Decoherence of Hydrodynamic Histories: A Simple Spin Model
Abstract
In the decoherent histories approach to the quantum mechanics of closed systems, the existence of a quasiclassical domain in a large complex system is not postulated, but is a calculable emergent feature contingent on the Hamiltonian and initial conditions. Gell-Mann and Hartle have argued that the variables typically characterizing the quasiclassical domain are the integrals over small volumes of locally conserved densities. These variables are singled out because they are approximately conserved, and hence they will be approximately decoherent. The aim of this paper is to exhibit some simple models in which the phenomenon of approximate decoherence through approximate conservation may be seen explicitly. We derive a formula which shows the explicit connection between approximate conservation and approximate decoherence. We then consider a class of models consisting of a large number of weakly interacting components, in which the projections onto local densities may be decomposed into projections onto one of two alternatives of the individual components. One example is a box of weakly interacting particles divided into two sections. The projections are onto the number of particles in one section and may be written in terms of projections asking whether each individual particle is in the left or right section. Another example, which we consider in some detail, is a one-dimensional chain of locally coupled spins, and the projections are onto the total spin in a subsection of the chain. We compute the decoherence functional for histories of local densities, in the limit when the number of components is very large. We thus obtain an explicit expression for the degree of decoherence as a function of the coarse-graining. We find that decoherence requires two things: the smearing volumes must beKeywords
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