Forum Replies Created
July 8, 2015 at 5:40 pm #2536
Here are some general remarks on the practical usefulness of Bohmian mechanics (in a rather random order). They don’t address Maximilian’s specific question but contribute to the overall theme of ways in which Bohmian mechanics is helpful.
* Consider two theories making the same predictions: one clear, precise, and simple, the other unclear, vague, and complicated. Which one is better? There is no doubt that Bohmian mechanics is clear, precise, and simple (a complete definition fits on a single slide). Copenhagen QM is not completely clear (Feynman: “Nobody understands quantum mechanics”), and it is vague and complicated because of the measurement axiom that refers to observers and measurements (vague concepts, as nobody has a precise definition of which physical objects count as “observers”).
* It is one of the goals of physics to find out how the world works; it would seem odd to say, now that Bohmian mechanics provides a coherent answer that standard QM could not provide, that we are no longer interested in this goal.
* In the same vein, it is one of the goals of physics to find explanations of the observed phenomena. Again, it would seem strange to say we are no longer interested in explanations of, say, the double slit experiment–even more so in view of the idea, introduced by Bohr and eloquently elucidated for the double slit in Feynman’s lectures, that such an explanation be impossible. Let me add, I agree that it is one of the goals of physics to make predictions (and another to develop better technology), but there are also the goals to find out how the world works, and to find the explanations of the phenomena that call for explanation.
* Bohmian mechanics has inspired some discoveries, e.g., Bell’s nonlocality theorem. (My own recent work on multi-time wave functions and interior-boundary conditions was inspired by Bohmian mechanics.) It may also be useful in the search for, e.g., a theory of quantum gravity.
* Bohmian mechanics is easier to learn for students than standard quantum mechanics.
* Bohmian mechanics has applications to numerical methods for solving the Schrodinger equation. (Experts find a higher efficiency if grid points are not evenly spaced but |psi|^2 distributed, and a higher efficiency if the grid points “move with the flow,” which is what Bohmian trajectories do.)
* Let me draw some parallels with the following questions: Do we need mathematicians? Should students learn proofs (say, of the Gauss integral theorem)? Well, for practical physics computations it is usually not relevant to know the proof of the Gauss theorem, while it is very relevant to know the theorem itself. That is, a limited level of rigor is often sufficient for getting the right answer and efficient for getting it quickly. Nevertheless, sometimes math can get very confusing, and then it is useful to know the details of math facts. (E.g., how exactly is the delta function defined? And what exactly does it mean to say that the Laplacian of 1/r is -4 pi delta?) So, it is good that there are mathematicians who are very careful when formulating statements and proofs. (And, after all, what they do is correct.) The situation is a bit similar with Bohmian mechanics: Even if it is usually not necessary for finding the correct predictions, it can be useful to have a precise version of QM, particularly when QM gets confusing.
* Bohmian mechanics provides some useful approximations for the computation of predictions, e.g., concerning the statistics of arrival times (when will the detector click?) and semi-classical approximations. While the predictions are the same in Bohmian mechanics and standard QM, and can be computed also without Bohmian mechanics, certain approximations are suggested by the Bohmian approach. To be sure, the approximation can equally be used if Bohmian mechanics is wrong (say, if collapse theories are right).
* Bohmian mechanics permits an analysis of quantum measurements, while they are taken as primitive and/or unanalyzable in standard QM. In Bohmian mechanics, one can prove theorems about measurement. E.g., positive-operator-valued measures (POVMs, also known as “generalized observables”) arise from Bohmian mechanics through an analysis but have to be postulated in QM as an extension of the theory. As another example, I once used Bohmian mechanics to give a simple, clean, and clear-cut proof of a superselection rule (i.e., that certain superpositions are indistinguishable from mixtures), while standard QM could offer only hand-waving talk in support of this rule.
