Dual Relativistic Quantum Mechanics I

Abstract

In \cite{2} it was shown that Einstein's special theory of relativity and Maxwell's field theory have mathematically equivalent dual versions. The dual versions arise from an identity relating observer time to proper time as a contact transformation on configuration space, which leaves phase space invariant. The special theory has a dual version in the sense that, for any set of $n$ particles, every observer has two unique sets of global variables $({\bf{X}}, t)$ and $({\bf{X}}, \tau)$ to describe the dynamics, where ${\bf{X}}$ is the (unique) canonical center of mass. In the $({\bf{X}}, t)$ variables, time is relative and the speed of light is unique, while in the $({\bf{X}}, \tau)$ variables, time is unique and the speed of light is relative with no upper bound. In the Maxwell case, the two sets of particle wave equations are not equivalent. The dual version contains an additional longitudinal (dissipative) radiation term that appears instantaneously with acceleration, leading to the prediction that radiation from a betatron (of any frequency) will not produce photoelectrons. A major outcome is the dual unification of Newtonian mechanics and classical electrodynamics with Einstein's special theory of relativity, without a self-energy divergency, or need of the problematic Lorentz-Dirac equation or any assumptions about the size or structure of a particle. The purpose of this paper is to introduce and develop the dual theory of relativistic quantum mechanics. We obtain three distinct dual relativistic wave equations that reduce to the Schr{\"o}dinger equation when minimal coupling is turned off. We show that the dual Dirac equation provides a new formula for the anomalous magnetic moment of a charged particle. We can obtain the exact value for the electron g-factor and phenomenological values for the muon and proton g-factors.