TY - JOUR
T1 - Quantum field theory in Lorentzian universes from nothing
AU - Friedman, John L
AU - Higuchi, Atsushi
PY - 1995
Y1 - 1995
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0009096623&partnerID=8YFLogxK
U2 - 10.1103/PhysRevD.52.5687
DO - 10.1103/PhysRevD.52.5687
M3 - Article
AN - SCOPUS:0009096623
SN - 0556-2821
VL - 52
SP - 5687
EP - 5697
JO - Physical Review D
JF - Physical Review D
IS - 10
ER -