Bisolutions to the Klein-Gordon Equation and Quantum Field Theory on 2-dimensional Cylinder Spacetimes

TL;DR

This paper classifies quantum field algebras for Klein-Gordon theory on 2D cylinder spacetimes, showing that timelike cylinders support a unique F-local algebra, while spacelike cylinders do not support any, highlighting differences in quantum compatibility.

Contribution

It provides a classification of F-local quantum field algebras on 2D cylinder spacetimes, revealing conditions under which these algebras exist or are unique.

Findings

Timelike cylinders admit a unique F-local algebra.

Spacelike cylinders do not admit any F-local algebras.

The usual Klein-Gordon algebra is unique on timelike cylinders.

Abstract

We consider 2-dimensional cylinder spacetimes whose metrics differ from the flat Minkowskian metric within a compact region. By choice of time orientation, these spacetimes may be regarded as either globally hyperbolic timelike cylinders or nonglobally hyperbolic spacelike cylinders. For generic metrics in our class, we classify all possible candidate quantum field algebras for massive Klein-Gordon theory which obey the F-locality condition introduced by Kay. This condition requires each point of spacetime to have an intrinsically globally hyperbolic neighbourhood, N, such that the commutator (in the candidate algebra) of fields smeared with test functions supported in N agrees with the value obtained in the usual construction of Klein-Gordon theory on N. By considering bisolutions to the Klein-Gordon equation, we prove that generic timelike cylinders admit a unique F-local algebra -- namely the algebra obtained by the usual construction -- and that generic spacelike cylinders do not admit any F-local algebras, and are therefore non F-quantum compatible. Refined versions of our results are obtained for subclasses of metrics invariant under a symmetry group. Thus F-local field theory on 2-dimensional cylinder spacetimes essentially coincides with the usual globally hyperbolic theory. In particular the result of the author and Higuchi that the Minkowskian spacelike cylinder admits infinitely many F-local algebras is now seen to represent an anomalous case.