Quaternionic Gauge Transformations and Yang-Mills Fields in Weyl Type Geometries

TL;DR

This paper extends Weyl geometry to quaternionic gauge transformations, developing a formalism that incorporates Yang-Mills fields, torsion, and nonmetricity, with potential implications for particle physics and gravity.

Contribution

It introduces quaternionic gauge transformations into Weyl geometry, linking torsion and nonmetricity to energy-momentum and generalizing the action principle.

Findings

Quaternionic gauge transformations require shifting nonmetricity into torsion.

Gauge invariant variables are suitable for physics and general relativity.

Ricci tensor effects become significant at scales much larger than Planck length.

Abstract

This elementary discussion generalizes a Weyl geometry to allow quaternion valued gauge transformations and classical Yang-Mills geometric fields. This development will assume that the symmetric metric tensor is real in some gauge, and will develop the left and right handed approaches to quaternionic gauge transformations. Quaternionic gauge transformations are shown actually to require the shifting of some of Weyl's nonmetricity into torsion to define a properly transforming gauge field full curvature tensor, which is constructed as an asymmetric sum of left and right handed forms. Natural, gauge invariant, dimensionless variables are defined suitable for physics, and for use as a general formalism to describe these geometries, including General Relativity, in rather general circumstances. The geometry "self measures" these variables. Weyl's original action principle provides an example of an action rephrased in these gauge invariant variables, along with some unexpected possible insights on mechanics promoted by such a formulation of that action. Those include the torsion tensor and nonmetricity being constructed from mechanical energy-momentum. The Weyl form of action is then generalized to a quaternionic gauge field. The insights on mechanics now include spin 1/2 Dirac free fields. For physically reasonable choices of free parameters, the dimensionless Ricci tensor becomes nonnegligible in particle physics at distances much greater than the Planck length, along with limited general relativistic effects.