Saturday, October 17, 2020

David Bohm’s Pilot Wave Interpretation of Quantum Mechanics

Today I want to take on a topic many of you requested, repeatedly. That is David Bohm’s approach to Quantum Mechanics, also known as the Pilot Wave Interpretation, or sometimes just Bohmian Mechanics. In this video, I want to tell you what Bohmian mechanics is, how it works, and what’s good and bad about it.

Ahead, I want to tell you a little about David Bohm himself, because I think the historical context is relevant to understand today’s situation with Bohmian Mechanics. David Bohm was born in 1917 in Pennsylvania, in the Eastern United States. His early work in physics was in the areas we would now call plasma physics and nuclear physics. In 1951, he published a textbook about quantum mechanics. In the course of writing it, he became dissatisfied with the then prevailing standard interpretation of quantum mechanics.

The standard interpretation at the time was that pioneered by the Copenhagen group – notably Bohr, Heisenberg, and Schrödinger – and is today usually referred to as the Copenhagen Interpretation. It works as follows. In quantum mechanics, everything is described by a wave-function, usually denoted Psi. Psi is a function of time. One can calculate how it changes in time with a differential equation known as the Schrödinger equation. When one makes a measurement, one calculates probabilities for the measurement outcomes from the wave-function. The equation by help of which one calculates these probabilities is known as Born’s Rule. I explained in an earlier video how this works.

The peculiar thing about the Copenhagen Interpretation is now that it does not tell you what happens before you make a measurement. If you have a particle described by a wave-function that says the particle is in two places at once, then the Copenhagen Interpretation merely says, at the moment you measure the particle it’s either here or there, with a certain probability that follows from the wave-function. But how the particle transitioned from being in two places at once to suddenly being in only one place, the Copenhagen Interpretation does not tell you. Those who advocate this interpretation would say that’s a question you are not supposed to ask because, by definition, what happens before the measurement is not measureable.

Bohm was not the only one dismayed that the Copenhagen people would answer a question by saying you’re not supposed to ask it. Albert Einstein didn’t like it either. If you remember, Einstein famously said “God does not throw dice”, by which he meant he does not believe that the probabilistic nature of quantum mechanics is fundamental. In contrast to what is often claimed, Einstein did not think quantum mechanics was wrong. He just thought it is probabilistic the same way classical physics is probabilistic, namely, that our inability to predict the outcome of a measurement in quantum mechanics comes from our lack of information. Einstein thought, in a nutshell, there must be some more information, some information that is missing in quantum mechanics, which is why it appears random.

This missing information in quantum mechanics is usually called “hidden variables”. If you knew the hidden variables, you could predict the outcome of a measurement. But the variables are “hidden”, so you can only calculate the probability of getting a particular outcome.

Back to Bohm. In 1952, he published two papers in which he laid out his idea for how to make sense of quantum mechanics. According to Bohm, the wave-function in quantum mechanics is not what we actually observe. Instead, what we observe are particles, which are guided by the wave-function. One can arrive at this interpretation in a few lines of calculation. I will not go through this in detail because it’s probably not so interesting for most of you. Let me just say you take the wave-function apart into an absolute value and a phase, insert it into the Schrödinger equation, and then separate the resulting equation into its real and imaginary part. That’s pretty much it.

The result is that in Bohmian mechanics the Schrödinger equation falls apart into two equations. One describes the conservation of probability and determines what the guiding field does. The other determines the position of the particle, and it depends on the guiding field. This second equation is usually called the “guiding equation.” So this is how Bohmian mechanics works. You have particles, and they are guided by a field which in return depends on the particle.

To use Bohm’s theory, you then need one further assumption, one that tells what the probability is for the particle to be at a certain place in the guiding field. This adds another equation, usually called the “quantum equilibrium hypothesis”. It is basically equivalent to Born’s rule and says that the probability for finding the particle in a particular place in the guiding field is given by the absolute square of the wave-function at that place. Taken together, these equations – the conservation of probability, the guiding equation, and the quantum equilibrium hypothesis – give the exact same predictions as quantum mechanics. The important difference is that in Bohmian mechanics, the particle is really always in only one place, which is not the case in quantum mechanics.

