The algebraic quantisation of the Birkhoff-Gustavson normal form

Abstract

The author develops an algebraic quantisation method for the Birkhoff-Gustavson normal form. For this purpose the Weyl quantisation rule is used. The method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities. The author gives a brief review of the normal form, derives some of its general properties, and finds a general analytic solution for the fourth-degree normal form for Hamiltonians of two degrees of freedom. In particular, this includes the Henon-Heiles system. The author compares the results of specific examples with other works. The question of canonically invariant quantisation, the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed. The author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals. If the normal form accidentally terminates, so that the classical system is integrable, then this implies quantum integrability of the normal-form Hamiltonian.