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?