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So I recently read a chapter in a math book that vaguely describe how connections on bundles occur in particle physics, but they are very cryptic about the physics part and I just want to know a little bit more about it. So I'll tell you what I read and then follow up with some questions I have.

The way the text is structured, they start by making the observation that by defining a real-valued differential 2-form [itex] F [/itex] on R^4 in a certain way in terms of the components of the E and B fields, Maxwell's equations take the elegant form [itex]dF=d*F=0[/itex]. Then they remark that if one writes [itex]F=dA[/itex] by virtue of Poincaré's lemma, then the "potential" [itex]A[/itex] is defined up to the addition of an exact 1-form [itex]df[/itex] which we may write [itex]d(ln(g))=g^{-1}dg[/itex] for [itex]g=e^f[/itex]. In particular, by commutativity, the "gauge transformation" [itex]A\mapsto A+df[/itex] can be written [itex]A\mapsto g^{-1}Ag+g^{-1}dg[/itex] which we recognize as the way the connection 1-forms of a connection on a line bundle transform under a change of local trivialisation with transition function [itex]g:\mathbb{R}^4\rightarrow \mathbb{K}^*[/itex]. As to [itex]F[/itex], it can be written [itex]F=dA+A\wedge A[/itex] since the exterior product of real-valued 1-forms vanish. As such, we recognize [itex]F[/itex] as the curvature 2-form of the connection.

Here, the line bundle could be real, or it could be complex with structure group [itex]U(1)[/itex]. In this case, we admit that [itex]A[/itex] and [itex]F[/itex] are really [itex]i\mathbb{R}[/itex]-valued.

In the more general context of a connection on a vector bundle with (matrix) structure group [itex] G [/itex] on a space-time 4-manifold [itex] M [/itex], the Maxwell equation [itex]dF=0[/itex] becomes [itex]dF = F\wedge A - A\wedge F[/itex], or, regarding [itex]F[/itex] as the curvature form of the corresponding principal [itex] G [/itex]-bundle, [itex]DF=0[/itex], where [itex]D[/itex] stands for exterior covariant differentiation. The other Maxwell equation [itex]d*F=0[/itex] generalizes to [itex]D*F=0[/itex] and this is called the Yang-Mills equation.

I also read on wikipedia that in the standard model, the structure group [itex] G [/itex] is [itex] U(1)\times SU(2)\times SU(3) [/itex] where each one of these 3 groups is a symetry group for some "internal structure" that a particle may have.

K so what is the internal structure that is related to U(1) exactly?

How do particles enter the picture? Are particles assumed to be represented by a wave function [itex] \psi:M\rightarrow\mathbb{C} [/itex] as in QM? Is U(1) related to the fact that the probability density [itex] |\psi|^2 [/itex] is invariant under phase transformation [itex] \psi\mapsto e^{i\phi}\psi [/itex] ?? But then what is the relationship with the EM field?!

PS I only have an undergraduate background in physics.

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# I Help me understand gauge theory?

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