Intrinsic Dirac Behavior of Scalar Curvature in a Quaternionic Weyl-Cartan Geometry

TL;DR

This paper explores a quaternionic geometric formulation of Dirac theory, linking spin, curvature, and electromagnetic phenomena, and extends it to complex and Weyl-Cartan geometries with potential atomic-scale implications.

Contribution

It introduces a quaternionic scalar curvature framework for Dirac theory, connecting spin projections to geometric structures and generalizing gauge and curvature concepts.

Findings

Quaternionic Dirac theory projects into complex solutions for scalar curvature.

Eigenvalues and eigenfunctions are consistent with known Dirac solutions.

Solutions include atomic and subatomic scale phenomena, with electromagnetic quanta emerging.

Abstract

The "spin-up" and "spin-down" projections of the second order, chiral form of Dirac Theory are shown to fit a superposition of forms predicted in an earlier classical, complex scalar gauge theory (April, 1992 Class. Quantum Grav.). In some sense, it appears to be possible to view the two component Dirac spinor as a single component, quaternionic, spacetime scalar. "Spin space" transformations can be considered transformations of the internal quaternion basis. Essentially, quaternionic Dirac Theory projects into the complex plane neatly, where spin becomes related to the self-dual antisymmetric part of the metric. The correct Dirac eigenvalues and well-behaved eigenfunctions project intact into a pair of complex solutions for the scalar curvature in the earlier theory's Weyl-Cartan type geometry. Some estimates are made for predicted, interesting atomic and subatomic scale phenomena. A form of electromagnetic quanta appears. A generalization of the complex geometric structure is then sketched in an appendix that allows quaternionic gauges and curvatures, and has some Weyl nonmetricity mixed with torsion. It appears to be a well defined structure, and leads to the full, second order, quaternionic Dirac Equation form, and a first order equation for a closely related, auxiliary wavefunction. A family of "free particle" solutions is examined in the Lorentzian limit of the symmetric part of the metric. More generally, when limited to two quaternion dimensions (just two components), reasonably similar solutions can be superposed linearly into new solutions, and separate into two families with different commutation characteristics. The integrability conditions for the equation for the auxiliary wavefunction impose six conditions on the original wavefunction, satisfied for the "free particle" solutions examined. Covariance is examined. The Darwin solution for the hydrogen atom is examined.