Quantum-field-theoretical approach to phase-space techniques: Generalizing the positive-P representation

We present an introduction to phase-space techniques ~PST! based on a quantum-field-theoretical ~QFT! approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation ~even after phase-space doubling! and hence fail to produce a stochastic differential equation ~SDE!, we show how the system in question may be approximated via stochastic difference equations (SDE). Second, we show that introducing sources into the SDE’s ~or SDE’s! generalizes them to a full quantum nonlinear stochastic response problem ~thus generalizing Kubo’s linear reaction theory to a quantum nonlinear stochastic response theory!. Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.