Timeline for From classical to quantum mechanics
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18 events
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| Jan 9, 2017 at 9:33 | comment | added | mathphys | @IgorKhavkine Thanks for your links. It looks promising stuff. Hope that it becomes clear to me what is going on. | |
| Jan 8, 2017 at 23:00 | comment | added | Igor Khavkine | @mathphys, you might want to check out some explicit examples worked out for instance at here. Otherwise, I think you should just pick an introductory reference on deformation quantization (of where there are many) and just start reading. | |
| Jan 8, 2017 at 22:59 | comment | added | Igor Khavkine | @jjcale, I meant that in that case the OP's $\mathbf{A}$ would be the CCR algebra (at least morally). The reason I need to add qualifications is that one needs to pick a specific subalgebra of $C^\infty(X)$ to deform to get precisely the CCR algebra. Other choices of subalgebras are also possible and could be related to each other by playing with limits, completions, etc. | |
| Jan 8, 2017 at 16:38 | comment | added | mathphys | @IgorKhavkine Thank you for the link! Do you know what is the $\mathbf{\Phi}(H)$ where $H$ is the Hamiltonian (the easiest case). I'm using notations of your link. | |
| Jan 8, 2017 at 12:10 | comment | added | jjcale | @IgorKhavkine What does "CCR algebra for $\mathbf{A}$" mean ? | |
| Jan 8, 2017 at 10:44 | comment | added | Alexander Chervov | My impression is the same as Igor's - it seems "deformation quantization" answers all yours questions | |
| Jan 7, 2017 at 23:45 | comment | added | Igor Khavkine | The standard $(\mathbb{R}^2, dx_1 \wedge dx_2)$ case is answered by the Wigner-Weyl-Moyal formula for the deformed "star" product, which then essentially gives the CCR algebra for $\mathbf{A}$. The CCR algebra has a known representation theory, though its irreducible representations are not of the form that you have guessed. This is all standard in the literature on deformation quantization. Reading up on that might answer many of your questions, or at least help you sharpen them. | |
| Jan 7, 2017 at 21:51 | history | edited | mathphys | CC BY-SA 3.0 |
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| Jan 7, 2017 at 19:19 | history | edited | mathphys | CC BY-SA 3.0 |
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| Jan 7, 2017 at 13:55 | comment | added | mathphys | @IgorKhavkine I think your comment clarifies my first question. It seams that we need a riemanian metric also to make things canonically, right ? After all in classical mechanics the symplectic form is needed and not just its corresponding element in the second de rham cohomology. I hope my seconde question will get answers | |
| Jan 7, 2017 at 10:44 | comment | added | Igor Khavkine | The inverse $\Pi = \omega^{-1}$ of a symplectic form is a Poisson bivector. A Poisson bivector, via deformation quantization, gives rise to a non-commutative deformation of $C^\infty(X)$ and hence a cocycle in $\mathrm{HH}^2$. But I don't know if this correspondence respects de Rham cohomology classes. | |
| Jan 7, 2017 at 9:54 | comment | added | mathphys | @IgorKhavkine I have edited my question. The point is that I don't know really how to formulate the question. I hope that someone can provide a nice interpretation. | |
| Jan 7, 2017 at 9:51 | history | edited | mathphys | CC BY-SA 3.0 |
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| Jan 6, 2017 at 23:52 | comment | added | Igor Khavkine | Is your question basically asking to describe deformation quantization? | |
| Jan 6, 2017 at 21:00 | history | edited | mathphys | CC BY-SA 3.0 |
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| Jan 6, 2017 at 20:37 | history | edited | mathphys | CC BY-SA 3.0 |
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| Jan 6, 2017 at 20:18 | review | First posts | |||
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| Jan 6, 2017 at 20:14 | history | asked | mathphys | CC BY-SA 3.0 |