Derivation of the relativistic "proper-time" quantum evolution equations from Canonical Invariance

Abstract

Based on 1) the spectral resolution of the energy operator; 2) the linearity of correspondence between physical observables and quantum Hermitian operators; 3) the definition of conjugate coordinate-momentum variables in classical mechanics; and 4) the fact that the physical point in phase space remains unchanged under (canonical) transformations between one pair of conjugate variables to another, we are able to show that <t_s|E_s>, the proper-time rest-energy transformation matrices, are given as a*exp[-iE_s t_s/\hbar], from which we obtain the proper-time rest -energy evolution equation i\hbar{\partial/\partial t_s} |Psi>= \hat{E_s}|Psi>. For special relativistic situations this equation can be reduced to the usual i\hbar{\partial/\partial t}|Psi>=\hat{E}|Psi> dynamical equations, where t is the "reference time" and E is the total energy. Extension of these equations to accelerating frames is then provided.