Timeline for Formal series convergence in deformation quantization and $C^*$-condition
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2013 at 16:53 | comment | added | Abdelmalek Abdesselam | This seems to be related to issue of divergence of perturbation theory in QFT. For a model like phi^4 the gradings by the coupling or hbar are essentially the same. Typically these series diverge but in some examples one can prove the next best thing which is Borel summability. I am not too familiar with the DQ framework, but I suppose you could apply it to the quartic oscillator in 1d. That may be a good test example for this problem of summing the series in hbar. | |
| Sep 6, 2013 at 9:47 | comment | added | Issam Ibnouhsein | Ok, thanks for this answer. It gives some perspective on the comments that seemed to suggest that the link between convergence and $C^*$-condition was trivial. Do you have a reference for the details of the first non-trivial step (getting an algebra over $\mathbb{C}$ by imposing convergence)? | |
| Sep 6, 2013 at 9:38 | vote | accept | Issam Ibnouhsein | ||
| Sep 6, 2013 at 9:27 | comment | added | Urs Schreiber | Thanks, that's already good to have this confirmed! I wasn't sure. For what it's worth, here is my feeling about the issue: in simple cases where everything is under control, such as the quantization of the 2-sphere, the strict C*-deformation produces the algebras of endomorphisms of the spaces of states that are produced by the geometric quantization process. But in geometric quantization we know fully well and in generality how to quantize the observables: find a maximal subgroup of the classical group of observables which preserves the polarization, then prequantize that. | |
| Sep 6, 2013 at 8:41 | comment | added | Stefan Waldmann | @Urs: haha, you got me. There are not so many examples around. The trivial one is to consider as subalgebra of smooth functions on $\mathbb{R}^{2n}$ the span of exponentials. They form a sub-algebra for the (formal) Weyl star product and give the $C^*$-algebraic Weyl algebra as completion. But beyond...? | |
| Sep 6, 2013 at 8:01 | comment | added | Urs Schreiber | Can you point to one explicit example that discusses C*-algebras which are explicitly the completion of algebras of converging formal power series inside a formal deformation quantization? I thought that's what Issam is asking for. | |
| Sep 6, 2013 at 6:55 | history | answered | Stefan Waldmann | CC BY-SA 3.0 |