Lorentzian universes from nothing

Lorentzian universes from nothing, are spacetimes with a single spacelike boundary that nevertheless have smooth Lorentzian metrics. They are the Lorentzian counterpart of spacetimes with no past boundary that appear in the Hartle-Hawking prescription for a wavefunction of the universe. One can always choose metrics for which these Lorentzian spacetimes have no closed timelike curves; time nonorientability is then their only causal pathology. Classically, such spacetimes are locally indistinguishable from their globally hyperbolic covering spaces, and the initial-value problem for classical fields is globally well defined.

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. The difficulty can be overcome by restricting the size of neighbourhoods 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 the theory allows too little information to fix the global evolution of a state, because correlations between field operators at a pair of points are defined only if a curve joining the points lies in a single neighbourhood.

One could hope that the difficulties are restrictions on the observables in a generalized sum-over-histories approach, but the conjecture remains unexplored.