Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

TL;DR

This paper develops a noncommutative Fourier transform for wave functions on Lie groups, enabling analysis of quantum systems with noncommuting momenta and laying groundwork for Wigner functions and path integrals on group manifolds.

Contribution

It introduces an invertible, group-theoretic Fourier transform for Lie groups, incorporating star products and noncommutative geometry, extending quantum analysis tools to group manifolds.

Findings

Constructed an isometric noncommutative Fourier transform for Lie groups.

Derived a noncommutative Poisson summation formula for compact Lie groups.

Facilitated the computation of Wigner functions and path integrals on group manifolds.

Abstract

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.