Representations of the Weyl group and Wigner functions for SU(3)

TL;DR

This paper constructs SU(3) irreducible representations using three-particle harmonic oscillator bases, representing the Weyl group as permutations, and expresses SU(3) Wigner functions via SU(2) functions and Weyl transformations, with applications in optics.

Contribution

It introduces a novel construction of SU(3) representations and Wigner functions using harmonic oscillator bases and permutation symmetries, highlighting dual reductive pairs in quantum optics.

Findings

Explicit SU(3) irreps on harmonic oscillator bases

Wigner functions expressed as products of SU(2) functions

Relevance of dual reductive pairs in quantum optics

Abstract

Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of permutations of the particle coordinates of these space. Wigner functions for SU(3) are expressed as products of SU(2) Wigner functions and matrix elements of Weyl transformations. The constructions make explicit use of dual reductive pairs which are shown to be particularly relevant to problems in optics and quantum interferometry.