Thermal Quantum Fields with Spatially Cut-off Interactions in 1+1 Space-time Dimensions

TL;DR

This paper constructs and analyzes interacting quantum fields in 1+1 dimensions at positive temperature, demonstrating that the interacting state remains normal relative to the free system and describing the algebraic structure of the interaction.

Contribution

It introduces a method to incorporate spatially cut-off interactions into quantum fields in 1+1 dimensions using generalized path space techniques, extending prior foundational work.

Findings

Interacting KMS state is normal with respect to the free Araki-Woods representation.

Observable algebra and modular conjugation remain unchanged by interaction.

Interacting Liouvillean expressed in terms of free Liouvillean and interaction.

Abstract

We construct interacting quantum fields in 1+1 space-time dimensions, representing charged or neutral scalar bosons at positive temperature and zero chemical potential. Our work is based on prior work by Klein and Landau and Hoegh-Krohn. Generalized path space methods are used to add a spatially cut-off interaction to the free system, which is described in the Araki-Woods representation. It is shown that the interacting KMS state is normal w.r.t. the Araki-Woods representation. The observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting Liouvillean is described in terms of the free Liouvillean and the interaction.