Dual Relativistic Quantum Mechanics I

It was shown in Dirac (Proc. R. Soc. (Lond.) A117, 610; A118, 351, 1928) that the ultra-violet divergence in quantum electrodynamics (QED) is caused by a violation of the time-energy uncertainly relationship, due to the implicit assumption of infinitesimal time information (Dyson’s conjecture). In Wheeler et al. (Rev Mod Phys 17:157–181, 1949) 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 (X, t) and (X, τ) to describe the dynamics, where X is the (unique) canonical center of mass. In the (X, t) variables, time is relative and the speed of light is unique, while in the (X, τ) 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 cyclotron 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 the need for point particles, without a self-energy divergency and, without need for the problematic Lorentz–Dirac equation. The purpose of this paper is to introduce the dual theory of relativistic quantum mechanics. In our approach, we obtain three distinct dual relativistic wave equations that reduce to the Schrödinger 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.