Timeline for Deformation quantization of Poisson bracket without star-product
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| May 30, 2016 at 12:43 | comment | added | issoroloap | thank you @Dilaton, i'm going to reply to that answer. here i just remark that, while i agree with Arnold and found his reference very useful, it doesn't quite solve my problem, yet. | |
| May 29, 2016 at 21:16 | comment | added | Dilaton | See also Arnold Neumaier's answer to the same question which says that Rieffel quantization could be relevant. | |
| May 25, 2016 at 21:38 | history | edited | issoroloap | CC BY-SA 3.0 |
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| May 25, 2016 at 8:20 | history | edited | issoroloap |
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| May 25, 2016 at 6:43 | comment | added | issoroloap | @Igor Yes I was not very specific in the motivation, I wanted just to to give an idea. What I had in mind was more the non-relativistic hydrodynamic Poisson bracket $\{\overline{f},\overline{g}\} = \int \frac{\delta \overline{f}}{\delta u } K( \frac{\delta \overline{g}}{\delta u} )dx$, with $K = \sum K_i \partial_x^i$ a differential operator with coefficients $K_i(u,u_x,u_{xx},\ldots)$ that are diff. polynomials and $\overline{f}=\int f(u,u_x,u_{xx},\ldots)dx$, with $x\in S^1$. People consider quantization of such systems (KdV, etc) all the time and I have been wandering about its unicity. | |
| May 25, 2016 at 1:23 | comment | added | Igor Khavkine | A note about your motivation. You were not specific about what you mean by local functional. If one takes it to mean "spacetime local", like $A[\phi] = \int_M f(x) a(\phi,\partial\phi,\ldots)$ with $f(x)$ having compact support on the spacetime $M$, then your insistence on local functionals is moot. The Poisson bracket of two local functionals, given by the Peierls formula $\{A,B\} = \int_{M\times M} \frac{\delta A}{\delta\phi(x)} G(x,y) \frac{\delta B}{\delta\phi(y)} dx \, dy$, is already only bi-local since the causal Green function $G(x,y)$ only vanishes when $x,y$ are spacelike separated. | |
| May 24, 2016 at 23:34 | history | edited | issoroloap | CC BY-SA 3.0 |
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| May 24, 2016 at 23:27 | history | asked | issoroloap | CC BY-SA 3.0 |