5.Situation of the Proper-time in the Quantum Theory

Moshe Shapiro

Based on (1) the spectral resolution of the energy operator; (2) the linearity of correspondence between physical observables and quantum self-adjoint 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 , the proper-time rest-energy transformation matrices, are given as a exp(−i E s t s /ℏ), from which we obtain the proper-time rest-energy evolution equation . For special relativistic situations this equation can be reduced to the usual dynamical equations, where t is the 'reference time' and E is the total energy. Extension of these equations to accelerating frames is then provided.

Philippe Droz-Vincent

Tepper L. Gill, Trey Morris, S. K. Kurtz

This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time theory. For completeness, we give a brief outline of the canonical proper-time approach to electrodynamics and mechanics, and then introduce the canonical proper-time approach to relativistic quantum theory. This theory leads to three new relativistic wave equations. In each case, the canonical generator of proper-time translations is strictly positive definite, so that it represents a particle. We show that the canonical proper-time extension of the Dirac equation for Hydrogen gives results that are consistently closer to the experimental data, when compared to the Dirac equation. However, these results are not sufficient to account for either the Lamb shift or the anomalous magnetic moment.

H. D. Zeh

Hou Y. Yau

A quantum harmonic oscillator oscillates in space. As we have learned from general relativity, spacetime is weaved as unity. If space and time are supposed to be treated on an equal footing, can a quantum harmonic oscillator oscillate in proper time? In this paper, we examine this possibility by assuming matter can oscillate in proper time. We demonstrate that a field with oscillations of matter in proper time can mimic the properties of a zero-spin bosonic field. A particle observed in this field is a proper time oscillator. The assumption also gives rise to properties that can reduce differences between quantum theory and general relativity, e.g., self-adjoint internal time operator and proper time uncertainty relation. Neglecting all quantum effects, a proper time oscillator can mimic a point mass at rest in general relativity. The spacetime outside a 'stationary' proper time oscillator is a Schwarzschild field. To test the idea, we propose to study a particle's arrival time. In motion, the proper time oscillation translates to oscillations in both time and space. These oscillations lead to uncertainty in the arrival time of a particle. Specifically, we study the effects on a neutrino's arrival time. The possible detection of a neutrino's arrival time uncertainty can test the theory proposed in this paper.