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started motives in physics with text that I posted as an answer to this Physics.SE question.
Needs to be polished and expanded. But I have to run now.
Isn’t the idea of cosmic Galois group due Cartier ? (I am not sure) Also “The proposal that the natural domain for geometric quantization are Lagrangian correspondences is due to Lars Hörmander, Fourier Integral Operators I., Acta Math. 127 (1971), Alan Weinstein” is I think not giving the right credit to Maslov who earlier, in 1960-s, wrote several papers and several books on the subject of quantization (including geometric quantization, and, much more, on WKB method/series to any order and any number of dimensions) using Lagrangian geometry; those may be in a little more old-fashioned language but I think they have huge overlap, at least with Hoermander’s work (including deep parts on pseudodifferential operators). For example, Maslov index may be considered (people say) as a characteristic class in Lagrangian cobordism, just the right thing for your purposes (I’d like if you explain us the details of that interpretation one day soon).
I think only the name “cosmic Galois group” is due to Cartier, not the idea.
Does it appear earlier than here
At a deeper level, I was more prophetic than I had thought, and there are many reasons to believe in a “cosmic Galois group” acting on the fundamental constants of the physical theories?
@Zoran,
thanks for the link to the lecture by Bloch, have added that to the entry (is there a text document that goes with this?)
concerning Maslov: can you point me to page and verse where Maslov talks about Lagrangian correspondences?
Concerning your last question: say again, on what would you like me to give details?
Concerning your last question: say again, on what would you like me to give details?
Maslov index has tens of different definitions, most of them axiomatic. An example approach is here. So how to precisely justify the phrase that Maslov index/class is a “characteristic class in Lagrangian cobordism” which is used by insiders?
concerning Maslov: can you point me to page and verse where Maslov talks about Lagrangian correspondences?
Maslov defined the Maslov canonical operator, which is giving the cocycle for what is now called the Maslov bundle; he was, to my knowledge, not talking in invariant terms. He does not use the invariant language, but the language of cocycles (say in his 1965 book on asymptotic analysis; he uses the same Maslov cocycle method over and over for many applications in several books and many papers). The Maslov cocycle is precisely interesting because we do not have global polarization so we can not express things in terms of p or x, but need to do the general case of correspondences. Look at section 5.13 (and specially 5.13.4) in the recent book
for restatement of the Maslov 1965 cocycle/Maslov canonical operator in modern invariant language. Edit: I think, it is quite inspirative (and related to the issue) to read the example on generating functions on page 9 of the book.
I can not remember the Russia book in whose introduction very clearly the Maslov idea is sketched comparing local and global language in several sentences. I spent couple of hours looking at candidate references and could not find the page which is in my recent memory, if I find it, I will send it.
the lecture by Bloch, have added that to the entry (is there a text document that goes with this?)
The aspect of motives in the study of Feynman diagrams are studied in a series of papers by Bloch, Esnault, Kreimer, Broadhurst etc. e.g.
The subject has risen significantly since the discovery or relations of this research to the phenomena of total positivity and cluster algebras, see the recent papers by Alexander Postnikov, Alexander Goncharov, Anastasia Volovich and collaborators. State of the art was expounded on the
In some special SUSY models this line of research lead to some nonperturbative results (including what was discussed under amplituhedron).
I put the above references to the entry motives and physics. Also
to the Maslov index. By the way, the link http://www.ams.org/online_bks/surv14 to Geometric asymptotics is broken. You recall, I was writing couple of years ago to AMS Webmaster who gave bad explanation (like next year we will have new site for free books so we have removed them from the server). Now even the search for the file fails (I spent 20 minutes trying). If somebody finds the file on some AMS site, please put the correct link into Maslov index and semiclassical approximation.
Added
to semiclassical approximation. I also added
Relation to quantum integrable systems is in a series of works of Vũ Ngọc, e.g.
Thanks for the motivic references! I have added more of that to motives in physics.
Concerning Maslov: thanks; i can try to look into that. But in #2 you said it’s not correct to say that Hörmander/Weinstein were the first to consider Lagrangian correspondences. Where did Maslov talk about Lagrangian correspondences first, if he did?
No, I am saying that it is not correct to say that they were the first to do the quantization via Lagrangian correspondences. You do not need to use word “Lagrangian correspondence” to use it. His achievement was exactly to go to the Lagrangian correspondence case via the corresponding cocycle. It is generally understood that Hoermander’s work and Maslov work are essentially equivalent (of course, Maslov went further in physical applications and Hoermander further in fine analytic properties) and of course, Hoermander done his work already in late 1960s.
Much overlap in the field is also with work of Sato and Kashiwara in late 1960s, but they did not look at Maslov cocycle and quantization via correspondences though they had deeply used other aspects of Lagrangian geometry in microlocal and semiclassical analysis.
New stub Lagrangian cobordism to have the references handy.
On the other hand, Maslov originated the very notion and name for the Lagrangian submanifold if not the concept and the name for Lagrangian correspondence which are implicit in his work. A short historical account is here.
Zoran, thanks for the links. Here is a maybe naive question: you seem to be thinking of a close relation between Lagrangian cobordism and Lagrangian correspondences. Could you elaborate on what you have in mind?
Or else, could you elaborate on how you think of Lagrangian cobordism to be related to the topic of this thread?
I don’t know. I mentioned Lagrangian cobordism only as a setup which Arnold allegedly connected with Maslov method of quantization; the method closely connected with Hoermander’s work. I know close to nothing about Lagrangian cobordism. I think that somebody who works on modern setup like you understands these big words, unlike me, and will once enlighten me.
