Photon: Electromagnetic particle-wave solution to the homogenous classical wave equation in a real geometric algebra formalism

Abstract

This paper presents a deterministic quantum theory of radiation which is a consequence of Maxwell's classical electrodynamic theory. The formalism used in this theory is the geometric algebra G₄, which is generated by four orthonormal vectors. The first part of the paper reformulates Special Relativity through an axiom which describes the differential element of Minkowski space. The Minkowski metric is derived by squaring this element while keeping the square of spatial and temporal vectors positive definite, unlike in Hestenes' space-time algebra where the square of the spatial vectors is negative definite. The second part of the paper rederives the results of Classical Electrodynamics in a linear and isotropic medium. Maxwell's four differential laws are united into a single equation which states that the space-time derivative of the electromagnetic field is proportional to the charge-current density. This may be transformed into the formulation of Riestz by multiplying both sides of the equation from the left by the temporal vector. The space-time derivative of the Maxwell Equation is the Charge Conservation-Wave Equation. Multiplying both sides of the Maxwell Equation from the left by the electromagnetic field yields the Energy-Momentum Equation. And the third part of the paper presents a new solution to the homogenous wave equation in vacuum: a point singularity which travels rectilinearly at the speed of light through the interplay of its electric and magnetic field intensities. This expression for the particle-wave is similar to that of the plane wave except that the imaginary number j is now interpreted as the unit spatial trivector e₁e₂e₃ and the k⋅r dependence is replaced by kr. The electric field of the wave is circularly polarized. This particle-wave satisfies the de Broglie quantization relations since its energy is proportional to its frequency and its momentum is proportional to its wave vector by the same constant. The new model for the photon, which has simultaneously a definite position and momentum, does not contradict Heisenberg's Uncertainty Principle since its relations only hold for plane waves and not for particle-waves which we postulate in this paper.