Quantum field theory on certain non-globally hyperbolic spacetimes

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 two- and four-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the four-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 two-dimensional spacelike cylinder with both massive and massless fields. In each case, F-quantum compatibility is proved by constructing a suitable F-local algebra.