On ergodic limits of normal states in quantum statistical mechanics

The asymptotic behavior for $t→±∞ of S(t)=exp(−i H t)Sexp(+i H t)$ and its time average $S̄ (t)=t−1∫0tdu S(u)$ is discussed. Here is S an element of the Banach space $B1$⁠, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space $H$⁠, and H is the Hamiltonian, i.e., a bounded or unbounded self‐adjoint operator on $H$⁠. Necessary and sufficient conditions are given for the existence of the limits $S̄(± ∞)$ and S(± ∞) with respect to the weak topology on $B1$⁠, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, $ρ̄(t)$ has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].