Symmetry breaking via internal geometry

Andrew Vansickle Talmadge

doi:10.1155/ijmms.2005.2023

Abstract

Gauge theories commonly employ complex vector-valued fields to reduce symmetry groups through the Higgs mechanism of spontaneous symmetry breaking. The geometry of the internal spaceVis tacitly assumed to be the metric geometry of some static, nondynamical hermitian metrick. In this paper, we considerG-principal bundle gauge theories, whereGis a subgroup ofU(V,k)(the unitary transformations on the internal vector spaceVwith hermitian metrick) and we consider allowing the hermitian metric on the internal spaceVto become an additional dynamical element of the theory. We find a mechanism for interpreting the Higgs scalar field as a feature of the geometry of the internal space while retaining the successful aspects of the Higgs mechanism and spontaneous symmetry breaking

Highlights

In this paper, we consider G-principal bundle gauge theories, where G is a subgroup of U(V,k) as models of particle interactions that employ a complex vector space V -valued field, known as a Higgs field
We allow the hermitian metric on the internal space V to become a dynamic element of the theory and we find a way to realize the Higgs scalar field as a feature of the geometry of the internal space while retaining the successful aspects of the Higgs mechanism and spontaneous symmetry breaking
Particle fields are represented by equivariant mappings from the principal bundle into a vector space V upon which G acts on the left

Summary

Introduction

We consider G-principal bundle gauge theories, where G is a subgroup of U(V ,k) (the unitary transformations on the internal vector space V with hermitian metric k) as models of particle interactions that employ a complex vector space V -valued field, known as a Higgs field. We allow the hermitian metric on the internal space V to become a dynamic element of the theory and we find a way to realize the Higgs scalar field as a feature of the geometry of the internal space while retaining the successful aspects of the Higgs mechanism and spontaneous symmetry breaking. This interpretation is obtained using a Lagrangian procedure to identify a nonzero vacuum value for a special class of internal metric fields. The last section is a summary of the results contained in this paper

Background and notation

Symmetry-breaking metric fields

Dynamics of the internal hermitian form

Conclusions