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Hi everyone,
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 F 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 dF=d*F=0. Then they remark that if one writes F=dA by virtue of Poincaré's lemma, then the "potential" A is defined up to the addition of an exact 1-form df which we may write d(ln(g))=g^{-1}dg for g=e^f. In particular, by commutativity, the "gauge transformation" A\mapsto A+df can be written A\mapsto g^{-1}Ag+g^{-1}dg 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 g:\mathbb{R}^4\rightarrow \mathbb{K}^*. As to F, it can be written F=dA+A\wedge A since the exterior product of real-valued 1-forms vanish. As such, we recognize F as the curvature 2-form of the connection.
Here, the line bundle could be real, or it could be complex with structure group U(1). In this case, we admit that A and F are really i\mathbb{R}-valued.
In the more general context of a connection on a vector bundle with (matrix) structure group G on a space-time 4-manifold M, the Maxwell equation dF=0 becomes dF = F\wedge A - A\wedge F, or, regarding F as the curvature form of the corresponding principal G-bundle, DF=0, where D stands for exterior covariant differentiation. The other Maxwell equation d*F=0 generalizes to D*F=0 and this is called the Yang-Mills equation.
I also read on wikipedia that in the standard model, the structure group G is U(1)\times SU(2)\times SU(3) 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 \psi:M\rightarrow\mathbb{C} as in QM? Is U(1) related to the fact that the probability density |\psi|^2 is invariant under phase transformation \psi\mapsto e^{i\phi}\psi ?? But then what is the relationship with the EM field?!
PS I only have an undergraduate background in physics.
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 F 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 dF=d*F=0. Then they remark that if one writes F=dA by virtue of Poincaré's lemma, then the "potential" A is defined up to the addition of an exact 1-form df which we may write d(ln(g))=g^{-1}dg for g=e^f. In particular, by commutativity, the "gauge transformation" A\mapsto A+df can be written A\mapsto g^{-1}Ag+g^{-1}dg 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 g:\mathbb{R}^4\rightarrow \mathbb{K}^*. As to F, it can be written F=dA+A\wedge A since the exterior product of real-valued 1-forms vanish. As such, we recognize F as the curvature 2-form of the connection.
Here, the line bundle could be real, or it could be complex with structure group U(1). In this case, we admit that A and F are really i\mathbb{R}-valued.
In the more general context of a connection on a vector bundle with (matrix) structure group G on a space-time 4-manifold M, the Maxwell equation dF=0 becomes dF = F\wedge A - A\wedge F, or, regarding F as the curvature form of the corresponding principal G-bundle, DF=0, where D stands for exterior covariant differentiation. The other Maxwell equation d*F=0 generalizes to D*F=0 and this is called the Yang-Mills equation.
I also read on wikipedia that in the standard model, the structure group G is U(1)\times SU(2)\times SU(3) 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 \psi:M\rightarrow\mathbb{C} as in QM? Is U(1) related to the fact that the probability density |\psi|^2 is invariant under phase transformation \psi\mapsto e^{i\phi}\psi ?? But then what is the relationship with the EM field?!
PS I only have an undergraduate background in physics.