Thursday, May 29, 2014

The measurement problem in QM

In 1976, when I delivered the John Locke Lectures at Oxford, I often spent time with Peter Strawson, and one day at lunch he made a remark I have never been able to forget. He said, "Surely half the pleasure of life is sardonic comment on the passing show".  This blog is devoted to comments, not all of them sardonic, on the passing philosophical show.
                                                                                                                 HILARY PUTNAM


This first post is an exchange (which I have received permission from all the participants to include) on the measurement problem in QM.

For background, see my
  1. "A Philosopher Looks at Quantum Mechanics (Again)" (2005), 126-147 in my Philosophy in an Age of Science, and
  2. "Quantum Mechanics and Ontology" (2012), 148-161, in the same volume

Hilary Putnam <hilary.putnam@gmail.com>
Apr 16


to David Albert, Shelly Goldstein, Tim Maudlin and Roderich Tumulka

Dear David.
  I recall that you have said (more than once, in fact) that I give too much weight to (the failure of) Lorentz invariance as an objection to Bohmian mechanics (BM). If Special Relativity (SR) were the state of the art description of space-time, the fact that the violations of Lorentz invariance in BM are undetectable according to BM itself, might be a way of mitigating, if not wholly disposing of, that objection.  But Special Relativity is not  the state of the art description; the best description we have (pending a theory that reconciles General Relativity (GR) with QM) is GR. The standard model in cosmology is soaked with GR—expansion, inflation, black holes, etc., etc. are all GR phenomena. And GR implies that while SR is wrong globally, it is right locally; as we all know, in infinitesimal regions, the space-time is MInkowskian.   And BM is wedded to Euclidean space-time. It derived, after all, from De Broglie's attempt to see the psi-function as the description of a wave in flat 3+1 space-time.
    I have been following the apparent detection of gravitational waves (caused by quantum fluctuations of the vacuum during inflation, assuming the BICEP2 observations hold up, as seems very likely) with fascination.  This leads me to wonder if the difficulty over many decades of finding a compelling scientific realist picture that answers  the "collapse" dilemmas is not, at bottom, due to the fact that we are prematurely trying to say what the psi-function is and how it figures in the dynamics before physics has come up with a good account of what its "theater of operation" really is; prematurely, as long as we don't know how to unify GR and QM.   One thing is sure, I think: the space-time in which cosmological processes take place is general relativistic; the "theater" can't be flat 3+1 space, or even Minkowski space-time. This makes me suspicious not just of BM, but of GRW as well. Even if Roderich has produced an SR version of GRW for a special case, the "theater"  is a Minkowski space-time in which there are bunches of "flashes". As I said (typed) a moment ago, that can't be the true theater of operations, and it may well be premature to try to resolve the "collapse/no collapse" problem until we have a theory that fits GR, at the very least.
     Remarks welcome from all.
Best,
    Hilary





Roderich Tumulka <tumulka@math.rutgers.edu>
Apr 17


to amiel, me, da5, Shelly, Tim, Abraham 




Dear Hilary,

You invited remarks, and I would have a few. It's not a big difference for
the foundations of QM whether space-time is flat or curved. GRW theory
with flashes works just as well in curved space-time, and Bohmian
mechanics does, too, once we allow that a certain foliation of space-time
into spacelike hypersurfaces, possibly selected in a covariant way, plays
a special dynamical role.

(The use of a preferred foliation may be regarded as going against the
spirit of (special or general) relativity, but that is largely a matter
of taste. Other things being equal, I'd prefer a theory that doesn't
involve a preferred foliation, but in this case other things are not
equal. Also, note that inhabitants of a Bohmian world can't find out which
foliation is the preferred one.)

Of course, using a curved space-time (i.e., a given nonflat 4-metric) is
not yet full GR, where matter influences curvature, and furthermore it is
unclear in which sense we will have a 4-metric in the correct theory of
quantum gravity. Nevertheless, I expect that, once we have a full quantum
gravity theory, the options for the foundations will be similar to the
ones we have today: a Bohmian version that involves a preferred foliation
and a collapse version that does not (but is more complex).

