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Thermal time as an unsharp observable

Jan van Neerven, Pierre Portal

Abstract

We show that the Connes-Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.

Thermal time as an unsharp observable

Abstract

We show that the Connes-Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.
Paper Structure (14 sections, 9 theorems, 96 equations)

This paper contains 14 sections, 9 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Thermal time
  3. Mathematical preliminaries
  4. Positive operator-valued measures
  5. The Tomita--Takesaki flow
  6. Noncommutative $L^2$-spaces
  7. Thermal time for the quantum harmonic oscillator
  8. Thermal time as a POVM in $H^2(\mathbb{D})$
  9. Modular time
  10. Modular time for the free massless relativistic particle
  11. Modular time as a POVM in $H^2(\mathbb{U})$
  12. Modular time for the noncommutative integral
  13. Modular time as a POVM
  14. Open problems

Key Result

Theorem 3.1

If $\Phi: \mathscr{S}(H) \to M_1^+(\Omega)$ is a convexity preserving mapping, then there exists a unique POVM $E:\mathscr{F}\to\mathscr{E}(H)$ such that for all $T\in\mathscr{S}(H)$ we have

Theorems & Definitions (17)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Definition 5.1
  • Lemma 5.2
  • proof
  • Definition 5.3
  • ...and 7 more