Deformation quantization: observable algebras, states and representation theory. (English) Zbl 1083.53087

Waldmann, Stefan

Dragovich, B.(ed.) et al., Proceedings of the 2nd summer school in modern mathematical physics, Kopaonik, Yugoslavia, September 1–12, 2002. Belgrade: Institute of Physics (ISBN 86-82441-13-6). Sveske Fizičkikh Nauka. Series A: Conferences 16, 1, 83-107 (2003).

In this lecture notes the author gives a nice introduction to deformation quantization. The quantization problem is discussed in detail (canonical quantization, ordering prescription, etc.) which motivate the notion of different star products. Some interesting examples of deformation quantization are given. Applications beyond quantization theory are found in noncommutative field theories. Starting from a star product algebra, physical applications require to study representation of this algebra. The representation theory is developed from the beginning to the recent development of it including techniques like Rieffel induction and Morita equivalence.
The exposition of the paper is well motivated, well organized and clear.
For the entire collection see [Zbl 1058.81003].

Reviewer: Zoran Rakić (Beograd)

MSC:

53D55	Deformation quantization, star products
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
81S10	Geometry and quantization, symplectic methods
47L55	Representations of (nonselfadjoint) operator algebras

Keywords:

Poisson manifold; Poisson algebra; star product algebra; classical limit; Rieffel induction; Morita equivalence; noncommutative field theory