The Relation Between KMS-states for Different Temperatures

TL;DR

This paper explores the relationship between KMS states at different temperatures in thermal field theories, constructing local approximations and analyzing conditions for global state convergence based on surface energy considerations.

Contribution

It adapts a construction by Buchholz and Junglas to thermal field theories to relate KMS states at different temperatures, introducing a method to analyze their convergence.

Findings

Existence of convergent subnet of states as regions expand.

Surface energy influences whether local states form global KMS states.

A generalized cluster condition controls surface energy.

Abstract

Given a thermal field theory for some temperature $\beta^{-1}$, we construct the theory at an arbitrary temperature $ 1 / \beta'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region $\O_\circ \subset \rr^4$, but coincide with the given KMS state in the space-like complement of a slightly larger region $\hat{\O}$. By a weak*-compactness argument there always exists a convergent subnet of states as the size of $ \O_\circ$ and $ \hat{\O}$ tends towards $ \rr^4$. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of $ \O_\circ$ and $ \hat{\O}$. We show that this surface energy can be controlled by a generalized cluster condition.