Intrinsic wave mechanical behaviour in a Weyl-like geometry which conserves lengths

Abstract

Einstein's lambda transformation and Weyl's gauge transformation are coupled to produce an asymmetric Weyl-like geometry. In the models of interest, lengths do not change under parallel transport. Automatic internal constraints of such geometries very closely resemble the equations of wave mechanics. Allowing an asymmetric metric tensor appears to add phenomena suggesting spin, although the equations are still scalar equations, not spinor ones. However, correspondence with Dirac theory exists for non-null, constant, uniform, electromagnetic fields in the limit that the symmetric part of the metric is Lorentzian.

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

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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 [1]. In some sense, it appears to be possible to view the two component Dirac spinor as a single component, quaternionic, spacetime scalar.

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“…and it is clearly complex valued. The appendix in reference [1] demonstrates that solutions of equation ( 1) match solutions to Dirac's Equation for the case of uniform, constant, non-null electromagnetic fields. In that proof, the Dirac Equation itself is taken to be the (chiral) second order form of the equation, ηµν ψ ,µ,ν + ıqA µ,ν ψ + 2ıqA µ ψ ,ν − q 2 A µ A ν ψ…”

“…This work presents results from an extended effort to identify spin 1/2, quantum mechanical wavefunctions with the scalar curvature of a Weyl-like Cartan geometry with a self-dual antisymmetric part to the metric [1][2][3]. The earlier papers demonstrated that the natural geometric identifications made therein imply as a simplest case that in the limit that the spacetime is Lorentzian (ĝ µν ≈ ηµν ), there exists a geometric wavefunction ψ which obeys ηµν ψ ,µ,ν + ıqA µ,ν ψ + 2ıqA µ ψ ,ν − q 2 A µ A ν ψ…”

“…As a quick reference summary [1][2][3] (or see the appendix of this paper for a more detailed, quaternion generalization of these steps), equation (1) follows from a Weyl-like Cartan geometric model with gauge invariant variables defined for cases in which the curvature B = 0. For the metric, that definition is ĝµν = (B/C) g µν (4) where the constant C = ±1.…”

“…In the sections to follow, the scalar nature of B will lead to the conclusion that at least in some sense, the full Dirac wavefunction ψ of equation ( 3) can be treated as a quaternionic spacetime scalar rather than necessarily always having spacetime spinor properties. It will be seen that this scalar property follows by projecting ("spin-up" and "spin-down") very general solutions to the equivalent quaternionic form of Dirac Equation (3) into the complex scalar solutions of equation (1) in the complex plane. This projecting preserves the full content of the Dirac ψ, with no information lost, and so it can be considered an embedding of Dirac Theory.…”