Quantum Mechanics on Lie Groups: II. Path Integrals

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 $L<sup>2(G)$. 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.

• The paper constructs a phase-space and Lagrangian path integral for quantum mechanics on arbitrary Lie groups, resolving compactness through sums over logarithms and non-unimodularity through modular-function factors.
• The authors derive one-loop propagators using group-based fluctuation determinants and prove cancellation of all two-loop contact divergences, with Haar-measure ghosts playing an essential role.
• The framework yields semiclassical partition functions for quantum Euler-Arnold systems and reproduces the Ricci-scalar term in the high-temperature heat-kernel expansion, providing a nontrivial consistency check.

This paper, the second in a series by Beauvillain, Oblak and Petropoulos (2607.16029), constructs the Feynman path integral governing transition amplitudes on an arbitrary Lie group $G$, building on the noncommutative Fourier analysis developed in the first paper of the series. The construction handles two obstacles that previous treatments—largely motivated by loop quantum gravity—left unresolved: compact directions in $G$, which require a sum over winding sectors, and non-unimodularity, which forces a careful distinction between left and right Haar measures throughout. As applications, the authors derive semiclassical (one- and two-loop) expressions for propagators and partition functions of Euler-Arnold systems, whose quantum Hamiltonians are anisotropic Laplacians on $G$.

Classical setting: Lie-Poisson and Euler-Arnold dynamics

The phase space is $T^*G \cong G \times \mathfrak{g}^*$ with symplectic form $-\mathrm{d}\mathcal{A}$, where $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$ is built from the right Maurer-Cartan form $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$. The induced Poisson brackets contain both Lie derivatives along $\mathfrak{g}$ and the coadjoint term $c^k{}_{ij}p_k$. For Hamiltonians depending only on momenta—the Lie-Poisson systems—the momentum equation decouples as $\dot p = \mathrm{ad}^*_{\partial_p H}p$ and evolves on a coadjoint orbit, with conserved charge $G$0 obtained by reconstruction of $G$1. The quadratic subclass, with inertia tensor $G$2, comprises the Euler-Arnold systems; their classical trajectories are affinely parametrized geodesics of the one-sided invariant metric $G$3, and the on-shell action satisfies $G$4.

A point worth emphasizing is that the relevant Hilbert space is $G$5 itself—the quantization of the full cotangent bundle—not the carrier space of a single irreducible representation obtained from quantizing a coadjoint orbit. The authors note that quantum Euler-Arnold systems are generally neither integrable nor one-loop exact, despite classical integrability of many such systems; this motivates genuine multi-loop computations.

Noncommutative Fourier machinery

The operator algebra has commuting position operators $G$6 and momenta satisfying $G$7, with symmetric ordering ensuring that exponentials of momenta map to exponential operators composed via the global Baker-Campbell-Hausdorff formula $G$8. The Hilbert space $G$9 is embedded into periodic functions on $G$0 via the inclusion $G$1, generalizing the identification of $G$2 with $G$3-periodic functions on $G$4. For almost every $G$5, the set of logarithms $G$6 is a discrete lattice $G$7 along the maximal torus through $G$8—the structural fact underlying the winding sums below.

The momentum representation lives on $G$9 with the Gutt star product, defined so that plane waves multiply according to the Baker-Campbell-Hausdorff formula. The noncommutative Fourier transform is a bijective isometry intertwining position and momentum representations, and both position-space ($T^*G \cong G \times \mathfrak{g}^*$0) and momentum-space ($T^*G \cong G \times \mathfrak{g}^*$1) resolutions of the identity are available—precisely what the time-slicing derivation requires. Fourier coefficients on $T^*G \cong G \times \mathfrak{g}^*$2 arise by projecting with $T^*G \cong G \times \mathfrak{g}^*$3, the projector onto translations by logarithms of the identity.

Construction of the path integral

The propagator on $T^*G \cong G \times \mathfrak{g}^*$4 is first rewritten as a sum over decompactified kernels on $T^*G \cong G \times \mathfrak{g}^*$5: because any admissible Hamiltonian commutes with translations by elements of $T^*G \cong G \times \mathfrak{g}^*$6, the double sum over initial and final logarithms collapses to a single sum over final logarithms. Time-slicing with alternating insertions of the two identity resolutions yields a discretized phase-space path integral with two non-Abelian features: a Jacobian $T^*G \cong G \times \mathfrak{g}^*$7 at each time step, and star products entering the short-time kernel. In the continuum limit the star products contribute only subleading terms, and the result is

$T^*G \cong G \times \mathfrak{g}^*$8

with action $T^*G \cong G \times \mathfrak{g}^*$9. The sum over logarithms of $-\mathrm{d}\mathcal{A}$0 is the direct generalization of the winding-number sum on the circle: it labels branches of the logarithm, i.e., windings along maximal torus directions.

