Quantum Mechanics on SO(3) via Non-commutative Dual Variables

TL;DR

This paper develops a non-commutative phase space formulation of quantum mechanics on SO(3), deriving a path integral and analyzing quantum corrections, with implications for quantum gravity models.

Contribution

It introduces a novel non-commutative dual space representation for quantum states on SO(3), providing a new path integral formulation and insights into quantum corrections.

Findings

Derived the first order path integral in non-commutative variables.

Identified quantum corrections to the classical action.

Analyzed semi-classical limit and free particle on SO(3).

Abstract

We formulate quantum mechanics on SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semi-classical approximation of the first order path integral and the example of a free particle on SO(3). On the basis of these results, we comment on the relevance of similar structures and methods for more complicated theories with group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam models.