Completeness principle and quantum field theory on nonglobally hyperbolic spacetimes

TL;DR

This paper investigates the challenges of quantizing scalar fields on spacetimes with closed timelike curves, revealing the incompleteness of traditional solutions and proposing a modified quantization approach based on a new completeness principle.

Contribution

It introduces a completeness principle for quantum field theory on nonglobally hyperbolic spacetimes and develops a modified quantization method applicable in such contexts.

Findings

Standard positive and negative frequency solutions are incomplete on spacetimes with closed timelike curves.

The modified quantization procedure yields consistent results for the Hadamard function and <φ^2>.

Naive image method results align with the proposed approach.

Abstract

We analyse in details the problems which one faces trying to quantize a scalar field on the spacelike cylinder being the simple example of a spacetime with closed timelike curves. Our analysis brings to light the fact that the usual set of positive and negative frequency solutions of the field equation turns out to be incomplete. The consequence of this fact is that the usual formulation of quantum field theory breaks down on such a spacetime. We postulate the completeness principle and build on its basis the modified quantization procedure. As an example, the Hadamard function and $<\phi^2>$ for the scalar field on the spacelike cylinder are calculated. It is shown that the ``naive'' method of images gives the same results of calculation.