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

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.

29 pages; matches the published version plus footnote 7, a journal reference included

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• Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds
• Anomalous dimension in semiclassical gravity
• Quantum cosmology of (loop) quantum gravity condensates: An example
• Relational evolution of effectively interacting GFT quantum gravity condensates
• Non-commutative Fourier transform for the Lorentz group via the Duflo map
• Classical Group Field Theory
• On the space of generalized fluxes for loop quantum gravity
• Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
• New insights in quantum geometry