Relativistic quantum kinematics in the Moyal representation

Abstract

In this paper, we obtain the phase-space quantization for relativistic spinning particles. The main tool is what we call a "Stratonovich-Weyl quantizer" 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. Our phase spaces are coadjoint orbits of the restricted Poincaré group; we compute and explicitly coordinatize the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincaré group; we show that there is a unique Stratonovich-Weyl quantizer from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.

We develop the formalism by computing relativistic Wigner functions and twisted products for Klein-Gordon particles; these Wigner functions are supported on the mass shell. We thereby obtain an expression for the position probability density which is local, that is, free from the difficulty of supraluminal propagation of the usual position probability density. It is shown explicitly how observables on phase space may be quantized; for example, we prove that the canonical position coordinate corresponds to the Newton-Wigner position operator, irrespective of spin.

We show how relativistic phase-space quantization applies to particles governed by the Dirac equation. In effect, we construct a Stratonovich-Weyl quantizer whose associated Hilbert space is the space of positive-energy solutions of the Dirac equation.