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started an Examples-section at geometric quantization
Hey, I looked into your transparancies given at cafe. It is too hi level for my understanding yet, and certainly interesting. However, I would disagree with the first line: that Isbell duality or whatever duality interchanges deformation quantization and geometric quantization. Deformation quantizaton makes sense both at algebra level and space/manifold/variety level; the side of Isbell duality is not essential. Also the geometric quantization could be described in terms of coordinate algebras if you like (and hence extended to noncommutative manifolds, for example, supermanifolds). But the idea of both is very different. The second is about quantization line bundle and its sections, and produces true Hilbert space. The deformation quantizuation rather has a formal parameter and is not producing a representation at a Hilbert space. This is why Connes was criticising deformation quantization very much. There is also recent paper by Witten which explains why geometric quantization gives much more when applicable.
I would disagree with the first line: that Isbell duality or whatever duality interchanges deformation quantization and geometric quantization.
Oh, yes, I didn’t mean that. The duality arrow is meant only to apply to the first line. Ah, but I see now that it is misleading. Maybe I should change it.
The point that I felt like hinting at in that table is that it is not entirely a coincidence that there are two formalized concepts of quantization, because they correspond to the two different sides of reality: algebra, and geometry.
Actually it parallels that other duality: Heisenberg picture $\leftrightarrow$ Schrödinger picture
The deformation quantizuation rather has a formal parameter and is not producing a representation at a Hilbert space.
Yes, but the idea is that one can improve on that. There is “$C^\ast$-algebraic deformation quantization” which precisely studies the corrections to this deficiency (I forget if we have an $n$Lab entry about that. It is discussed on the $n$Café somewhere, you find it by searching for “Eli Hawkin”, for I once reported on a series of talks that he gave on this)
There is “$C^\ast$-algebraic deformation quantization” which precisely studies the corrections to this deficiency
Right, that is interesting. I forgot about this.
On the other hand, I would like to remind you that it is not entirely true that there are only two approaches to quantization, there are so many more major types of quantization, e.g. Weyl quantization, Fedosov quantization, path integral quantization etc.
Heisenberg picture ↔ Schrödinger picture
This duality is in a way, and more closely (sometimes even literally, e.g. in the case of free theory) achieved via the Segal-Bargmann transform between the coherent state quantization (coherent states evolve just like the classical equations for operators) and geometric quantization for the polarization which corresponds to the choice of L^2(configuration space).
I would still think that the deformation quantization is NOT necessarily, and not even generically, about algebra – the infinitesimal deformations apply to manifolds as well, and the main case of Kontsevich quantization PRECISELY does not work so well for algebraic Poisson varieties, while it works for smooth Poisson manifolds. So Kontsevich, in the case of varieties, a posteriori suggested a different framework which I think he called the semialgebraic deformation quantization.
Also there is a thought – the coherent states, which come out from geometric quantization (as evaluation functionals) are used to define the Berezin symbols of operators. Berezin quantization is contentwise so much closer to geometric quantization than to the deformation quantization (to latter almost no connection in my knowledge), while it is about symbols of operators, hence closer to Heisenberg picture. I’d think that the geometry of various quantizations is not that closely parallel to time/space and function/space dualities.
P.S. in fact, many of these quantizations can be considered as ordering rules which make isomorphism between noncommutative and commutative. Sometimes, the ordering is determined by polarization, or some other line bundle tricks (like Fedosov). The isomorphism is in the definition of the star product. Weyl quantization is about specific ordering prescription which is “symmetric”. This is an interesting viewpoint.
P.S. II Of course, there is a short memo on deformation quantization – it is formal. So it is just a formal precursor to quantization, just like formal group or formal scheme is to an algebraic/Lie group or algebraic scheme. If we extend to C-star algebras, this is interesting, though still, being deformational it restricts in a way which geometric does not.
I would still think that the deformation quantization is NOT necessarily, and not even generically, about algebra
But it is manifestly about deforming algebras. I am not sure I understand what you have in mind.
In deformation quantization one takes the algebras of observables as the basic datum of a physical system and discusses how the commutative algebra of classical observables is deformed to a non-commutative algebra of quantum observables.
But it is manifestly about deforming algebras.
Yes, you are right. I meant the following. Deforming Poisson structure on manifold, keeps the manifolds and changes the bracket. So you are right in this sense. On the other hand, there are many examples where the dual picture is concerned. Like the deformation quantization of Lie groups leading to quantum groups. Poisson structure there corresponds to the classical r-matrix. Now if we take the point of view of universal enveloping algebra then we deform the coproduct while the algebra is isomorphic, and if take the dual point of view of function algebra then we deform the product while the coproduct is undeformed. Both is deformation quantization. Regarding that it is affine it looks like both is done (co)algebraically, but if we were defining homogeneous spaces, than the situation on one side can be more geometric while both are deformation quantization. There are also sheaf and stack versions implied as well.
