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

TL;DR

This paper derives relativistic proper-time quantum evolution equations from canonical invariance principles, connecting classical and quantum mechanics, and extends these equations to accelerating frames.

Contribution

It introduces a novel derivation of proper-time quantum evolution equations based on canonical invariance, linking classical phase space invariance to quantum dynamics.

Findings

Derived proper-time rest-energy evolution equations from classical invariance principles.

Reduced equations to standard relativistic quantum dynamics in inertial frames.

Extended the equations to include accelerating reference frames.

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.