[zaph0d_]

I once took an advanced seminar course on the mathematical foundations of electrodynamics in parallel to my theoretical electrodynamics course during my third bachelor semester. I did not have any clue about differential geometry and did not understand the advanced formalism the lecturer introduced in the seminar. But I was quite shocked how easily Maxwells equations can be derived and how compact the formula was. The article suggests that Gauge theory and fiber bundels are subjects, where math and theoretical physics seem to help each other, which is absolutely facinating!

[espeed]

Note the article is by Nobel Laureate C.N. Yang [1] who also worked with James Simons and co-authored what has become known as the "Wu-Yang Dictionary" [2].

[1] Yang-Mills Theory: https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory

[2] "Wu-Yang Dictionary" http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20non...

...The mathematics of these results is in fact well known to the mathematicians in fiber bundle theory. An identification table of terminologies is given in Sec. V. We should emphasize that our interest in this paper does not lie in the beautiful, deep, and general mathematical development in fiber bundle theory. Rather we are concerned with the necessary concepts to describe the physics of gauge theories. It is remarkable that these concepts have already been intensively studied as mathematical constructs.

"Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application" https://www.youtube.com/watch?v=h5gnATQMtPg

[zaph0d_]

Thank you for the interessting material! I have heard a lecture on the geometric and topological applications to solid-state physics. This is one of the things that excites me a lot that whole areas of physics can be "geometrized".

[Retra]

Part of me believes that geometry is really just a subset of human thinking that comes very naturally/quickly to us, and thus we are more successful studying physics through geometry than through less 'algebraic' frameworks. So we made the most advances there just because we are biased toward doing so.

But then I think that we may be so strongly biased toward doing so because there's something fundamentally easy about evolving a brain that comprehends geometry. That information with geometric representations are fundamentally easier to evolve good mental models for than other kinds of information.

[emmelaich]

You've reminded me of Plato's quotes:

"God ever geometrizes."

and

"Geometry existed before the creation."

[cgmg]

My favorite formulation of Maxwell's equations is

◻²A = J

where A is the 4-potential, J is the 4-current, and ◻² is the 4-Laplacian or d'Alembertian (https://en.wikipedia.org/wiki/Classical_electromagnetism_and...). The reason this is so elegant is because it is both manifestly covariant and manifestly a wave equation. See https://physics.stackexchange.com/questions/201847/why-is-th.... Furthermore, conservation of 4-current is given by

◻·J = 0

where ◻· is the 4-divergence. Again, the equation is manifestly covariant and very elegant. There are reasons to believe that the electromagnetic potential is in a sense more fundamental than the electromagnetic field:

https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect

[Arnavion]

(To save others time, that symbol really is a box and not a glyph missing from your browser's fonts.)

[dboreham]

Nice to see this article available on the Internet: I read a PDF that was passed around by email at the time it was published (Frank Yang is a relative of one of my neighbors and I once tried to chat with him about Maxwell over dinner...).

If you haven't read Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" it's worth a look : https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

[zaph0d_]

Thank you. Awesome! What did he say at dinner?

[dboreham]

It was pretty amazing and I was so such awe combined with fear of saying something stupid that I don't remember everything. He mostly talked about non-Physics subjects. He did talk about Fermi and working for Oppenheimer. I asked about Teller...do not now remember what he said to that. I had attempted to understand something about Yang-Mills Gauge Theory in preparation for dinner, but completely failed, so instead I figured as a Scot and card-carrying EE I'd ask him what he thought about the apparent quantum jump in progress made by Maxwell -- how was Maxwell able to come up with such modern looking physics in the age of steam, for example. At the time I did not know that the history of Maxwell was one of his subjects of interest. He talked about some of the themes that you can see in the article above (which was written 10 years later). He also mentioned, in a joking way, his prediction years earlier that "In the next ten years, the most important discovery in high-energy physics is that `the party's over'.".

[zaph0d_]

Thank you for sharing that story! It must be really amazing to hear storys from a Nobel laureate about working for other laureates like Fermi.