Canonical Perturbation Theory for Nonlinear Quantum Oscillators and Fields

Abstract

The method of successive canonical transformation is developed to study both the classical and quantum versions of a system of n weakly interacting nonlinear oscillators. The method reproduces the essential features of the solutions to these nonlinear differential equations. A general quartic polynomial in the oscillator coordinates is taken as the interaction Hamiltonian. After the canonical transformations to the desired order, the classical system is quantized, resulting in immediate identification of the raising and lowering operators for the perturbed eigenstates in the nonresonant case. In the resonant case (commensurable frequencies) the transformed operators lead to finite-dimensional subspaces within which eigenstates lie, and are obtained by matrix methods. In each case the Heisenberg equations for the operators are solved as fully as presently possible. A particular case of two quantum oscillators is studied in detail, both for the resonant and nonresonant cases. Finally, the method is generalized to the continuum and applied to the Lee model. The usual results for the lowest sector are obtained to second order: the physical V particle and the mass shift of the V. Instability of this particle is related to the occurrence of nonlinear resonance in the lowest-order ``interaction.''