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The Bisognano-Wichmann property for non-unitary Wightman conformal field theories

James E. Tener

Abstract

The Bisognano-Wichmann and Haag duality properties for algebraic quantum field theories are often studied using the powerful tools of Tomita-Takesaki modular theory for nets of operator algebras. In this article, we study analogous properties of nets of algebras generated by smeared Wightman fields, for potentially non-unitary theories. In light of recent work constructing Wightman field theories for (non-unitary) Möbius vertex algebras, we obtain a broadly applicable non-unitary version of the Bisognano-Wichmann property. In this setting we do not have access to the traditional tools of Hilbert space functional analysis, like functional calculus. Instead, results analogous to those of Tomita-Takesaki theory are derived `by hand' from the Wightman axioms. As an application, we demonstrate Haag duality for nets of smeared Wightman fields.

The Bisognano-Wichmann property for non-unitary Wightman conformal field theories

Abstract

The Bisognano-Wichmann and Haag duality properties for algebraic quantum field theories are often studied using the powerful tools of Tomita-Takesaki modular theory for nets of operator algebras. In this article, we study analogous properties of nets of algebras generated by smeared Wightman fields, for potentially non-unitary theories. In light of recent work constructing Wightman field theories for (non-unitary) Möbius vertex algebras, we obtain a broadly applicable non-unitary version of the Bisognano-Wichmann property. In this setting we do not have access to the traditional tools of Hilbert space functional analysis, like functional calculus. Instead, results analogous to those of Tomita-Takesaki theory are derived `by hand' from the Wightman axioms. As an application, we demonstrate Haag duality for nets of smeared Wightman fields.

Paper Structure

This paper contains 8 sections, 29 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Conformal nets and Tomita-Takesaki theory
  4. Wightman CFTs and vertex algebras
  5. Analytic continuation for non-unitary Wightman CFTs
  6. Invariant Hermitian forms, Bisognano-Wichmann, and the KMS condition
  7. Haag duality for Wightman CFTs with invariant Hermitian forms
  8. Applications for unitary CFTs

Key Result

Theorem 1

Let $({\mathcal{F}},{\mathcal{D}},U,\Omega)$ be a potentially non-unitary Wightman CFT on $S^1$. Then ${\mathcal{P}}(I_+)\Omega \subset D(\widetilde{V}_{i\pi})$. If $\varphi_1, \ldots, \varphi_k \in {\mathcal{F}}$ with conformal dimension $d_j$ and $f_j \in C^\infty(S^1)$ with $\operatorname{supp} f The same holds with $I_+$ replaced by $I_-$ and $\widetilde{V}_{i\pi}$ replaced by $\widetilde{V}_{

Theorems & Definitions (74)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 64 more