Unifying Geometrical Representations of Gauge Theory

2013/08/31 by Scott T Alsid, Scott Alsid, Mario Serna +1
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #BRST quantization #Embedding #Field (mathematics) #Gauge (firearms) #Gauge theory #Introduction to gauge theory #Metric (unit) #Quantum and Classical Electrodynamics #Relativity and Gravitational Theory #Supersymmetric gauge theory #Surface (topology) #hep-th #math-ph #math.MP #msc:14M15 #msc:30L05 #msc:51M35 #msc:53B10 #msc:53C07 #msc:70S15 #msc:83E15

paper · pdf · doi:10.1007/s10701-014-9841-x

published in Foundations of Physics 45(1), 75-103 (Springer Science+Business Media) · Updated with referee suggestions. 28 pages, 7 figures. Accepted to Foundations of Physics

openalex publication_date 2014/10/16 · arxiv created 2014/10/28 · arxiv updated 2015/01/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We unify three approaches within the vast body of gauge-theory research that have independently developed distinct representations of a geometrical surface-like structure underlying the vector-potential. The three approaches that we unify are: those who use the compactified dimensions of Kaluza-Klein theory, those who use Grassmannian models (also called gauge theory embedding or CP^N-1 models) to represent gauge fields, and those who use a hidden spatial metric to replace the gauge fields. In this paper we identify a correspondence between the geometrical representations of the three schools.Each school was mostly independently developed, does not compete with other schools, and attempts to isolate the gauge-invariant geometrical surface-like structures that are responsible for the resulting physics. By providing a mapping between geometrical representations, we hope physicists can now isolate representation-dependent physics from gauge-invariant physical results and share results between each school. We provide visual examples of the geometrical relationships between each school for U(1) electric and magnetic fields. We highlight a first new result: in all three representations a static electric field (electric field from a fixed ring of charge or a sphere of charge) has a hidden gauge-invariant time dependent surface that is underlying the vector potential.

Citations

• Yang-Mills fields and Riemannian geometry
• Construction of instantons
• Appearance of Gauge Structure in Simple Dynamical Systems
• The Grassmannian sigma model in SU(2) Yang–Mills theory
• Quantum mechanics on manifolds embedded in Euclidean space
• Existence of Universal Connections
• Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase
• Canonical surfaces associated with projectors in Grassmannian sigma models
• Riemannian gauge theory and charge quantization
• On Kaluza-Klein theory
• Quantentheorie und f�nfdimensionale Relativit�tstheorie