nLab
Weyl functional calculus

Contents

Idea

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.

References

  • 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

Discussion of quantization of Chern-Simons theory in terms of Weyl quantization is in

Revised on July 18, 2015 09:56:20 by Urs Schreiber (94.118.176.238)