* Bohmian mechanics also permits an analysis of issues with philosophical subtleties, e.g., (i) limitations to knowledge, (ii) quantum non-locality, (iii) tunneling times. (i) This is the phenomenon that some facts in the world cannot be completely revealed by any experiment; e.g., wave functions cannot be measured. (ii) Bell’s theorem says that entangled particles must undergo some action-at-a-distance (which cannot, however, be used for sending messages). (iii) Bohmian trajectories provide an obvious definition for how long a particle stayed inside the barrier during tunneling and a deconstruction of the allegation of faster-than-light motion inside the barrier. All of these examples have aspects that go beyond mere operational statements (“if we set up an experiment like this …, then the outcome will be x=… with probability p(x)=…”). And in all of these examples, the clear picture provided by Bohmian mechanics allows us to understand and deal with these aspects.
* Bohmian mechanics takes away the need for philosophical contortions when explaining QM or talking about what happens in certain experiments or what happens out there in the world. (As an example of such a contortion, Bell mentions: “Was the world wave function waiting to jump for thousands of millions of years until a single-celled living creature appeared?”)July 8, 2015 at 5:37 pm #2535
Hi Maximilian, I have a few examples. I am not sure whether they are of the kind you are interested in, but I thought I mention them nevertheless.
Bell has a paper [Intl.J.Quant.Chem. 14 (1980) 155; reprinted in “Speakable and unspeakable in quantum mechanics” p. 111] on how Bohmian mechanics answers the question of whether a delayed-choice experiment involves retrocausation.
There is quite some literature on how Bohmian mechanics helps for scattering theory; here is a selection of references:
M. Daumer et al., J. Stat. Phys. 88 (1997) 967, arXiv:quant-ph/9512016.
D. Durr et al., Lett. Math. Phys. 93 (2010) 253, arXiv:1002.0984.
T. Norsen, Am. J. Phys. 82 (2014) 337, arXiv:1305.1280.
People have also used Bohmian mechanics to consider the questions of how long a tunneling particle remains inside the barrier and whether it moves there faster than light. Two references:
C.R. Leavens, in: J.T. Cushing, A. Fine, S. Goldstein (eds.), Bohmian Mechanics and Quantum Theory: An Appraisal (Kluwer, Dordrecht, 1996).
And, again, T. Norsen, Am. J. Phys. 82 (2014) 337, arXiv:1305.1280.
I personally found the Bohmian picture useful for studying the probability distribution of the time at which a detector clicks. I hope to write a paper about it one day.
I have used Bohmian mechanics in the analysis of systems in thermal equilibrium, which has led to the use of the so-called GAP measure, a probability distribution over wave functions appropriate in thermal equilibrium [S. Goldstein et al., J. Stat. Phys. 125 (2006) 1193, arXiv:quant-ph/0309021]. The GAP measure was introduced without Bohmian mechanics (and termed “scrooge measure”) by R. Jozsa et al., Phys. Rev. A 49 (1994) 668, but no connection to thermal equilibrium was made.
I am presently working on a paper with S. Goldstein and W. Struyve using Bohmian mechanics to evaluate the question whether Boltzmann brains will appear numerously in the late universe, assuming the late universe will be in the Bunch-Davies state on a de Sitter space-time.
Best, RodiJuly 8, 2015 at 4:34 pm #2532
Hi Rainer, here are three remarks on the problem you raised.
First, the situation is in a certain respect the same in Bohmian mechanics and in standard quantum mechanics. That is because if we consider the wave function of object and apparatus together (or of the whole solar system) then Bohmian mechanics and standard quantum mechanics both use the Schroedinger equation, so they use the same wave function.
Second, your question is ultimately one about the form of that wave function. Considering two systems, the x-system and the y-system (the environment of the x-system), for the x-system to be an “effective system,” i.e., to have an effective wave function, we need that Psi(x,y) is locally a product, i.e., Psi(x,y) = psi(x)phi(y) + Phi(x,y) where Phi has y-support macroscopically disjoint from the actual (Bohmian) Y. So the question is, why should the wave function of object and apparatus before a quantum experiment be of such local product form? Why should the object be initially unentangled with the apparatus? In fact, random wave functions of two systems typically have a high degree of entanglement. For example, if the x-system and the y-system are jointly in thermal equilibrium then they are highly entangled.
