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Pilot wave theory, fundamental forces

  1. Jan 3, 2010 #1
    I read a short high-level article about the pilot wave interpretation of quantum mechanics and I have some questions.

    Is there a good way to formulate that theory so that the only force on a particle is from the pilot wave (inertia, gravity, EM, ... move/effect the wave which in turn effects the particle)? Seems like people would have tried this, but I can't find anything when searching the web.

    Also, the article claimed that pilot wave theory provides new, testable predictions. Where I can find more information about that?
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  3. Jan 4, 2010 #2


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    According to the pilot wave theory, the only force on a particle is from the pilot wave. It is quite obvious from most treatments of the theory, but I don't know any reference in which this point is particularly emphasized.

    It would help if you could specify which article are you talking about.
  4. Jan 4, 2010 #3
    Hi Demystifier,

    But surely when you analyze the force equations there is a [tex]-\nabla V[/tex] term as well as the quantum force term [tex]-\nabla Q[/tex] (where V and Q are the respectively the classical and quantum potentials). This implies that the particles attract/repel each other as well as being pushed around by the pilot-wave, no?

  5. Jan 4, 2010 #4
    Regarding question 1: As Zenith said, that article seemed to imply that gravity, ... act on the particle (mathematically through a potential V). Demystifier, do you know where I can look to find the formulations in which the only force on the particle is the pilot wave?

    Regarding question 2: The article was by Mike Towler at Cambridge University, but I can't find the link now. However, I don't know if that's relevant -- the article just mentioned (in a bullet) that pilot wave theory provides new, testable predictions, but it did not say what they were. I would like a reference to find out what they are.

  6. Jan 4, 2010 #5
    I've referred to the article that I think you mean in recent threads. You can find it at :

    http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/towler_pilot_waves.pdf" [Broken]

    He also has a full on-line graduate course in pilot-wave theory at:

    http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken]

    If you look in the sidebar link "Further Reading" in Towler's course there are links to hundreds of relevant papers. As regards testable predictions, presumably he means Valentini's non-equilibrium stuff leading to observable consequences in the cosmic microwave background etc. (though there are some other more flaky ones such as detecting possible violations of Pauli's exclusion principle, and/or using "lasers" - mounted on the head of a shark? - to detect whether particles held in traps are absolutely at rest in violation of Heisenberg uncertainty principle).

    Looking at Towler's list you might read Valentini's recent "Beyond the quantum" article in Physics World, or the following three articles:

    Inflationary cosmology as a probe of primordial quantum mechanics A. Valentini (2008).
    De Broglie-Bohm prediction of quantum violations for cosmological super-Hubble modes, A. Valentini (2008).
    Astrophysical and cosmological tests of quantum theory, A. Valentini (2007).

    For the laser stuff, see the book "Quantum Cauasality" by Rigg.
    Last edited by a moderator: May 4, 2017
  7. Jan 4, 2010 #6
    Great, thanks for the info Zenith.

    Also, if the above refs don't include a formulation in which the only force on a particle is the pilot wave, I would still like to know if anyone knows of a ref for that.
  8. Jan 5, 2010 #7


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    That is certainly true. However, when I think about the pilot wave theory, I like to think of the wave function, and not of the quantum potential, as the fundamental quantity. The wave function guides the particle and the wave function by itself does not distinguish between classical and quantum force. All "force" is described by the wave function. (See however my next post which clarifies it more carefully.)
    Last edited: Jan 5, 2010
  9. Jan 5, 2010 #8


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    ANY paper on pilot wave theory describes how the motion of the particle is described only by the pilot wave (and the initial position of the particle). However, it is actually incorrect to say that the pilot wave determines the force. Namely, by definition a force is a quantity that determines acceleration, while the pilot wave determines the velocity. The initial velocity is not arbitrary in pilot wave theory, which is why it is somewhat misleading to formulate pilot wave theory in terms of forces and quantum potentials. The quantum potential is useful only to demonstrate similarity between classical mechanics and pilot wave mechanics, but the quantum potential does not have a fundamental role in pilot wave theory.

