# 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.