A relevant mechanism is the stretching of wave functions under the free evolution. It is known from scattering theory that the wave function of a single particle, when evolving freely (zero potential), becomes locally a plane wave. Let us apply this to, for example, a double-slit experiment, in which the electron used for the experiment gets “boiled off” from a piece of metal; initially, that electron is in thermal equilibrium with the metal, and thus highly entangled with other electrons in the metal. But once it leaves the metal, it propagates freely, so its wave function approaches a spherical shape, locally a plane wave. A collimation slit allows only the part of the wave propagating in the right direction to enter the double-slit setup. If the electron (the x-system) actually passes through the collimation slit, and gets used in the experiment, then the wave function Psi(x,y) of electron and apparatus is, in a neighborhood of (X,Y) in configuration space, a plane wave in the x-variable (and complicated in y). Since plane waves are not entangled (in the spinless case), Psi is locally a product psi(x)phi(y). (In the presence of spin, the spin of the electron in the double-slit setup may well be entangled with the spin of electrons in the metal piece, but that does not matter for the experiment.)
Third, the measurement rule can be formulated in Bohmian mechanics as follows: *IF* the object is initially unentangled with the apparatus *THEN* the probability distribution of the outcome is given by the standard formula. However, Bohmian mechanics is still meaningful, and still makes unambiguous predictions, in case the object *IS* initially entangled (or not perfectly unentangled) with the apparatus. The prediction is just whatever follows from the Schroedinger equation and Bohm’s equation of motion for object and apparatus together. For standard quantum mechanics, in contrast, this situation would be problematical because it is not covered by the standard axioms.March 2, 2015 at 8:53 pm #1986
Perhaps I understand better now the root of our disagreement. It seems to have something to do with whether the definition of locality refers to actions of an agent on one side having consequences on the other side, or whether it refers to events on one side having consequences on the other side, where events may include random events not controlled by an agent. When I use the term “locality,” I mean the latter, and so I did not mention any agent when formulating the locality condition (L) in my paper, while I called the variant with an agent “control locality” (CL). As I say near the end of my paper, CL alone is not sufficient to imply Bell’s inequality; in fact, it’s not clear what CL would mean for a stochastic theory. In the passage from Bell’s 1964 paper that I quoted in the last section of my paper, he describes the assumption of EPR’s argument as CL, and I’d say Bell is inaccurate in this passage; the EPR argument requires L. Maybe you mean CL by “locality,” and therefore find that determinism can’t be concluded from it.
Best, RoderichFebruary 4, 2015 at 2:08 pm #1946
Yes, let’s agree to disagree. Thank you for the discussion.
All the best, RoderichFebruary 2, 2015 at 2:59 pm #1944
Dear Richard, (in response to your #1901)
I agree that “we can’t decide whether B is a function of events on all or part of H without applying some theory” if “theory” means laws governing a possible reality (i.e., what I called a “scenario” in my paper). Quantum theory is often understood as not talking about reality but only about empirical predictions; but if we understand “quantum theory” as saying that, for an EPR pair of particles, there are no further variables in addition to the wave function (i.e., what I called “Bohr’s scenario” in my paper), then I agree that “Prob(B/S(H))=1 but Prob(B/S(HP(R_B)))=1/2.” These relations show that Bohr’s scenario is nonlocal (because they show that the state-of-affairs on H outside P(R_B) has an influence on B). I agree in particular that, in this scenario, B is certain conditional on S(H) and not certain conditional on S(HP(R_B)). I further agree that my formulation of locality implies that events outside P(R_B) cannot influence events in R_B. However, I disagree with your last statement, and maintain that the conjunction of Prob(B/S(H))=1 and Prob(B/S(HP(R_B)))=1/2 entails a failure of locality.
Your statement before seems not relevant. To explain, let me use the notation Y for S(HP(R_B)) and X for the remainder of S(H). The condition that X has no influence on B can be expressed as Prob(B/X,Y)=Prob(B/Y). I agree with your remark that specifying X does not alter the fact that Prob(B/X,Y)=1 but Prob(B/Y)=1/2; but the relevant fact for whether X has an influence on B is that specifying X alters the conditional probability of B; that is, that Prob(B/X,Y)=Prob(B/Y).
