Problem of quantization: state of the art

The "problem of quantization":

Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate functions $p_j$ and $q_j$ $(j = 1, . . . , n)$, and a mapping $Q : f → Q_f$ from $Obs$ into self-adjoint operators on $L^2(R^n)$ such that (q1)–(q5)* are satisfied.

(*Please refer to the paper for the conditions (q1) - (q5).)

Ref: Quantization Methods: A Guide for Physicists and Analysts, pp. 2-3, [math-ph/0405065]

To researchers in this area:

What is the current state-of-the-art in this area?

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Requirement 1.4 is not enlighting \hat {f,g} = [\hat f ,\hat g ] , this should not hold on any large class of observables... The correct definition is notion of deformation quantization - [\hat f ,\hat g ] = h \hat {f,g} + O(h^2) , where by \hat f I mean correspondence from classical to quantum. – Alexander Chervov Sep 19 '12 at 12:22
Under this relaxed condition the case of R^2n is more or less covered by standard correspondence p-> d/dx q->x. Moreover this problem can be generalized to symplectic manifolds - see section 5. Berezin and Berezin-Toeplitz quantization on K¨ahler manifolds in the reference. Actually one hopes something similar for Poisson manifolds - to each symplectic leave should correspond an operator representation. But this is subtle project even in the case of Lie algebras - where this reduces to the "orbit method" there are many problems. – Alexander Chervov Sep 19 '12 at 12:29
– Joel Reyes Noche Sep 19 '12 at 13:27
This is essentially the same question as the one Joel linked to, made slightly more narrow. Why not just edit the original question and ask for it to be reopened? – Igor Khavkine Sep 19 '12 at 13:56
No go theorem is called van Hove theorem - I remember it in Hurt's Geometric quantization, i do not have this book now. google gives some references - I am not sure what is the best.... – Alexander Chervov Sep 19 '12 at 16:12

Dear Sadiq; Interesting question. It is the general question of representation of what is often called the Weyl algebra & quantization. There is a huge literature on it, including some of my own work. I looked at your very nice paper, and the references in it. My feeling is that the papers you cite offer a pretty good picture of what is known on your question. There is also work by Marc Rieffel. Regards, Palle Palle Jorgensen palle-jorgensen@uiowa.edu

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Thank you sir. Much appreciated. – Sadiq Ahmed Sep 20 '12 at 17:17
What would you suggest be a good primer / intro. to the mathematical foundations of quantum mechanics? - I found this helpful, but I'm wondering what you would suggest. – Sadiq Ahmed Sep 20 '12 at 17:24
An excellent introduction (which takes you very far in a remarkably efficient way) is the book by Jonathan Dimock "Quantum Mechanics and Quantum Field Theory, A Mathematical Primer" at Cambridge University Press, 2011. – Abdelmalek Abdesselam Sep 20 '12 at 18:08
@Sadiq: That's funny, after posting my previous comment, I just saw that you answered a similar question on MO (3 hours ago) with precisely the book I mentioned. – Abdelmalek Abdesselam Sep 20 '12 at 18:13

Quantization is big area, so let me concentrate on some mathematical aspect which are close to me and somewhat related with representation theory and algebraic geometry. Undoubtedly this is biased and incomplete answer. Hope others would add more.

Morally quantization is a bridge between commutative and non-commutative worlds. Let me say that the hope is to built such a fundamental bridge that everyone can put on a truck everything he wants on the commutative side and the truck can succsesfully go to the non-commutative side, while the current state of art that heavy trucks cannot go, and sometimes you should sit on a horse and be very experienced to reach the other side...

1) Construction of non-commutative algebra.

On the commutative side we have an algebra and Poisson bracket. We want to construct non-commutative algebra which is deformation quantization. Great achievement is due to M. Kontsevich 1998 here, which shows that smooth algebra of functions can be quantized in the sense of formal power series over formal parameter.

