The Wigner-weyl Formalism and the Relativistic Semi-classical Approximation 1

The relativistic semi-classical approximation for a free massive particle is studied using the Wigner-Weyl formalism. A non-covariant Wigner function is defined using the Newton-Wigner position operator. The perturbative solution for the time evolution is found. Causality is found to be perturbatively respected. 1 1 Introduction. The Wigner-Weyl formalism [1] offers a classical-like formulation of quantum mechanics using phase space functions as observables and the Wigner function as an analogue of the Liouville density function. It has been used to study semi-classical non-relativistic quantum mechanics for time evolution [2] and spectra determination [3]. Relativistic generalizations of the Wigner function have been proposed [4], they are manifestly covariant and are not suitable to study the semi-classical approximation. The problem lies in the fact that the natural position operator in a manifestly covariant representation of a wave equation (as is the Dirac representation for a spin 1/2 particle) does not have a clear semi-classical limit; the Dirac position operator, for instance, gives rise to a velocity that is always equal to the speed of light. Another drawback of the manifestly covariant representation that is related to the preceding one is that it has more than one irreducible representation of the Poincaré group corresponding to more than one particle (an electron and a positron for the Dirac case). In this letter we propose a new definition of the Wigner function that uses a Foldy-Wouthuysen representation [5] or equivalently the Newton-Wigner position operator [6]. We solve pertubatively its time evolution and conclude with some remarks.