I worked on Maslov method in local coordinates as physics undergraduate student (sophomore), and all I know (better to say knew, having in mind the elapsed time) is in local coordinates (and mostly in elaborate examples; Maslov attacked many equations with this method); people in references use modern terminology. Weinstein’s paper is called Maslov gerbe, for example. Arnold talks on characteristic class in Lagrangian cobordism, but his paper is mainly on the very index appearing in the story, not the fully fledged semiclassical analysis…I hope you understand that the method of Maslov cocycle is a deeper thing than the used index in that theory. No wonder he had a reason to invent Lagrangian submanifolds in his work.
Okay. I do appreciate that Maslov’s work is deep and important (and he is linked to now from Lagrangian submanifold and other related entries).
But given what we have so far here I do not see evidence that we can attribute to him the notion of Lagrangian correspondences and their role in classical mechanics.
I appreciate that Maslov introduced the notion of Lagrangian submanifold, which enters into that of Lagrangian correspondences, but this alone does not seem good reason to include his name in a list of precursors of motivic structures. (We might go on and include Riemann for the notion of manifold itself, and then eventually we include everyone…)
I hope it is clear that already listing Lagrangian correspondences in a list on motivic concepts is something non-standard, which is justified only by some recent developments. If we now include contributions in which even the notion of correspondence did not appear, then I think that means overdoing it.
So unless or until you remember a point in Maslov’s writing which connects directly to Lagrangian correspondences the way later used by Weinstein, I would like to remove the citation to him that you added to motives in physics.
The link list there is already long. It should not be diluted by links that do not really pertain to the topic at hand.
Wait for me, I have the most important deadline in last 3 years this Sunday and spent already about 4 hours trying to help on your current topic. It is very difficult to dig the exact lines in references I read two decades ago on your urge.
Okay, sure, there is no rush.
I can, however, before that cite the modern textbooks
Hormander, vol. 4 Chapter 25, Lagrangian distributions and Fourier integral operators
page 10
In section 25.1 we generalize the notion of conormal distribution by defining the space of Lagrangian distributions associated with an arbitrary conic Lagrangian $\Lambda\subset T^*(X)\backslash 0$
…
In section 25.2 we introduce the notion of Fourier integral operators; this is the class of operators having Lagrangian distribution kernels.
page 52, notes on that chapter
A systematic discussion of a global theory was given in Hoermander [26]… As emphasised by Maslov the theory has much in common with his canonical operators.
…
The definition in Hoermander [26] of Lagrangian distributions – called Fourier integral distributions there – was based on representations with non-degenarate phase functions. Following a suggestion by Melrose (cf. Melrose [1]) we have chosen a different definition which is obviously global and invariant. It is of course equivalent to the one using phase functions.
Second source, Francois Treves, Introduction to the pseudodifferential operators and Fourier integral operators, Plenum 1982
From introduction (I have a translation so I am back traslating):
From the other side, I stayed faithful to the established term Fourier integral operators, though I am ready to agree that this term is not very well chosen, and, it could be, that it is more right to call these operators Maslov operators, like it is called by many Russian authors. Though, outside of Soviet Union the term FIO is customary and fighting with this habit is already too late.
So in my interpretation (Z.Š), Maslov has been talking on conormal distributions in various generalities, including in generality in which he successfully, in the language of local phases defined (FO) operators from Lagrangian distributions defined with help of certain Lagrangian submanifolds. This is the main point of majority of his works 1965-1967. Hoermander had in addition to local phase function approach also an invariant approach (in the second paper). Essentially it is the same theory at different stages (and aspects of development, Hoermander more on mathematical and Maslov more on physical side of the story; isn’t this an entry on physics).
If you want to erase because it is not in line with your ambition in the page, as only of historical value, you could always create a new page, say historical notes on motives in physics or historical notes on Fourier integral operators and delegate historical side information there. But a genuine content should never be entirely erased.
Thanks, Zoran.
We should maybe just add a link, where Maslov is mentioned, to an entry like Lagrangian distributions and Fourier integral operators
I can look into it later, not now…
Surely.
I am thinking that every major contributor should keep a special reminder page for things to do or leftovers, where one could also delegate unsure paragraphs (like the one I contributed without full justification) and links to pages which are still orphaned and are parts of a bigger plan. I think, when I work on some circle of entries I do not like too much when somebody else puts the links in haste just to make them non-orphaned in a non-planned way. On the other hand, I admit, if not bookkeeping, I can myself forget that I left some page orphaned and not return to it for a year or more, being busy. As soon as I get out of deadline, I will start something of this sort of page for myself, if I have a pointer from some page for pages under construction then I am safe that people will consider the page non-orphaned if it is linked from such a page. Of course, for you this would be non-appropriate as you have plan to work on almost anything in the $n$Lab :)
where Maslov is mentioned
It is not that much about the name, as much about the whole huge literature, and lots of physics applications worked out, of Maslov and Karasev school, which has a bit different terminology (for example Karasev found the concept of symplectic groupoid). I myself consider Hoermander the greatest analyst of 20th century.
Added this reference here to motives in physics and to period:
Good part of the thesis is on loop quantum gravity, e.g. chapter 5.3 is about the Freidel-Levine work (group field theory, related star product, spin foam models etc.) which I put into the references somewhere in nLab months ago and you were strongly objecting against the relevance and correctness of those papers, and appropriateness for nLab. Did you change your mind ?
You are right, thanks for catching me here. I have removed that reference again.
It looked promising, but you are right, it doesn’t actually discuss what one would expect it discusses, instead there is that last section…
Is there not anywhere a good review of motivic periods in correlation functions/scattering amplitudes? So far the best account I have seen is still Kontsevich’s original article. Which means something…
I do not know if there is such. For specifically N=4 SYM state of the art is about
but this is not a survey.
Thanks! I have added a pointer to that to scattering amplitude and also to motives in physics (even if it is not a review).
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