DECOHERENCE OF HYDRODYNAMIC HISTORIES - A SIMPLE SPIN MODEL

In the context of the decoherent histories approach to the quantum mec hanics of closed systems, Gell-Mann and Hartle have argued that the va riables typically characterizing the quasiclassical domain of a large complex system are the integrals over small volumes of locally conserv ed densities-hydrodynamic variables. The aim of this paper is to exhib it some simple models in which approximate decoherence arises as a res ult of local conservation. We derive a formula which shows the explici t connection between local conservation and approximate decoherence. W e then consider a class of models consisting of a large number of weak ly interacting components, in which the projections onto local densiti es may be decomposed into projections onto one of two alternatives of the individual components. The main example we consider is a one-dimen sional chain of locally coupled spins, and the projections are onto th e total spin in a subsection of the chain. We compute the decoherence functional for histories of local densities, in the limit when the num ber of components is very large. We find that decoherence requires two things: the smearing volumes must be sufficiently large to ensure app roximate conservation, and the local densities must be partitioned int o sufficiently large ranges to ensure protection against quantum fluct uations.