On ergodic limits of normal states in quantum statistical mechanics

The asymptotic behavior for $\mathrm{t\to \pm \infty \hspace{0.17em}}\mathrm{of}\mathrm{\hspace{0.17em}S(t)=}\exp \mathrm{(-i\hspace{0.17em}H\hspace{0.17em}t)S}\exp \mathrm{(+i\hspace{0.17em}H\hspace{0.17em}t)}$ and its time average ${\mathrm{S̄\hspace{0.17em}(t)=t}}^{\text{−1}}{\int }_0^t\mathrm{du\hspace{0.17em}S(u)}$ is discussed. Here is S an element of the Banach space ${\mathcal{B}}_1$ , constituted by the trace class of operators on the (separable or nonseparable) Hilbert space $\mathcal{H}$ , and H is the Hamiltonian, i.e., a bounded or unbounded self‐adjoint operator on $\mathcal{H}$ . Necessary and sufficient conditions are given for the existence of the limits $\mathrm{S̄(\pm \hspace{0.17em}\infty )}$ and S(± ∞) with respect to the weak topology on ${\mathcal{B}}_1$ , 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, $\mathrm{\rho ̄(t)}$ 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)].