TY - JOUR

T1 - Quantum ergodicity of C* dynamical systems

AU - Zelditch, Steven

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 1996

Y1 - 1996

N2 - We define a notion of quantum, or non-commutative, ergodicity for a class of C*-dynamical systems (script A sign, G, α) which we call quantized GNS systems. Such a system possesses a natural classical limit state ω, which induces a classical limit system by the GNS construction. The criterion for quantum ergodicity is that the time average 〈A〉 of an observable A ∈ script A sign equals the "space average" ω(A)I plus an error K which is negligible in the classical limit. We prove that ergodicity of ω is a sufficient condition for quantum ergodicity of (script A sign, G, α.) if the classical limit system is abelian, give a conditional converse, and discuss a number of applications.

AB - We define a notion of quantum, or non-commutative, ergodicity for a class of C*-dynamical systems (script A sign, G, α) which we call quantized GNS systems. Such a system possesses a natural classical limit state ω, which induces a classical limit system by the GNS construction. The criterion for quantum ergodicity is that the time average 〈A〉 of an observable A ∈ script A sign equals the "space average" ω(A)I plus an error K which is negligible in the classical limit. We prove that ergodicity of ω is a sufficient condition for quantum ergodicity of (script A sign, G, α.) if the classical limit system is abelian, give a conditional converse, and discuss a number of applications.

UR - http://www.scopus.com/inward/record.url?scp=0030550843&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030550843&partnerID=8YFLogxK

U2 - 10.1007/BF02101904

DO - 10.1007/BF02101904

M3 - Article

AN - SCOPUS:0030550843

SN - 0010-3616

VL - 177

SP - 507

EP - 528

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -