On the diagonalization of quantum Birkhoff–Gustavson normal form

Abstract

An application to quantum mechanics of one of classical perturbation theory methods, the Birkhoff–Gustavson normal form (BGNF), is described. In the quantum case it results in the Van Vleck perturbation theory performed upon Wick normal ordered operators. Algebraic aspects of this procedure and formal construction of invariants (integrals of motion) for a perturbed system are considered. It turned out that a larger set of such operators existed in the quantum mechanics, rather than in the classical one. It is demonstrated that, according to general results of the quantum mechanical perturbation theory, the quantum BGNF may always be diagonalized, and two formal processes for such diagonalization are constructed. In the opposite case, the classical BGNF is, in general, nondiagonalizable. This reflects the fact that the classical perturbation theory cannot handle a system with two or more resonances. Possible reasons for such different behavior of two very close, in spirit, perturbation procedures are discussed. Results of the described procedure, entirely performed upon the Wick normal ordered operators, are equivalent to those of Rayleigh–Schrödinger perturbation expansion.