The thermal and two-particle stress-energy must be ill-defined on the 2-d Misner space chronology horizon

TL;DR

This paper demonstrates that in two-dimensional Misner space, two-particle and thermal states can have finite stress-energy in the initial region, but these states inevitably become ill-defined on the Cauchy horizon due to singularities.

Contribution

It extends the understanding of quantum states in Misner space by showing the inevitability of stress-energy divergence on the horizon despite finite initial values.

Findings

Stress-energy tensor vanishes in initial globally hyperbolic region.

Two-particle and thermal states share singularities with the conformal vacuum.

Stress-energy tensor becomes ill-defined on the Cauchy horizon.

Abstract

We show that an analogue of the (four dimensional) image sum method can be used to reproduce the results, due to Krasnikov, that for the model of a real massless scalar field on the initial globally hyperbolic region IGH of two-dimensional Misner space there exist two-particle and thermal Hadamard states (built on the conformal vacuum) such that the (expectation value of the renormalised) stress-energy tensor in these states vanishes on IGH. However, we shall prove that the conclusions of a general theorem by Kay, Radzikowski and Wald still apply for these states. That is, in any of these states, for any point b on the Cauchy horizon and any neighbourhood N of b, there exists at least one pair of non-null related points (x,x'), with x and x' in the intersection of IGH with N, such that (a suitably differentiated form of) its two-point function is singular. (We prove this by showing that the two-point functions of these states share the same singularities as the conformal vacuum on which they are built.) In other words, the stress-energy tensor in any of these states is necessarily ill-defined on the Cauchy horizon.