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The quantization problem is one of the most well-known current problems of theoretical and mathematical physics. It is also part of Hilbert's sixth problem (on the axiomatization of physics - see here and here) and contains the non-perturbative quantization of Yang-Mills theories, which constitute one of the Millennium problems. By the quantization problem I mean the following vague steps:

  1. defining formally what is a classical theory, whose "space" I denote by $\mathbf{Class}$;
  2. defining formally what is a quantum theory, whose "space" I denote by $\mathbf{Quant}$;
  3. building some kind of globally defined rule $Q:\mathbf{Class}\rightarrow \mathbf{Quant}$ preserving expected physical properties and mapping standard classical theories into their well-know quantum version.

There are some standard quantization procedures (such as geometric quantization), which can be applied to the classical theories whose phase space is a symplectic manifold. On the other hand, even in these cases, due to Groenewold-type examples (here), the quantization procedure, if it exists, then it cannot be functorial, in the sense of mapping all classical observables into corresponding quantum observables.

There are other approaches (such as formal deformation quantization) which can be applied more generally to Poisson manifolds, but building only a perturbative quantum theory. More recently, there are some approaches which intent to extend geometric quantization to "higher symplectic geometric" in the sense of "higher geometry", allowing to quantize a much more ample class of classical theories. This is the case, e.g., of cohomological (or motivic, or pull-push) quantization - see here and here.

On the other hand, there are also approaches to "higher deformation quantization" which extend it from Poisson algebras (i.e, $P_1$-algebras) to $P_n$-algebras, assigning to them Factorization algebras (here and here) and its variations, such as vertex operator algebras, chiral algebras, blob homology, etc. See here.

My questions are:

  1. what are the limitations of the "higher" quantizations described above?
  2. there are other examples of them?
  3. what is the current status of the quantization problem and what are the most recent reviews about that?

Thank you very much. Any contribution or comment will be most appreciated.

P.S: some discussion should also appear here: https://www.researchgate.net/post/The_Quantization_Problem_Modern_Quantization_Procedures_and_Current_Status

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    $\begingroup$ the discussion at Researchgate linked to seems very broad and open-ended; I'm unsure there is a focused answer possible here. $\endgroup$
    – Carlo Beenakker
    Commented Oct 6, 2021 at 10:10
  • $\begingroup$ Probably this helps. $\endgroup$
    – user90189
    Commented Oct 7, 2021 at 16:20
  • $\begingroup$ I dont think having a globally defined rule $Q:Class\to Quant$ is a reasonnable thing to ask. I rather think one should have a globally defined "classical approximation" rule $A:Quant \to Class$. The quantization problem is then about finding a pre-image of a specific (class of) classical theory(ies) by $A$. $\endgroup$
    – DamienC
    Commented Sep 15, 2022 at 8:59

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