I can give you references if you like.

Best,   Rodi








David Z Albert <da5@columbia.edu>
Apr 17


to Roderich, me, Shelly, Tim, Abraham, amiel 




Hi Hillary,

Just saw your note, and Roddy's response - and what Roddy says sounds just right to me.





Hilary Putnam <hilary.putnam@gmail.com>
Apr 17


to Roderich, da5, Shelly, Tim




Dear Rodi, 
    You write, "Of course, using a curved space-time (i.e., a given nonflat 4-metric) is not yet full GR, where matter influences curvature, and furthermore it is unclear in which sense we will have a 4-metric in the correct theory of
quantum gravity. Nevertheless (sic), I expect that, once we have a full quantum

gravity theory, the options for the foundations will be similar to the
ones we have today: a Bohmian version that involves a preferred foliation
and a collapse version that does not (but is more complex)."
     The "nevertheless" is just where I have my doubts. I am reminded of how Lorentz thought his transformations would be explained by the action of velocity relative to the ether on "inter-molecular forces". The sense in which "we will have a 4-metric" in the future "full (quantum mechanical) GR" may be very surprising.
       I would happy to receive references.  Is one of them the Callender-Weingard "The Bohmian Model of Quantum Cosmology"? What do you think of it?
       Best wishes,
         Hilary





Tim Maudlin <twm3@nyu.edu>
Apr 17


to David, Roderich, me, Shelly, Abraham, amiel 




HI All,

I’m with Rodi and David here. It is, of course, not possible to say anything rigorous about how quantum theory and GR are going to come together, but one can take the attitude that the real problem is with the part of Einstein’s field equation that he himself characterized as “low-grade wood”, i.e. the Stress-energy tensor as the mathematical representation of the matter. IF some decent (local!) representation of the matter distribution (or energy distribution, if you like) is forthcoming, then the space-time geometry can be made to depend on that in something like the usual way.One has to focus, as Bell would say, on the local beables here. That four-dimensional geometry plus a foliation would give the resources to specify a Bohm-type dynamics, and we could get dynamical and variable space-time geometry.

If the combination of QM and GR demands more extensive revision of the GR picture (e.g. if space-time becomes discrete rather than continuous), then there have to be more adjustments to the GR side. (I have some ideas about how to do this, but you have to rewrite all the physics using the Theory of Linear Structures, so that’s a long story). But even there, it looks to me that space-time geometry + foliation provides the resources to write down the dynamics, and having the geometry itself be a product of the dynamics is not problematic (if the dynamics is Markov).

I also would not really demand that the foliation in such an approach be unobservable. I would rather try to find the mathematically most natural way to write a theory that has the right limiting behavior. I would not, having done that, be very surprised if the foliation turned out to be unobservable, but also not surprised if it had, in principle, observable features.

Cheers,

Tim





Hilary Putnam <hilary.putnam@gmail.com>
Apr 17


to Tim, David, Roderich, Shelly




Thanks all!  
On the plus side, the existence of Bohmian dynamics (and GRW if Rodi succeeds in extending what he was working on to a full QM) does show that we don't need to take seriously the idea of "changing  the logic" (as I once  thought) or invoking consciousness - no need for "romantic" interpretations, as Bell called them, to get at least one "realist" interpretation.
But lacking a theory with testable predictions, I am still inclined to think it is premature to speculate on what the right story will turn out to be.  (What with the "multiverse" talk, the present state of string theory, etc.., there is enough and more than enough non-empirical speculation around. But thanks for your responses.
Warm regards,
Hilary





Shelly Goldstein <oldstein@math.rutgers.edu>
Apr 17


to me, Tim, David, Roderich, Abraham, amiel 




Hi Hilary. I agree more or less with everything that Rodi, David and Tim wrote, and don't know whether there is anything more that needs to be said. But I'll say the following, though I'm not sure whether it's "anything more":

For me, a Bohmian version of a quantum theory is more or less a version in which one takes structure in space-time, including space-time itself, seriously. Thus there should be local beables in some sort of space-time, which could well be discrete and dynamical. I can't imagine why, if we live in a quantum world, it could not be Bohmian in this sense. (Of course, the way I've described Bohmian here, probably too broadly, it's hard to imagine it could fail to be Bohmian.) It seems to me that the burden of proof should be on someone who claims that a Bohmian version is impossible, and not on someone who maintains it should be possible.