Two further results refine this formula. First, the symbol of a purely kinetic operator is $-\mathrm{d}\mathcal{A}$1—momenta and positions mix even though the operator depends only on momenta. Second, the change of variables $-\mathrm{d}\mathcal{A}$2 converts everything to the right Haar measure and recovers the standard Lagrangian splitting $-\mathrm{d}\mathcal{A}$3, at the cost of a prefactor $-\mathrm{d}\mathcal{A}$4 involving the modular function $-\mathrm{d}\mathcal{A}$5. This factor traces to the unpaired momentum integration at the initial time slice. Gaussian integration over momenta then gives the Lagrangian path integral with curved measure $-\mathrm{d}\mathcal{A}$6, valid for quadratic kinetic terms. The authors argue explicitly that writing the path integral directly over paths in $-\mathrm{d}\mathcal{A}$7 obscures both why perturbations must be integrated over the decompactified $-\mathrm{d}\mathcal{A}$8 and how classical trajectories should be counted—both made transparent here by separating topological data (the log sum) from local fluctuations.

Semiclassical propagators at one and two loops

Right-translation invariance of Lie-Poisson Hamiltonians implies $-\mathrm{d}\mathcal{A}$9, so only the propagator from the identity need be computed. Expanding around each classical geodesic via $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$0 introduces the comoving inertia tensor $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$1 and the conserved charge $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$2 into the fluctuation Lagrangian. The curved Haar measure is handled by a Faddeev-Popov trick: the Jacobian determinant is exponentiated in terms of Grassmann-odd ghost fields $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$3 coupled to the Maurer-Cartan form. Crucially, these measure terms carry a factor of $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$4 and therefore enter only at two loops and beyond.

At one loop, the fluctuation operator factors as $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$5 with $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$6 built from $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$7, the structure constants, and $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$8. Its Green's function is constructed à la Gelfand-Yaglom/Forman from time-ordered exponentials, and the functional determinant follows from a trace-log argument with a carefully justified equal-time prescription (only the $$\mathcal{A}_{(g,p)} = \langle p,\,\Omega^{\mathrm{R}}_g\rangle$$9 branch appears, fixed by the time discretization). The determinant ratio evaluates to $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$0, where $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$1 is a path-dependent matrix primitive of the time-ordered exponential of $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$2. After cancellations between the formal $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$3 normalization and the modular-function products, the one-loop propagator takes the compact form

$\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$4

where the sum runs over unit-time reference geodesics. The prefactor $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$5 is identified as the Van Vleck-Morette determinant. Notably, the expansion does not use Riemann normal coordinates but exploits the group structure directly, yielding an explicit catalog of vertices usable at all loop orders.

At two loops, the cubic and quartic vertices are written out explicitly in terms of structure constants, the charge $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$6, and the comoving inertia. Equal-time contractions generate $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$7 divergences from three sources: coincident derivative legs at one vertex, coincident ghost legs, and cross-contractions between distinct vertices. The paper proves that all such divergences cancel at two loops, with the ghosts—encoding the curvature of the Haar measure—essential to the cancellation. The complete two-loop propagator is given in an appendix; its coefficient $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$8 fits the DeWitt ansatz for heat kernels. The authors state the reasonable expectation, without proof, that analogous cancellations hold at higher orders.

Partition functions and characters

For right-invariant Hamiltonians, the canonical partition function reduces to the group volume times the diagonal kernel at the identity. A subtle point arises: the identity has a continuum of logarithms rather than the discrete set generic elements possess, but since this occurs on a measure-zero set in the defining trace, the correct prescription is to restrict the winding sum to logarithms lying in a fixed Cartan subalgebra. Wick rotation $\Omega^{\mathrm{R}}_g = \mathrm{d}g\, g^{-1}$9 then transplants the propagator technology verbatim.

In the high-temperature limit only the constant saddle survives, giving $\mathfrak{g}$0, where

$\mathfrak{g}$1

is shown to be the Ricci scalar of the invariant metric, in agreement with standard heat-kernel results. This agreement constitutes a nontrivial check of the entire framework, including the ghost regularization.

Finally, the character of the left regular representation is computed as a path integral with linear Hamiltonian $\mathfrak{g}$2: the momentum integral produces a functional delta function forcing $\mathfrak{g}$3, whose unique solution exists only when the endpoint lies in the appropriate winding sector. The path integral thus reproduces the Peter-Weyl expectation $\mathfrak{g}$4, providing an independent derivation consistent with the Frobenius formula for induced representations.

Limitations and open questions

Several caveats qualify the results. The exponential map is assumed surjective up to a measure-zero set, and the treatment of the identity's exceptional logarithm structure requires the Cartan-subalgebra prescription, whose general necessity the authors flag explicitly. The divergence cancellation at two loops is proven, while cancellation at higher loop orders is conjectured on diagrammatic grounds rather than established. The two-loop coefficient $\mathfrak{g}$5 is given in raw form; extracting closed-form expressions for specific groups or comparing against known heat-kernel expansions beyond the Ricci term remains to be done. The framework also presumes knowledge of all classical geodesics connecting the endpoints—an assumption that is benign for integrable systems but restrictive in general.

Conclusion

The paper supplies a complete, measure-correct path-integral formalism for quantum mechanics on arbitrary Lie groups, resolving the compactness problem through sums over logarithms in maximal tori and the non-unimodularity problem through systematic tracking of modular functions. The resulting one-loop propagators, two-loop heat-kernel coefficients, and high-temperature partition functions—with the Ricci-scalar check—provide a concrete perturbative toolkit for quantum Euler-Arnold systems, including cases that are neither integrable nor one-loop exact.