Deformation Quantization: Observable Algebras, States and Representation Theory

TL;DR

This paper introduces deformation quantization, focusing on star products and the representation theory of deformed observable algebras, with applications in noncommutative field theories.

Contribution

It reviews recent advances in the representation theory of star product algebras, including Rieffel induction and Morita equivalence, within the context of deformation quantization.

Findings

Development of a comprehensive framework for representations of star product algebras

Application of Rieffel induction techniques in deformation quantization

Connections established between deformation quantization and noncommutative field theories

Abstract

In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star product algebra, physical applications require to study representations of this algebra. I review the recent development of a representation theory including techniques like Rieffel induction and Morita equivalence. Applications beyond quantization theory are found in noncommutative field theories.