XXVI Workshop on Geometric Methods in Physics 1-7.07.2007

Alfonso Gracia-Saz

Bohr-Sommerfeld rules and the symbol of a function of an operator

Weyl Quantization provides a correspondence between quantum observables (i.e. operators on a Hilbert space) and classical observables (i.e. ``symbols'' or smooth functions on a Poisson manifold) in the description of a physical system. Bohr-Sommerfeld rules are an asymptotic method to calculate the eigenvalues of an operator via its symbol. To develop Bohr-Sommerfeld rules for an operator, we first need to solve the following problem: given an operator $A$ and a function $f$, what is the symbol of $f(A)$ in terms of the symbol of $A$? We will explain the context of this problem and give a solution in the form of a combinatorial formula ``a la Feynmann'', which has other applications in quantum mechanics.