By no dualization whatsoever you will get from this geometric quantization. I hope you agree.
By no dualization whatsoever you will get from this geometric quantization. I hope you agree.
I did agre with this in #4. But now that I thought about it, I am not so sure anymore if I don’t want to disagree after all :-)
So if you look at the geometric quantization of symplectic groupoids and C-star algebraic deformation quantization, at least, you see that both quantization proceudres do end up precisely dual to each other: one constructs an actual centrally extended groupoid (geometric) the other precisely its algebra of functions (algebraic).
But I agree that even if there is this duality at the horizon, it is far from being formalized. Nevertheless, I think it is very useful to make it explicit, for it contains a whole bunch of other aspects. To reflect this, I have now created a table:
Check it out and let me know what you think.
Isbell duality - table
What might a ’higher deformation quantization’ be?
You know, I was wondering about the same question when writing that entry.
The Poisson n-algebras appearing for instance in the context of quantization via factorization algebras are certainly an aspect of this. But I am not aware that an attempt at a more comprehensive approach exists yet.
Well, looking at n-plectic geometry there are various evident definitions to make, concering deformation quantization of Poisson bracket Lie $n$-algebras. But I don’t think to date anyone has seriously thought about this.
On the other hand, if we strictly take the view of $C^\ast$-algebraid deformation quantization in the sense of looking at algebras of function on the geometric quantization of symplectic groupoids then one could say the case is better understood: just consider the $\infty$-algebras of functions on geometric quantization of symplectic infinity-groupoids.
Anyway, all this needs to be studied more.
So if you look at the geometric quantization of symplectic groupoids and C-star algebraic deformation quantization, at least, you see that both quantization proceudres do end up precisely dual to each other
It seems we are converging to the essential point: if you add specifics to Rieffel strict quantization, then you can make picture rich enough to enable true Hilbert space and hence you are in the setup of geometric quantization essentially. In any case, it is a matter of terminology to some extent. You are right that deformation quantization is usually on the dual side, though I find it more essential that the construction and requirements are stronger to enable trie HIlbert space, unlike the generic deformation quantization which is weaker and only formal. I’ll dig the Rieffel reference soon.
Re #8
both quantization procedures do end up precisely dual to each other: one constructs an actual centrally extended groupoid (geometric) the other precisely its algebra of functions (algebraic).
Is there a name for the algebraic dual of forming a central extension?
Not sure what the right word would be, but the general mechanism here is one known from many other situations (such as $C^\ast$-algebraic K-theory, for instance): one deforms an algebra of functions on some space by replacing functions by sections of a non-trivial bundle over the space.
In the present case that non-trivial bundle happens to be a line 2-bundle over a groupoid (this is the central extension of the groupoid), but otherwise it’s the same kind of phenomenon.
I realize that I should spell this out more in detail in the respective entries. I need to see when to find the time. Right now I have some other things that feel more urgent. But this may change eventually…
brief paragraph geometric quantization – Example – 2-sphere.
I have finally started to work on filling the section on the actual geometric quantization step:
After a brief survey, the discussion proceeds in four steps
traditional formulation by polarization and metaplectic correction of prequantum bundle;
formulation in complex geometric as Euler characteristic of abelian sheaf cohomology of metaplectically corrected prequantum bundle;
formulation as the Dolbeault-Dirac index of the prequantum bundle with the metaplectic correction now understood as nothing but the Spin-structure;
finally the full truth: formulation as the spin^c index of the prequantum bundle.
This is still just a start though.
have been expanding a good bit the section on geometric quantization by Spin^c-structure and push-forward in K-theory, with plenty of pedagogical (I hope) detail. here
Added pointer to Souriau 74 and added the flow chart diagram from that article to to geometric quantization – History and variants. This needs more accompanying text, but I have to run now.
I have expanded the section geometric quantization – Examples – Schrödinger representation, making all signs, all conventions and all identifications fully explicit.
The page geometric quantization has an error that prevents it from displaying. I don’t know how to fix this.
Thanks for the alert.
I tried re-saving the entry after adding a trivial whitespace somewhere, but problem remains.
Just saw a similar problem with the page thesis Wellen (schreiber), but there re-saving did solve the issue.
Will contact Richard.
Richard kindly fixed it (here).
Thanks to both of you!
In the original statement “the actual step of genuine geometric quantization consists first of forming half its space of sections in a certain sense. Physically this means passing to the space of wavefunctions that depend only on canonical coordinates but not on canonical momenta”, changed “canonical coordinates” to “canonical positions”, since often both canonical position and momentum are referred to as canonical coordinates.
Anonymous
This point would best be discussed in the entry polarization, where the link canonical coordinates was pointing to.
I have made canonical position also redirect there, so that the link works.
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