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FQXi BLOGS
CATEGORY: Blog
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TOPIC: Is the world made of wave-vectors?
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It seems appropriate to start my contribution to quantum foundational debates at fqxi with one of the most contentious issues in the subject - the status of the wave-vector. There are essentially two common views: either the wavefunction is ontological or it is epistemic. The ontological view holds that the wave-vector should be interpreted realistically, i.e. whatever things can actually exist in reality - the possible ontological states - are in one to one correspondence with wave-vectors. In contrast, the epistemic view holds that the wave-vector is a state of knowledge/information/belief (delete as appropriate) and its role more analogous to that of a probability distribution in classical physics rather than that of a point in phase space.
Let me briefly review the arguments for each position, with the underlying assumption that a realist view of quantum theory is desirable.
The Case for Ontological Wave-vectors:
- We know that any "completion" of quantum mechanics, i.e. a hidden-variable theory, has to have very bizzare properties, e.g. nonlocality, so it seems that there is no mathematical object other than the wave-vector that we could reasonably attach to reality.
- Almost all interpretations of quantum theory that make well-defined statements about what exists in reality, e.g. many-worlds, Bohmian mechanics and spontaneous collapse models, take the wave-vector to be an ontological object. There are simply no viable realist interpretations that don't take this view.
- Interference would be very difficult to explain if the wave-vector were a purely epistemic object. It seems that we must conceive of it as more like a real wave in 3d space in order to understand this, despite the fact that it is actually defined on configuration space.
The Case for Epistemic Wave-vectors:
- The measurement problem is arguably the most important interpretational question, but it poses no problem if the wave-vector is epistemic. We are very used to the idea that epistemic states can change radically when new information is aquired without having any major implications for what is going on in reality, e.g. updating a probability distribution via Bayes' rule.
- Many of the phenomena of quantum information theory have perfectly straightforward explanations if the wave-vector is epistemic, but seem bizzare if it is ontological. For example, teleportation is closely analogous to the transfer of a probability distribution from one point to another using classical correlations (a.k.a. a secret key), but seems to require instantaneous transfer of an infinite amount of information if the wave-vector is ontological.
- The reason why there are no realist interpretations in which the wave-vector is epistemic is just that we simply haven't thought hard enough about what the possible ontologies underlying quantum theory are.
In my opinion, the case for epistemic wave-vectors is gathering pace, but the question I want to ask here is whether this is really an all-or-nothing issue? Could it not be the case that some parameters of the wavefunction are ontological, i.e. enough of them to explain interference, and others are epistemic, i.e. enough to give a neat resolution of the measurement problem and to explain the analogies to classical probability theory?
The most simplistic version of this might take the modulus squared of the wavefunction expressed in the position basis to be epistemic and the phase to be ontological. Of course, this is not particularly compelling because it breaks the symmetry between position and momentum. Both interference and the analogies to probability theory can exist with respect to all bases, so this hardly solves the problem.
Still, it suggests that perhaps we should ask whether we can define in a more rigorous way what it means for a parameter of a theory to be ontological or epistemic. This might suggest new possible interpretations of quantum theory or, if not, at least it might lead to some no-go theorems that explain why our current realist interpretations look the way they do.
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't Hooft has published some interesting articles about viable local deterministic theories underlying quantum theory, See e.g. here
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Interesting way to frame the issue -- in my opinion it's probably *not* all-or-nothing, and determining where the wavefunction picture does (and doesn't) match up with reality is a very important question to be asking.
Here's an example of a middle ground I've been playing around with lately: Suppose there's a realistic wavefunction that is a solution to the Klein-Gordon equation, and *not* the Schrodinger equation. A single position-space measurement then can't determine the momentum-space wavefunction (because one needs a second boundary condition), but we can still derive the (incorrect) momentum-space wavefunction using the Schrodinger equation. In this case, the measured position space of the wavefunction would be ontological but the calculated momentum space would be epistemic. For a momentum measurement, it would be the other way around.
At the very least, such a picture wouldn't break the fundamental symmetry between position and momentum -- the symmetry is broken by the physical measurement(s), not the interpretation.
