Moyal quantisation

Gerald V. Dunne

The quasiclassical generalised Moyal quantisation procedure is applied to the problem of quantum canonical invariance. Simple new formulae for the Moyal bracket corresponding to various correspondence rules are given, and are used to calculate the 'additional potential terms' arising from a point canonical transformation made on a free Hamiltonian. The status of canonical transformations in other equivalent quantum phase space formulations is also reviewed.

Wolfgang Nolting, William Dean Brewer

E. Gozzi

In this paper we explore the idea of looking at the Dirac quantisation conditions as ¯h-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure. 1.

José F. Cariñena, JoséM. Garcia-Bondia, Joseph C. Várilly

The authors obtain the phase-space quantisation for relativistic spinning particles. The main tool is what they call a 'Stratonovich-Weyl quantiser' which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Their phase spaces are coadjoint orbits of the restricted Poincare group; they compute and explicitly coordinatise the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincare group; they show that there is a unique Stratonovich-Weyl quantiser from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.