Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

TL;DR

This work develops a unitary, noncommutative Fourier transform between square-integrable functions on a Lie group $G$ and momentum-space wavefunctions on the dual Lie algebra $\mathfrak{g}^*$, where momentum variables do not commute. By introducing a symmetric quantization and a quotient by the identity subset $\mathcal{I}$, the authors construct consistent position and momentum representations linked by a transform $F$ that preserves inner products and intertwines operator actions. They define an $\mathcal{I}$-invariant momentum space and a corresponding star product (the Gutt product) to realize a faithful noncommutative Fourier analysis, including an explicit inverse transform and a robust Fourier-series formalism. The framework yields a noncommutative Poisson summation formula for any compact Lie group and is demonstrated in detail for $U(1)$ and $SU(2)$, paving the way for Wigner functions and path integrals on group manifolds. Overall, the paper provides foundational tools to perform quantum mechanics on group-configured spaces with noncommuting momenta, bridging representation theory, deformation quantization, and geometric quantization in a coherent operational setting.

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.

Figures (3)

• Figure 1: A cartoon of the principal branch (blue disk) of the logarithm from a Lie algebra $\mathfrak{g}$ (the plane) to a Lie group $G$ (the sphere). On the principal branch, the exponential is injective, and its image in the group is dense. In the picture, the exponential of the principal branch only misses the north pole. The exponential is not injective on the boundary of the principal branch; this boundary is a circle in the present case, wholly sent on the north pole by the exponential. In fact, this exact picture is stricly valid for $G=\text{SU(2)}$, up to the replacement of $S^2$ by $S^3$ and $\mathbb{R}^2$ by $\mathbb{R}^3$: see section \ref{['sec: examples']}.
• Figure 2: Logarithms of the identity in the $\mathfrak{u}(1)$ Lie algebra (left) and the $\mathfrak{su}(2)$ Lie algebra (right). In the U(1) case, logarithms of the identity form a one-dimensional lattice of $2\pi$-separated points in the Lie algebra $\mathfrak{u}(1)\cong\mathbb{R}$. This is because the exponential is $\exp: x\mapsto e^{ix}$. In the SU(2) case, we take the exponential map to be $\exp:\vec{X} \mapsto e^{i\vec{X}\cdot \vec{\sigma}}$ in terms of Pauli matrices (see section \ref{['sec: examples']}). With this convention, the logarithms of the identity in $\mathfrak{su}(2)\cong\mathbb{R}^3$ are concentric spheres with radii $2\pi n$, $n=0,1,2,...$, as predicted in general by eq. \ref{['eq: sum over I unpacked']}.
• Figure 3: Localization of ${\cal I}$-invariant plane waves \ref{['e103']} in $\vec{X}$ at fixed $p$ (left), and in $p$ at fixed $\vec{X}$ (right). On the left panel, we chose $\|p\|=2.2$ so that only five values of $m$ are allowed in \ref{['e103']}. On the right panel, the plane waves are localized on planes perpendicular to $\vec{u}_X$ that are evenly spaced by integer multiples of $\vec{u}_X$.