Here's a possible relevant link:
Best, Shelly





Hilary Putnam <hilary.putnam@gmail.com>
Apr 17


to Shelly, Tim, David, Roderich, Abraham, amiel 




Thanks to you as well!
I certainly think a Bohmian version is possible. But Newtonian space-time was very different from Minkowskian space-time, and GR from Minkowskian, and it is possible that (GR+QM) space-time will turn out to be very different as well. I admit that I am suspicious of the idea of non-relativistic trajectories hidden from us so they can't be observed, which is what historically Bohmian trajectories are. 
Best,
     Hilary





Shelly Goldstein <oldstein@math.rutgers.edu>
Apr 17


to me, Tim, David, Roderich, Abraham, amiel 




Yes, hidden shmidden, but can you give some indication of what it is about the new space-time and its very different nature that should be relevant to making a Bohmian version unlikely. Exactly what sort of worry do you have in mind?





Hilary Putnam <hilary.putnam@gmail.com>
Apr 17


to Shelly, Tim, David, Roderich, Abraham, amiel 




Briefly, that we don't even know how space time metrics are to be superimposed (apart from ideas that go back to Wheeler, et al in 1973), whether this will involve a background super-space and a cosmic time or not, whether the Bohmian theory appropriate to this future theory (if there is one) 
will still be based on position-representation and probability currents, etc., etc., I think that you guys are betting that the picture re interpretations of QM will not be changed drastically, but I am reluctant to bet on that, although I do expect that in time physics will make sense of what is going on. Reading Robert DeSalle's fine Understanding Space-Time recently brought back to me how unanticipated the next picture of space time and causality was at each of the stages:Galileo-Descartes-Leibnitz, Newton, Einstein-SR, and Einstein-GR. I see no reason to think the next such picture won't be as orthogonal to the picture we have now as GR  is to Newton plus Maxwell. And I suspect theories [string theory in particular] for which the arguments are all apriori (as of now, of course).
Warm regards,
  Hilary





Shelly Goldstein <oldstein@math.rutgers.edu>
Apr 17


to me, Tim, David, Roderich, Abraham, amiel 




I guess I'm unclear about what you mean by a theory that is Bohmian or a theory that is not Bohmian. For me one that is not Bohmian would involve only wave functions---or would involve fundamental axioms about measurement or observation. So for me GRW  is a kind of Bohmian theory. At least, relative to this discussion, I don't see a relevant difference.





Hilary Putnam <hilary.putnam@gmail.com>
Apr 17


to Shelly, Tim, David, Roderich, Abraham, amiel 




Then for you "Bohmian" just means "sensible non-romantic interpretation". I wasn't expressing skepticism about the idea that that is what we need.





Sheldon Goldstein oldstein1@verizon.net via yahoo.com 
Apr 17


to me, Shelly, Tim, David, Roderich, Abraham, amiel 




On 4/17/2014 8:59 PM, Hilary Putnam wrote:
Then for you "Bohmian" just means "sensible non-romantic interpretation".
Yes. It's surprising that that is so very accurate.















Roderich Tumulka <tumulka@math.rutgers.edu>
Apr 23


to me, Shelly, Tim, David, Abraham, amiel 




Dear Hilary,

I agree that we don't know whether the final theory of quantum gravity
will have a 4-metric and, if not, what replaces it. But at least it seems
clear that on the macroscopic scale, and outside of extreme conditions
such as in black holes, it is a good approximation to pretend there is a
Lorentzian 4-metric.

