Deformation Quantization and Group Actions

Abstract

This set of notes corresponds to a mini-course given in Villa de Leyva in July 2015. It does not contain any new result and is meant to be an elementary first introduction to formal Deformation Quantization, hoping it will be an incentive to learn more advanced topics in the subject. Quantization of a classical system is a way to pass from classical to quantum results. There exist several mathematical attempts to formulate possible quantization methods. Formal deformation quantization was introduced in the seventies by Flato et al. and understands quantization as a deformation (called a star product) of the structure of the algebra of classical observables. After an introduction to the concept of quantization in Sect. 2.1, we introduce formal deformation quantization in Sect. 2.2, the description of Fedosov’s construction of a star product on a symplectic manifold in Sect. 2.3, an introduction to classifications of star products in Sect. 2.4 and a brief introduction to the notion of formality and its link with star products on a Poisson manifold in Sect. 2.5. Various notions of group actions in the context of deformation quantization are given in Sect. 2.6, along with the study of the invariance of a Fedosov’s star product, and classifications of invariant star products on a manifold endowed with an invariant connection. We present in Sect. 2.7 the concept of reduction in the formal deformation quantization setting, and show how quantization commutes with reduction, considering here only the simplest form of reduction and following a simplified version of Bordemann–Waldmann’s approach. We conclude by briefly mentioning in Sect. 2.8 convergence issues in the deformation quantization programme.