A technique for the analysis of the invariance identities of classical gauge field theory by means of functional equations.

Persistent Link:
http://hdl.handle.net/10150/185191
Title:
A technique for the analysis of the invariance identities of classical gauge field theory by means of functional equations.
Author:
Stapleton, David Paul.
Issue Date:
1990
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
In order to obtain the equations of motion for a particle in a classical gauge field, a variational principle is considered. The theory is general in that the structural group is an arbitrary r-dimensional Lie group and the base space is an arbitrary n-dimensional psuedo-Riemannian manifold. An n + r dimensional principal fiber bundle is constructed in order to introduce the usual gauge potentials and field strengths. In addition, a set of r quantities (called "coupling parameters") which transform as the components of an adjoint type (0,1) object and also depend upon the parameter of the particle's trajectory are constructed. The gauge potentials and coupling parameters are evaluated on the identity section of the principle bundle, and the Lagrangian is assumed to be a C³ scalar function of these and of the components of the metric tensor and tangent vector on the base space. The Lagrangian is not gauge-invariant, but it is stipulated that when the arguments of the Euler-Lagrange vector (evaluated on the identity section) are replaced by their counterparts (which may be evaluated on an arbitrary section) the resulting vector must be gauge-invariant. A novel application of methods from the theory of functional equations is applied together with standard techniques inherent in the theory of differential equations to show that the arguments of the Lagrangian must occur together in certain prescribed combinations. The invariance postulates uniquely determine the Lagrangian in terms of its arguments other than the coupling parameters and r functions of the coupling parameters. The Lagrangian is shown to separate into a free-field term and an interaction term, and the functions of the coupling parameters are found to be the components of an adjoint type (0,1) quantity whose adjoint absolute derivative vanishes. This agrees with the equations of certain approaches to the Yang-Mills theory for isotopic spin particles.¹ Standard initial conditions are shown to determine a unique (local) solution to the derived equations of motion. ftn¹ The equations have the same formal structure as systems obtained in the classical limit of quantum mechanical results found by Wong (1), pp. 691-693.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Rund, Hanno

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleA technique for the analysis of the invariance identities of classical gauge field theory by means of functional equations.en_US
dc.creatorStapleton, David Paul.en_US
dc.contributor.authorStapleton, David Paul.en_US
dc.date.issued1990en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractIn order to obtain the equations of motion for a particle in a classical gauge field, a variational principle is considered. The theory is general in that the structural group is an arbitrary r-dimensional Lie group and the base space is an arbitrary n-dimensional psuedo-Riemannian manifold. An n + r dimensional principal fiber bundle is constructed in order to introduce the usual gauge potentials and field strengths. In addition, a set of r quantities (called "coupling parameters") which transform as the components of an adjoint type (0,1) object and also depend upon the parameter of the particle's trajectory are constructed. The gauge potentials and coupling parameters are evaluated on the identity section of the principle bundle, and the Lagrangian is assumed to be a C³ scalar function of these and of the components of the metric tensor and tangent vector on the base space. The Lagrangian is not gauge-invariant, but it is stipulated that when the arguments of the Euler-Lagrange vector (evaluated on the identity section) are replaced by their counterparts (which may be evaluated on an arbitrary section) the resulting vector must be gauge-invariant. A novel application of methods from the theory of functional equations is applied together with standard techniques inherent in the theory of differential equations to show that the arguments of the Lagrangian must occur together in certain prescribed combinations. The invariance postulates uniquely determine the Lagrangian in terms of its arguments other than the coupling parameters and r functions of the coupling parameters. The Lagrangian is shown to separate into a free-field term and an interaction term, and the functions of the coupling parameters are found to be the components of an adjoint type (0,1) quantity whose adjoint absolute derivative vanishes. This agrees with the equations of certain approaches to the Yang-Mills theory for isotopic spin particles.¹ Standard initial conditions are shown to determine a unique (local) solution to the derived equations of motion. ftn¹ The equations have the same formal structure as systems obtained in the classical limit of quantum mechanical results found by Wong (1), pp. 691-693.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematicsen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorRund, Hannoen_US
dc.contributor.committeememberGreenlee, W.M.en_US
dc.contributor.committeememberHumara, Omaen_US
dc.identifier.proquest9103052en_US
dc.identifier.oclc709777863en_US
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