The Four Dimensional Dirac Equation in Five Dimensions

TL;DR

This paper explores a five-dimensional formulation of the Dirac equation, deriving four-dimensional theories and identifying new spin-electric interactions with potential applications in quantum mechanics and superconductivity.

Contribution

It introduces a novel 5D Clifford algebra and a null propagation formalism that unify relativistic quantum mechanics and statistical mechanics in four dimensions.

Findings

Derived four-dimensional Dirac and statistical mechanics from 5D formalism

Identified a new spin-electric interaction relevant to magnetic resonance and superconductivity

Explored non-relativistic limits revealing potential physical applications

Abstract

The Dirac equation may be thought as originating from a theory of five-dimensional (5D) space-time. We define a special 5D Clifford algebra and introduce a spin-1/2 constraint equation to describe null propagation in a 5D space-time manifold. We explain how the 5D null formalism breaks down to four dimensions to recover two single-particle theories. Namely, we obtain Dirac's relativistic quantum mechanics and a formulation of statistical mechanics. Exploring the non-relativistic limit in five and four dimensions, we identify a new spin-electric interaction with possible applications to magnetic resonance spectroscopy (within quantum mechanics) and superconductivity (within statistical mechanics).