arXiv subjects

J. E. Rankin

Publications and source records attributed to J. E. Rankin.

4 recordsLinked to original sources

arXiv2019

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

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.

gr-qc↗

arXiv2018

Higher order gravity, gauge invariant variables, and quantum behavior in Weyl-like geometries

This paper presents the detailed, standard treatment of a simple, gauge invariant action for Weyl and Weyl-like Cartan geometries outlined in a previous paper. In addition to the familiar scalar curvature squared and Maxwell terms, the action chosen contains the logarithmic derivative of the scalar curvature combined with the intrinsic four vector (Weyl vector) in a gauge invariant fashion. This introduces higher order derivative terms directly into the action. No separate, "matter" fields are introduced. As the usual Weyl metric and four vector are varied, certain gauge invariant combinations of quantities arise naturally as the results are collected, provided the scalar curvature is nonzero. This paper demonstrates the general validity of these results for any gauge choice. Additionally, "matter" terms appear in the field equations. Furthermore, the resulting forms isolate the familiar mathematical structure of a coupled Einstein-Maxwell-Schroedinger (relativistic) system of classical fields, with the exception of additional, second derivative terms in the stress tensor for the Schroedinger field, and the algebraic independence of the conjugate wavefunction. This independence is found to be equivalent to the presence of a second, negative energy, Schroedinger field. A detailed comparison is made between this model, and the standard Einstein-Maxwell-Schroedinger field theory. The possible use of such continuum models as a basis for quantum phenomena, and some generalizations of the model are discussed.

gr-qc↗

arXiv2017

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

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.

gr-qc↗

arXiv1994

Quantum Behavior in Asymmetric, Weyl-Like Cartan Geometries

This discussion examines recent developments in the theory of a Weyl-like, Cartan geometry with natural Schrödinger field behavior proposed previously. In that model, very nearly exactly a coupled Einstein-Maxwell- Schrödinger, classical field theory emerges from a gauge invariant, purely geometric action based solely on variations of the electromagnetic potentials and the metric. In spite of this, only slight differences appear between the resulting Schrödinger part, and the conventional theory of the Schrödinger field. Close examination of the differences reveals that most are general relativistic effects which are unobservable in flat spacetime, and which are estimated to interact significantly only via their gravitational fields, or on scales comparable with neutrino interaction cross sections. The only remaining difference is that the wavefunction obeying the conjugate wave equation is not always restricted to be exactly the complex conjugate of the primary wavefunction. Generalizations of the model lead naturally to spinlike phenomena, a possible new mechanism for a theory of rest mass, and spinor connections containing the form of an SU(2) potential.

gr-qc↗