Aspects of Quantum Fields on Causal Sets

TL;DR

This paper investigates quantum fields on causal sets, constructing Green functions, analyzing the Sorkin-Johnston vacuum in de Sitter space, and exploring entanglement entropy, revealing volume law behavior and conditions for area law emergence.

Contribution

It introduces a method to construct Green functions and the SJ vacuum on causal sets in curved spacetimes, and studies entanglement entropy behavior in de Sitter horizons.

Findings

Causal set SJ vacuum differs from known de Sitter vacua.

Entanglement entropy follows a volume law, with an area law emerging under mode truncation.

Constructed Green functions beyond Riemann normal neighborhoods in curved spacetimes.

Abstract

We study some kinematical aspects of quantum fields on causal sets. In particular, we are interested in free scalar fields on a fixed background causal set. We present various results building up to the study of the entanglement entropy of de Sitter horizons using causal sets. We begin by obtaining causal set analogs of Green functions for this field. First we construct the retarded Green function in a Riemann normal neighborhood (RNN) of an arbitrary curved spacetime. Then, we show that in de Sitter and patches of anti-de Sitter spacetimes the construction can be done beyond the RNN. This allows us to construct the QFT vacuum on the causal set using the Sorkin-Johnston construction. We calculate the SJ vacuum on a causal set approximated by de Sitter spacetime, using numerical techniques. We find that the causal set SJ vacuum does not correspond to any of the known Mottola-Allen vacua of de Sitter spacetime. This has potential phenomenological consequences for early universe physics. Finally, we study the spacetime entanglement entropy for causal set de Sitter horizons. The entanglement entropy of de Sitter horizons is of particular interest. As in the case of nested causal diamonds in 2d Minkowski spacetime, we find that the causal set naturally gives a volume law of entropy, both for nested causal diamonds in 4d Minkowski spacetime as well as 2d and 4d de Sitter spacetimes. However, an area law emerges when the high frequency modes in the SJ spectrum are truncated. The choice of truncation turns out to be non-trivial and we end with several interesting questions.