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I have started a table geometric quantization - contents and added it as a floating TOC to the relevant entries.
Parts of this remain a bit unfinished. The $n$Lab is pretty much unusable in the last hours. I’ll give up now, have wasted too much time with this already. Maybe later it has recovered.
Suggestions for the contents of geometric quantization: coherent state (and eventually Berezin quantization), Maslov index.
Okay, I need to think about this a bit, actually. What’s the specific perspective of geometric quantization on coherent states?
Do you have an electronic copy of this article
Anatol Odzijewicz, Coherent State Method in Geometric Quantization, TWENTY YEARS OF BIALOWIEZA: A MATHEMATICAL ANTHOLOGY Aspects of Differential Geometric Methods in Physics (pp 47-78)
?
Okay, I have created an entry coherent state (in geometric quantization) in order to collect some references.
I could do much more. And would have liked to. But the $n$Lab is out of business. I’ll quit now, this is too frustrating.
No, I do not have, I have some article of this author (maybe that one) in printed form, somewhere in my boxes on the topic. I think the one I have is from Reports in math. physics and is only about the case related to Lie groups.
For coherent states in greater generality of Kahler manifolds, look the article of Rawnsley to see how it fits with the quantization line bundle (one can do rudiments of the basic framework there in symplectic context, already) or, in the case of homogeneous spaces, the summary in a chapter of my article, arxiv version.
(or look in summary in my paper on coherent states), I do not think that it is appropriate to separate coherent state from a version “in geometric quantization”. It is almost like to separate subset and subset (in set theory). It is tautologically about the same thing and any separation is artificial, regardless weather the geometric quantization is mentioned in the title or not. Hardly any important article on geometry of coherent states is not about a geometric quantization in disguise and separating would mean creating confusion. By the definition a coherent state is a projective line of the Riesz dual vector $e_q$ to the functional of evaluation in a fixed point $q$ of a quantization line bundle (take a vector as a section evaluate the section at the projection of $q$ and divide by $q$, i.e. $s(\pi(q))/q$). Berezin quantization produces covariant and contravariant symbols of operators from the evaluation functionals, or more general smeared out evaluation functionals, the coherent state projective measure.
Thanks for the references. I have added more of them to coherent state (in geometric quantization).
I think it’s useful to have a separate entry. After all, standard textbooks will discuss coherent states, but not even mention geometric quantization. The first line of
William D. Kirwin, Coherent States in Geometric Quantization (arXiv:0502026)
is
Coherent states are ubiquitous in the mathematical physics literature. Yet there seems to be a lack of general theory in the context of geometric quantization.
Many physics texts do some gauge theory in textbooks without ever mentioning fiber bundles. So one can do say SU(2) coherent states without EVER mentioning the line bundle, which is the essence of the geometry of coherent states. Typically the line bundle is the quantization line bundle, but the point is that books and articles usually do not mention ANY line bundle. So heck with them ! I think all good references do (I worked in that subject for about 3-4 years and read partly through few hundred references).
Urs, topology is full of homotopy theory without a context of infinity-topoi. When doing such entries you typically take infinity-point of view without an excuse. What do you think ?
Also the orbit method may be viewed just a special case of geometric quantization.
After a couple days of looking at this, I decided that the page name really should be simply coherent state in geometric quantization. This has triggered the cache bug.
Zoran,
three thoughts:
a) my impression is that, as opposed to the cases where we give the $\infty$-categorical point of view, the universality of the geometric quantization perspective in general and on coherent states in particular has not been established;
b) if you do feel this is the universal point of view (I’d be happy if it were, geometric quantization is more up my alley than algebraic deformation quantization is) then I won’t object if you make that clear in the main entry coherent state. After all, you wrote that once, in the first place.
c) I don’t quite see why why we need to have such a long discussion about this issue. What is at stake? Currently we have two entries, both stubby, none with any substantial content. We can maybe worry about how to better organize the material once there is any?
Toby: okay, thanks. I have fought the cache bug now.
We can maybe worry about how to better organize the material once there is any?
I am sorry but I do not enjoy making effort contributing to something what is organized in what I consider a misleading way. Especially in a moment when I am not working on the subject (I did so many times turned to the topic others including you currently do on the $n$Lab just to aid in the peak of effort, but it is always waste of time in my main pursuits).
the universality of the geometric quantization perspective in general and on coherent states in particular has not been established
I do not understand what is “universality of geometric quantization perspective”. The coherent states and quantization line bundle are two sides of the same coin. Not in formal sense like in the logic or category theory but in physical content and methods yes (this is a topic in mathematical physics). I mean there are hardly any counterexamples – coherent states in a context which is not naturally explained via a quantization-like line bundle ?
We can maybe worry about how to better organize the material once there is any?
There is some under coherent state.
In the case of Lie groups, the geometric quantization has two main classical cases. One is in the extreme of solvable groups, the orbit method, which with some modifications can give ideas for general reductive groups. Another is the case of semisimple Lie groups with Borel-Weil theorem. For both of them one can associate the corresponding coherent states. Not all infinite-dimensional representations of reductive groups are coherent state representations however.
why why we need to have such a long discussion
I was trying to quickly teach something on how we who worked in the field (geometric theory of coherent states) look at this subject, at the place where I think is some misunderstanding.
Urs, thanks for another reference by Spera (the one with djvu file). I have read two papers by Spera before but not that one.
I mean there are hardly any counterexamples – coherent states in a context which is not naturally explained via a quantization-like line bundle ?
If you think that is so, then put this into the entry and explain.
And if you want to, by all means, feel free to merge the two entries.
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