KMS, etc

TL;DR

This paper develops a general framework for Wick rotation in quantum systems, clarifies the role of the KMS condition and modular conjugation, and explores their implications in quantum field theory and gravitational backgrounds.

Contribution

It introduces a unified approach to Wick rotation, elucidates the KMS condition's role in quantum field theory, and applies these concepts to gravitational backgrounds like Unruh and Hawking effects.

Findings

Established a general form of Wick rotation for quantum systems.

Clarified the relationship between KMS condition, spin-statistics, and PCT theorem.

Applied the framework to gravitational phenomena such as Unruh and Hawking effects.

Abstract

A general form of the ``Wick rotation'', starting from imaginary-time Green functions of quantum-mechanical systems in thermal equilibrium at positive temperature, is established. Extending work of H. Araki, the role of the KMS condition and of an associated anti-unitary symmetry operation, the ``modular conjugation'', in constructing analytic continuations of Green functions from real- to imaginary times, and back, is clarified. The relationship between the KMS condition for the vacuum with respect to Lorentz boosts, on one hand, and the spin-statistics connection and the PCT theorem, on the other hand, in local, relativistic quantum field theory is recalled. General results on the reconstruction of local quantum theories in various non-trivial gravitational backgrounds from ``Euclidian amplitudes'' are presented. In particular, a general form of the KMS condition is proposed and applied, e.g., to the Unruh- and the Hawking effects. This paper is dedicated to Huzihiro Araki on the occasion of his seventieth birthday, with admiration, affection and best wishes.