Proof of the ergodic theorem and the H-theorem in quantum mechanics – notes on von Neumann’s paper

In this post I will work through von Neumann’s paper “Besweis des Ergodensatzes und des H-Theorems in der neuen Mechanik” from 1929 (“Proof of the ergodic theorem and the H-theorem in quantum mechanics”). This paper was brought to my attention in the introduction to a very nice talk by Guy Salomon in IWOTA about “stability” (this is not one of von Neumann’s papers that one usually sees cited in my area). Guy mentioned that von Neumann starts by considering the canonical position and momentum operators that almost commute in some sense (they satisfy and is a tiny number) and contemplates that if one could approximate these operators with a pair of commuting operators then one could do some calculations and derive some results. That’s what got me interested in looking up the paper, since this is very closely related to my recent work with Malte Gerhold on bounded perturbations of the canonical commutation relations. However, when I tried to read the paper, I quickly saw that it doesn’t have much to say on the issue of stability, however it does seem to have some elaborate considerations with physical consequences.

What is it that von Neumann was trying to do in that paper? I would like to understand this paper – the math as well as the physical consequences. It turns out that the only way I can really understand something is to try to explain it – hence this blog post. Luckily, this paper, which was originally written in German, was translated to English by Roderich Tumulka and published in the European Physical Journal H (and one can also find it on the arxiv). So following Tumulka’s translation I will now try to produce an annotated summary of the main part (Sections 0-2) of von Neumann’s paper.

With hindsight, after reading through the paper, I found that it very little to do with the stability questions that interested me, but it was a nice exercise.

UPDATE: After publishing the post, I continued looking and somehow I reached the following commentary on von Neumann’s paper which was published by four mathematical/theoretical physicists including the translator (ironically, I wasn’t aware of it and did not think of looking for something like this until I finished wrestling with the paper by myself). I am sure that it will be more useful than this post for people who are interested in understanding von Neumann’s work:

“Long time behavior of macroscopic quantum systems“, by Goldstein, Lebowitz, Tumulka dn Zanghi, The European Journal of Mathematics, 2010.

Below is my work-through the main parts of the paper. Sections (and subsections) as well as their numbering follow the sections in the paper.