Two approaches to coupling classical and quantum variables

Abstract

We address the issue of coupling variables whichare essentially classical to variables that are quantum.Two approaches are discussed. In the first, continuousquantum measurement theory is used to construct a phenomenological description of theinteraction of a quasiclassical variable X with aquantum variable x, where the quasiclassical nature ofX is assumed to have come about as a result ofdecoherence. The state of the quantum subsystem evolvesaccording to the stochastic nonlinear Schrodingerequation of a continuously measured system, and theclassical system couples to a stochastic c-number\(\overline x \left( t \right)\) representing the imprecisely measured value of x. The theorygives intuitively sensible results even when the quantumsystem starts out in a superposition of well-separatedlocalized states. The second approach involves a derivation of an effective theory from theunderlying quantum theory of the combinedquasiclassical-quantum system, and uses the decoherenthistories approach to quantum theory.