The Real Dirac Equation

• URL: http://arxiv.org/abs/2212.13568v3
• Date: Thu, 21 Sep 2023 18:59:20 GMT
• Title: The Real Dirac Equation
• Authors: Sokol Andoni
• Abstract summary: I derive a manifestly covariant first order equation from the direct quantization of the classical 4-momentum vector. Relative to a Lorentz frame, the classical spacetime frame vectors emu appear instead of the gammamu matrices.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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