In Weyl quantization of the flat space , the classical observables of the form are replaced by suitable operators which in the case when is a polynomial correspond to writing with and replaced by noncommutative variables and in symmetric or Weyl ordering. This means that all possible orderings between and are summed with an equal weight. More generally, one can extend this rule to more general functions via integral formulas due Weyl and Wigner. This is also useful in fundations of the theory of pseudodifferential operators.
E. M. Stein, Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals, Princeton University Press 1993 M. W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Annals Global Anal. Geom. 28 (2005) 271–283
Discussion of quantization of Chern-Simons theory in terms of Weyl quantization is in
Jørgen Andersen, Deformation quantization and geometric quantization of abelian moduli spaces, Commun. Math. Phys., 255 (2005), 727–745
Razvan Gelca, Alejandro Uribe, The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same, Commun.Math.Phys. 233 (2003) 493-512 (arXiv:math-ph/0201059)