Long-time behavior of macroscopic quantum systems: Commentary accompanying the English translation of John von Neumann’s 1929 article on the quantum ergodic theorem

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the “quantum H -theorem”, is actually a much weaker statement than Boltzmann’s classical H -theorem, the other theorem, which he calls the “quantum ergodic theorem”, is a beautiful and very non-trivial result. It expresses a fact we call “normal typicality” and can be summarized as follows: for a “typical” finite family of commuting macroscopic observables, every initial wave function ψ 0 from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψ t is close to their micro-canonical distribution.