The references I had in mind are mainly about the formulation of Bohmian
mechanics and GRW in curved space-time; they spell out the equations and
prove that the theories work.

Bohmian mechanics:
o Sections 4.5 and 4.6 of R. Tumulka:
  Closed 3-Forms and Random World Lines.
  Ph. D. thesis, Ludwig-Maximilians University, Munich (2001).
o Section 2 of R. Tumulka:
  Bohmian Mechanics at Space-Time Singularities. II. Spacelike
  Singularities.
  General Relativity and Gravitation 42: 303--346 (2010).

GRWf:
o Section 3 of R.Tumulka:
  A relativistic version of the Ghirardi-Rimini-Weber model.
  Journal of Statistical Physics 125: 821--840 (2006).
o Sections 4.1 and 4.2 of R. Tumulka:
  The Point Processes of the GRW Theory of Wave Function Collapse.
  Reviews in Mathematical Physics 21: 155--227 (2009).

GRWm:
o D. Bedingham, D. Durr, G.C. Ghirardi, S. Goldstein, R. Tumulka, and
  N. Zanghi:
  Matter Density and Relativistic Models of Wave Function Collapse.
  Journal of Statistical Physics 154: 623--631 (2014).

About whether a preferred foliation violates the spirit of relativity:
o D. Durr, S. Goldstein, T. Norsen, W. Struyve, and N. Zanghi:
  Can Bohmian mechanics be made relativistic?
  Proceedings of the Royal Society A 470: 20130699 (2014).

I don't know the paper of Callender and Weingard that you mentioned.


Best,   Rodi


11 comments:

  1. Welcome to the blogosphere! Now you just need to add some Everettians to your list of correspondents.

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  2. An interesting read. IMHO the compelling scientific realist picture that answers the collapse dilemma is akin to the optical Fourier transform. A photon is an extended-entity wave, when you detect it with an electron which is another wave, you see a dot on the screen. But photons and electrons are as pointlike as seismic waves and hurricanes.

    As for what psi-wavefunction is, see weak-measurement work by Aephraim Steinberg et al and Jeff Lundeen et al. Mindful of LIGO think "displacement current", think "spacewarp".

    Unifying GR and QM is nice, but a red herring. A photon conveys inertia from the emitting body to the absorbing body, and inertial mass equates to active gravitational mass, so the virtual photon is a virtual graviton too. Besides, remember the current in the wire? The electromagnetic forces largely cancel, but not quite. Then when you stop that current, they still don't.

    Cosmological processes don't take place in spacetime, they take place in space. Spacetime is static. The map is not the territory. And curved spacetime relates to inhomogeneous space. Remember Wheeler's geons? The strong curvature regime isn't down near the event horizon. It's nothing to do with gravity. It's everything to do with electromagnetism. I kid ye not. Check out Percy Hammond.

    John Duffield

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  3. My objection for BM starts with its treatment of the Hydrogen atom: the electron is stationary at a fixed distance from the proton (otherwise the electron would radiate energy). Now in BM trajectories are surreal, so this is not a problem, but why is this distance distinguished from any other distances? Because God made it so? Something is fundamentally wrong here.

    I think the problem of quantum mechanics interpretation is ill-posed: what is needed is to reconstruct quantum mechanics from first principles. Then the right QM interpretation will follow naturally. What would those principle be? I have a proposal (which I am in the process of writing up for publication):

    1. laws of nature are invariant under time evolution
    2. laws of nature are invariant under tensor composition (if system A is described by QM, and system B is described by QM then the composite system A tensor B is described by quantum mechanics as well)
    3. positivity (ability to have a state space)

    4. nature violated Bell's inequalities (technical postulate needed to distinguish between classical and quantum mechanics)

    The full QM in the C* algebra formalism follows as a mathematical theorem from those 4 postulates, nothing more, nothing less.

