A covariant approach to the quantisation of a rigid body

TL;DR

This paper develops a covariant quantum mechanics framework for rigid bodies on curved spacetimes, identifying two quantum structures and interpreting two-valued wavefunctions on SO(3) in a novel geometric way.

Contribution

It introduces a covariant approach to quantising rigid bodies, revealing two inequivalent quantum structures and providing a new geometric interpretation of two-valued wavefunctions.

Findings

Identifies two inequivalent quantum structures for rigid bodies.

Provides a new geometric interpretation of two-valued wavefunctions on SO(3).

Calculates quantum energy and momentum spectra for the rigid body.

Abstract

This paper concerns the quantisation of a rigid body in the framework of ``covariant quantum mechanics'' on a curved spacetime with absolute time. The basic idea is to consider the multi-configuration space, i.e. the configuration space for $n$ particles, as the $n$-fold product of the configuration space for one particle. Then we impose a rigid constraint on the multi-configuration space. The resulting space is then dealt with as a configuration space of a single abstract `particle'. The same idea is applied to all geometric and dynamical structures. We show that the above configuration space fits into the general framework of ``covariant quantum mechanics''. Hence, the methods of this theory can be applied to the rigid body. Accordingly, we find exactly two inequivalent choices of quantum structures for the rigid body. Then, we evaluate the quantum energy and momentum operators and the `rotational part' of their spectra. We provide a new mathematical interpretation of two-valued wavefunctions on SO(3) in terms of single-valued sections of a new non-trivial quantum bundle. These results have clear analogies with spin.