Quantum mechanics on $SO(3)$ via noncommutative dual variables

We formulate quantum mechanics on the group $SO(3)$ using a noncommutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new noncommutative 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 noncommutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the nontrivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semiclassical 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.