    ReplyDelete
    Replies
    1. Hi Florin,

      the hydrogen situation in BM is only so for the ground state. In different states the electron would - according to BM - move e.g. in circles or even quite chaotically (e.g. in superpositions of different eigenstates, see the illustration at http://bohm-c705.uibk.ac.at/ ).
      So if you want to model a situation in BM in which the electron moves around the core, you should better choose the wave function accordingly. Otherwise you just refer to a different physical situation than you inteded to.

      The possibility to radiate and fall into the core depends on a mechanicsm that would allow this. The normal Schrödinger equation for the hydrogen atom simply does not include such a term. So an electron that is modeled by this equation also cannot do this.
      Don't apply the intuition about a different (and classical) theory here!

      Best, Matthias

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  4. My first thought on this question was that Bell's theorem works on a sufficiently neutral basis (mostly independent of actual QM theory) so as to let us reflect on the measurement problem without bothering too much of issues such as the reconciliation of QM and GR.
    However after a second thought... Bell's theorem involves at least space time.
    Moreover recent research seems to establish a conceptual link between quantum entanglement and wormholes in GR. See
    http://mobile.extremetech.com/#/extreme/575-wormholes-are-just-quantum-entangled-black-holes-says-new-research
    I do not know what it's worth but it might occur that a deeper connection exists between spacetime geometry and the measurement problem after all, which a new physical theory would uncover?

    ReplyDelete
  5. Thanks for posting this interesting discussion. Maybe someone here can answer a question that's been bugging me ever since I read von Neumann's Mathematical Foundations of Quantum Mechanics.

    Von Neumann uses a "filter argument" in his derivation of the quantum mechanical entropy. The filter argument says that if there is any physical difference between two systems, then a filter can be created that will let one pass through but not the other.

    Applied to Bohmian QM, the filter argument says that if the Bohmian position variables have any physical effect, then we can create a filter that will sweep up a subset of the systems that are represented by a particular pilot wave. But sweeping up a subset in this way would lead to an ensemble that has a non-quantum mechanical distribution, so Bohmian QM would make different predictions from standard QM.

    On the other hand, if the Bohmian position variables have absolutely no physical effect, then they are unphysical by definition, and can be dropped from the theory.

    This seems to put an end to Bohmian QM. But none of the discussions I've read of Bohm's approach make any mention of von Neumann's filter argument. (Note this argument is completely different from the one set out in von Neumann's "no hidden variables proof.")

    Does anyone know of a response to this objection?

    ReplyDelete
    Replies
    1. Dear Robert,

      I am not a supporter of BM, but I will attempt to give an answer in BM spirit (the real Bohmians please jump in and correct me if I am saying something wrong). The way the question is posed is unclear. What does "create a filter that will sweep up a subset of the systems that are represented by a particular pilot wave." actually mean? How is the filter going to be implemented in practice?

      Let me speculate on the filter implementation in a particular case: in a double slit experiment. Let us block one of the slits (filtering out the paths that go thorough there). Then what happens? The interference pattern vanishes. What is the BM explanation? The quantum potential changes. Therefore the assertion "But sweeping up a subset in this way would lead to an ensemble that has a non-quantum mechanical distribution" is false.

      Therefore the solution to your objection is to consider contextuality: whenever one changes the experimental setting, the quantum potential in BM changes as well and no predictions contradicting standard QM predictions are possible. In fact it is a mathematical theorem that BM recovers exactly the standard QM predictions.

      Delete
    2. Thanks for your response, Florin, but I think you missed the point. BM asks us to consider an ensemble of systems which all have the same pilot wave but different initial values of the position variable (let's call it q). If the q values has any physical effect, it must be possible to separate out a subset of the original ensemble: let's say all those initial q values that result in the particle going through the top slit. Then we have a new ensemble that violates the predictions of QM (100% probability of top slit vs. 50%).

      If it is not possible to use the q values to create such a sub-ensemble, then the q values have no physical reality - they have no physical effect.

      As far as the theorem you mention, it holds under the assumption that the initial distribution of q values corresponds to the square modulus of the pilot wave. If the filtering process is possible, then this assumption is violated and the so the theorem doesn't hold.

