Relativistic probability waves

A canonical structure compatible with the action of the Lorentz group can be obtained considering the energy and time as conjugate variables of an extended phase space. Scalar probability waves, describing free relativistic particles, are associated with functional coherent states for an extended Liouville equation. Relativistic action waves are provided by distributions localized in the momentum space, evolving according to the continuity and Hamilton-Jacobi equations. Presuming the existence of minimum space and time intervals, the action distributions take the form of relativistic Wigner functions. The nonrelativistic quantum dynamics is retrieved approximating the time distribution function by a Gaussian wave packet.