Quantum systems in the hyperbolic phase-space: Explicit maps, differential form of the star product and their applications

We obtain a closed-form expression for the $s$-parametrized mapping kernels for the phase-space representation of quantum systems with the SU(1,1) symmetry group, in terms of an expansion over the appropriate tensor operators, which enables us to express the kernels in terms of continuous dual Hahn polynomials. Using this representation, we derive an explicit differential form of the star-product for the SU(1,1) map, which is expandable in the inverse Bargmann index and plays the role of the semiclassical parameter in hyperbolic phase space. This formulation allows us to analyze quantum dynamics through evolved phase-space distributions that obey a Moyal-like equation. We illustrate the star-product by deriving the Bopp operators, which represent the action of the group generators on the kernels, and by discussing the symplectic-like structure of the Moyal equation for Hamiltonians that are linear or quadratic in the SU(1,1) generators. We analyze the semiclassical limit of the Moyal evolution equation and show that, over short-time quantum dynamics, the evolution of the initial distribution along classical trajectories reproduces the quantum behavior; the magnitude of quantum corrections depends on the chosen map and is minimal for the Wigner map. Examples of applications of the developed formalism to the analysis of quantum dynamics governed by non-linear Hamiltonians are discussed.