Timeline for Can one classify irreducible unitary representations of the Weyl algebra?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 18, 2014 at 22:22 | comment | added | yuggib | @AlexZorn Sorry, I come from a physical background, and I misunderstood. Again because of my background, I find your requirements for a unitary irrep a bit confusing. One of them is that the operators must be everywhere defined. So the "Fock space" representation that you are considering is not what the physicists usually call a Fock space, with $a_+$ and $a_-$ being the annihilation and creation operators: with the "physical" definition, these operators are unbounded and thus only densely defined. | |
| Nov 18, 2014 at 18:23 | comment | added | Alex Zorn | Thanks yuggib, I'm specifically curious about the representations of the algebra which are not necessarily representations of the group. I've added an edit so this is more clear. | |
| Nov 18, 2014 at 18:22 | history | edited | Alex Zorn | CC BY-SA 3.0 |
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| Nov 18, 2014 at 15:53 | comment | added | yuggib | ... Haag's Theorem of QFT. Anyways, a brief review of Mackey's work, with a (very sketchy) reference to the problems in QFT can be found in this pdf by Varadarajan. Don't know if this could be useful for you ;-) | |
| Nov 18, 2014 at 15:50 | comment | added | yuggib | The Stone-von Neumann theorem referred above have been extended by Mackey to characterize the irreducible representations of locally compact groups. In quantum field theories however, the Weyl algebra is no longer the algebra of a locally compact group. I don't know if there are ways of extending the Mackey results to groups that are not locally compact (I am not an expert in the field), however I know (but maybe also you) that the existence of inequivalent irreducible representations of CCR with infinite degrees of freedom is central in the so-called... | |
| Nov 17, 2014 at 22:26 | comment | added | Alex Zorn | But those are only the ones that come from representations of the Heisenberg group (see, for example, "the Weyl form of the CCR" on that page). | |
| Nov 17, 2014 at 22:07 | comment | added | Qiaochu Yuan | en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem | |
| Nov 17, 2014 at 21:54 | history | asked | Alex Zorn | CC BY-SA 3.0 |