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arXiv:2607.16029

Quantum Mechanics on Lie Groups: II. Path Integrals

Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos

quant-phhep-thmath-phmath.CASubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
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Abstract

We continue our study of quantum dynamics on a Lie group $G$, initiated in arXiv:2512.19840, by building the path integral that governs transition amplitudes in the Hilbert space $L^2(G)$. This relies on the proper handling of both a noncommutative momentum space and the presence of compact directions in $G$. We show that compactness can be handled through a sum over winding numbers in maximal tori of $G$, generalizing the similar sum commonly encountered for path integrals on a circle. As applications, we compute semiclassical approximations of propagators and partition functions of Euler-Arnold systems, up to (and including) two-loop order.

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We continue our study of quantum dynamics on a Lie group GGG, initiated in arXiv:2512.19840, by building the path integral that governs transition amplitudes in the Hilbert space L2(G)L^2(G)L2(G). This relies on the proper handling of both a noncommutative momentum space and the presence of compact directions in GGG. We show that compactness can be handled through a sum over winding numbers in maximal tori of GGG, generalizing the similar sum commonly encountered for path integrals on a circle. As applications, we compute semiclassical approximations of propagators and partition functions of Euler-Arnold systems, up to (and including) two-loop order.
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.
The quantization of physical systems moving on group and symmetric spaces has been an area of active research over the past three decades. This book shows that it is possible to introduce a representation-independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables.Furthermore, the book shows that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations, and although the configuration space is in general a multidimensional curved manifold, it is shown that the resulting lattice phase space path integral has the form of a lattice phase space path integral on a multidimensional flat manifold. Hence, a novel and extremely natural phase space path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phase space path integral quantization is (a) exact, (b) more general than, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds.To illustrate the general theory, a representation-independent propagator is explicitly constructed for SU(2) and the affine group
The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group GGG 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∗GT^*GT∗G and the Hilbert space of square integrable functions on GGG. 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.
A c-number path integral representation is constructed for the solution of the Dirac equation. The integration is over the real trajectories in the continuous three-space and other two canonical pairs of compact variables controlling the dynamics of the spin and the chirality flips. The problems to overcome in order to find a c-number path integral representation for the solution of the Dirac equation i¯h∂tΨ = HΨ = [ (−ic¯h ∂ + eA) · α + mc 2 β − eV] Ψ (1) is (i) to generate the first order derivatives in the space coordinates and (ii) to treat the spinor indices. The traditional method to tackle these problems is to use space-time lattice and sum over the lattice trajectories [1], and the application of Grassman variables [2]. The inherent limitation of this solution lies in the difficulty in arriving at the propagation in continuous space and in developing analytic non-perturbative approximations. We present