Foundations of Physics

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

Unifying Geometrical Representations of Gauge Theory

  • Scott Alsid
  • Mario Serna


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 



The authors would like to thank Laura Serna, Kevin Cahill, Richard Cook, Matt Robinson, Christian Wohlwend, Ricardo Schiappa, and Yang–Hui He for helpful comments after reviewing the manuscript. We would also like to thank the reviewers for helpful contributions increasing the quality of the final paper. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government. Distribution A: Approved for public release. Distribution unlimited.


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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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