University of Toronto's Symplectic Geometry Seminar

August 11, 2006, 11:10pm
Bahen 6183

Alfonso Gracia-Saz

Keio University

The symbol of a function of an operator

Abstract: In the quantum description of a physical system, the observables are represented by operators on a Hilbert space. In the classical description, they are represented by functions on a Poisson manifold. Weyl quantization provides a bijection between quantum and classical observables. To every (quantum) operator \widehat{A}, we associate a (classical) function A , called its symbol. We consider the following problem. Let \widehat{A} be an operator with symbol A and let f be a smooth function. Then \widehat{B} := f(\widehat{A}) is another operator, with symbol B . What is B in terms of A ? We will provide an answer to this question in the form of a formula ``\`a la Feynman'', i.e., a power series whose terms are labeled by diagrams. As an application, we will discuss an asymptotic method to obtain the eigenvalues of a hamiltonian which is the sum of kinetic and potential energy. The talk will come with a moral: There are no difficult calculations, only unfortunate notations.