What’s wrong with the wave function?

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    The call to supplement the wave function with local beables is almost as old as quantum mechanics. But what exactly is the problem with the wave function as the representation of a quantum system? I canvass three potential problems with the wave function: the well-known problems of incompleteness and dimensionality, and the lesser known problem of non-locality introduced recently by Myrvold. Building on Myrvold’s insight, I show that the standard ways of introducing local beables into quantum mechanics are unsuccessful. I consider whether we really need local beables, and assess the prospects for a new theory of local beables.


    A couple of comments:

    “But epistemic construals of the wave function face formidable ob-
    stacles, most notably a number of no-go theorems (Bell 1964; Kochen and
    Specker 1967; Pusey, Barrett and Rudolph 2012).”

    What is it about Bell and Kochen-Specker that you think an obstacle to epistemic construals of the wave function? Those theorems make no assumptions about the status of the wave function, so they apply equally well to theories in which the wave function is ontic. Why is there any special difficulty for epistemic approaches?

    “The sticking place, as one might expect, is interference: if the wave function is purely epistemic, how can it exhibit interference effects?”

    I don’t know how many times I have said this, but there is no problem of principle with interference in theories where the wave function is epistemic, even without retrocausality. Spekkens’ toy theory paper (http://arxiv.org/abs/quant-ph/0401052) contains a section on interference, which, given how often interference is pointed to as a problem, it is obvious that few people have read.

    OK, you might say, but this only gives a qualitative account of interference and does not reproduce the cos^2 distribution of a real interference experiment. To which I would respond that it is usually the qualitative aspects that people are referring to when they argue that interference is problematic. But fine, we are still working on that.

    Robert Griffiths

    Speaking as a physicist, the problem with thinking of the wavefunction as representing a quantum system goes back to the days when Born proposed that Schrodinger’s wave be interpreted not as something physically real but as a device for assigning probabilities. Thus a spherical wave emerging from a scattering event can be reconciled with the fact that the particles tend to come out in well-defined directions. The consistent histories approach is in some ways an updated version of Born’s idea. An important part of the update is is being precise about what the probabilities refer to: subspaces of the Hilbert space. These, and not Bell’s classical hidden variables, are the quantum mechanical “beables”.

    In your discussion of EPRB in Sec. 6 you seem unaware of the fact that competent experimentalists regularly interpret their experiments as revealing properties possessed by measured systems before the measurement took place. This is retrodiction, not retrocausation. If Alice has a good piece of apparatus, and it indicates that the z component of spin of the spin half particle was +1/2 and not -1/2, she is perfectly justified in saying that the particle had precisely this property before the measurement took place, and that fact is the cause of the measurement outcome. There is no need whatsoever to invoke retrocausation. For a detailed discussion, see my “EPR, Bell, and Quantum Locality”, Am. J. Phys. 79 (2011) 954. arXiv:1007.4281.


    Thanks, Matt, that’s helpful.

    About Bell and K&S: You’re right that the theorems apply equally to theories in which the wave function is ontic. I should make that clear. They constrain property ascription in Everett, for example. But Everettians can shrug their shoulders – of course you can’t ascribe pre-measurement properties to systems corresponding to their unique outcomes, because measurements typically don’t have unique outcomes. But even if there’s no special obstacle for epistemic views (compared to ontic views), the theorems still constrain epistemic views. What’s more, epistemic views invite the picture (although they don’t mandate it) that the wave function represents our knowledge of pre-existing properties that are revealed on measurement. This is precisely the picture that the theorems make trouble for.

    About interference: I confess that I don’t really know what to make of Spekkens’ toy models. Sure, you get interference, but how exactly? You start with a principle about knowledge – that knowing more than half the information is ruled out. But without knowing why that principle holds, it’s hard to judge whether this is really an epistemic view. Bohm’s theory has restrictions on knowledge too, but the way in which those restrictions are brought about involves the wave function pushing the particles around (on a literal reading at least). I’m not sure that Spekkens can simply assert that this kind of restriction on knowledge can arise without the state being a causal entity in some sense. So I guess I’m not convinced yet that Spekkens has shown that you can get interference out of a wave function epistemic theory.


    Thanks for your helpful comments, Robert. I should make it clear that Bell’s view of what the beables are is not the only view, or the best. Spontaneous collapse and Everettian theories I think agree with the consistent histories approach that subspaces of the Hilbert space are the beables.

    And you are right that I should clearly distinguish retrocausation from retrodiction. The latter certainly doesn’t entail the former, and I don’t want to seem to be implying otherwise. I will read the article you suggest.

    Jiri Soucek

    Dear Matthew,

    I am afraid that the problem with the superposition may be harder. In fact, the superposition problem must be consider as a part of the problem of the which way information. The problem stays in the fact that the existence resp. not existence of the which way information can influence the real physical processes. The superposition is only one side of this problem. I am almost sure that there exists only one rational explanation which is based on the concept of a context introduced in my paper attached below. This explanation is based on the following considerations: there are incompatible events, which cannot be observed simultaneously in the given experiment; events observable in the given experiment make together the context of this experiment (the experimental setting defines the context); thus the existence (resp. not existence) of the which way information defines the context of the eperiment; thus with the which way information we observe different events then in the situation when the which way information is not available. Then when we have not available the which way information we can observe the interference. This means that the phenomenon of the interference is primarily associates with the context of the experiment. All this is explained in details in my paper on the extended probability theory and quantum mechanics attached to this note. In conclusion I think that the problem of interference is only a part of the larger problem of understanding well the role of the which way information (of information in general) in quantum physics.

    Your Jiri Soucek

    Jiri Soucek

    Dear Matthew,

    I would like to comment your effort to consider seriously the psi-epistemic position. I think that there are not only two options but three options – besides the psi-ontic and psi epistemic positions there exists also psi-hybrid position introduced in my paper on modified QM. This psi-hybrid position means that some wave functions represent individual states and others represent states of ensembles. Typically the set of individual states forms the orthogonal base of the Hilbert space. The psi-hybrid position offers more advantages and less disadvantages then the two other positions. Details can be found in attached papers.

    Your Jiri Soucek

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