Deformation in Phase Space

TL;DR

This paper reviews various phase space quantization methods, framing quantum mechanics as a deformation of classical mechanics, and explores the resulting quantum groups as deformations of Poisson-Lie groups.

Contribution

It provides a comprehensive overview of phase space quantization procedures and their connection to quantum groups, highlighting the deformation perspective.

Findings

Quantization viewed as deformation of classical algebra

Connection between phase space methods and quantum groups

Analysis of quantum groups as deformations of Poisson-Lie groups

Abstract

We review several procedures of quantization formulated in the framework of (classical) phase space M. These quantization methods consider Quantum Mechanics as a "deformation" of Classical Mechanics by means of the "transformation" of the commutative algebra of smooth functions on M in a new non-commutative algebra. These ideas lead in a natural way to Quantum Groups as deformation (or quantization, in a broad sense) of Poisson-Lie groups, which is also analysed here.