Modular Groups of Quantum Fields in Thermal States

TL;DR

This paper explores the structure of modular groups in quantum fields at thermal equilibrium, revealing geometric descriptions of modular flows and their relation to temperature and domain boundaries.

Contribution

It provides a detailed analysis of modular flows in thermal quantum fields, including geometric interpretations and the connection to ground states and the Unruh effect.

Findings

Modular flows in thermal states have simple geometric descriptions in 2D models.

At large distances, the flow pattern resembles time translations.

Near domain boundaries, the flow approximates zero-temperature results.

Abstract

For a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half-sided translations. The modular flows associated with the algebras of the forward light cone and a space-like wedge admit a simple geometric description in two dimensional models that factorize in light-cone coordinates. At large distances from the domain boundary compared to the inverse temperature the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. Associated with each domain there is also a one parameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh-effect.