    See also Section 4 in
  10. Jan 5, 2010 #9
    I see your point, but perhaps it's slightly misleading to present this as the settled view of the pilot-wave community. I know that the Goldstein group that you link to present it in this way, but many others (e.g. Peter Holland, Basil Hiley, and Peter Rigg, to name three authors of pilot-wave textbooks) argue quite vehemently the opposite position. This is particularly the case if one argues that the wave field is a repository of energy, along the lines I did in https://www.physicsforums.com/showthread.php?p=2369492#post2369492".

    Holland and Hiley in particular have some serious-sounding arguments in their recent papers - which I could look up if I could be bothered - in which they claim to prove that the quantum potential is fundamental.

    For the moment let's just say we don't know who's right - so I don't think it's true to say definitively, as you do, that the quantum potential does not have a fundamental role.
    Last edited by a moderator: Apr 24, 2017
  11. Jan 5, 2010 #10


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    I would like to see these papers if you know the exact references.

    Anyway, what is your opinion? What is more fundamental, wave function or quantum potential?
  12. Jan 9, 2010 #11

    I'd like to chime in on this question.

    It is known that the wavefunction and its corresponding Schroedinger equation imply the quantum potential (via the Madelung equations obtained from the polar decomposition of the Schroedinger equation), but that the converse is not true without an additional, ad-hoc constraint on the "phase function" (or "velocity potential" in hydrodynamics language), S(x,t), which couples to the probability density via the quantum potential. This additional constraint on S(x,t) turns out to be equivalent to the Bohr-Sommerfeld-Wilsion (BSW) quantization constraint, or, equivalently, the constraint that the derived wavefunctions encoding S(x,t) be single-valued. Without this ad-hoc constraint, there will be non-quantum solutions to the Madelung equations that do not corresponding to any single-valued wavefunction satisfying the Schroedinger equation. What this then implies is that the addition of the quantum potential to the otherwise classical Hamilton-Jacobi fluid equations, (which is essentially what the Madelung equations are), is not sufficient to establish a hydrodynamics that is equivalently expressible as the Schroedinger dynamics of a single-valued wavefunction. On the other hand, the single-valued wavefunction of QM and its dynamical equation (the Schroedinger equation) do contain all the physical information of the quantum potential, in addition to other essential physical information (the BSW quantization constraint), so as to allow for an equivalent reformulation via the hydrodynamic Madelung equations. Based on this established relation between the Schroedinger equation and Madelung equations, I think one is forced to conclude that the wavefunction is more fundamental than the quantum potential.

    As an historical aside, the inequivalence between the Schroedinger equation and the Madelung equations was actually discovered twice in different (but related) contexts; the first time was by Takehiko Takabayasi in 1952, who showed that Madelung's hydrodynamic equations are not equivalent to Schroedinger's equation without the (in his own words) "ad-hoc" BSW quantization constraint on the velocity potential S(x,t) in Madelung's equations. Takabayasi also tried to argue that Bohm's 1952 causal interpretation of QM, which made use of Madelung's equations, was also inequivalent to QM, but this turned out to be wrong as we now know. The second time was by Timothy Wallstrom in 1988, in the context of stochastic mechanical derivations of the Schroedinger equation. Wallstrom showed that even though stochastic mechanical theories such as Edward Nelson's can derive the Madelung equations (and, consequently, the quantum potential), they do not derive the Schroedinger dynamics for a single-valued wavefunction without also imposing the ad-hoc BSW constraint on the velocity potential S(x,t) in the stochastic mechanical equations of motion. You can read more about all this in Wallstrom's concise 1994 paper:

    Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations
    Phys. Rev. A 49, 1613–1617

    In my opinion, if one could find a dynamical justification for the BSW quantization constraint from the dynamics of the particles in stochastic mechanical theories, then one could reasonably claim that the quantum potential is more fundamental than the wavefunction in the context of such theories. In fact, if stochastic mechanical theories could successfully derive the Schroedinger equation, then even the deterministic pilot-wave theories would be "coarse-grained" approximations to the stochastic mechanical theories, and it would only appear on the coarse-grained level that the dynamics of the pilot-wave (wavefunction) and particles are Aristotelian. Moreover, the wavefunction would have to then be interpreted as an epistemic mathematical construct, rather than an ontic field. The quantum potential, on the other hand, would still be interpreted as an ontic potential energy field. So the success or failure of stochastic mechanical derivations of the Schroedinger equation clearly has direct and significant implications for your (Demystifier's) question.
    Last edited: Jan 9, 2010
  13. Jan 13, 2010 #12

    Hi Demystifier,

    Sorry for the slight delay. I was out of town for a few days and the thread slipped off the bottom of the page..

