Spectral Spacetime Entropy for Quasifree Theories

TL;DR

This paper introduces a covariant spectral method for calculating entanglement entropy in quasifree quantum field theories, applicable in curved spacetimes and quantum gravity contexts, with novel results on entropy scaling in causal sets.

Contribution

It presents a new spacetime-based spectral approach for entanglement entropy calculation, enabling covariant regularisation and application to non-Cauchy settings like causal set theory.

Findings

Reproduces known entanglement entropy results

Provides a covariant regularisation method

Finds a modified entropy scaling coefficient in causal sets

Abstract

Motivated by the necessity to UV-regularise entanglement entropy, we present a spectral method for calculating the entropy of quasifree states, for both bosonic and fermionic field theories. This construction is defined in spacetime rather than on a hypersurface, enabling the covariant regularisation of entropies, and its calculation in generic spacetime regions. We derive these formulae, which have previously appeared in the literature, in a new manner and highlight certain aspects of them, such as their connection to the density matrix and its eigenvalues. The spacetime nature of the formulation makes it particularly apt in the context of semiclassical and quantum gravity and in connection to black hole entropy. Another useful property of the formulation is its application to settings where no notion of a Cauchy surface exists, such as in the causal set theory approach to quantum gravity. We show example applications of the formulae which demonstrate their ability to reproduce known results. We also show a calculation in a causal set in $1+1$ dimensions which makes use of several of the unique and useful features of the formalism. In this last example, we obtain a novel result of a slightly modified entanglement entropy scaling coefficient, giving a possible signature of spacetime discreteness.