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I wanted to restart this discussion b/c the previous thread got sidetracked, and something about it has left me deeply confused, and I think my confusion is similar to the original posters confusion.
In Wigners theory, particles (like say an electron) are unitary irreducible representations of the 2 fold (universal) cover of the Poincare group Sl(2,C). Ok good!
But,
Shouldn't an electron be considered a section of the fiber bundle U(1) over flat spacetime. Classically this is true, upon quantization it should also remain true.. Correct?
In other words is the representation of an electron under SL(2,C) identical to the representation of an electron under SL(2,C) * U(1). Locally that is.
What has me confused is basically how you add gauge groups to spacetime, and how it meshes with the particle point of view. Theres also the added subtletly of Z2 factors showing up all over the place that has me confused.
Also, U(1) is an abelian group, it seems to me more complicated topological gauge groups would exhibit global features that clash with Wigners point of view. I mean is it even appropriate to tensor a spacetime group with an internal symmetry group and then ask for representations thereof upon quantization.
Finally, I can't find good references where the fiber bundle point of view is used in a quantum sense, its always developed classicaly.
In Wigners theory, particles (like say an electron) are unitary irreducible representations of the 2 fold (universal) cover of the Poincare group Sl(2,C). Ok good!
But,
Shouldn't an electron be considered a section of the fiber bundle U(1) over flat spacetime. Classically this is true, upon quantization it should also remain true.. Correct?
In other words is the representation of an electron under SL(2,C) identical to the representation of an electron under SL(2,C) * U(1). Locally that is.
What has me confused is basically how you add gauge groups to spacetime, and how it meshes with the particle point of view. Theres also the added subtletly of Z2 factors showing up all over the place that has me confused.
Also, U(1) is an abelian group, it seems to me more complicated topological gauge groups would exhibit global features that clash with Wigners point of view. I mean is it even appropriate to tensor a spacetime group with an internal symmetry group and then ask for representations thereof upon quantization.
Finally, I can't find good references where the fiber bundle point of view is used in a quantum sense, its always developed classicaly.