As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to relate electrodynamics and complex line bundles and connections (anything to do with quantum theory, perhaps?). I know Yang-Mills theory has to do with all this idea of studying bundles and connections, but I hope it has physical meaning and not just 'geometrical convenience'. Can anyone explain me intuitively this mathematics-physics interplay?

More concisely, suppose our ‘electromagnetic world’ is modelled by means of a complex line bundle with base the 4-dimensional spacetime.

-In Quantum Mechanics, a complex valued function over spacetime should be understood as a ‘phased’-distribution of the probability of a particle being in a certain place. How should be understood a section over our complex line bundle?

-And in this setting, which is the role of a connection?

As it may be difficult to answer this in a few words, it would be also helpful if somebody pointed out some small fragment of a book where this questions are studied. All ideas are welcomed.

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    $\begingroup$ Have you read en.wikipedia.org/wiki/Introduction_to_gauge_theory ? $\endgroup$ – S. Carnahan Nov 29 '14 at 8:49
  • $\begingroup$ Thank you, I have just read it. But the problem is still there: which is the intimate relation between Electrodynamics and U(1)? $\endgroup$ – Jjm Nov 29 '14 at 9:55
  • $\begingroup$ Can you point to a case when you have a satisfactory (for you) answer to a similar question? Why is mechanics linked to ODE's and symplectic geometry? $\endgroup$ – Peter Dalakov Nov 29 '14 at 10:09
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    $\begingroup$ @Jjm, from your own "sample answers", it seems that your question is not really about mathematical intuition but physical intuition. As such, it might be a better fit for physics.SE. One thing you should keep in mind, though, is that the U(1) of quantum mechanics and the U(1) of electromagnetism have different origins, so similarities between them are more of a coincidence than not. The U(1) in EM appears even classically, when considering the differences between the field-strength and the vector-potential. $\endgroup$ – Igor Khavkine Nov 29 '14 at 11:47
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    $\begingroup$ @IgorKhavkine: I'm not sure that the U(1) appears classically if you just look at the electromagnetic field alone. Classical gauge transformations in Minkowski spacetime, say, are simply $A \mapsto A + d\Theta$, where $\Theta$ is a real function. There is no indication here that $A$ is a connection on a circle bundle, as opposed to an $\mathbb{R}$ bundle. It's only when you couple electromagnetism to matter that the gauge invariance of the classical action will tell you that the matter is in a unitary representation of the gauge group and, being finite-dimensional, this picks out U(1)... $\endgroup$ – José Figueroa-O'Farrill Nov 29 '14 at 15:48