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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics