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Another, perhaps more important observation is that in quantization in the Schrödinger approach you consider wave functions on configuration space, depending on the position $x$ of dimension length. … In particular in quantization theory this turned out to be a very useful tool. …

One should definitely take a look at the work of Bordemann, Meinrenken, and Schlichenmaier: they provide a Berezin-Toeplitz inspired deformation quantization for all compact quantizable (i.e. the Kähler … The asymptotics of this was discussed before by Cahen, Gutt, and Rawnsley in their 4 papers of quantization on Kähler manifolds. …

Yes, it's just putting signs correctly. Martin Bordemann has a preprint from the 90s where he adapted Fedosov's construction in the graded setting. If you are only interested in the flat situation thi …

to 1) A $\mathbb{k}[[\hbar]]$-linear map between $A[[\hbar]]$ and $B[[\hbar]]$ is necessarily of the form $T = T_0 + \hbar T_1 + \cdots$ with $T_r\colon A \longrightarrow B$ being $\mathbb{k}$-linear …

For this (and many related reasons) formal deformation quantization is not the final answer. … So this is the problem of quantization: given a classical physical system, one wants to guess its quantum description. …

In deformation quantization there is a full classification available: let us first focus on the symplectic case which is easier. … choose a quantization machinery. …

Physically, this is important for the quantization of Dirac's magnetic monopole. …