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arXiv:1503.02579 · Journal of Physics Conference Series

Foundations for proper-time relativistic quantum theory

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

Published May 14, 2015• 1 citation

Abstract

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.

DOI: 10.1088/1742-6596/615/1/012013

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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.

In this paper, as suggested by the classical canonical equations, a new set of the corresponding relativistic equations is set up. Therefrom a relativistic form of Heisenberg's equation is deduced. The rclalivistic Hamiltonian system of a particle according to the formulation here established has been fully discussed and by following it, however, Diracs equation appears naturally as a necessary form of relativistic wave equation tor electron in quantum mechanics. The process of taking square root in Dirac's theory is seen to have its classical analogy. Finally, some applications of the relativistic Heisenberg's equation to Dirac's theory have been discussed and it has thereby been pointed out that this equation brings some quantities to being more symmetrical in the relativity sense and also some more general than those the non-relativistic equation can introduce.

In the theory of the Dirac equation in quantum mechanics, the Dirac matrix beta is taken to connect 'proper time' and coordinate time rates of change of operators. The usual Heisenberg equation for the coordinate time rate of change of an operator then leads to a Heisenberg-like equation for its proper time rate (the 'Hamiltonian' being analogous to the generator of proper time translations in classical mechanics). Certain terms have to be set to zero by the use of the Dirac equation, and this is here justified. The equivalence of the formalisms under these conditions is shown and the method entitles us to derive some well known results for the Dirac equation in a comparatively effortless manner. The present work justifies methods introduced by H.C. Corben (1961).

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.

The year 2025 marked the centennial of quantum mechanics, inaugurated by Heisenberg's matrix formulation and the foundational contributions of Pauli, Schrodinger, and Dirac. Concurrently, 2026 marks the centennial of the Klein - Gordon equation, the second-order relativistic wave equation from which both the Schrodinger and Dirac equations were derived. This article supplements the recent review published in J.Phys. A: Math.Theor.,58 (2025) 053001 by providing a more detailed examination of the formative period 1925 - 1928, with particular attention to contributions that have received insufficient recognition in the standard narrative. We reconstruct Kramers' independent derivation of the Dirac equation - obtained essentially simultaneously with Dirac's own result yet unpublished for seven years - and discuss its relation to Van der Waerden's group-theoretical approach. The role of Charles Galton Darwin in elucidating the physical content of the Dirac equation is also highlighted. In addition, we present two modern derivations not catalogued in the earlier review: one based on Operational Dynamical Modeling, which deduces the Dirac equation from relativistic Ehrenfest relations and the canonical commutation algebra, and one rooted in the Madelung hydrodynamic formulation. Three broad periods of quantum theory development -- foundational, consolidation, and the modern era of quantum information -- are briefly surveyed.