Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)

Logic, Geometry And Probability Theory

“…(2) was crucial for the interpretation of µ(A) and µ(B) as relative frequencies [35] in a frequentistic interpretation. But as explained in [35,8] one of the main dissatisfactions of von Neumann was that Eq. (2) was not generally valid in the quantum case, making the frequentistic interpretation untenable.…”

“…(2) was not generally valid in the quantum case, making the frequentistic interpretation untenable. This was one of the reasons that led him to search for generalizations of the algebra of projections in Hilbert space, and type II 1 factors were good candidates for this objective [8].…”

Structural Realism, Scientific Change, and Partial Structures

“…As is well known, in 1932, von Neumann provided a unified formulation of quantum mechanics in terms of Hilbert spaces (see von Neumann [1932]). One of the major advantages of this approach is that not only it rigorously established the equivalence between Heisenberg's and Schrödinger's mechanics, but it also provided a way of introducing probability in quantum mechanics without recourse to Dirac's inconsistent δ-functions (for a discussion, see Muller [1997] and Rédei [1997]). What is less well known is that right after the publication of his book on quantum mechanics, von Neumann started to have doubts about the Hilbert spaces formalism (see Rédei [1997] and [1998]).…”

“…By introducing this kind of structure, a unified approach to the introduction of probability in quantum mechanics (encompassing both finite and infinite systems) is provided. And according to von Neumann, this is the kind of structure that we should use in quantum mechanics (for references and details, see Rédei [1997]). …”

Quantum measurements without sums

“…Ever since John von Neumann denounced, back in 1935 [34], his own foundation of quantum mechanics in terms of Hilbert spaces, there has been an ongoing search for a high-level, fully abstract formalism of quantum mechanics. With the emergence of quantum information technology, this quest became more important than ever.…”