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I must admit I have never been able to understand 't Hooft's theories, but I am convinced that Bell's theorem is correct. Therefore, I deduce that either he is using a very different definition of locality than Bell, which would not violate my claim since I meant Bell-type nonlocality, or his theories cannot account for known quantum mechanical phenomena. Given my respect for 't Hooft, I suspect he just means something different by locality, but I haven't been able to figure this out from his papers.
Regarding Ken Wharton's comment, I must admit that I was mainly thinking about nonrelativistic quantum mechanics, since I find that the majority of quantum foundational questions can be framed in that context. Having said that, I do suspect that a no-go theorem is possible in the nonrelativistic context. Scott Aaronson has the beginnings of such an argument here, but only for a very restricted class of ontological models that are similar in spirit to Bohm's theory. If relativistic wave equations really do offer new possibilities for this then that would be very interesting. However, I'm not entirely clear about the physical interpretation of the solutions you are talking about, so it would be nice to see more details.
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Hi Matthew, thanks for laying this issue out so clearly. I used to hold the ontological view, but have gradually come around (collapsed?) more to the epistemic side, mostly guided by Occam's razor. Quantum mechanics is economically described, mathematically, by an observer assigning a complex probability to each history (path) -- weighted by its action and constrained by her knowledge. As you've very clearly pointed out, the tricky part is that -- unlike the real probabilities we're naturally familiar with after visiting Vegas -- these probabilities interfere. This implies these possibilities must, in some sense, coexist.
So, at this point, for me anyway, the ontological vs. epistemic question comes down to how to interpret this coexistence. It's not naturally understandable as a single, classical reality like we perceive, but we don't experience a fractured existence either. The mathematics of quantum mechanics tells us it's both, and we just have to get used to it. We need to take our understanding of epistemic probability and mix in the ontological aspect of these possibilities interfering -- the many worlds picture seems a decent attempt at doing this, as long as too much "existence" isn't attributed to the worlds. I don't think it's possible to successfully disentangle the epistemic and ontological aspects of quantum mechanics -- I think we just have to come to grips with the fact that our reality is best understood as a collection of possibilities that interfere with one another.
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't Hooft assumes that the "small print" in Bell's theorem will save his theories, specifically the so-called pre-determinism loophole :) Bell himself identified this and other loopholes that one needs to assume won't cause trouble.
Let me explain a simple case of Bell's theorem a bit for readers who are not familiar with it. When deriving Bell type inequalities one considers counterfactual experimental situations and by demanding that these experiments must be consistent with local hidden variables you obtain limits on the behavior of correlation functions. E.g. in the Aspect type experiment with the two polarizers at an angle of alpha one records the fraction of times both photons of an entangled pair pass through or are blocked.
Here you could reason as follows (this is similar to how Heinz Pagels explains it in one of his books): one minus the correlation measures the fraction of instances where the photon on one side does something different from the photon on the other side. Local hidden variable theories predict that 1 minus the correlation evaluated at two times alpha must be less than or equal to twice the value of this function at alpha.
This follows from the assumption that what the photon does doesn't depend on the setting of the other polarizer, so one minus the correlation function will give you the fraction of times the photon would have made a different decision had the polarizor setting be different by an angle alpha. The fraction of times the photon will do something different at two alpha difference must be less than or equal to twice the fraction at alpha, because making the same fraction of differences twice to a set of data specifying whether or not the photon passed through can never result in more than twice the result you get when you make the changes once. You would expect that from time to time you change the same data point twice, putting it back to the original value, so the result can only be less than or equal to twice the value at alpha.
The pre-determinism loophole is an objection to the above reasoning on the grounds of absolute determinism. The derivation of the Bell inequalities depends on the assumption that the setting of the polarizors could have been different while leaving everything else unchanged. This is not necessarily true in a deterministic setting. The choices the experimenter makes are just as deterministic as the photons. Clearly the state in which the experimenter makes a different choice regarding the setting of the polarizers must have evolved from different initial conditions of the early universe...
Experimental violation of the Bell inequalities could thus just as well be interpreted as experimental proof that it is impossible to make different decisions regarding the experimental set up while keeping everything else unchanged.
