Some Aspects of Operator Algebras in Quantum Physics

TL;DR

This paper introduces operator algebras and their representation theory in quantum physics, emphasizing their role in understanding quantum phase transitions through examples bridging quantum information, algebraic quantum physics, and statistical mechanics.

Contribution

It provides an accessible introduction to operator algebras in quantum physics with a focus on their application to quantum phase transitions and related physical phenomena.

Findings

Illustrates the role of operator algebras in quantum phase transitions

Connects concepts from quantum information, algebraic quantum physics, and statistical mechanics

Provides explicit examples demonstrating the convergence of diverse theoretical strands

Abstract

Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.