arXiv:2010.15481

Time asymptotics for interacting systems

Heide Narnhofer

Published Oct 29, 2020• 0 citations

Abstract

We argue that for Fermi systems with Galilei invariant interaction the time evolution is weakly asymptotically abelian in time invariant states but not norm asymptotically abelian.Consequences for the existence of invariant states are discussed.

DOI: 10.48550/arxiv.2010.15481

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arXiv:2010.15481

Time asymptotics for interacting systems

Heide Narnhofer

Published Oct 29, 2020• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2010.15481

Related papers

Interacting Fermi systems in the tracial state

Heide Narnhofer

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 $η$ -abelian in the tracial state but not norm asymptotically abelian.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 $η$-abelian in the tracial state but not norm asymptotically abelian.

arXiv:2406.04977Jun 7, 20240 citations

Weak asymptotic abelianess for Galilei invariant time evolution

Heide Narnhofer

Journal of Physics A Mathematical and TheoreticalJun 30, 20084 citations

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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.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.

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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

η

-abelian in the tracial state but not norm asymptotically abelian.

A one-dimensional interacting Fermi system with weak-coupling parameters

{g}_{1} (<0)

and

{g}_{2}

is reexamined in the third-order renormalization-group approach. Our result indicates that the value of the invariant coupling ${g}_{1}^{\ensuremath{'}}$ varies, depending on whether the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$,

q=0

or $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$ is taken. For $\ensuremath{\omega}\ensuremath{\rightarrow}0$,

q=0

, the invariant coupling ${g}_{1}^{\ensuremath{'}}$ scales to $\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}1.24$, and for $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$, it scales to $\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}0.716$. We also argue that the invariant coupling ${g}_{1}^{\ensuremath{'}}$ obtained from neither limit is appropriate for calculating the critical exponents of various response functions.