As they say, a picture speaks a thousand words, so let me just show you how this looks like for the double slit experiment. These thin black curves you see here are the possible ways that the particle could go from the double slit to the screen where it is measured by following the guiding field. Just which way the particle goes is determined by the place it started from. The randomness in the observed outcome is simply due to not knowing exactly where the particle came from.

What is it good for? The great thing about Bohmian mechanics is that it explains what happens in a quantum measurement. Bohmian mechanics says that the reason we can only make probabilistic predictions in quantum mechanics is just that we did not exactly know where the particle initially was. If we measure it, we find out where it is. Nothing mysterious about this. Bohm’s theory, therefore, says that probabilities in quantum mechanics are of the same type as in classical mechanics. The reason we can only predict probabilities for outcomes is because we are missing information. Bohmian mechanics is a hidden variables theory, and the hidden variables are the positions of those particles.

So, that’s the big benefit of Bohmian mechanics. I should add that while Bohm was working on his papers, it was brought to his attention that a very similar idea had previously been put forward in 1927 by De Broglie. This is why, in the literature, the theory is often more accurately referred to as “De Broglie Bohm”. But de Broglie’s proposal did, at the time, not attract much attention. So how did physicists react to Bohm’s proposal in fifty-two. Not very kindly. Niels Bohr called it “very foolish”. Leon Rosenfeld called it “very ingenious, but basically wrong”. Oppenheimer put it down as “juvenile deviationism”. And Einstein, too, was not convinced. He called it “a physical fairy-tale for children” and “not very hopeful.”

Why the criticism? One of the big disadvantages of Bohmian mechanics, that Einstein in particular disliked, is that it is even more non-local than quantum mechanics already is. That’s because the guiding field depends on all the particles you want to measure. This means, if you have a system of entangled particles, then the guiding equation says the velocity of one particle depends on the velocity of the other particles, regardless of how far away they are from each other.

That’s a problem because we know that quantum mechanics is strictly speaking only an approximation. The correct theory is really a more complicated version of quantum mechanics, known as quantum field theory. Quantum field theory is the type of theory that we use for the standard model of particle physics. It’s what people at CERN use to make predictions for their experiments. And in quantum field theory, locality and the speed of light limit, are super-important. They are built very deeply into the math.

The problem is now that since Bohmian mechanics is not local, it has turned out to be very difficult to make a quantum field theory out of it. Some have made attempts, but currently there is simply no Pilot Wave alternative for the Standard Model of Particle Physics. And for many physicists, me included, this is a game stopper. It means the Bohmian approach cannot reproduce the achievements of the Copenhagen Interpretation.

Bohmian mechanics has another odd feature that seems to have perplexed Albert Einstein and John Bell in particular. It’s that, depending on the exact initial position of the particle, the guiding field tells the particle to go either one way or another. But the guiding field has a lot of valleys where particles could be going. So what happens with the empty valleys if you make a measurement? In principle, these empty valleys continue to exist. David Deutsch has claimed this means “pilot-wave theories are parallel-universes theories in a state of chronic denial.”

Bohm himself, interestingly enough, seems to have changed his attitude towards his own theory. He originally thought it would in some cases give predictions different from quantum mechanics. I only learned this recently from a Biography of Bohm written by David Peat. Peat writes

“Bohm told Einstein… his only hope was that conventional quantum theory would not apply to very rapid processes. Experiments done in a rapid succession would, he hoped, show divergences from the conventional theory and give clues as to what lies at a deeper level.”

However, Bohm had pretty much the whole community against him. After a particularly hefty criticism by Heisenberg, Bohm changed course and claimed that his theory made the same predictions as quantum mechanics. But it did not help. After this, they just complained that the theory did not make new predictions. And in the end, they just ignored him.

So is Bohmian mechanics in the end just a way of making you feel better about the predictions of quantum mechanics? Depends on whether or not you think the “quantum equilibrium hypothesis” is always fulfilled. If it is always fulfilled, the two theories give the same predictions. But if the equilibrium is actually a state the system must first settle in, as the name certainly suggests, then there might be cases when this assumption is not fulfilled. And then, Bohmian mechanics is really a different theory. Physicists still debate today whether such deviations from quantum equilibrium can happen, and whether we can therefore find out that Bohm was right."" This video was sponsored by Brilliant which is a website that offers interactive courses on a large variety of topics in science and mathematics. I always try to show you some of the key equations, but if you really want to understand how to use them, then Brilliant is a great starting point. For this video, for example, I would recommend their courses on differential equations, linear algebra, and quantum objects. To support this channel and learn more about Brilliant, go to and sign up for free. The first 200 subscribers using this link will get 20 percent off the annual premium subscription.

You can join the chats on this week’s topic using the Converseful app in the bottom right corner:


  1. Dear Sabine, please consider this a first request for you to consider AlgKoopman in such a way as this. Perhaps enough people might repeat that request that you might, but the larger reason is that its mathematics somewhat furthers the superdeterminism project, so you could make your own request.
    So far, nobody has said that it's “very foolish”, “very ingenious, but basically wrong”, “juvenile deviationism”, “a physical fairy-tale for children”, nor “not very hopeful.” On the other hand, Annals of Physics has only just selected it as a highlighted article, so nobody as serious as the people who said such scathing things about Bohm's work have yet weighed in. From correspondence, however, it seems that some physicists are starting to think carefully about what aspects of AlgKoopman they will use in their own work (not all aspects, but some!) Mathematically, AlgKoopman's motivation comes largely from a field theoretic background so that there is a moderately clear path to Lorentz invariance already in the literature in Physica Scripta 2019, and AlgKoopman more-or-less reconciles Bohr and Einstein by showing that it is natural to extend classical mechanics to include “hidden” measurements that derive from the Poisson bracket structure to make CM equivalent to quantum mechanics, instead of the usual idea that “hidden” variables must be added to QM. In contrast to deBB approaches, which have usually been associated with the Hamilton-Jacobi formalism for CM, AlgKoopman is, unsurprisingly, associated with Koopman's Hilbert space formalism for CM, which is a mathematically more natural starting point for characterizing differences between CM and QM.
    It seems to be the case that only some fraction even of physicists understand how AlgKoopman changes the game, but I'm very curious who amongst well-known physicists will make the first attempt to show how that is or, in demolishing it, will clarify aspects I have not the cleverness to understand, which I hope will be constructive.

    1. I have a friend who, on seeing the above, promptly said "how is algkoopman not a physical fairy tale for children peter !!", so please consider that particular entry half-removed from the list above. He's going through the rest on Messenger as I write this now, so AlgKoopman is well roasted.

  2. It is not clear to me that quantum field theory (QFT) is more fundamental than plain vanilla QM. Quantum mechanics is represented in space, usually with |r⟩ or ⟨r| so the state |ψ⟩ or ⟨ψ| is contracted with this so ψ(r) = ⟨r|ψ⟩. This is not difficult in nonrelativistic QM, but with spacetime things become complicated. In QFT the wave function is determined by the action of a field operator on a Fock space of states. The Wightman conditions that field operators on a spatial surface commute, or locality, is then imposed. The merging of spacetime with QM in this special relativistic setting always struck me as having bolts and seams visibly apparent.

    Of course, we know this cannot be fundamental with gravitation. A system falling into a black hole can be observed close to the horizon, say be a near horizon observer, and its quantum states appearing outside as Hawking radiation. The locality of a field in some point of space is not generally true. QM with its full nonlocality of fields and waves is rearing its head.

    The Bohm QM when applied to the Klein-Gordon equation has for the Hamilton-Jacobi equation a solution that is faster than light. There is a correction term to the classical relativistic mechanics, related to the quantum potential, that gives this funny result. This appears to hint as the more general nonlocality of Bohm QM, but it is troubling if you want to do much with this. Interacting field theory is impossible to work. There is the guy at Rutgers, Sheldon, who insists on Bohm and some years back announced with great fanfare they were able to do what Feynman did with QED in 1949.

    1. Lawrence, can you point me toward any papers fleshing out these views?

    2. “QM with its full nonlocality of fields and waves is rearing its head.”

      “The Bohm QM when applied to the Klein-Gordon equation has for the Hamilton-Jacobi equation a solution that is faster than light.”

      With the mathematical formalism of both Copenhagen QM and the de Broglie-Bohm version of QM implying superluminal connectivity (but not information transfer), perhaps this is telling us something profound about the inner workings of these descriptions of nature at the very smallest scales. The early Universe went from a diameter of one nanometer (10^-9 m) at 10^-36 seconds after the initiation of the Big Bang to 10.6 light years at between 10^-33 or 10^-32 seconds after the Big Bang due to the hypothesized inflaton field. Perhaps, (though I personally don’t have a clue), this capacity for space to expand at phenomenal rates is somehow connected to the non-locality aspect of QM, and its variants, that is built into their math. One thought that comes to mind are extra-dimensional theories like Randall-Sundrum 1 and 2, where it might be imagined that things like instantaneous (or perhaps near instantaneous?) spin correlation of photons measured, for example, 143 kilometers apart in the Canary Islands by Anton Zeilinger’s group is happening through hypothesized extra dimensions of space.

      Granted there is no evidence of extra dimensions in any physics experiment, either high energy at CERN, or low energy table top experiments such as those conducted by Eric Adelberger’s group at the University of Washington. But as a long-time UFO buff I’ve often pondered about reports where these (alleged) physical objects will shrink down to a point and just blink out like the image on an old CRT television when the set is turned off.

    3. @ Tam Hunt: The following below by Susskind make this argument. The first of these is Black Hole Complementarity vs Locality, and the second is his old Holography article of 1994. The simple argument I give though is the physics in a nutshell. Quantum black holes challenge ideas of quantum field locality.

    4. @ David Schroeder: The Bohm QM can be cast into a path integral. I did this over 10 years ago and never really did anything with it. The real part of the Schrodinger equation gives a Hamilton-Jacobi equation with a so called quantum potential. Now one can take a symplectic transformation or cananical change of variables. This apparent faster than light part changes as well. Then what I did was to take a summation over all possible symplectic transformations and with some algebra I derived a form of path integral. The theory is based on unitary-symplectic Lie algebras, or USp(n), so the integration measure is mod diffoemorphisms based on Usp(n). This also means this faster than light part really plays no actual physics.

      Bohm's QM is just QM in a different form. Bohm himself came around to admitting this. To do things with it though is very clumsy. I think it is useful for quantum chaos due to its close affiliation with standard classical mechanics.

  3. Dear Sabine,
    As you know there have been many improvements in Bohm's original work e.g. by Hailey and more recently by Sutherland. In your opinion have they been able to overcome problems with original Bohm model?

    1. I have written this transcript after consultation with several people who work in the field and to my best knowledge it summarizes the current state of research. Ie, the answer to your question is no.

    2. Who are those “people?”

    3. Thanks. I agree. I never liked the idea of a separate guided wave any way. This leads to conflict with relativity which it seems that Hailey or Sutherland have not resolved. Also I understand one can criticize Bohmian mechanics from both angles. If it agrees with expts. then why go through all this tortuous non linear mathematics when simple linear Quantum mechanics works so well. If it does not then it should be discarded anyway. It seems that it is not just interpretation which one can tolerate as a philosophical interpretation. It actually modifies QM. Do you agree?

    4. Dear Kashyap,

      I seem to have missed the nonlinearity you talk about. Where do you find a nonlinearity in the maths for the Bohmian mechanics?

