Coherent distributions for the rigid rotator

TL;DR

The paper develops coherent, positive phase-space distributions for the rigid rotator by solving the classical Liouville equation on $T^*SO(3)$ and connecting these to Wigner-type quasiprobabilities through discretization of left-invariant vector fields. It shows how action-wave solutions on Hamilton–Jacobi manifolds yield classical coherent distributions $f_0$, and how a quantum extension $f_\Psi$ obtained from a wavefunction $\Psi\in L^2(SO(3))$ produces a Wigner-like map with a zero-point energy term in the Hamiltonian, $\hat{H}=\sum_k (\hat{L}'_k)^2/(2I_k) + \hbar^2/(12I_k)$; the formalism ensures that Liouville evolution of $f_\Psi$ corresponds to the time-dependent Schrödinger evolution of $\Psi$. The work highlights the role of $SO(3)$ geometry in shaping quantum corrections and provides a structured bridge between classical phase-space distributions and quantum dynamics for rotational degrees of freedom. This approach clarifies how curvature and compactness of the rotation group induce a finite zero-point energy and yields a coherent framework for interpreting intrinsic angular-momentum observables within a phase-space formalism.

Abstract

Coherent solutions of the classical Liouville equation for the rigid rotator are presented as positive phase-space distributions associated with the Lagrangian submanifolds of Hamilton-Jacobi theory. These solutions become Wigner-type quasiprobability distributions by a formal discretization of the left-invariant vector fields from their Fourier transform in angular momentum. The results are consistent with the usual quantization of the anisotropic rotator, but the expected value of the Hamiltonian contains a finite "zero point" energy term. It is shown that during the time when a quasiprobability distribution evolves according to the Liouville equation, the related quantum wave function should satisfy the time-dependent Schroedinger equation.