However questions remains. a) We want not the formal power series, we want actual non-commutative algebra. This is not really understood (imho). Moreover I am not sure it is clear what kind of algebra we should get - C^*-algebra, fon-Neaumann algebra or what ? Nick Landsman worked on this, and at least he has some proposals what should we get in analytical setup, but it seems the goals are not achieved, yet.

b) It is also not so clear about the uniqueness of the construction - ideally we want unique algebra up to isomorphism. However Kontsevich construction depends on two things 1) coordinate choice 2) choice of the "propagotor" (formality morphism) (and Galois-Teichmuller group expected to act on the space of quantizations). While the (1) is addressed, second seems to me obscure...

2) Basic naive desire f-> \hat f

To define quantum dynamical system from classical one, we need to "f" in commutative algebra associate "\hat f" in non-commutative algebra. E.g. for classical Hamiltonian to write down quantum Hamiltonian. Actually this desire is too naive, however in some sense it is very basic, all what follows would follow if such natural map exists.

Different versions of quantization address this issue. However in all versions there are some additional choices and it is not clear how to deal with them. In deformation quantization naively this map can be taked as identity - since we just introduce the new product on the old algebra, however everything depends on the choice of the coordinates. Berezin-Toeplitz quantization needs to choose complex structure on symplectic manifold to processed. Geometric quantization needs choice of the polarization. Newly introduced "brane quantization" of Gukov and Witten, needs choice of the "complexifaction of the manifold".

So undoubtly something non clear is here.

The only thing which I believe that if "f" belongs to the Poisson center, then the correct "\hat f" is given by Duflo map (Duflo-Kirillov-Kontsevich) ( http://arxiv.org/abs/hep-th/0409005 ).

3) Now we can discuss correspondence between various structures on the commutative side with various structures on non-commutative side.

3a) Consider classical integrable system: defined by some set of H_i : {H_i, H_j } = 0. We want to construct quantum integrable system [\hat H_i , \hat H_j ] = 0. The "practical experience" is that great number of integrable systems people know how to do it, but still there is not universal recipe and I do not know general results.

3b) Generalization of the example above are Lagrangian and coisotropic manifolds, which means that {H_i, H_j } = F(H_k) i.e. ideal is closed with respect to Poisson bracket. One may want to contruct one sided ideal in non-comummuative algebra corresponding. Here is big progress by Felder, Cattaneo &K however it might not be the last word.

3c) Automorphisms. "quantization is a functor ? ". To each Poisson automorphism one may want to contstuct automorphism of the non-commutative algebra. Even in the case of R^2n this is very difficult conjecture related to the Jacobion conjecture - see papers by Kontsevich and Kanel-Belov in arXiv.

3d) "All in one". Probably the most comprehensive point should be something like this. "All classical" data should be encoded in something like Fukay category on the commutative side. While on the non-commuative side we have "only" category of modules. The "hope" is that Fukay category is in some sense isomorphic to the category of modules of quantized algebra. See paper by Bressler-Soibelman.

The list of objects on commutative and non-commutative side which should correspond to each other is not short. (Some years ago Picard group of Poisson manifold has been introduced and studied by analogy with the quantum Picard group; what should be an analogue of fundamental group in quantum case ? ; "quantum groups" - quantizing the group like structures; quantization of algebras given by quadratic relations, Koszul duality and so on; Boht-Sommerfeld quantization conditions and so on...)

If it would be successful it should have various applications, let me mention that "orbit method" in representation theory is a particular case of quantization ideology - which hopes for each symplectic leaf to construct a representaion of the quantum algebra... Another application is to Langlands correspondence over complex numbers. The Hecke eigensheaves - should be understood as quantizations of the Liouville tori of the Hitchin integrable system and "eigen" property is more simple on the classical side.

Huh, actually may be it is not good idea to cover in short words such a big theory, any way may be you can find something helpful...

(from: Alexander Chervov)

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