The resolvent algebra of non-relativistic Bose fields: sectors, morphisms, fields and dynamics

TL;DR

This paper develops a C*-algebraic framework for non-relativistic Bose fields, extending the resolvent algebra to include fields and morphisms, and demonstrating stability under interacting dynamics, aiding the study of long-range phenomena.

Contribution

It introduces a field algebra extension with particle number changing morphisms and shows its stability under dynamics, advancing the algebraic approach to Bose systems.

Findings

Extended the observable algebra to a field algebra with isometries.

Proved stability of the field algebra under interacting dynamics.

Established a framework for analyzing phase transitions and condensates.

Abstract

It was recently shown [2] that the resolvent algebra of a non-relativistic Bose field determines a gauge invariant (particle number preserving) kinematical algebra of observables which is stable under the automorphic action of a large family of interacting dynamics involving pair potentials. In the present article, this observable algebra is extended to a field algebra by adding to it isometries, which transform as tensors under gauge transformations and induce particle number changing morphisms of the observables. Different morphisms are linked by intertwiners in the observable algebra. It is shown that such intertwiners also induce time translations of the morphisms. As a consequence, the field algebra is stable under the automorphic action of the interacting dynamics as well. These results establish a concrete C*-algebraic framework for interacting non-relativistic Bose systems in infinite space. It provides an adequate basis for studies of long range phenomena, such as phase transitions, stability properties of equilibrium states, condensates, and the breakdown of symmetries.