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Haag Duality in the Thermal Sector

Stefano Galanda, Leonardo Sangaletti

Abstract

We prove that the net of localised von Neumann algebras associated with a real scalar field propagating on Minkowski spacetime, in the KMS representation, satisfies a generalised version of Haag duality. Our proof combines ideas from existing arguments for the ground-state representation with purification techniques.

Haag Duality in the Thermal Sector

Abstract

We prove that the net of localised von Neumann algebras associated with a real scalar field propagating on Minkowski spacetime, in the KMS representation, satisfies a generalised version of Haag duality. Our proof combines ideas from existing arguments for the ground-state representation with purification techniques.
Paper Structure (14 sections, 13 theorems, 126 equations)

This paper contains 14 sections, 13 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Setup and results
  4. Real scalar field
  5. Main result
  6. General results for the thermal one-particle structure
  7. Weyl and Segal formulation of the CCR
  8. Results on one-particle Hilbert spaces and duality
  9. Scalar field on Minkowski spacetime in the KMS representation
  10. Fock-KMS representation of local net
  11. Haag duality for causal diamonds
  12. Eckmann-Osterwalder theorem
  13. Technical Results
  14. Pre-cyclicity in the thermal sector

Key Result

Theorem 1

Let $\mathcal{O} \subset \mathbb{M}$ be a generic open causal diamondGiven a generic $\mathcal{O}_1 \subset \mathbb{M}$ the interior of its Cauchy development is a causal diamond. on Minkowski spacetime $\mathbb{M}$ and let $\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})$ be the abstract net of Weyl $ Then, generalised Haag duality holds in the following sense:

Theorems & Definitions (30)

  • Definition 1
  • Definition 2: Kay:1985yx Definition $1\textbf{a}$
  • Definition 3: Kay:1985yx Definition $1\textbf{b}$
  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3
  • ...and 20 more