      Von Neumann wrote a paper showing that the filtering process must be possible if the quantity has any physical effect. (He describes the process as creating a membrane that is permeable to systems with some values of the variable (q) and impermeable to others.) I don't have the reference at hand but you can find it in a footnote in his book.

      Delete
    3. Hi Robert,

      " If the q values has any physical effect, it must be possible to separate out a subset of the original ensemble"

      I think this is not correct for BM. Let me illustrate why:
      "Having a physical effect" could mean e.g. that the initial position influences where the particle ends up in the experiment. This is so in BM.
      Why should it therefore be possible by the experimental (and statistical means) that you can *control* the position directly? In fact, one can prove that for BM this is not possible in the generic situation of quantum equilibrium. See http://arxiv.org/pdf/quant-ph/0308039v1.pdf, ch. 11 "Absolute Uncertainty" (pp. 46).

      The simple argument that particle positions are physical in BM is that you need something to represent what is happening in physical space in your theory. Otherwise (strictly speaking and not accepting tacit assumptions or conventions) the theory is an empty mathematical formalism.

      Best, Matthias

      Delete
  6. Hilary Putnam: “This leads me to wonder if the difficulty over many decades of finding a compelling scientific realist picture that answers the "collapse" dilemmas is not, at bottom, due to the fact that we are prematurely trying to say what the psi-function is and how it figures in the dynamics before physics has come up with a good account of what its "theater of operation" really is; … As I said (typed) a moment ago, that can't be the true theater of operations, and it may well be premature to try to resolve the "collapse/no collapse" problem until …”


    Indeed, ‘theater’ before anything else.

    Let us consider only 3-player-theater (time, space and particles). While all the three have the duality-personality, I would like to ‘simplify’ the issue as below.
    Space, 99% wave.
    Time, 99% wave.
    Particle (such as electron), 99% particle.

    That is, the wave-function of electron is actually describing a ‘particle’ floating on the space-time-wave. Electron as a ‘particle’ is always as a ‘whole’ while its ‘position and momentum’ are depending upon the space-time-wave (described by the electron-wave function). Thus, there is no ‘collapse’ issue as there is nothing to collapse about. The visualized collapse of electron in our measurement is caused by the space-time wave which is ‘structured’ by the space-time force.

    F (space-time Force) = K ħ/ (delta S x delta T); the ‘theater’.
    K (coupling constant, dimensionless); ħ (Planck constant); S (space); T (time).
    Then, delta P (linear momentum) = F x delta T = K ħ/ (delta S)
    So, delta P x delta S = K ħ
    When, K is near to 1 (but a bit smaller than 1), then delta P x delta S > ħ (the uncertainty principle).
    When K ħ is near to (0 ħ), the F is ‘gravity’.

    ReplyDelete
  7. Dear authors of the blog,

    is it really so clear that one needs a preferred (albeit dynamical) foliation of spacetime into spacelike hypersurfaces for relativistic BM?

    To me, the situation seems to be more complicated: the foliation is not necessary to formulate a Lorentz covariant and nonlocal law of motion (this could e.g. be done by forward and backward light cones as well, see http://arxiv.org/abs/quant-ph/0105040 for a similar idea).
    The crucial argument in favor of a foliation then seems to be that one has no idea how to statistically analyze laws of motion different from ones generated by velocity vector fields.
    This argument is, however, just a "human" one: one does not know better at the moment. Could it not be that for relativistic physics of N non-independent particles, the way how to do statistics has to be changed in general?

    An indication that this might be so is Wheeler-Feynman electrodynamics (http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21.425, http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.17.157): Its laws of motion are very natural from a certain perspective but can also not be formulated using a velocity vector field. Consequently, no way of analyzing the statistics is known.
    In the expectation that many people might object to the theory because of its unusual view towards "causality", it is in any case an example for how the question of the analyzability of law that are not of a velocity vector field type arises, also in a context different from relativistic BM.

    I'm sometimes wondering why this line of thought is not discussed more intensely when it comes to relativistic BM...

    Best, Matthias

    ReplyDelete