Quantum Field Theory on Certain Non-Globally Hyperbolic Spacetimes

TL;DR

This paper investigates algebraic quantum field theory on specific non-globally hyperbolic spacetimes with closed timelike curves, demonstrating F-quantum compatibility for various fields on spacelike cylinders.

Contribution

It extends the concept of F-quantum compatibility to massive fields and lower-dimensional spacetimes, constructing suitable F-local algebras in these cases.

Findings

4D spacelike cylinder is F-quantum compatible with massive fields

2D spacelike cylinder is F-quantum compatible with both massive and massless fields

Constructs explicit F-local algebras demonstrating compatibility

Abstract

We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes containing closed timelike curves within the framework proposed by Kay for algebraic quantum field theory on non-globally hyperbolic spacetimes. In this context, a spacetime (M,g) is said to be `F-quantum compatible' with a field theory if it admits a *-algebra of local observables for that theory which satisfies a locality condition known as `F-locality'. Kay's proposal is that, in formulating algebraic quantum field theory on $(M,g)$, F-locality should be imposed as a necessary condition on the *-algebra of observables. The spacetimes studied are the 2- and 4-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the 4-dimensional spacelike cylinder is F-quantum compatible with massless fields. We prove that it is also F-quantum compatible with massive fields and prove the F-quantum compatibility of the 2-dimensional spacelike cylinder with both massive and massless fields. In each case, F-quantum compatibility is proved by constructing a suitable F-local algebra.