Decoherent histories and hydrodynamic equations

Abstract

For a system consisting of a large collection of particles, a set of variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent, and that their probabilites will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. This system has the appealing feature that the microscopic dynamics of each individual foriegn particle is readily obtained and the approach to local equilibrium may be seen explicitly. It is shown that, for certain physically reasonable initial states, the probabilities for histories of number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes non-trivial superpositions of number density. Histories of phase space densities are also discussed. The case of histories of number, momentum and energy density for more general systems, such as a dilute gas, is also discussed in outline. When the initial state is a local equilibrium state, it is shown that the histories are trivally decoherent, and that the probabilities for histories are peaked about hydrodynamic equations. An argument for decoherence of more general initial states is given.

Macroscopic observables

“…We demonstrate that the histories generated by any sequence of macroscopic observables of accuracy σ ≫ ξ/N are consistent, where ξ is the quantum correlation length-scale of the system. This generalizes some ideas introduced by Halliwell [10] on how to achieve classicality in closed quantum systems. Finally, Section VI discusses the role of macroscopic observables in NMR quantum information processing.…”

“…As was observed by Halliwell [10], histories corresponding to a sequence of finite accuracy measurement of macroscopic observables generate a consistent family if the system is a sufficiently large sample of identically prepared molecules, i.e. when ρ = ν ⊗N .…”

Decoherence Properties of Arbitrarily Long Histories

“…It has been successfully applied in various fields of quantum theory including quantum cosmology [8], derivation of effective classical dynamics from the fundamental quantum dynamical laws [6,7,9], and the study of the coarse-grained evolution of iterated quantum maps [10]. Recently the formalism of decoherent histories has also been applied for investigating classicality in quantum information processing [11].…”

On the possibility of decoherence due to a relativistic effect

“…Sources of decoherence such as scattering [4,5] and quantum gravity [6,7] and both internal [8,9] and external [10,11] environment are taken into consideration. Omnes [12] recently presented a general theory of the effect of decoherence that includes both the harmonic model of Caldeira-Legett [10] and the external environment considered by Joos and Zeh [11] and is related to the quantum state [13] diffusion model.…”

“…However, we wish to signal a possible case of decoherence, with suppression of correlations in the total momentum, for a system of particles even if isolated, provided that we consider relativistic corrections to the non-relativistic Hamiltonian. It would pointed out that de-coherence of isolated systems has been discussed before in various contexts: Halliwell [8] investigated the variables that will generally become effectively classical as the local densities (of number of particles, momentum, energy), Calzetta and Hu [14] the decoherence of histories of certain correlation functions in a field theory. We should also mention that the diagonalization in the square of momentum basis has been argued for a system of (non-relativistic) particles interacting with the linearized gravitational field [15].…”