Timeline for Quantization and noncommutative deformations
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2011 at 15:54 | history | edited | Stefan Waldmann | CC BY-SA 3.0 |
typos typos typos...
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| Aug 3, 2011 at 8:33 | vote | accept | amine | ||
| Jul 31, 2011 at 12:57 | vote | accept | amine | ||
| Jul 31, 2011 at 12:57 | |||||
| Jul 29, 2011 at 15:54 | comment | added | Stefan Waldmann | ... making $\mathbb{R}[[\hbar]]$ an ordered ring such that $\hbar > 0$. (the other one is $\hbar < 0$) | |
| Jul 29, 2011 at 15:53 | comment | added | Stefan Waldmann | As for the second comment: this is a long story, what to best notion of positivity is: the one you mention is the "SOS" (sum of squares) positivity which you can always formulate when you have an involution. However, it does not make $\mathbb{R}[[\hbar]]$ an ordered ring, a feature which is desirable. The point is that $\hbar$ itself is not positive (only $\hbar^2$ is) by your definition but physicists will definitely insist on that ;) So the one I'm using is to say a real formal series in $\hbar$ is positive, if the lowes non-vanishing term is. This is the unique one... | |
| Jul 29, 2011 at 15:50 | comment | added | Stefan Waldmann | Hi Theo. Usually I don't like ot advertise for my own stuff too much, but since you asked for: in the review MR2130623 I gave a rather large panorama of what has been done (mainly by people around me in Freiburg as well as by Henrique Bursztyn) in the recent past. It is already quite old, but gives some first hints. On my homepage you can also find the more recent stuff ;) | |
| Jul 29, 2011 at 15:40 | comment | added | Theo Johnson-Freyd | There's another possible notion of positivity, which is I think the one Richard Borcherds uses in his recent work on QFT. Namely, make a choice as to what you want the complex conjugate of $\hbar$ to be (it doesn't matter, of course --- you can rescale it away), and define a positive element of $A=\mathbb C\llbracket\hbar\rrbracket$ to be any element of the form $aa^\ast$ for $a\in A$. Note then that $\lambda\hbar$ for $\lambda$ invertible is never positive, so this is a different choice from using, say, lexicographical ordering, which is I think the one you advocate. | |
| Jul 29, 2011 at 15:36 | comment | added | Theo Johnson-Freyd | Oh, for my sake if nothing else, can you include a few references? For example, when you say "we have worked out many things like the GNS construction of representations etc" do you mean I should look in some of your papers? (which ones?) | |
| Jul 29, 2011 at 7:57 | history | answered | Stefan Waldmann | CC BY-SA 3.0 |