Moyal dynamics and trajectories

Abstract

We give first an approximation of the operator δ h : f → δ h f := h * h f −f * h h in terms of h2n , n 0, where h ≡ h(p, q), (p, q) ∈ R 2n , is a Hamilton function and * h denotes the star product. The operator, which is the generator of time translations in a * h-algebra, can be considered as a canonical extension of the Liouville operatorUsing 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.

A classical bound on quantum entropy

“…Braunss [5] has argued that, for S cl defined by S q + ln h in the limit that the Planck constant → 0,…”

“…The bounding classical entropy then reduces by standard thermodynamic evaluation to be just (1), Note Added A referee has identified a technical gap in Braunss' formal proof of his inequality in [5], which is, nevertheless, assumed here.…”

“…Approximate counting of quantum microstates, however, is normally toilsome, and can be approximated heuristically by semiclassical proposals [3], which, ultimately, should devolve to a bona fide classical limit, despite occasional ambiguities and complications along the way [4]. However, a more systematic approach was initiated by Braunss [5], who appreciated the underlying simplicity of phase space in taking a classical limit of intricate quantum systems. He thus tracked the information loss involved in smearing away quantum effects, to argue that the entropy of a quantum system is majorized by that of its classical limit, asinformation of the former is forfeited in the latter, an intuitively plausible relation.…”