    Just based on a quick search in "Further Reading" on http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken], the following papers indicate what I mean (links on the page):

    Schroedinger revisited: an algebraic approach, M.R. Brown and B.J. Hiley (2004).
    See p. 9, paragraph 4

    From the Heisenberg picture to Bohm, B. Hiley (2002)
    Section 3, p. 7 onwards

    Hamiltonian theory of wave and particle in quantum mechanics I: Liouville's theorem and the interpretation of de Broglie-Bohm theory, P.R. Holland (2001).
    Section 1.2, p.6 "The role of the quantum potential"

    Plus see the book by Rigg in the Textbook section at the top.
    Last edited by a moderator: May 4, 2017
  14. Jan 19, 2010 #13
    On my behalf, thanks for these refs, Zenith.

    By the way, not to be pushy, but are either of you (Zenith and Demystifier) interested at all in discussing the question (about the fundamentality of the wavefunction vs quantum potential) that I suggested an answer to? I was really expecting that it would be discussed.
    Last edited by a moderator: May 4, 2017
  15. Jan 19, 2010 #14


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    Zenith, thanks for the references.

    Maaneli, #11 was a great post. I mostly agree with it.
  16. Jan 21, 2010 #15
    Thanks. Out of curiosity, which parts do you disagree with?
  17. Jan 21, 2010 #16


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    Well, I think it is not completely clear whether the Walstrom argument is correct or not. See Sec. IV of
  18. Jan 21, 2010 #17
    Ah, I've studied that paper recently, and I found several problems with Smolin's arguments. First, his arguments are only applied to the artificial and trivial case of the Schroedinger equation on a circle, whereas the arguments of Takabayasi and Wallstrom apply to the Schroedinger equation in 2 dimensions or greater. And for even just 2 dimensions, Smolin's claim of a well defined mapping between solutions of the Nelson equations and solutions of the Schroedinger equation, is problematic. For example, Valentini and Bacciagaluppi have pointed out that for just one node in a 2 dimensional wavefunction, moving the line across which psi is discontinuous will in general produce a different wavefunction, so that the mapping between solutions of the Schroedinger and Nelson equations is not well defined; and for the case of more than one node, the mapping seems even more ill-defined.

    Additionally, Valentini and Bacciagaluppi pointed out that even for the case of the circle, it is problematic to allow discontinuous wavefunctions to be physical wavefunctions, since, as is well known, discontinuous wavefunctions can have divergent values of observables such as the variance of the total energy, the mean kinetic energy, etc.. This is why physical wavefunctions are required to be continuous, or more precisely, that their first derivatives be square-integrable, so that the wavefunctions form a Sobolev space. Smolin does not address this point, aside from a brief comment on page 9 where he asserts that the expectation value of the Nelsonian energy is well defined. But even if so (and he doesn't explicitly show this for the general case), how is this to be reconciled with the fact that the standard definitions of operator expectation values (using the derived discontinuous wavefunctions) are divergent for the aforementioned observables? And if Smolin is going to use the Nelsonian definition of energy expectation values, instead of the standard quantum mechanical definitions, how can he claim that Nelson's theory derives standard quantum mechanics? Smolin does not address any of these inconsistencies.

    Lastly, Wallstrom explicitly showed in his 1994 paper that if one allows S(x,t) to be arbitrarily multi-valued in Nelson's equations (so that the derived wavefunctions are arbitrarily multi-valued, as Smolin wants to allow), then this leads to non-quantized values of angular momentum for the case of a 2-dimensional central force problem. In other words, Nelson's stochastic mechanics would not be empirically equivalent to standard quantum mechanics, because it would predict non-quantum values of angular momentum for a well established quantum mechanical situation.
  19. Jan 22, 2010 #18


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    Thanks Maaneli. Are you talking about the book by Valentini and Bacciagaluppi, or about another reference I am not aware of?
  20. Jan 22, 2010 #19
    Not the book, private communications. But Valentini did tell me that he plans to publish these criticisms in his next book.
  21. Jan 22, 2010 #20


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