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Bekenstein Bound for Approximately Local Charged States

Stefan Hollands, Roberto Longo

TL;DR

The paper broadens the Bekenstein-type energy–entropy bound in quantum field theory beyond strictly localized vacuum vectors to include charged and approximately localized states. It develops a sum-rule framework linking relative entropies on a region and its commutant and leverages DHR endomorphisms, left inverses, and Connes cocycles to bound the region-restricted relative entropy by energy in the charged representation, plus the logarithm of the sector's statistical dimension and a small tolerance. Key results include $S(\varphi||\omega)_{\mathcal{A}(B)} \le 2\pi R \,(\Phi, P_\rho \Phi) + \log d(\rho)$ for charged localized states and the extended bound $S(\varphi||\Omega)_{\mathcal{A}(B)} \le 2\pi R \,(\Phi, P\Phi) + \log d(\rho) + \varepsilon$ for charged, approximately localized states, under suitable domination conditions. By combining modular theory, DR reconstruction, and left inverses, the work extends entropy–energy inequalities to a broader class of quantum field-theoretic states, with potential implications for conformal theories, holography, and horizon criteria in quantum gravity.

Abstract

We generalize the energy-entropy ratio inequality in quantum field theory (QFT) established by one of us from localized states to a larger class of states. The states considered in this paper can be in a charged (non-vacuum) representation of the QFT or may be only approximately localized in the region under consideration. Our inequality is $S(Ψ|\!| Ω) \le 2πR \, ( Ψ, H_ρΨ) + \log d(ρ) + \varepsilon$, where $S$ is the relative entropy, where $R$ is a "radius" (width) characterizing the size of the region, $d(ρ)$ is the statistical (quantum) dimension of the given charged sector $ρ$ hosting the quantum state $Ψ$, $Ω$ is the vacuum state, $H_ρ$ is the Hamiltonian in the charged sector, and $\varepsilon$ is a tolerance measuring the deviation of $Ψ$ from the vacuum according to observers in the causal complement of the region.

Bekenstein Bound for Approximately Local Charged States

TL;DR

The paper broadens the Bekenstein-type energy–entropy bound in quantum field theory beyond strictly localized vacuum vectors to include charged and approximately localized states. It develops a sum-rule framework linking relative entropies on a region and its commutant and leverages DHR endomorphisms, left inverses, and Connes cocycles to bound the region-restricted relative entropy by energy in the charged representation, plus the logarithm of the sector's statistical dimension and a small tolerance. Key results include for charged localized states and the extended bound for charged, approximately localized states, under suitable domination conditions. By combining modular theory, DR reconstruction, and left inverses, the work extends entropy–energy inequalities to a broader class of quantum field-theoretic states, with potential implications for conformal theories, holography, and horizon criteria in quantum gravity.

Abstract

We generalize the energy-entropy ratio inequality in quantum field theory (QFT) established by one of us from localized states to a larger class of states. The states considered in this paper can be in a charged (non-vacuum) representation of the QFT or may be only approximately localized in the region under consideration. Our inequality is , where is the relative entropy, where is a "radius" (width) characterizing the size of the region, is the statistical (quantum) dimension of the given charged sector hosting the quantum state , is the vacuum state, is the Hamiltonian in the charged sector, and is a tolerance measuring the deviation of from the vacuum according to observers in the causal complement of the region.
Paper Structure (11 sections, 17 theorems, 86 equations)

This paper contains 11 sections, 17 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. General background
  3. Unitary families and Leibniz rule
  4. Relative modular operator
  5. Relative entropy
  6. Inner endomorphisms
  7. Entropy-energy bounds in QFT
  8. Approximately localized vector states
  9. Charged localized states
  10. Charged, approximately localized states
  11. Outlook

Key Result

Lemma 2.1

Let $V(s)$, $s\in (-a,a)$, $a >0$, be a family of unitaries on the Hilbert space ${\mathcal{H}}$, and $\Phi\in{\mathcal{H}}$ a unit vector. Then $-i\partial_s (\Phi, V(s)\Phi) = \alpha \in\mathbb R\cup \{\pm\infty\}$ iff

Theorems & Definitions (20)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Remark 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 2.9
  • Lemma 3.1
  • ...and 10 more