nLab Weyl functional calculus



What is called Weyl quantization is a method of quantization applicable to symplectic manifolds which are symplectic vector spaces or quotients of these by discrete groups (tori).

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

  • 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

Discussion of the generalization to BV-quantization is in

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