The Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group G is developed in detail. Several features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a semiquantized phase space, a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space T*G and the Hilbert space of square integrable functions on G. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.The Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group G is developed in detail. Several features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a semiquantized phase space, a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space T*G and the Hilbert space of square integrable functions on G. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.

arXiv:quant-ph/0407257Journal of Mathematical PhysicsJan 1, 200538 citations

Wigner-Weyl isomorphism for Quantum Mechanics on Lie groups

Mukunda, N, Marmo, G, Zampini, A +2 more

The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space $T^*G$ and the Hilbert space of square integrable functions on $G$ . General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space $T^*G$ and the Hilbert space of square integrable functions on $G$. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.

Jan 1, 200421 citations

Lie Groups and Quantum Mechanics

D. J. Simms

Lecture notes in mathematicsJan 1, 196896 citations

Unitary representations of lie groups in quantum mechanics

Lochlainn O'Raifeartaigh

Springer eBooksApr 6, 20086 citations

Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos

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. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. 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.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. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. 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.

quant-pharXiv:2512.19840Dec 24, 2025

The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group

G

is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised phase space', a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space

T^*G

and the Hilbert space of square integrable functions on

G

. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.