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I think the wave function is *both* epistemological *and* ontological. As far as I can tell it's true that the wave function represents knowledge/information. But surely, if we seek an objective physical basis for reality, and QM is the best theory we currently have, we have to accept that the wave function has real physical properties as well (which seems to imply MWI -Many Worlds Interpretation).
Normally map and territory are considered separate, but when it comes to concepts like 'QM wave function' it would seem that the distinction between the two is starting to break down. I was just discussing this on the everything-list.
The issue could possibly be clarified if we allow for different but equally valid 'levels of description' about reality. The QM wave function appears to describe reality at a high (abstract) level of description. This does not have to invalidate a lower level description in terms of (for instance) particles. Something like this was suggested by David Bohm, with his two-level model of reality. At one level was the guilding wave. At the other level was the particle.
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OK, a few comments on the comments.
Firstly, I don't personally view both ontic and epistemic as a viable solution and this has to do with my take on the foundations of probability theory. This may seem like a hopelessly abstract thing to worry about for a physicist, since we're used to just doing whatever works and then worrying about the implications later. However, the foundations of probability are a very contentious issue and I *hope* that they can be separated from the foundations of quantum theory. Although we may use one or other conception of probability in constructing our interpretations, I would like to find an interpretation for which it is possible to give an account from the point of view of all viable conceptions of probability (propensity, frequentist and subjective) that is at least as consistent as the corresponding accounts of classical probability. Part of the reason for this is laziness - if QM really does require a radically different view of probability then we really should have to redo every arguments in the foundations of probability and statistics and I'm not too keen on doing that myself.
Now, the subjective view is the most difficult to reconcile with an ontological wave-vector because it implies that probabilities are functions of an agent's belief/information/knowledge and thus are not wholly determined by some agent-independent thing that exists in reality. An ontological wave-vector is obviously such a thing, so it's difficult to reconcile with this view of probability.
Secondly, I'd like to state that I think the whole problem of the foundations of quantum theory boils down to coming up with a viable ontology. We know that a straightforward hidden-variable theory (in Bell's sense) won't work if we want to preserve *fundamental* Lorentz invariance, but this doesn't mean we should immediately jump to the conclusion that the wave-vector is the only possible ontological object. There may be many more wild and crazy ideas that we haven't thought of that turn out to make a lot of sens To be sure, I don't know what any of these are yet, but I'm not willing to admit defeat.
Finally, on the pre-determinism loophole in Bell's theorem. I think there are essentially two ways of exploiting this: either you have a crypto-deterministic theory or you have backwards-causation. The simplest possible crypto-deterministic theory would be to say that the setting of every measurement device that will ever be used to make a measurement was pre-determined at the instant of the big-bang. Then, this information could potentially be available at every point in the universe, enabling Bell inequality violations without nonlocality. Obviously, this is an extreme example, but any crypto-deterministic theory must postulate correlations between things that we have no physical grounds to believe should be correlated. The backwards-causation idea seems more viable to me, but it awaits the formulation of a non ad-hoc theory that exploits it. I once discussed this with 't Hooft at a conference and he admitted (under duress) that he thought something like backwards causation must be the answer, but haven't figured out if his theory can be interpreted in this way.
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I find it hard to believe that 't Hooft likes backward causation... The best first-principles argument for including backward causation is time symmetry (or CPT symmetry), but I believe 't Hooft is advocating an explicitly time-asymmetric theory, where different initial states converge onto the same final state (thereby removing all possibility of running the laws of physics in reverse). That's up there with Many Worlds as the precise opposite of what a backward causation interpretation would look like.
I am almost certain that backward causation is the key to all this, and the only reason that it's taken 80+ years to incorporate it is because it is so contrary to how we humans experience time and causality.
(As for the earlier comment concerning the Klein Gordon equation, I'll just mention that the non-relativistic limit of the KGE is not the Schrodinger Equation, because the KGE has twice as many solutions -- so what I said earlier would also apply to non-relativistic QM. I think I'll be able to post a more complete story on arxiv within the month -- for now, if anyone's interested in this general approach, I have a related paper in the January Foundations of Physics.)
