THE WEYL CORRESPONDENCE AS A FUNCTIONAL CALCULUS FOR NON-COMMUTING OPERATORS

In this expository paper, we describe the Weyl calculus for bounded, self-adjoint operators acting on a Hilbert space as well as the original Weyl correspondence for the position Q and momentum P operators on S(R-n). We describe some classes of functions for which the calculus is well defined and give a representation for the action of the calculus in these separate cases. In particular, we verify that the Weyl calculus is well defined for polynomials and give results consistent with the natural algebraic definition. The proof of this result for the original Weyl correspondence is obtained via an analysis of the commutator of P and Q on S(R-2n). We also discuss the connection of the Weyl calculus with some recent developments in functional calculi.