Causality and propagation in the Wigner, Husimi, Glauber, and Kirkwood phase-space representations

Time evolution is considered in phase space in terms of evolution kernels for various phase-space quasi-distributions. The propagators for the Wigner function, the standard-ordered function, the Kirkwood (antistandard-ordered) function, the Glauber P and Q functions, and the Husimi function are explicitly written as bilinear transforms of the evolution operator. Free propagation, propagation in dispersive media, and scattering, are studied, and manifestations of causality and interference are analyzed. It is shown that free propagation and scattering in the Husimi, Glauber, and Kirkwood representations with the underlying dynamics of the Schrödinger equation involve divergent evolution kernels connecting distant phase-space points at all times. The time evolution is much simpler in the Wigner representation where (i) free propagation is a simple classical translation involving no interference, and (ii) analytical properties of the scattering matrix restrict the velocities of propagation so that no information can travel due to scattering faster than free motion. As an example, a correlation is found between the coordinate and momentum of particles detected after they are released from a box. Propagators with relativistic dispersion relations of free photons or Klein-Gordon particles are briefly discussed in an Appendix.