Best, RoderichFebruary 2, 2015 at 2:59 pm #1943
Dear Bob, (at #1900)
Yes, it would good to find out what exactly we disagree about. So let me answer your questions about my position. By “English properties” I did not mean macroscopic properties. In your example, I assume that S is 1-dimensional. I agree that with “cat is alive” is associated a (high-dimensional) subspace A of the Hilbert space of the relevant particles, and similarly for D, and that AD=DA=0, whereas [A,S] and [D,S] are significantly nonzero. I also agree that |x+> and |z+> are distinct as mathematical objects, and that Carol can produce either of these reliably with her apparatus. I also agree that Donald, if not given additional information, has no reliable way to distinguish them by some sort of measurement. How do I understand this inability? I conclude that, in this example, nature knows the actual quantum state of every particle (as does Carol), and the laws of nature are such that Donald has no access to this information. Nature can keep a secret. There are limitations to knowledge. There is a fact about the quantum state of this particular particle, and Donald has no way of finding out. That may be shocking, but we better get used to it. There are things we cannot measure. And once we think of it, that seems obvious: You may not remember what you had for breakfast on February 1 ten years ago, but there was a definite fact about what you had. But now there is no experiment that could answer what you had. So it may seem very natural that there are things we cannot measure.
Best, RoderichJanuary 28, 2015 at 5:48 pm #1899
Thank you for your replies #1860 and 1861. There are two relevant meanings of the word “property”: let me call them “English property” (the ordinary meaning of “property” in English) and “quantum property.” An English property of a system is something that a system either has or has not, while a “quantum property” of a system is a subspace of the system’s Hilbert space (or, equivalently, a projector); of course, a system may be in a superposition of a subspace S and its orthogonal complement, in which case one is neither justified in saying that the system has the quantum property S nor that it does not have the quantum property S. For a harmonic oscillator, “the energy is less than 2 omega” is a quantum property. For a macroscopic system such as Schroedinger’s cat, “alive” is a quantum property, and may also be an English property depending on the chosen solution to the quantum measurement problem. In fact, the measurement problem can be phrased as saying that for the theory to make sense we need that “alive” is also an English property of cats (except in a many-worlds framework). In quantum physics we usually focus on quantum properties, but for investigating locality we need to consider English properties.
Concerning classical pictures of spin, I would not tell my students they should think of |z+> as a top whose spin axis is random with some constraints, exactly because this picture is bound to mislead. So I am perhaps actually more inclined against classical pictures than you.
But the salient point is that my reasoning does not make use of any classical picture of spin. It only makes use of the 2-dimensional Hilbert space of a spin-1/2, of the quantum rules for prediction, and of the mathematical fact that the 1-dimensional subspaces in that Hilbert space are in a one-to-one correspondence to the unit vectors in 3-dimensional physical space. I label vectors in Hilbert space by directions in 3-space, but do not assume spinning classical tops. Now you need to read my Carol-Donald argument again. By the way, I agree that entropy is the log of the dimension of the subspace corresponding to a given macro-state, and that does not affect my Carol-Donald argument.
Best, RoderichJanuary 28, 2015 at 5:47 pm #1898
Dear Richard (at #1859 and 1894),
I like that you point specifically to the step in the reasoning that you are objecting to. That helps for a good discussion.
I see that the word “pre-determines” has connotations that are irrelevant to the argument. For my purposes, “x pre-determines y” just means “y is a function of x.” It is not necessary to refer to your definitions 1, 2, 1a, or 1b. So let me go through the reasoning again.
Let R_A and R_B be the space-time regions in which Alice’s and Bob’s experiment, respectively, are carried out, let A and B be Alice’s and Bob’s outcomes, respectively, let H be a spacelike hypersurface after R_A and before R_B, let P(R_B) denote the relativistic past of R_B (interior of the past light cone), and HP(R_B) the intersection of H and P(R_B). Let S(H) be the state-of-affairs on H. (People often find it difficult to understand what that means, we may talk more about this later.) Then B=f(S(H)) with some function f. Now assume locality. Then B cannot depend on local beables outside P(R_B) (nor on nonlocal beables involving both sides), so B=g(S(HP(R_B))) with some function g. That is what I concluded.
