Any resources you recommend that go through these derivations in a reasonably easy to follow way?
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Well there's no real "derivation" of the SM gauge group.. It's just what works after trial and error haha
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U(1) symmetry represents EM gauge invariance ?
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Not quite EM.. EM gauge invariance comes from breaking SU(2) x U(1) --> U(1)_EM
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So, if I'm counting internal degrees of freedom that's 8 for SU(3), 3 for SU(2) and 1 for U(1) for a total of 12 ? Not counting their canonical conjugates.
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SU(3)_c is the gauge symmetry that corresponds to the strong interaction (i.e. the "color force")