On the (Non)Hadamard Property of the SJ State in a $1+1$D Causal Diamond

TL;DR

This paper investigates whether the Sorkin-Johnston (SJ) state in a 1+1D causal diamond satisfies the Hadamard condition, revealing it does not at the boundary, and explores a modified softened SJ state to address this issue.

Contribution

It demonstrates the non-Hadamard nature of the SJ state at the boundary and introduces a softened version to achieve Hadamard properties, linking to entanglement entropy features.

Findings

SJ state is non-Hadamard at the boundary

Softened SJ state can be made Hadamard

Potential connection to entanglement entropy in causal set theory

Abstract

The Sorkin-Johnston (SJ) state is a candidate physical vacuum state for a scalar field in a generic curved spacetime. It has the attractive feature that it is covariantly and uniquely defined in any globally hyperbolic spacetime, often reflecting the underlying symmetries if there are any. A potential drawback of the SJ state is that it does not always satisfy the Hadamard condition. In this work, we study the extent to which the SJ state in a $1+1$D causal diamond is Hadamard, finding that it is not Hadamard at the boundary. We then study the softened SJ state, which is a slight modification of the original state to make it Hadamard. We use the softened SJ state to investigate whether some peculiar features of entanglement entropy in causal set theory may be linked to its non-Hadamard nature.