Maybe these remarks provide some clarification.
Best, RoderichJanuary 20, 2015 at 3:36 pm #1858
Indeed one might suspect that statements about probability, or even fixity, must be relativized in the way you suggest. So let me go through the reasoning in more detail. The state-of-affairs (i.e., the “real, factual situation,” the “beables”) on that hypersurface includes Alice’s outcome, so it pre-determines Bob’s outcome (as Bob will choose the z direction). Now, if we assume locality then non-local beables and the local beables outside Bob’s past light cone cannot have any influence on Bob’s outcome. Therefore, Bob’s outcome is pre-determined by the local beables on the intersection of that hypersurface with Bob’s past light cone. I see no way around this conclusion.
Best, RoderichJanuary 20, 2015 at 3:34 pm #1857
Let me first address the Carol-Donald issue. I was, in fact, paying attention to the distinction you emphasize; let me elaborate some more. Let n_k be the unit vector in the direction in which Carol has prepared the spin of particle k. She predicts that if one were to make a Stern-Gerlach experiment on particle k in the direction n_k, the result will be “up.” According to the rules of quantum mechanics, her prediction is correct in 100% of the cases. According to the same rules, the prediction would be falsified at least sometimes if she had named other directions. Therefore, nature must, in order to produce results in agreement with the rules of quantum mechanics, remember for each particle k what the direction n_k was. Therefore, nature must remember for each particle in this experiment its quantum state up to a phase factor.
On the use of projections: Like you, I guess that the predictions of quantum mechanics are correct from the quarks to the quasars. Projections are a mathematical tool for computing these predictions. However, projections and properties are not the same concept. When contemplating Bell’s proof, we need to consider, besides observables and outcomes of experiments, possibilities for what the reality might be like, and to this end it is necessary to consider properties in the ordinary English meaning, as distinct from projections.
RoderichJanuary 18, 2015 at 3:22 am #1835
Thank you to all of the commentators for your remarks!
To Dieter at 1674:
You may be surprised but I completely agree with you that Everett’s many-worlds proposal is a realist theory. In my paper, I considered the conditions (R1) through (R4) for their potential relevance to Bell’s theorem, not for categorizing interpretations of quantum mechanics as realist ones or others. Also, I do not imply that “realism” is a good name for any of (R1) through (R4); I formulate these conditions as possible interpretations of what people might mean when mentioning realism as an assumption of Bell’s proof.
On the question whether the many-worlds picture is local or non-local I have commented in Brit.J.Phil.Sci. 62:1 (arxiv:0903.2211), Section 5. Maybe we agree more than you think.
To Richard at 1737:
ad 1. Mutual counterfactual dependence is one possibility, but not the only one. Mutual stochastic dependence is another. Think of the flashes in GRWf, for example; they occur in a fundamentally stochastic way with a non-local joint distribution.
ad 2. and 3. I am happy to change my statement to “Bob’s outcome was already fixed on some spacelike hypersurface before his experiment.” Would that take care of your concern?
To Bob at 1776:
Thank you for pointing to your article. I think that your definition of Einstein locality does not capture Einstein’s, or Bell’s, or my idea of locality. I am happy with your formulation “Objective properties of isolated individual systems do not change when something is done to another non-interacting system,” but only if “property” is understood in the ordinary English meaning of the word as something that a system either has or does not have, not if “property” means projection operator. That is because locality, as I understand it and as I believe Einstein and Bell understood it, refers to reality, to “the real factual situation” (Einstein), to the variables that do have values.
Also, I disagree with your objection to my Carol-Donald example. The fact that Carol knows the quantum state of each of the particles she prepared (and can prove this by predicting the outcome with certainty for the corresponding direction in 3-space) shows that there is a matter of fact in nature about the quantum state of each of these particles. I guess our disagreement is related to the previous point: You need to think in terms of the real factual situation, not in terms of projection operators.