGH)\times (N\cap IGH)$suchthattherenormalizedtwo-pointfunctionofanautomorphicfield$G^\alpha_{\rm ren}(x,x’)$intheSushkovstateissingular\mathrm{.}Inconsequence$\langle\alpha|T_{ab}|\alpha\rangle$ (as well as other renormalized expectation values such as $⟨\alpha \mathrm{\mid }{\varphi }^2\mathrm{\mid }\alpha ⟩$) 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.