The Bekenstein Bound

TL;DR

This paper discusses the Bekenstein entropy bound, highlighting counterexamples under various definitions and presenting a proof for a specific formulation proposed by Casini that resolves previous issues.

Contribution

It clarifies the conditions under which the Bekenstein bound holds and presents a proof for Casini's formulation that avoids earlier counterexamples.

Findings

Counterexamples exist for many definitions of the bound.

Casini's formulation allows a rigorous proof of the bound.

The proof applies to a specific, consistent set of definitions.

Abstract

Bekenstein's conjectured entropy bound for a system of linear size $R$ and energy $E$, namely $S \leq 2 \pi E R$, has counterexamples for many of the ways in which the "system," $R$, $E$, and $S$ may be defined. One consistent set of definitions for these quantities in flat Minkowski spacetime is that $S$ is the total von Neumann entropy and $E$ is the expectation value of the energy in a "vacuum-outside-$R$" quantum state that has the the vacuum expectation values for all operators entirely outside a sphere of radius $R$. However, there are counterexamples to the Bekenstein bound for this set of definitions. Nevertheless, an alternative formulation ten years ago by Horacio Casini for the definitions of $S$ and of $2 \pi E R$ have finally enabled a proof for this particular formulation of the Bekenstein bound.