I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the different parts of Qiaochu Yuan's post on noncommutative probability. Motivating von Neumann algebras as a noncommutative analog of Peter Whittle's "Probability via Expectation" approach to axiomatic probability theory, I came across the SEP article Quantum Theory: von Neumann vs. Dirac. It paints a subtle picture of how the role and use of von Neumann algebras changed over time in algebraic quantum field theories.

My feeling that I would understand von Neumann algebras evaporated, and I started to wonder whether von Neumann's 1954 ICM address was really as outdated as hinted at by George Dyson in "Turing's Cathedral," where he wrote:

Von Neumann never returned to pure mathematics, and even his attention to computing was distracted by his duties at the AEC. At the International Congress of Mathematicians held in Amsterdam on September 2-9, 1954, he was invited to give the opening lecture, billed as a survey of "Unsolved Problems in Mathematics" that would update David Hilbert's famous 1900 Paris address. The talk, instead, was largely a rehash of some of von Neumann's own early work. "The lecture was about rings of operators, a subject that was new and fashionable in the 1930s," remembers Freeman Dyson. "Nothing about unsolved problems. Nothing about the future. Nothing about computers, the subject that we knew was dearest to von Neumann's heart. Somebody said in a voice loud enough to be heard all over the hall, 'Aufgewärmte Suppe,' which is German for 'warmed-up soup.'"

The address in not contained in the proceedings of the congress, but M. Rédei (1999) in "Unsolved Problems in Mathematics" J. von Neumann's address to the ICM ... 1954 describes the content of both planed and actual address based on unpublished material (John von Neumann Archive, Library of Congress, Washington, D.C.). The sketch of the planed talk mentioned:

The unsolved problems to which attention is called fall into three groups.

  1. Problems involving the algebraic structure of rings of operators.
  2. The role and meaning of these in view of the present difficulties and uncertainties in quantum theory.
  3. Problems of reformulation and unification in logics and probability theory based on this approach.

The plan contains a list of 24 more specific issues, like the isomorphism problem of finite von Neumann algebras, the characterization problem of infinite von Neumann algebras, or the infinite direct product. But he must have realized that he would be unable to do justice to the complexity of these problems in the given time frame:

What von Neumann decided to do in the lecture was to concentrate on the second and third group of problems: He gave a general motivation for the theory of operator rings and, in particular, he discussed the potential conceptual significance of a particular ring, the finite, continuous ring (the type $\mathbf {II}_1$ von Neumann algebra) both for the mathematical theory of operators and for a better understanding of quantum mechanics, quantum logic and (quantum) probability.

Question: Was von Neumann's address really outdated? Was it really out of fashion? Hindsight tells us that the call for a unification of the theory of operators, logic and probability theory remained unanswered. The guess that the type $\mathbf {II}_1$ von Neumann algebra would be crucial also turned out to be false. But the SEP article gave me the impression that von Neumann algebras themselves were still actively researched in 1954, and significant developments like the fine structure of type $\mathbf {III}$ von Neumann algebra were only discovered much later (1973), and proved crucial for certain applications to quantum field theory. And around 1954, we have the Kaplansky density theorem in 1952, Kaplansky's observation that $u^*u$ is always positive (hence is not needed as an axiom for $C^*$-algebras) in 1953, Sakai's abstract characterization of a von Neumann algebra (as a $C^*$-algebra isometrically isomorph to the dual of a Banach space) in 1956.

Is it valid to draw a connection between von Neumann's address and those developments around 1954? Were there other similar developments "vaguely" connected to his address? How to judge for example Segal's work from 1959 on quantum field theory using Segal algebras (a slight generalization of $C^*$-algebras) in that context? What about the "Probability via Expectation" approach to quantum probability theory in general?

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Thomas Klimpel
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  • Is that text "Probability via Expectation" also dealing with non-commutative probability? I seem to remember learning years ago that there is a parallel between geometry with Hyperbolic/Euclidean/Spherical geometry on one hand and probability with Independence/Free independence/??? . I don't remember what the third type of independence was. Does it ring a bell to you? Maybe I just dreamed that up though. – Raskolnikov Jul 14 '21 at 16:14
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    @Raskolnikov Only the last two sections 16.5 "QM: the Static Case" (6 pages) and 16.6 "QM: the Dynamic Case" (3 pages) talk about QM. The preface says "This book is intended as a first text in theory and application of probability, demanding a reasonable, but not extensive, knowledge of mathematics. It takes the reader to what one might describe as a good intermediate level." and "..., in the expectation approach, classical probability and the probability of quantum theory are seen to differ only in a modification of the axioms--a modification rich in consequences, but succinctly expressible." – Thomas Klimpel Jul 15 '21 at 09:31
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    The typescript of von Neumann's address is published in Rédei M., Stöltzner M. (eds) John von Neumann and the Foundations of Quantum Physics. Vienna Circle Institute Yearbook [2000] (Institut ‘Wiener Kreis’ Society for the Advancement of the Scientific World Conception), vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2012-0_16 – Loren Rosen Jul 14 '21 at 15:56

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