Star exponentials of the elements of the inhomogeneous symplectic Lie algebra

Ravinder Rupchand Puri

Jean-Marie Maillard

We compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2ℓ-dimensional phase space of a quantum system with ℓ degrees of freedom, and we show in the particular case of the Moyal star product that the Weyl transform of the Moyal star exponential of the one-dimensional harmonic oscillator Hamiltonian is the evolution operator of this quantum system.

Guy Burdet, M. Perrin

We interpret the one-element central extension of the Lie algebra of the group of real inhomogeneous linear canonical transformations of phase space as a time dependent invariance algebra in the Schrödinger picture. By limiting to a two-dimensional space quantum system, we exhibit all the nonconjugate Hamiltonians contained in the symplectic algebra which, in the Heisenberg picture, plays the role of dynamical algebra and contains several known degeneracy and spectrum generating algebras.

Jasel Berra-Montiel, Alberto Molgado, Mar Sánchez-Córdova

In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of $SU(2)$ and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on $S^2$ and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.

Guy Burdet, M. Perrin, Marcel Perroud