nLab
Weyl functional calculus

In Weyl quantization of the flat space $R^n$, the classical observables of the form $f(x,p)$ are replaced by suitable operators which in the case when $f$ is a polynomial correspond to writing $f$ with $x$ and $p$ replaced by noncommutative variables $x$ and $ih\frac{\partial }{\partial x}$ in symmetric or Weyl ordering. This means that all possible orderings between $x$ and $ih\frac{\partial }{\partial x}$ 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.

• Lars Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), no. 3, 360–444. MR80j:47060, doi
• Robert F. V. Anderson, The Weyl functional calculus, J. Functional Analysis 4:240–267, 1969, MR635128; On the Weyl functional calculus, J. Functional Analysis 6:110–115, 1970, MR262857