Dirac, Schroedinger, and Maxwell equations in scalar and vector field quantum mechanics

TL;DR

This paper revisits relativistic quantum mechanics using a photon-like dispersion relation, deriving scalar and vector field equations akin to Maxwell's, and redefines wave-particle duality as electromagnetic wave-particle duality.

Contribution

It introduces a novel derivation of the Dirac equation and formulates vector field quantum mechanics similar to Maxwell's equations.

Findings

Derived simple new form of the Dirac equation.

Established vector field quantum mechanics analogous to Maxwell's equations.

Reinterpreted wave-particle duality as electromagnetic wave-particle duality.

Abstract

The quantum theory of relativistic particles, based on the first quantization technique similar to that used by Schroedinger and Dirac in formulating quantum mechanics, is reconsidered on the basis of a photon-like dispersion relation corresponding to the energy conservation equation of Einstein's special relativity. First, scalar quantum mechanics of particles operating with their wave functions is discussed. Using the first quantization of the photon-like dispersion relation, very simple new derivation of the Dirac equation is given. Then, vector field quantum mechanics is introduced, which defines vector fields associated with the relativistic particle. Basic equations for the vector-field quantum mechanics, similar to the source-free Maxwell equations, are derived. Following these equations, the particle's de Broglie wave can be considered as the transversal electromagnetic wave. Therefore, the wave-particle duality can be redefined as the electromagnetic wave-particle duality. Relationships between the scalar and vector field quantum mechanics are analyzed.