Moyal dynamics and trajectories

Abstract

We give first an approximation of the operator δh: f → δhf ≔ h*ℏf − f*ℏh in terms of ℏ2n, n ⩾ 0, where , is a Hamilton function and *ℏ denotes the star product. The operator, which is the generator of time translations in a *ℏ-algebra, can be considered as a canonical extension of the Liouville operator Lh: f → Lhf ≔ {h, f}Poisson. Using this operator we investigate the dynamics and trajectories of some examples with a scheme that extends the Hamilton–Jacobi method for classical dynamics to Moyal dynamics. The examples we have chosen are Hamiltonians with a one-dimensional quartic potential and two-dimensional radially symmetric nonrelativistic and relativistic Coulomb potentials, and the Hamiltonian for a Schwarzschild metric. We further state a conjecture concerning an extension of the Bohr–Sommerfeld formula for the calculation of the exact eigenvalues for systems with classically periodic trajectories.