Quantum mechanics of a generalised rigid body

TL;DR

This paper develops a quantum framework for generalized rigid bodies on Lie groups, simplifying energy spectrum derivation and extending to cosets, with applications to fluid quantization.

Contribution

It introduces a quantum model for rigid bodies on arbitrary Lie groups, utilizing automorphisms and harmonic analysis, and extends the approach to cosets, including new exactly-solvable models.

Findings

Derived energy spectra for various Lie groups

Extended methods to cosets and rigid rotors

Presented new exactly-solvable quantum models

Abstract

We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any one-sided-invariant metric. We show how the derivation of the spectrum of energy eigenstates can be simplified by making use of automorphisms of the Lie algebra and (for groups of Type I) by methods of harmonic analysis. We show how the method can be extended to cosets, generalising the linear rigid rotor. As examples, we consider all connected and simply-connected Lie groups up to dimension 3. This includes the universal cover of the archetypical rigid body, along with a number of new exactly-solvable models. We also discuss a possible application to the topical problem of quantising a perfect fluid.