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A Bekenstein-type bound in QFT

Roberto Longo

Abstract

Let B be a spacetime region of width 2R > 0, and φa vector state localized in B. We show that the vacuum relative entropy of φ, on the local von Neumann algebra of B, is bounded by 2πR-times the energy of the state φin B. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation covariant, local Quantum Field Theory on the Minkowski spacetime.

A Bekenstein-type bound in QFT

Abstract

Let B be a spacetime region of width 2R > 0, and φa vector state localized in B. We show that the vacuum relative entropy of φ, on the local von Neumann algebra of B, is bounded by 2πR-times the energy of the state φin B. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation covariant, local Quantum Field Theory on the Minkowski spacetime.
Paper Structure (9 sections, 16 theorems, 54 equations)

This paper contains 9 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Order for unbounded operators
  4. Standard subspaces
  5. Abstract entropy formulas
  6. Translation covariant nets of standard subspaces
  7. Relative entropy/energy ratio bound
  8. Final comments
  9. Appendix

Key Result

Lemma 2.1

Let $A$ be positive selfadjoint and injective on ${\mathcal{H}}$, and $\xi\in{\mathcal{H}}$. If $(\xi , \log A\,\xi)$ is well-defined, then If $(\xi , \log A\,\xi) < \infty$, then

Theorems & Definitions (16)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • Theorem 2.10
  • ...and 6 more