# E&M and geometry - a historical perspective

Recently, I was contemplating the beautiful formulation of electromagnetism (specifically Maxwell's equations) in terms of differential forms: $$F=\mathrm{d} A\implies \mathrm{d}F=0 \hspace{1cm}\text{and}\hspace{1cm} \mathrm{d}\star\mathrm{d}F=\mu_0 J$$ I started thinking about the history of this way of looking at things, and realized that I don't know much about it at all. My first question was therefore: Was it known already at the time of Maxwell (or soon after) that electromagnetism could be cast in this geometric form? How was this first introduced and who did it?

After consulting Maxwell's treatise, it became clear that at least Maxwell himself was not aware of this formulation. But maybe someone else immediately recognized the geometric formulation once Maxwell published his results...

In modern times, one is - at least as a physicist - usually first introduced to the field strength tensor $F$ through the covariant formulation of Maxwell's equation using tensor calculus, where it is defined as $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$. When one then learns about differential forms etc. it is then obvious that $F=\mathrm{d}A$ and the geometric formulation follows quite naturally. However, was this also the case historically? Did 'they' come up with the tensor calculus formulation of $F$ first, and did they only then recognize the geometric description? Or was the geometric description discovered first? Another possibility is that it took the introduction of Einstein's general relativity for anyone to realize that fields can be interpreted in terms of geometry.

In conclusion, I am interested in a chronological description of the development of the different formulations of electromagnetism, with emphasis on the following points:

1. Who first came up with the geometric formulation in terms of differential forms?
2. Is it known at all how this person arrived at this?
3. Was the geometric interpretation discovered before tensor calculus became popular, or only after it was know that $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$? Was this after the introduction of GR, and was it at all influenced by Einstein's work?
• You might also find geometric calculus, based upon clifford algebra, interesting here. It manages to take those free space equations for the EM field and marry them into one equation. – Muphrid Jul 3 '14 at 15:52
• Did 'they' come up with the tensor calculus formulation of F first, and did they only then recognize the geometric description? I would say that a tensor formulation is geometric. – Daniel Mahler Jul 7 '14 at 5:41
• Great question. I linked here from another site. – Tom Au Oct 28 '14 at 23:47

Going by a magic 8-ball a brief web search, the most important steps towards the geometrization of electromagnetism (ie its formulation as a classical Yang-Mills theory in terms of principal connections) should be:

• Maxwell's equations: James Clerk Maxwell, A dynamical theory of the electromagnetic field (1865)

• differential forms: Élie Cartan, Sur certaines expressions différentielles et le problème de Pfaff (1899)

• special relativity: Albert Einstein, Zur Elektrodynamik bewegter Körper (1905)

• gauge invariance: Hermann Weyl, Elektron und Gravitation I (1929)

I'm not sure about the next one:

• principal bundles: Henri Cartan, Séminaire Henri Cartan, 2 (1949-1950)

• Yang-Mills theory: Chen Ning Yang and Robert Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance (1954)

• Wong's equation: S.K. Wong, Field and particle equations for the classical Yang-Mills field and particles with isotopic spin (1970)

I actually don't know who has to be blamed for classical Yang-Mills theory, ie putting it all together.

This is a wiki answer, so feel free to add to or modify the list as you see fit.

• I'm not sure one has to think of it as a gauge theory to come to the formulation in terms of differential forms. Maxwell didn't even have our notation for vector calculus available, so he was a long way from differential forms. It may have been possible to formulate it this way by around 1900 but I would guess that it didn't happen until after special relativity appeared, when people knew to look for Lorentz invariance (covariance?). – gn0m0n Jul 3 '14 at 19:58
• @gn0m0n: sure, no need for gauge theory just to use differential forms; but the question (or at least its title) was concerned with EM and geometry - and geometrically, vector potential and field strength are not just some arbitrary forms, but principal connection and corresponding curvature – Christoph Jul 3 '14 at 20:12
• Sure they are... I just took the question to be asking when the formulation in terms of diff. forms occurred, and I was speculating that it could have happened before people were thinking of the vector potential and field strength in terms of fiber bundles. Whether that actually happened or not, I don't know. – gn0m0n Jul 4 '14 at 2:25
• According to mers.byu.edu/docs/thesis/phddiss_warnick_lib.pdf ("A DIFFERENTIAL FORMS APPROACH TO ELECTROMAGNETICS IN ANISOTROPIC MEDIA" by Warnick), "Weyl and Poincare expressed Maxwell’s laws using differential forms early this century" (top of p. 103). He might say more than that. Flanders's book appeared in 1963 and certainly included a treatment of E&M. He might say something about the history in it. – gn0m0n Jul 4 '14 at 3:28
• You might also be interested in sciencedirect.com/science/article/pii/0315086081900276 (The history of differential forms from Clairaut to Poincaré) or math.toronto.edu/mgualt/wiki/samelson_forms_history.pdf (Differential Forms, the Early Days) but they don't seem to address when they were first used for E&M. – gn0m0n Jul 4 '14 at 3:29