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Does any one know how to write the Maxwell equations as an equation on a principal $U(1)$-bundle?

In Freed & Uhlenbeck's Instantons and Four manifolds, the authors claim that the Maxwell equations can be written in a gauge theoretic way: the electro-magnetic field can be viewed as the curvature $F$ of a $U(1)$-bundle on the 4-dimensional Lorentz manifold, and the Maxwell equations are equivalent to $d^∗_A(F)=0$, where A is the connection. Does anyone have a reference for this statement? I just want to have a look at the proof and see how everything matches.

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    $\begingroup$ Please, please don't use dollar signs and \textit{} to make text italic. Simple underscores will work here; use \emph{} if you are writing a paper. $\endgroup$
    – David Roberts ♦
    Commented Sep 8, 2014 at 23:29
  • $\begingroup$ You mean a principal $\text{U}(1)$-bundle with connection. A bare principal $\text{U}(1)$-bundle doesn't have a connection. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 9, 2014 at 1:09
  • $\begingroup$ @DavidRoberts Thank you so much for telling me the underscore trick! I tried \emph{} and * but they didn't work, and I didn't know other ways to make it italic. Thanks again! $\endgroup$
    – Boyu Zhang
    Commented Sep 9, 2014 at 3:22
  • $\begingroup$ @QiaochuYuan Yeah, sure. Thanks. I thought the equation is about the connection, so the $U(1)$ bundle is not a priorily endowed with a connection. Anyway, I didn't intend to write down all the details. I believe anyone familiar with this result can recognize it, and that's probably enough. $\endgroup$
    – Boyu Zhang
    Commented Sep 9, 2014 at 3:28
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    $\begingroup$ See the literature cited in mathoverflow.net/questions/72160/… $\endgroup$
    – Zurab Silagadze
    Commented Sep 9, 2014 at 4:04

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I don't have my copy handy, but I think this is all worked out in chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". The book is essentially a textbook on differential geometry with a view toward physics, so the exposition is very detailed; I just can't remember how much time is spent on Maxwell's equations.

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There is a quick explanation of the translation of Maxwell's equations into connection/curvature language in this bulletin article by Edward Witten: http://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01167-6/

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  • $\begingroup$ Thank you so much for helping! But this paper simply states the result, without giving a proof. It seems that everyone knows the result, but no one bothers to write down the proof... $\endgroup$
    – Boyu Zhang
    Commented Sep 9, 2014 at 3:26
  • $\begingroup$ Could you say what exactly you would want to see proven? There is the original Maxwell's equations from the 1850s not nvolving any principal bundles. Then there is Dirac's charge quantization argument from the 1930s which argues that this needs to be refined to the version where the electromagnetic field is a connection on a principal bundle. What one may derive is that this is the right structure to produce U(1)-valued line holonomies, which are the gauge coupling action functionals of charged particles (electrons). But apart from that... $\endgroup$
    – Urs Schreiber
    Commented Sep 9, 2014 at 5:58
  • $\begingroup$ ...apart from that saying that the EM-field is a U(1)-principal connection is part of mathematical theory building in physics and not something one may derive from first principles. (Well, one may give some general arguments about the need for gauge fields to be modeled in differential cohomology, but I guess that's not what you are after.) $\endgroup$
    – Urs Schreiber
    Commented Sep 9, 2014 at 5:59
  • $\begingroup$ @Urs I gather the OP is after a calculation that takes d*F=0 and dF=0 and outputs what a physicist would recognise as Maxwell's equations. $\endgroup$
    – David Roberts ♦
    Commented Sep 9, 2014 at 22:20

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