Quantum field theory in Lorentzian universes from nothing

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. A time-nonorientable, spacelike hypersurface serves as a generalized Cauchy surface, a surface on which freely specified initial data for wave equations have unique global time evolutions. A simple example is antipodally identified de Sitter space. Classically, such spacetimes are locally indistinguishable from their globally hyperbolic covering spaces. The construction of a quantum field theory is more problematic. Time nonorientability precludes the existence of a global algebra of obsevables, and hence of global states, regarded as positive linear functions on a global algebra. One can, however, define a family of local algebras on an atlas of globally hyperbolic subspacetimes, with overlap conditions on the intersections of neighborhoods. This family locally coincides with the family of algebras on a globally hyperbolic spacetime; and one can ask whether a sensible quantum field theory is obtained if one defines a state as an assignment of a positive linear function to every local algebra. We show, however, that the extension of a generic positive linear function from a single algebra to the collection of all local algebras violates positivity: one cannot find a colleciton of quantum states satisfying the physically appropriate overlap conditions. One can overcome this difficulty by artificially restricting the size of neighborhoods in a way that has no classical counterpart.

Neighborhoods in the atlas must be small enough that the union of any pair is time orientable. Correlations between field operators at a pair of points are then defined only if a curve joining the points lies in a single neighborhood. Any state on one neighborhood of an atlas can be extended to a collection of states on the atlas, and the structure of local algebras and states is thus locally indistinguishable from quantum field theory on a globally hyperbolic spacetime. But the artificiality of the size restriction on neighborhoods means that the structure is not a satisfactory global field theory. The structure is not unique, because there is no unique maximal atlas. The resulting theory allows less information than quantum field theory in a globally hyperbolic spacetime, because there are always sets of points in the spacetime for which no correlation function is defined. Finally, in showing that one can 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.