Entropy distribution of localised states

TL;DR

This paper analyzes the geometric distribution of relative entropy for localized charged states in Quantum Field Theory, revealing entropy behaviors, bounds, and energy conditions across various models and deformations.

Contribution

It provides a rigorous operator algebraic framework for entropy distribution, establishes the Averaged Null Energy Condition, and explicitly describes entropy in a conformal U(1)-current model.

Findings

Second derivative of relative entropy is zero outside charge support.

Asymptotic mean entropy density equals π times charge energy.

Quantum Null Energy Condition holds and is not saturated in the studied models.

Abstract

We study the geometric distribution of the relative entropy of a charged localised state in Quantum Field Theory. With respect to translations, the second derivative of the vacuum relative entropy is zero out of the charge localisation support and positive in mean over the support of any single charge. For a spatial strip, the asymptotic mean entropy density is $\pi E$, with $E$ the corresponding vacuum charge energy. In a conformal QFT, for a charge in a ball of radius $r$, the relative entropy is non linear, the asymptotic mean radial entropy density is $\pi E$ and Bekenstein's bound is satisfied. We also study the null deformation case. We construct, operator algebraically, a positive selfadjoint operator that may be interpreted as the deformation generator, we thus get a rigorous form of the Averaged Null Energy Condition that holds in full generality. In the one dimensional conformal $U(1)$-current model, we give a complete and explicit description of the entropy distribution of a localised charged state in all points of the real line; in particular, the second derivative of the relative entropy is strictly positive in all points where the charge density is non zero, thus the Quantum Null Energy Condition holds here for these states and is not saturated in these points.