Foundations of Physics

, Volume 45, Issue 1, pp 75–103 | Cite as

Unifying Geometrical Representations of Gauge Theory



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.


Kaluza Klein Gauge field theory: composite Field theoretical model: \(CP^{N-1}\) Gauge geometry embedding Grassmannian models Hidden-spatial geometry 


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© Springer Science+Business Media New York (outside the USA)  2014

Authors and Affiliations

  1. 1.Department of PhysicsUnited States Air Force AcademyUSAF AcademyUSA

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