Deformation quantization in the teaching of Lie group representations

TL;DR

This paper demonstrates how deformation quantization can be used to explicitly compute unitary irreducible representations of the Euclidean motion group, illustrating the method's utility in mathematical physics education.

Contribution

It introduces a straightforward application of deformation quantization to Lie group representations, linking mathematical concepts with physical quantization methods.

Findings

Concrete computations of representations using deformation quantization

Illustration of the method of orbits in a pedagogical context

Highlighting the unity of mathematical physics topics

Abstract

In this work, we present straightforward and concrete computations of the unitary irreducible representations of the Euclidean motion group $M(2)$ employing the methods of deformation quantization. Deformation quantization is a quantization method of classical mechanics and is an autonomous approach to quantum mechanics, arising from the Wigner quasiprobability distributions and Weyl correspondence. We advertise the utility and power of deformation theory in Lie group representations. In implementing this idea, many aspects of the method of orbits is also learned, thus further adding to the mathematical toolkit of the beginning graduate student of physics. Furthermore, the essential unity of many topics in mathematics and physics (such as Lie groups and Lie algebras, quantization, functional analysis and symplectic geometry) is witnessed, an aspect seldom encountered in textbooks, in an elementary way.