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Journal of Physics A Mathematical and Theoretical

Weak asymptotic abelianess for Galilei invariant time evolution

Heide Narnhofer

Published Jun 30, 2008• 4 citations
Publisher

Abstract

We extend the result of weak asymptotic abelianess for Galilei invariant time evolution from the tracial state to a set of time-invariant states which are uniformly clustering with respect to space translation.
DOI
10.1088/1751-8113/41/31/315201

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We argue that for Fermi systems on lattices or the continuum with interaction invariant under a kind of Galilei transformation the time evolution is either weakly asymptotically abelian or at least ηηη-abelian in the tracial state but not norm asymptotically abelian.
General conditions are derived for the essential self-adjointness of –∆ + V, where V is a translation-invariant random potential, and for the existence of the perturbation expansion. A sequence of graphs is exhibited violating Dell'Antonio's bound for skeleton graphs. For a translation-invariant and clustering Gaussian random potential V, and a translation-invariant and clustering initial state S of the Fermi gas, uncorrelated with the random potential, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, convergence of the state for sufficiently small values of the parameter τ to a gauge-invariant and quasi-free asymptotic state, and the semigroup describing the evolution of the two-point function. The asymptotic system is Bernoulli. Results are obtained not only for the average over the random potential but also with probability one. If the random potential V′ is absolutely continuous with respect to V, and if the state S′ is given by a density matrix in the GNS representation for S, then the weak coupling limit is the same as for V and S.
For Fermi systems interacting with a Galilei invariant pair potential with a cut-off for particles with highly different velocities the time evolution corresponds to an automorphism. We prove that all states satisfying the KMS-condition are locally normal with respect to the representation in Fock-space. Removing the cut-off limit states remain locally normal provided the interaction is repulsive or of positive type. The modular automorphism of the von Neumann-algebra of the limit state coincides with the time evolution.