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Matthew, why don't you just embrace your personal subjective view as the objective, ontological reality? Then all the problems go away. :D I've always found solipsism to have a certain charm, but you may feel differently.
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About realist interpretations in which the wavefunctions is epistemic, does this mean theories in which the quantum fields would appear as ghosts?
Is it possible to say that we cannot observe particles in the same way as classical objects because they don't really exist? You could magine that if the Feynman rules had been discovered by experimentalist without a firm theoretical basis, then you could have had discussions about the ontological status of virtual particles and ghosts etc....
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I should emphasize that as far as I know there are no realist interpretations that have been worked out in full detail in which the wavefunction plays a purely epistemic role. It's more of an idea for an interpretation at the moment, and one that I would be very pleased to see developed. Rovelli might be inclined to argue otherwise, but I would say that his relational view doesn't quite qualify as "fully worked out" at the present time.
I would say that you can have debates about the ontological status of anything you like in quantum field theory, or indeed any other theory, regardless of the theoretical basis on which they were discovered. As far as I'm concerned the things you mention (i.e. virtual particles and ghosts) don't have such a status, but are just mathematical objects that provide a convenient way of conceptualizing the calculations we do in perturbation theory and renormalization theory. However, arguments for the opposite position might have some merit and I'm not a particular expert.
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It seems to me that this discussion has entirely neglected the fact that complex-valued wave-vector formulations of quantum mechanics and field theory are not fundamental. That is, it is well-known that schrodinger's wave equation (also the Feynman path integral) is just a Wick's rotated diffusion equation, and that it is also equivalent to a pair of diffusion equations in time-reversal duality. Most or all of you may be associating Nelson's stochastic mechanics with what I am saying; however, Nelson was the first to show that time-symmetric Markov processes are ALMOST equivalent to the complex valued wave-vector mechanics of Schrodinger and Feynman via the construction of the hamilton-jacobi-madelung equations. I say almost equivalent because the value of the action S in the Nelsonian drift velocity is ambiguous, and thus the full equivalence to the action in the complex-valued Kernel solution to the S.E. does not hold.
It was Masao Nagasawa who rigorously proved Schrodinger's 1931 conjecture that Schrodinger's equation is mathematically equivalent to a pair of diffusion equations in time-reversal duality. Moreover, Parisi and Wu's stochastic quantization formalism has also been shown by Huffel and Namiki to be formally equivalent to the path integral formulation of quantum field theory. Garnet Ord has also shown how one can derive the Schrodinger and Dirac equations from classical statistical mechanics, without analytic continuation.
In these cases, the Hilbert space of compelx-valued wave vectors is not at all unique either ontologically or epistemologically. In fact, contrary to questions about the ontology of the wave function, it is now a more well-defined question to ask about the particular form of the stochastic diffusion process on configuration space, and the physical sources of the stochastic noise. This I think is also the strongest reason for why one can undercut claims about the fundamentality of the Many-Worlds ontological interpretation of the Everettian's.
If there is any doubt about these claims, I can provide references.
~M
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Hi Maaneli,
I would be interested to read more about this. Do these theories predict deviations from ordinary quantum mechanics under certain circumstances? I mean, if there ae physical sources for the stochastic noise, then these could presumably be perturbed somehow...
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As to Matthew's backward causality and Lisi's collection of possibilities that interfere, the most basic principle underlying probability is that any given configuration before computation is equal to any other, and thus a larger group of some classifiable type of states is more probable than a smaller group. Generally in respect to time (and what is possible) the present divides the past from the future, which probabilistically means only that the present divides past-like configurations from future-like configurations. Since time has a distinct direction we should reasonably conclude that the set of all future-like configurations is larger than the whole of all past-like configurations. But we should also expect that past-like configs influence and thus interfere with that larger group, while the dominance of future-like configs cancel the possibility of time flowing backward.
