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I am in a small wave of activity along one of my principal lonegr goals in nlab: the connection between the operator theory and geometry. This is of extreme importance for physics if we ever want to go beyond the TQFTs in quantization program. As Tom Leinster has in his work seen, there are heat-kernel like expansions involved all around the place even when one takes categorical approach and the first terms are of topological nature. This is exactly so in the WKB-expansions where the zeroth term is often the exact value for topological or more general integrable models. Witten's calculation of Witten's index (related to tmf) is an example where such WKB aprpoximations are evaluated and in presence of supersymmetry there are no other terms. Thus I believe that the kantization preferred in nlab is limited to work exactly in simiklar cases and that in general we will have more terms of WKB-like nature involved. We need to develop a categorified WKB method which will unify both.
On the other hand, the WKB method is not just expansion like in quantum mechanics books, it does involve cocycles right away in usual symplectic geometry. There is so-called Maslov index related to the multidimensional WKB method which has been pioneered by V. Maslov. The quantity which is slowly changing is an analogue of the eikonal in geometric optics, so the whole thing is a generalization of the geometric optics approximation. One can see some aspects of that on (free online, on the AMS web site, under books, here) book on symplectic geometry by Guillemin and Sternberg.
Harmonic side of the stationary phase approximation (which is just a variant of WKB in fact) is studied for long under the name oscillatory integrals. This is studied especially by Lars Hörmander and the Japanese school of microlocal analysis (btw, that one is the number 3000 entry in nlab!); the differential equations are often decribed via D-modules and in nonlinear case D-schemes which Gorthendieck described via crystals.
Strangely enough Kashiwara who worked much in microlocal analysis and D-modules has created a notion of crystal bases and crystals of quantum groups but these are NOT related to crystals. Thus I created crystal basis to fix the opinions in the nlab before they go astray...
I created entry hyperfunction, one of the tools of microanalysis, by Japanese school, a neat version of generalized functions, more flexible than distributions of Laurent Schwarz. They are obtained as boundary values of holomorphic functions (reminds me of BV formalism :)).
Thanks for all this, Zoran. Very nice!
that one is the number 3000 entry in nlab!
All right, I was waiting for that to happen. Was thinking if I might make a blog post out of that. But not sure yet if that makes sense.
The entry on microlocal analysis is of course, still a stub. In particualr it does not have a definition of microlocalizaton, microlocal symbol etc. which involve dealing with the filtrations on the approproate algebra of symbols. But it is important to outline the setting and relevant mathematics to be able to incorporate these things.
that one is the number 3000 entry in nlab!
Out of curiosity, how do you tell which number an entry is?
the numbers are displayed here http://ncatlab.org/
I added more references to crystal basis, microlocal analysis (including a clarification of the concept, though still did not get to the exact math) and stub for a new entry algebraic microlocalization having just one reference for now.
Good delineation (more formal than now in our entry) is at Springer online enc. of math. : here
I have created entry Ioannis Vlassopoulos with an excerpt from his interesting program relating D-modules and equivariant localization techniques to get information on Floer/quantum cohomology. He did calculations for some toric varieties and also studied some general questions like the relation with Getzler-Jones-Petrack model for cyclic cohomology. This is related to the general philosophy I outlined above on the stationary phase approximation and on the other hand to the interesting line Urs and Domenico had on loop spaces etc.
Recent PhD student of Vinogradov (of diffieties) and of Cattaneo in Zurich genealogy page, Michael Johannes Bächtold, wrote a remark to the crystal that D-scheme and diffiety is the same thing (hence also equivalent by duality to a crystal of schemes). This was not clear to me, but I assume he is an expert according to the thesis title. I hope he helps us more with the D-modules, characteristics of PDEs and so on...
Great, I will contact you.
I added a page about pseudodifferential operators that contains the definition of both the operator and the class of symbols that I am familiar with. My motivation is different from yours, Zoran, however, because I’m primarily interested in understanding the characterization of Hadamard states on curved mainfolds in AQFT. All I know about this right now is that the spectrum condition on Minkowski space is replaced by a condition on the wavefront set of two point functions, which is celebrated by the people in the field as a major breakthrough :-) If I get to it, I may add content to the nLab as I learn about it.
Expanded hyperfunctions a little bit. That’s a really neat topic, thanks for pointing it out. Right now I try to wade through the book Goro Kato, Daniele C. Struppa:” Fundamentals of algebraic microlocal analysis” (reference added to the page algebraic microlocalization). It looks like a good way to better understand this paper: Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems.
Further expanded hyperfunctions and added wavefront sets. It would seem that one can construct a flabby sheaf of “microfunctions” for a sheaf of hyperfunctions that classifies the singularities, as soon as I think I understand that well enough I will add a page about that, too.
In general the study of singularities in microlocal analysis is a rather central concern, but its exact logic I do not have clear picture of. I am about to travel these days, so I can not help for few days in this though.
Tim,
thanks for all the recent edits you made! Great.
Here are some comments on formatting. You can format your entries whatever way you like, but the following is what – i think – is being followed mostly on the Lab (I am following it, anyway).
You can use the definition/theorem/proof-environments described here.
If you start the section hierarchy with a double “#” for main sections, triple “#” hash for their subsections and so on, then the automatic table of contents comes out as expected;
back in the old days I used to start all section headlines with lower case. I was eventually convinced that they should be uppercase, so that’s what I stick to now.
I have edited your wavefront set accordingly, so that you can see what I mean.
@Urs: Ok, thanks, will try to keep that in mind. I can’t remember all the things that are in the FAQ etc. Giving me feedback here about recent edits simplifies matters for me considerably.
Zoran wrote:
In general the study of singularities in microlocal analysis is a rather central concern, but its exact logic I do not have clear picture of.
I’m learning the material these days for the first time, too. The relevance of it is not completly clear to me yet, all I know is that the concept of “Hadamard states” is considered to be a good substitute for the “vacuum vector” in AQFT on general spacetimes, that one can characterize Hadamard states by their wavefront sets and that some people in the field like to consider real analytic spacetimes, for which hyperfunctions are the natural generalized functions, like distributions are for smooth manifolds.
Zoran wrote:
I am about to travel these days, so I can not help for few days in this though.
No problem, it will take me some more weeks to work my way through “Fundamentals of algebraic microlocal analysis” anyway.
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