Monday, August 30, 2021

Von Neumann's 1932 quantum proof was correct

A new paper starts:
The impossibility of theories with hidden variables as an alternative and replacement for quantum mechanics was discussed by J. von Neumann in 1932. His proof was criticized as being logically circular, by Grete Hermann soon after, and as fundamentally flawed, by John Bell in 1964. Bell's severe criticism of Neumann's proof and the explicit (counter) example of a hidden variable model for the measurement of a quantum spin are considered by most researchers, though not all, as the definitive demonstration that Neumann's proof is inadequate.
Yes, Bohm and Bell each have their cult followings, and they attack quantum mechanics conventional wisdom. They say that the theory went bad with von Neumann's work in 1932.

As this paper shows, von Neumann was right in 1932.

Bohm supposedly reinterpreted QM as a hidden variable theory, contradicting von Neumann. But as this article explains, the interpretation is unphysical and is only of obscure academic interest.

Bell did propose hidden variable theories, but they have been disproved by experiment.

This paper explains it all. A 2010 paper by Bub also argued that von Neumann's argument was correct. Those who work in Bohm or Bell theory have been making negative progress since 1932.


  1. Dear Roger,

    Thanks for pointing out the papers. Mildly interesting.

    I view these papers through the lens of my new approach / theory (yet to be published). Also, mainly due to my preoccupation with its development, I don't have the time to first gather the background which these papers assume (esp. Hermann's work, and also details of Bell's work), and only then to look into these papers.

    Personal note: It would have been both nice and convenient by me (Mumbai is quite nearby Pune) if I could discuss my new approach with Prof. Unnikrishnan in person. However, given my past experience of Indian physicists, that's not at all likely to happen. They would rather talk with you (and of course, two orders of magnitude more readily with Dr. Peter Morgan) but not with me. [Use the search string: "Brown coolies".] [I should find out if Unnikrishnan is an exception to this rule.]

    With those big caveats out of the way, let me still note a remark regarding Unnikrishnan's paper (because it's a bit of a more general interest).

    CSU (abbr. for Unnikrishnan) says:

    >> "I show that Neumann's assumption of the linear additivity of the expectation values, even for incompatible noncommuting) observables, is a necessary constraint related to the nature of observable physical variables and to the conservation laws. Therefore, any theory should necessarily obey it to qualify as a physically valid theory." <<

    I don't think CSU would be able to do that.

    What von Neumann did was a systematization of QM into as coherent a framework as possible. This framework does imply the linear additivity of the Exp(O) as a consequence.

    If CSU takes this consequence as the central fulcrum for his argument, then applying the proper epistemological principles, he still remains completely within the scope of the von Neumann framework. [And this framework, BTW, leaves the measurement problem as necessarily unresolved.] In particular, CSU cannot request us to accept that *every* physical theory obey the same feature. The feature is specific to just one effort (albeit the best effort thus far) at systematizing the otherwise disparate principles and items of knowledge, discovered by the QM founders using the experimental method.

    It would be possible to offer a more detailed explanation for what I say above, but I usually don't get into such matters. (I don't have to keep publishing papers.) But not to leave you / your readers entirely guessing (doing so would be irresponsible), let me note this much.

    (i) Following Bohr's insistence, von Neumann's framework still carries this feature: The Instrument (i.e. Detector) *must* be described *classically*, and the QM System quantum mechanically.

    (ii) The preceding feature, *as understood by all physicists so far*, leaves no room to be able to model any flows / exchanges *between* the System and the Detector, in any manner: either fully quantum mechanically or fully classically. (QMcal flows *are* permitted by the framework, but only *within* the System. Reason: Detector must remain classical.)

    It is the absence of such flows which implies that you can't have a purely quantum mechanical-level description of any flow phenomenon between the Instrument and the System, including nonlinear phenomena. (The first to strike me, as I began writing this comment was: the Navier-Stokes. Only then I realized that the constrain should apply to *all* nonlinear theories from every domain of physics... indeed to all flows, whether linear or nonlinear---between the Instrument and the System.)

    The much celebrated linearity of QM (by which I mean the linearity of the *differential equation* of Schrodinger's, and not that of the Exp(O)'s) indeed is also at the center of the reason why the mainstream QM cannot at all even hope to solve the measurement problem.

    I guess that's enough of comment for the paper.


  2. Roger,

    Another comment, regarding to your text:

    0. Yes, Bohm does seem to have some cult following, but not, perhaps, Bell.

    1. The main trouble with the Bell admirers isn't that they elevate him so much. They do that. But that's not the main trouble, really speaking.

    The main trouble is: They have almost made every one come to forget the fact that theoretically, Bell didn't add any new knowledge to the fundamentals of QM. He didn't alter a single postulate of the mainstream QM in any way---not even by way of interpretations, let alone by way of solving the measurement problem. So, QM stayed intact---I mean at the postulates level. Bell's best work---his eponymous theorem---is still just that: a theorem. Strictly speaking, it is just a *corollary* of the mainstream postulates---albeit formulated with an admirable degree of originality.

    Would you say that Feynman has a cult following? ... Realize, once again, that despite a complete reformulation of the QM (via the Lagrangian, path-integral, approach), Feynman still did *not* solve the measurement problem. [I doubt if he even wanted to do that, ever in his life.] ... Something similar applies also to Bell's work, and to his admirers.

    2. As to the Bohmians. I think not all of them qualify to be called cultists. Many might be. But at least *some* of them seem to be plain mistaken.

    First, hold the context. There is no proper solution to the measurement problem in the mainstream QM. There hasn't been any, since 1926 (Born's paper).

    Now, the theory of Bohmian mechanics is sufficiently simple that, it allows people to think that, with some new research---may be with a few fresh tweaks to the existing Bohmian mechanics---they may be able to pull off something new related to QM.

    The theory *also* is sufficiently complicated that once they are "into" BM, they can stay mistakenly stretching their research programs.

    But that's not quite the same as being in a cult, IMO. The description wouldn't fit all Bohmians. (It might fit many of their financial sponsors. But not to as many of the researchers themselves.)