An Effective Stochastic Semiclassical Theory for the Gravitational Field

Assuming that the mechanism proposed byGell-Mann and Hartle works as a mechanism fordecoherence and classicalization of the metric field, weformally derive the form of an effective theory for thegravitational field in a semiclassical regime. This effectivetheory takes the form of the usual semiclassical theoryof gravity, based on the semiclassical Einsteinequation, plus a stochastic correction which accounts for the backreaction of the lowest order matterstress-energy fluctuations.