Quantum Systems and Resolvent Algebras

TL;DR

This survey explores resolvent algebras as a robust algebraic framework for modeling quantum systems, simplifying analysis of finite and infinite systems, and handling interactions and constraints effectively.

Contribution

It introduces resolvent algebras as a new algebraic approach that overcomes conceptual and computational issues in traditional quantum system modeling.

Findings

Resolvent algebras encode system dimension and are nuclear.

They have a simple representation structure, especially in finite dimensions.

They facilitate analysis of interacting and constrained quantum systems.

Abstract

This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the Heisenberg picture it is based on the resolvents of the basic canonically conjugate operators and covers finite and infinite quantum systems. The resulting C*-algebras, the resolvent algebras, have many desirable properties. On one hand they encode specific information about the dimension of the respective quantum system and have the mathematically comfortable feature of being nuclear, and for finite dimensional systems they are even postliminal. This comes along with a surprisingly simple structure of their representations. On the other hand, they are a convenient framework for the study of interacting as well as constrained quantum systems since they allow the direct application of C*-algebraic methods which often simplify the analysis. Some pertinent facts are illustrated by instructive examples.