On a Recent Construction of "Vacuum-like" Quantum Field States in Curved Spacetime

TL;DR

This paper rigorously analyzes the S-J state construction for scalar fields in curved spacetimes, showing it is well-defined and pure in certain cases but generally not Hadamard or unitarily equivalent to Hadamard states, especially in ultrastatic spacetimes.

Contribution

It provides a rigorous foundation for the S-J state construction and identifies its limitations in representing physically acceptable quantum states in curved spacetime.

Findings

S-J states are well-defined and pure in globally hyperbolic spacetimes.

S-J states are generally not Hadamard states in ultrastatic spacetimes.

Representation induced by S-J states often not unitarily equivalent to Hadamard states.

Abstract

Afshordi, Aslanbeigi and Sorkin have recently proposed a construction of a distinguished "S-J state" for scalar field theory in (bounded regions of) general curved spacetimes. We establish rigorously that the proposal is well-defined on globally hyperbolic spacetimes or spacetime regions that can be embedded as relatively compact subsets of other globally hyperbolic spacetimes, and also show that, whenever the proposal is well-defined, it yields a pure quasifree state. However, by explicitly considering portions of ultrastatic spacetimes, we show that the S-J state is not in general a Hadamard state. In the specific case where the Cauchy surface is a round 3-sphere, we prove that the representation induced by the S-J state is generally not unitarily equivalent to that of a Hadamard state, and indeed that the representations induced by S-J states on nested regions of the ultrastatic spacetime also fail to be unitarily equivalent in general. The implications of these results are discussed.