Moyal Star-Product and Unitary Representations of the Euclidean Motion Group

TL;DR

This paper uses Moyal star-product quantization to construct unitary irreducible representations of the 3D Euclidean motion group, linking Lie algebra representations to unitary operators via exponentiation.

Contribution

It introduces a novel method to derive unitary representations of the Euclidean motion group using Moyal star-product quantization of its Lie algebra.

Findings

Constructed explicit unitary irreducible representations.

Linked Lie algebra representations to unitary operators.

Provided a new approach for group representation theory.

Abstract

In this paper, the Moyal star-product quantization is used to construct the unitary irreducible representations of the Euclidean motion group on 3-dimensions. These unitary representations will come from the representation of its Lie algebra whose operators are defined by the left Moyal star-product multiplication. In fact, these representations of the Lie algebra is the infinitisimal representation. Hence, the exponentiation of these operators gives rise to unitary operators that defines the desired unitary representations.