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the same. So without a clear distinction between a group's finite and infinitesimal elements, there's a tendency to explain spin reps with lots of hand-waving and physical nonsense. The truth is that we are exploiting the fact that reps in QM are inherently projective (since a vector and any multiple are physically indistinguishable). So we can pass to the double cover of SO(3) which is SU(2). But this description is likely to be lost on physicists . The pic is misleading with respect to the topology, but is the only (non-trivial) dimension in which a visual depiction of an entire group and its double-cover is possible.
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Fair enough... Basically my goal is to help physicists gain some intuition for spin reps. Having a very clear feeling for the difference between SO(3) and SU(2) is important, and it is equally important to understand why this difference is neglected. Obviously the problem is that their algebras are
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The "Set up" section has a bit of an oversight in it. It talks about definite forms and double covers of their special orthogonal group, but then when it gets to the case of indefinite forms it does not describe the analogous construction. The fundamental group of these special orthogonal groups
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looks like it could be relevant, especially the section on the
Clebsch-Gordan decomposition. Also there is some duplication with that article, which may not be a bad thing. Although I don't really have much of a grand vision for how this article and that one should look.
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isn't just cyclic anymore so you have to supply additional details to single-out a canonical 2-sheeted covering space. If I edited the page I'd probably go wild and change much of the exposition so I'd rather just leave it to the original authors.
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Some overlap is inevitable, as the articles concern closely related objects. The
Clebsch-Gordan decomposition is essentially covered by the section "Symmetry and the tensor square", although the latter could be expanded to make this more explicit.
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The picture is missleading. dim 1 is the only dimension in which spin is not simply connected, but this is one of the main proberties of spin.
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This article is about the spin representation and not the spin group. Probably the picture would clarify something there.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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Hi there, I added the pic in the intro in the hopes of helping readers visualize this concept.
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I got lost (among other places) where in says that the quadratic form identifies
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I mean something like in the case of the Dirac algebra where the Lie algebra of
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can be formed by taking commutators of the Dirac matrices.
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Any thoughts on what else needs to go in this article?
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