Timeline for Prequantization and Hilbert space
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 8, 2013 at 11:33 | comment | added | Urs Schreiber | In fact, on the contrary this is best known among representation theorists, who use this to study representations on their Hilbert spaces. Given a Hamiltonian G-action the push-forward is done in G-equivariant K-theory and hence lands in the representation ring. Notably Mathai-Zhang (following Landsman-Hochs) proved one of the deepest theorems in quantization in this context in full generality: quantization commutes with G-reduction (ncatlab.org/nlab/show/…) | |
| Jul 8, 2013 at 7:19 | comment | added | Alexander Chervov | @Urs my impression that "push-forward story" is more related to "metaplectic correction" - i.e. Hilbert space is not "half of functions" but "half-forms". This part of research does not consider a question of construction the representation on the Hilbert space (?). | |
| Jul 8, 2013 at 7:13 | comment | added | Urs Schreiber | References relating geometric quantization and deformation quantization are listed here: ncatlab.org/nlab/show/… . Notice that the modern view on geometric quantization is via push-forward of the prequantum bundle in K-theory, see ncatlab.org/nlab/show/geometric+quantization+by+push-forward | |
| Jul 8, 2013 at 6:48 | comment | added | Alexander Chervov | @Tobias Sorry, I do not know the reference. I do not think that it is fair to say that "both theories have a different ansatz to circument the Groenwald/van Hove no-go". Deformation quantization is natural and mature concept, but the requirment that { } should go to [ ] exactly for all/part observables (which is sometimes taken as part of g.q.) is rather artificial. | |
| Jul 7, 2013 at 23:06 | comment | added | Tobias Diez | Could you recommend some literature on the relationship between geometric and deformation quantization? I think both theories have a different ansatz to circument the Groenwald/van Hove no-go theorem about a proper quantization of all classical observables. Where geometric quantization only tries to quantizise only a subset of observables, deformation quantization relaxes the commutator correspondence to $\hat{ \{f, g\}} = \frac{i}{\hbar} [\hat f, \hat g] + O(\hbar^2)$. | |
| Jul 7, 2013 at 19:30 | vote | accept | CommunityBot | ||
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| Jul 7, 2013 at 18:50 | vote | accept | CommunityBot | ||
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| Jul 7, 2013 at 16:46 | history | answered | Alexander Chervov | CC BY-SA 3.0 |