Expressing entropy globally in terms of (4D) field-correlations

TL;DR

This paper derives a spacetime-based formula for scalar field entropy using correlation functions, applicable in both continuum and causal set frameworks, offering new insights into entanglement entropy across horizons.

Contribution

It introduces a novel expression for scalar field entropy directly in terms of spacetime correlations, valid in both continuum and causal set contexts.

Findings

Provides a new formula for entropy in terms of field correlations.

Applies to entanglement entropy across event horizons.

Works in both continuum spacetime and causal set models.

Abstract

We express the entropy of a scalar field phi directly in terms of its spacetime correlation function W(x,y) = <phi(x) phi(y)>, assuming that the higher correlators are of "Gaussian" form. The resulting formula associates an entropy S(R) to any spacetime region R; and when R is globally hyperbolic with Cauchy surface Sigma, S(R) can be interpreted as the entropy of the reduced density-matrix belonging to Sigma. One acquires in particular a new expression for the entropy of entanglement across an event-horizon. Thanks to its spacetime character, this expression makes sense in a causal set as well as in a continuum spacetime.