The Real Dirac Equation

TL;DR

This paper derives a covariant, frame-free form of the Dirac equation using Geometric Algebra, eliminating the need for matrices and internal degrees of freedom, and providing a more efficient computational framework.

Contribution

It introduces a novel, manifestly Lorentz-invariant Dirac equation derived from classical 4-momentum quantization, removing the need for ad hoc matrices and internal degrees of freedom.

Findings

The Dirac equation is reformulated in a covariant, matrix-free form.

The new formalism simplifies calculations and enhances computational efficiency.

Properties of electrons and positrons follow naturally from the new equation.

Abstract

Dirac's leaping insight that the normalized anti-commutator of the {\gamma}^{\mu} matrices must equal the timespace signature {\eta}^{\mu}{\nu} was decisive for the success of his equation. The {\gamma}^{\mu}-s are the same in all Lorentz frames and "describe some new degrees of freedom, belonging to some internal motion in the electron". Therefore, the imposed link to {\eta}^{\mu}{\nu} constitutes a separate postulate of Dirac's theory. I derive a manifestly covariant first order equation from the direct quantization of the classical 4-momentum vector using the formalism of Geometric Algebra. All properties of the Dirac electron & positron follow from the equation - preconceived 'internal degrees of freedom', ad hoc imposed signature and matrices unneeded. In the novel scheme, the Dirac operator is frame-free and manifestly Lorentz invariant. Relative to a Lorentz frame, the classical spacetime frame vectors e^{\mu} appear instead of the {\gamma}^{\mu} matrices. Axial frame vectors (without cross product) of the 3D orientation space defining spin and rotations appear instead of the Pauli matrices; polar frame vectors of the 3D position space naturally define boosts, etc. Not the least, the formalism shows a significantly higher computational efficiency compared to matrices.