Problem of quantization: state of the art [closed]

Question

As the title suggests, I'm interested in finding out the state-of-the-art in the problem of quantization.

Any suggestions and/or feedback would be greatly appreciated.

Regards.

Answer 1

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...