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I’ve been reading the “relevance for supersymmetry” of the nlab article on Deligne’s theorem, and had a few niggling doubts: doesn’t this only apply to finite-dimensional vector spaces?
More specifically: Hilbert spaces don’t categorify well, and I’m not aware of any ‘reasonable’ category built on them which abelian. Thus, I can’t see any natural way in which representations on them form a tensor category.
Since everything works out nicely in finite dimensions, what I can see is that Deligne’s theorem implies that the most general symmetries of a finite-dimensional quantum system are given by algebraic super-groups. (Well, almost – one also has to admit antilinear representations by Wigner/Bargmann, so even this doesn’t seem exactly true.)
So here is my question: am I missing something?
doesn’t this only apply to finite-dimensional vector spaces?
Yes, to have a complete argument, one needs to mention Wigner’s “little group method” (the “Mackey machine”) and apply Deligne’s theorem to that little group.
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