Understanding backward causation and non-locality involves simply accurately modeling the larger body of future-like configurations, to compare it with past-like states, and therein calculate the influence of the future. Presently we see backward into a distant past but we don’t yet see forward into the distant future very well. We do make one generalization. We define past-like configurations as being more ordered (dense, contracted) than more disordered future-like configs (diffused, expanded). I believe this vague and overly simple generalization of past and future based upon the second law is why science is not yet able to model and recognize the influence of future-like configs.
This may seem unfamiliar but the following model (of past and future-like states) is very simple. There are three evident fundamental axes in the space of all possible configurations. The most fundamental axis is of density. It begins at the singularity of the big bang and ends at zero density (or absolute zero Kelvin). At any point adjacent that axis and so adjacent our present there are two axes of possibility embedded in the first, the second range of possibilities is from a smooth configuration extreme to a lumpy configuration extreme, similar to contrast. Then a third axis exists embedded in the second and ranges from orderly extreme states to chaotic extreme states. This axis has led to Boltzmann's statistical side of the second law, however, this axis does not represent the past-like configurations divided apart from future-like configurations that exist along the first axis (of average density). Order to disorder does not synchronize with the increasing entropy along the most primary axis ranging from infinite density to zero density (or energy), along which the general course of the big bang can be plotted.
Our present state being rather empty and cold (-455F) is easily argued to be located nearer to absolute zero in the space of all possibilities than to the singularity of the big bang (Alpha). Should we still conclude that the measure of unique states between the present and zero (future-like) is larger than the quantity of states between the present and Alpha? (past-like). If so, then we would expect that the arrow of time is pointing to and moving toward some balance point/equilibrium where the group of all past-like states is equal to the group of all future-like states. Otherwise the arrow of time would point towards Alpha. In fact neither suggestion fits with observation. We know today that the expansion of the universe is accelerating. At present our universe appears to be accelerating toward absolute zero. If the expansion is not a temporary phenomenon, then perhaps zero is the ultimate balance of all possible states, as well as the balance between past-like and future-like configurations? As shown in the first attached image, this leads to a model of all possible states with positive and negative extremes balanced around zero and indicates two inverse time directions. The second image roughly models a cross section of the present, the second and third axis. Both images are taken from the essay below which presents a less generalized model of all possible states than how the second law currently leads most to imagine all states.
I don’t think we will understand how to merge the ontological and epistemic interpretations until we appreciate the nature of and the larger role of absolute zero in physics and cosmology. David Bohm started us in the right direction by introducing the concepts of explicate and implicate order.
Everything Moves Toward Balance
http://everythingforever.com/st_order7.htm
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The fact remains, the future expands, the past does not, it actually has a contraction function relative to the present and the future.
Backwards Causation, with a wave-vector solution that exists? ..because no particles can propergate againts the flow of Time. But curiously, a wave can propergate into the past, but it cannot travel very far,for the very fact that the past contracts away from the present,induce's any retarded "wave" to shrink as it propergates.
Close to the "present-now", a wave will exhibit expansion, because we are travelling along/into, the expanding future, but it's wave motion is constrained the further it travels against the flow of time.
There are no waves (detectable) that can stall the arrow of time.
One can ask this question about backwards causation, when we look across space we are looking across time?..we see light arriving that has taken time to arrive here, but from the same location, if there are observers existing at the light source, then they are seeing "us" in their past. So our past, is someone else's future?
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What can be said about the history of this state? (attached 1)
Any system in a state of imbalance must have a history that traces backwards to ever greater imbalance. (attached 2)
If we separate all positives from all negatives we have a kind of order, a grouping kind of order, like everything grouped together in a grocery store. If we combine positives and negatives we end up with neutral, or another kind of order, the order of balance. One kind of order exists in our past. Another kind of order exists in our future.
http://everythingforever.com
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This is what Bohm recognized with explicate and implicate order, he just didn't realize the extreme of explicate order exists literally in our past, and the extreme of implicate order exists literally in our future. The universe is an evolution of one type of order becoming another.
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Gevin:"One kind of order exists in our past. Another kind of order exists in our future."
The "past" order is fixed, cannot change, the "future" order has a vast number of configuration paramiters?
Knowing a past order configuration, does not guarantee a knowledge of a future configuration?
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