We generalize to a Lorentz covariant formalism (with proper-time dependence) the non-negative phase-space distribution proposed by Kuryshkin and make a specific proposal to define the corresponding correspondence rule between functions of phase-space and quantum operators on the basis of the introduction of a non-dispersive soliton wave surrounding particles in space-time.
The formal similarity of the Wigner-Moyal phase-space approach to quantum mechanics 1) to classical phase-space descriptions of stochastic processes clearly expresses some deep physical property. Indeed if the Wigner distribution F(X, P) had turned out to be non-negative for all states (which it did not) it would evidently have been considered as a real probability distribution in phase-space and many physicists would have accepted the existence of real physical particle trajectories in space-time. This explains the interest shown by many people 2-5) in the study of possible non-negative quantum distribution functions (NQDE), the discussions relating to the possible meaning of negative probabilities in quantum mechanics6), and the recent attempt made by the authors of this paper 7) to show how in the case of relativistic spin-zero particles the Wigner function should be replaced by a Lorentz vector distribution function F"(X, P) which always associates positive probability densities F Β° with the real motions of particles and antiparticles in phase space. Indeed if we start from a phase-space description of the Klein-Gordon equation (with Taylor expanded external potentials) based on new independent coordinates X ", P~ it is appropriate to define a distribution function by a Lorentz vector * * In units h= c= l.