Quantum field theory in Lorentzian universes from nothing.

Abstract

We examine quantum field theory in spacetimes that are time nonorientable but have no other causal pathology. These are Lorentzian universes-from-nothing, spacetimes with a single spacelike boundary that nevertheless have a smooth Lorentzian metric. Classically, such spacetimes are locally indistinguishable from their globally hyperbolic covering spaces. However, the construction of a quantum field theory (QFT) is more problematic. One can define a family of local algebras on an atlas of globally hyperbolic subspacetimes. But one cannot extend a generic positive linear function from a single algebra to the collection of all local algebras without violating positivity, while satisfying the physically appropriate overlap conditions. This difficulty can be overcome by restricting the size of neighborhoods so that the union of any pair is time- orientable. The structure of local algebras and states is then locally indistinguishable from that of QFT on a globally hyperbolic spacetime. But this size restriction on neighborhoods makes the structure unsatisfactory as a global field theory. The theory allows less information than QFT in a globally hyperbolic spacetime, because correlations between field operators at a pair of points are defined only if a curve joining the points lies in a single neigh- borhood. Moreover, to extend a local state to a collection of states, we use an antipodally symmetric state on the covering space, a state that would not yield a sensible state on the spacetime if all correlations could be measured.