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A note on the conceptual problems on the Unruh effect

Hideyasu Yamashita

Abstract

This brief note is written with a somewhat similar purpose to Earman's 2011 paper on the conceptual problems on the Unruh effect. However, we confine ourselves to Sewell's modular approach to the Unruh effect, which is based on the theorems of Tomita-Takesaki and Bisognano-Wichmann. This approach is rigorous, and has an advantage of being model-independent. However, we will see that a number of conceptual problems remain unsolved on this approach.

A note on the conceptual problems on the Unruh effect

Abstract

This brief note is written with a somewhat similar purpose to Earman's 2011 paper on the conceptual problems on the Unruh effect. However, we confine ourselves to Sewell's modular approach to the Unruh effect, which is based on the theorems of Tomita-Takesaki and Bisognano-Wichmann. This approach is rigorous, and has an advantage of being model-independent. However, we will see that a number of conceptual problems remain unsolved on this approach.
Paper Structure (9 sections, 2 theorems, 16 equations)

This paper contains 9 sections, 2 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Temperature and time
  3. KMS state and the Tomita– Takesaki theorem
  4. Connes– Rovelli modular temperature hypothesis
  5. Rindler coordinate
  6. The Bisognano– Wichmann theorem and the Unruh effect
  7. Axioms for QFTM
  8. The Bisognano– Wichmann theorem
  9. Conceptual difficulties of the modular approach

Key Result

Theorem 2.2

If $\omega$ is a faithful normal state on $\mathcal{A}$, there exists a unique one-parameter group of automorphisms $\{\sigma_{t}|t\in\mathbb{R}\}$ on $\mathcal{A}$ such that $\omega$ is a $(\sigma,1)$-KMS state. This $\sigma$ is called the group of modular automorphisms determined by $\omega$.

Theorems & Definitions (8)

  • Claim 1.1
  • Claim 1.2
  • Claim 1.3
  • Claim 1.4
  • Definition 2.1
  • Theorem 2.2: Tomita--Takesaki theorem
  • Claim 2.3
  • Theorem 4.3: Bisognano--Wichmann theorem (outline) BW1975, Theorem 4.1.1 in Haa96