Notions of Infinity in Quantum Physics

TL;DR

This paper reviews various notions of infinity in operator algebras and their relevance to quantum physics, including classifications, amenability, and specific algebraic structures like the CAR-algebra.

Contribution

It connects abstract mathematical concepts of infiniteness with their applications in quantum mechanics and quantum field theory, highlighting the CAR-algebra as Fölner.

Findings

CAR-algebra is a Fölner C*-algebra

Notions of infiniteness relate to quantum field theory

Classification of operator algebras informs quantum physics

Abstract

In this article we will review some notions of infiniteness that appear in Hilbert space operators and operator algebras. These include proper infiniteness, Murray von Neumann's classification into type I and type III factors and the class of F{/o} lner C*-algebras that capture some aspects of amenability. We will also mention how these notions reappear in the description of certain mathematical aspects of quantum mechanics, quantum field theory and the theory of superselection sectors. We also show that the algebra of the canonical anti-commutation relations (CAR-algebra) is in the class of F{/o} lner C*-algebras.