The Stochastic Limit of Quantum Theory: An introduction

The stochastic limit is not a new physical law, but a new technique toextract quickly, from nonlinear quantum systems, the dominatingcontribution of the interaction in some specified asymptotic regime.The surprising fact is that these dominating contributions can beexplicitly resummed leading to a new Hamiltonian theory.This new Hamiltonian theory is a manifestation of quantum chaos becausethe original fields become, in the stochastic limit, quantum whitenoises.These are independent increment quantum stochastic processes, i.e. thehighest degree in the hierarchy of chaos. Physically these ``masterfields'' describe the ``fast degrees of freedom'' of the originalquantum system in the limit in which they become ``infinitely fast,''i.e. a pure noise.The simplest asymptotic regime considered in the stochastic limit is theFriedrichs--van Hove time scaling t→t/λ^2, wheret is time and λ a small parameter (coupling constant in severalmodels).The main difference between the stochastic limit and the van Hove limitis that, in the latter, the field degrees of freedom are averaged outand therefore one obtains an irreversible evolution (master equation)while in the former the limit ``master field'' is explicitly determinedleading to a unitary evolution.In particular in the stochastic limit one preserves several informationson the statistics of the field (i.e. correlations, photon statistics,...) which are lost in the van Hove approach.