Quantum field theory in Lorentzian universes-from-nothing

TL;DR

This paper explores the challenges of defining quantum field theory in Lorentzian universes-from-nothing, highlighting issues with local algebras and states, and proposing restrictions to maintain consistency with global hyperbolic spacetimes.

Contribution

It demonstrates how to construct a consistent local algebraic QFT in nonorientable Lorentzian spacetimes by restricting neighborhood sizes, revealing limitations compared to globally hyperbolic cases.

Findings

Local algebras are indistinguishable from those in globally hyperbolic spacetimes.

Restrictions on neighborhood size are necessary for consistent state extension.

Global correlations are limited due to neighborhood restrictions.

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.