QUANTUM-FIELD THEORY IN LORENTZIAN UNIVERSES FROM NOTHING

We examine quantum held theory in spacetimes that are time nonorientab le but have no other causal pathology. These are Lorentzian universes from nothing, spacetimes with a single spacelike boundary that neverth eless have a smooth Lorentzian metric. A time-nonorientable, spacelike hypersurface serves as a generalized Cauchy surface, a surface on whi ch 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 fro m their globally hyperbolic covering spaces. The construction of a qua ntum field theory is more problematic. Time nonorientability precludes the existence of a global algebra of observables, and hence of global states, regarded as positive linear functions on a global algebra. On e can, however, define a family of local algebras on an atlas of globa lly hyperbolic subspacetimes, with overlap conditions on the intersect ions of neighborhoods. This family locally coincides with the family o f 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. W e show, however, that the extension of a generic positive Linear funct ion from a single algebra to the collection of all local algebras viol ates positivity: one cannot find a collection of quantum states satisf ying the physically appropriate overlap conditions. One can overcome t his difficulty by artificially restricting the size of neighborhoods i n a way that has no classical counterpart. Neighborhoods in the atlas must be small enough that the union of any pair is time orientable. Co rrelations between field operators at a pair of points are then define d only if a curve joining the points Lies in a single neighborhood. An y state on one neighborhood of an atlas can be extended to a collectio n of states on the atlas, and the structure of local algebras and stat es is thus locally indistinguishable from quantum held theory on a glo bally hyperbolic spacetime. But the artificiality of the size restrict ion on neighborhoods means that the structure is not a satisfactory gl obal field theory. The structure is not unique, because there is no un ique maximal atlas. The resulting theory allows less information than quantum held theory in a globally hyperbolic spacetime, because there are always sets of points in the spacetime for which no correlation fu nction is defined. Finally, in showing that one can extend a local sta te 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.