Timeline for Particle Physics and Representations of Groups
Current License: CC BY-SA 2.5
7 events
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| Jul 6, 2010 at 16:21 | comment | added | Jim Pivarski | On my way to work, I thought of a glib, two-word answer to the original question: "Noether's theorem." In the physics context, Noether's theorem states that symmetries in the Lagrangian of a theory correspond to conserved currents. Quantized conserved currents are particles, so it should be no surprise that Lie groups representing the gauge symmetries correspond to gauge particles (gluon, W, Z, photon). This doesn't say anything about the matter fields (quarks and leptons), though. | |
| Jul 6, 2010 at 16:19 | comment | added | Jim Pivarski | Thanks for the comments: I've fixed "torus" -> "cylinder" ("torus" would imply that two dimensions are curled up, not just the one). I should have emphasized more that the Kaluza-Klein picture is a historical precursor--- the right way to think about it is as a fiber bundle. The Göckeler and Schücker book looks perfect: thanks! I'll try to find one in my library or buy it online. | |
| Jul 6, 2010 at 16:13 | history | edited | Jim Pivarski | CC BY-SA 2.5 |
added 3 characters in body
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| Jul 6, 2010 at 8:36 | comment | added | jeremy | But the structure of everything in the classical case is still in terms of bundles with structure groups that are the gauge group. And it is the case that the classical structure of any yang-mills theory is basically the same as general relativity as you mention. But the Kaluza-Klein case isn't the right picture here. Also, the book I mentioned at the end of my answer above provides a reasonably detailed intro to the curvature of fiber bundles, from the physics POV, as well as several good references to the math POV. | |
| Jul 6, 2010 at 8:32 | comment | added | jeremy | In yang-mills we want to look at the already-broken phase ('macroscopic') of the would-be microscopic theory. But the procedure of: 'go from local gauge symmetry to a space that locally is MxG and then do gravity' is not well-defined. In particular, it fails if G=SU(3)xSU(2)xU(1) and we insist on chiral fermions (as was shown by Witten). The details of what's going on here are somewhat ugly, and I do not know of a nice exposition of them. Naively, it should have worked because the manifold MxG exists, but the correct fields do not, which is why I didn't talk about this in my answer above. | |
| Jul 6, 2010 at 8:26 | comment | added | jeremy | Kaluza-Klein is on a cylinder, not a torus. Also, the breaking of Diff(MxS^1) -> Diff(M) x U(1) is a symmetry-breaking kind of breaking. You can find discussions of this in many easy-to-find articles by Weinberg, Witten, and a bunch of others and should be in most reviews of KK theory. The reason the "extra" S^1 coordinate is not normally noticeable in the 'macroscopic' theory is because it is a low-energy theory and its dependence has been integrated out. The remnant of this dynamics is electromagnetism. But this isn't really quite the same picture as in yang-mills theory. | |
| Jul 6, 2010 at 7:10 | history | answered | Jim Pivarski | CC BY-SA 2.5 |