Dual geometric field theory

Rankin, J. E.

doi:10.1007/bf00670859

Cited by 3 publications

(13 citation statements)

References 13 publications

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“…Since g µν can now have a limited quaternionic nature, the expression g µν;γ = g µν,γ − g αν α µγ − g µα α νγ = 0 (11) is not necessarily the same as g µν;γ = g µν,γ − α µγ g αν − α νγ g µα = 0 (12) Thus, equation (11) defines the covariant derivative of g µν with respect to the "right handed Christoffel symbol" α µν , while equation (12) defines the covariant derivative of g µν with respect to the "left handed Christoffel symbol" α µν . Both equations actually define the associated Christoffel symbols, giving…”

Section: Left and Right Covariant Derivatives / Christoffel Symbolsmentioning

confidence: 99%

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

Rankin¹

2018

Preprint

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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.

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Section: Left and Right Covariant Derivatives / Christoffel Symbolsmentioning

confidence: 99%

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

Rankin¹

2018

Preprint

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“…However, the action of equation (32) does not give standard Dirac Theory. This is easily seen in a chiral representation, where the two chiral parts of ζ will each obey a second order Dirac Equation form, but without any first order Dirac Equations relating them to each other.…”

Section: Quaternion Form Of Dirac Theorymentioning

confidence: 99%

“…If the (chiral) Dirac ψ and φ are also related via the first order Dirac Equations [15], then the matching family of projected scalar functions ψ and ξ might be expected to have no more symptoms of negative energy density than the original Dirac solution has. Of course, the action of equation (32) does not require that the first order equations relate the chiral Dirac wavefunctions ψ and φ, as noted earlier. From its viewpoint, the first order equations would be additional conditions.…”

Section: B Complex/spin Projectionsmentioning

confidence: 99%

“…To get a feel for their magnitude, directly compare the forms generated for the "rest mass term" in the stress tensors from the action of equations ( 9), and the Klein-Gordon action in the appendix in reference [2]. The Klein-Gordon action can be used for this purpose since the rest mass term in it is the same magnitude as the analogous term in the action of equation (32), and the simple scalar nature of the earlier Klein-Gordon ψ is easier to handle. However, the Klein-Gordon case should be rendered dimensionless by multiplying the stress tensor by (1/b 0 ) to allow direct comparison of dimensionless quantities.…”

Section: B Scale Of Nonnegligible Scalar Curvaturementioning

confidence: 99%

“…It is only the quaternionic nature of B that pushes vµ and possibly ŵµν out of the complex plane and into the full quaternions, but nevertheless, equation (A.88) is substantially more complicated than the version in the complex plane. Most significantly, the basic Ricatti Equation change of variable, B = ψ −2 used in all the earlier versions of this model [1,32], no longer linearizes equation (A.88) into the "wave equation" without restricting B back into the complex plane. Instead,…”

Section: Integrability Conditionsmentioning

confidence: 99%

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

Rankin

2011

Preprint

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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. "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.

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

Rankin¹

1992

Class. Quantum Grav.

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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.

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Dual geometric field theory

Cited by 3 publications

References 13 publications

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

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

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

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

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