Deformation Quantization - a Brief Survey

Quantization is, most broadly, the process of forming a quantum mechanical system starting from a classical mechanical one. See (Be) for an early attempt to obtain a general definition of quantization. (AbM) also provides an introductory account of the subject. There are various methods of quantization; see (BW) for a general introduction to the geometry of quantization, and a specific geometric method (geometric quantization). In this survey we will be interested in deformation quantization. Intuitively a deformation of a mathematical object is a family of the same kind of objects depending on some parameter(s). The deformation of algebras is central to our problem, and in particular we are concerned with the deformations of function algebras. We use the Poisson bracket to ”deform” the ordinary commutative product of observables in classical mechanics, elements of our function algebra, and obtain a noncommutative product suitable for quantum mechanics. In the first section of this paper we will give a short overview of the main ideas in deformation quantization. In the second section we will go into more (historical) details about the directions in which the area developed in the last 20 years and in the last section we will try to give a very sketchy summary of recent results by Kontsevich.