Canonical perturbation theory for nonlinear systems

Abstract

We study perturbation expansions for quantum systems which are nonlinear in the sense that they may contain derivative couplings. The characteristic feature of these systems is the occurrence of products of noncommuting operators in the interaction Hamiltonian. In order to perform perturbation expansions in the Dyson-Wick manner it is in general necessary to order these operators according to some ordering scheme. We determine the most general scheme of this sort which allows a Dyson-Wick expansion with pair contractions. It involves a set $\ensuremath{\lambda}$ of three dimensionless ordering parameters which are related to the three $a$ priori undetermined equal-time contractions. Different $\ensuremath{\lambda}$-ordering schemes are related via an "equal-time" or "static" Wick theorem. In order to be able to contract operators which may be taken either at equal or at different times we introduce a time-ordering operation ${T}_{\ensuremath{\lambda}}$ which extends Dyson's time-ordering operation to equal times in a fashion dependent on the $\ensuremath{\lambda}$ ordering. We arrive at a "dynamic" Wick theorem which enables us to decompose the ${T}_{\ensuremath{\lambda}}$ product of a set of operators taken at arbitrary times into a sum of ${\ensuremath{\lambda}}^{\ensuremath{'}}$ products, where the most important case is that in which ${\ensuremath{\lambda}}^{\ensuremath{'}}$ represents normal ordering. Contrary to the usual Wick theorem ours works also in the case when two or more times coincide. Applying this theorem to the scattering operator expressed as a ${T}_{\ensuremath{\lambda}}$ product we obtain a class of Dyson-Wick expansions for nonlinear systems. Different expansions correspond to different choices of the ordering $\ensuremath{\lambda}$ and lead to different Feynman rules. The resulting diagrammatic technique for nonlinear systems is seen to have the following special features: (i) The interaction vertices include terms which arise from "quantum corrections," which are dependent on the ordering \ensuremath{\lambda} chosen for the noncommuting factors in the interaction Hamiltonian, and (ii) the equal-time contractions which occur in the closed-loop diagrams must be assigned corresponding $\ensuremath{\lambda}$-dependent values. In this way individual diagrams are $\ensuremath{\lambda}$ dependent. Nevertheless their sum is $\ensuremath{\lambda}$ independent, confirming the fact that the complete scattering operator is independent of the particular ordering chosen. The ordering multiplicity is also reflected in functional versions of the Dyson-Wick expansion. In the path-integral version this multiplicity lurks in the different possible discretization prescriptions. We discuss the relevance of our considerations to the question of the invariance of a quantum theory with respect to nonlinear point transformations. For the special case of a reducible system we present explicit calculations which show how this invariance is maintained in perturbation theory, for any ordering scheme belonging to the $\ensuremath{\lambda}$ class. In any particular scheme the invariance is ensured by a more or less intricate cancellation mechanism between the individual diagrams.