Bekenstein's bound for wave packets

TL;DR

The paper proves a local Bekenstein-type bound for Klein-Gordon wave packets in a ball of half-width $R$, namely $S(\Phi|B) \le 2\pi R E(\Phi|B)$ with $E(\Phi|B)=\int_B T_{00}(0,\mathbf{x})\,d\mathbf{x}$, and extends the result to a general local, Poincaré covariant net of standard subspaces, where the bound follows from $-\log \Delta_{H(B)} \le 2\pi R P$. It also analyzes the nonlocal case via a variational problem introducing a boundary term $\Gamma_\Phi$ and provides operator bounds for the local modular Hamiltonian in the massive, scalar setting; a fermionic (Majorana) analogue and an entropy-balance/ant formula for wave packets in 1+1 dimensions are discussed, highlighting the structure of quantum entropy under translations and boosts. The results connect local entropy, energy density, and modular data in QFT, offering quantitative Bekenstein-type bounds compatible with numerical studies and extending to one-particle Majorana states and U(1) current nets.

Abstract

Let $B$ be a spatial region of width $2R$ and $Φ$ a Klein-Gordon wave packet localized in $B$ at time zero. We show the inequality $S \leq 2πR E$; here, $S$ is the entropy of $Φ$ contained in a region $B$, and $E$ is the energy content of $Φ$ within $B$. We consider a wider setting and formulate a variational problem aimed at minimizing our bound when $Φ$ is not localized in $B$. Our inequality holds in more generality in the framework of local, Poincaré covariant nets of standard subspaces and is related to the Bekenstein inequality. We point out a general bound that is compatible with the recent numerical computations by Bostelmann, Cadamuro, and Minz concerning the one-particle modular Hamiltonian of a scalar massive quantum Klein-Gordon field. We also provide a version of the entropy balance and ant formulas for wave packets.

Theorems & Definitions (23)

• Proposition 2.1
• Proposition 3.1
• Corollary 3.2
• Corollary 3.3
• Corollary 3.4
• Lemma 4.1
• Lemma 4.2
• Proposition 4.3
• Corollary 4.4
• Proposition 4.5