      Sabine has not used the word nonlinear (or non linear) or chaos, etc. Note her summary:

      >> "Let me just say you take the wave-function apart into an absolute value and a phase, insert it into the Schrödinger equation, and then separate the resulting equation into its real and imaginary part. That’s pretty much it."

      This procedure would be unable to generate nonlinear differential equations if the original Schrodinger equation is linear---the way it actually is, in the mainstream QM as also in the Bohmian mechanics.


    5. Dear Ajit,
      Non linearity of Bohmian mechanics is rather well known. Just google! e.g.

      Bohmian Mechanics (Stanford Encyclopedia of Philosophy) › entries › qm-bohm
      Oct 26, 2001 — David Bohm (1952) rediscovered de Broglie's pilot-wave theory in 1952. Thus our deterministic Bohmian model yields the usual quantum predictions for the ... equation is somewhat complicated, and highly nonlinear.
      Kashyap Vasavada

  4. > The correct theory is really a more complicated version of quantum mechanics, known as quantum field theory. [...] The problem is now that since Bohmian mechanics is not local, it has turned out to be very difficult to make a quantum field theory out of it.

    I've always suspected that this is because QFT and Copenhagen are simply kicking the can down the road, and that John Bell was right to state that non-locality is the unsolved problem of integrating QM and GR.

  5. Sabine,

    Maybe you can help me with this question. The argument against Bohmian mechanics is that it is non-local, and QFT requires locality. But didn't Bell prove that the universe is non-local (for most physicists at least; I realize you have an alternative explanation for his results)?

    1. Scott,

      First, you cannot use a mathematical theorem to prove how the universe is. What Bell proved is that theories of a certain type obey an inequality. Experiment shows that this inequality is violated. It follows that one of the assumptions of Bell's theorem must be violated.

      A violation of one of these assumptions is qua definition what people in quantum foundations call "non-locality". It is an extremely misleading use of the word and has nothing to do with that particle physicists call "non-locality" which refers to non-local interactions.

      These two different types of non-locality have caused so much confusion I really think we should stop referring to quantum mechanics as "non-local". Some have suggested to instead use the term "non-separable" which makes much more sense indeed.

      In any case, Bohmian mechanics violates Bell's inequality and is thus non-local in Bell's sense. This is fine and not the problem I was talking about. The problem is that the ontology of Bohmian mechanics is non-local in the QFT sense (as I explained in the video). This is not necessarily a problem, but certainly one of the reasons why it's been hard to make a QFT out of it. The other problem is Lorenz-invariance (which I refer to as the "speed of light limit).

    2. well...there’s a topic for a future video....

    3. This is a great future topic. I thought about commenting on this here, but work, time limits and so forth have gotten in the way. It is the standard narrative that quantum nonlocality and field locality are completely independent. There are though I think subtle connections and QM and QFT nonlocality and locality have a connection. For most QFT and particle physics work it is just not important. Bring gravitation into the picture and it becomes very important and is one reason quantum gravitation, at least as canonical quantized and even string theory, has not worked well.

  6. Dr. Hossenfelder,

    I have always been partial to de Broglie - Bohm so thank you very much for this presentation. I have in the past mentioned my work history, so needless for me to say there were two things that stuck with me more than the rest of the information, "Don't ask questions" and "Hidden Variables." It seems to me that physics is all about asking questions and if things are hidden from us how can we really expect to correctly and accurately move forward? And, if we do not know or understand what is "hidden" how can we ask good questions? Doesn't this alone tell us that we still have so much more to learn and maybe we should be slowing down a bit so we can find answers to the questions we can't ask, and find some of the things that are hidden.

    And, you touched on a topic that I have been researching for a few weeks now because I have just a couple of basic questions that I cannot find an answer to, the double slit experiment. Say that I have a source that fires 1000 particles one at a time at the double slits. Do all 1000 particles pass through one of the slits to the back screen, or are some of the particles blocked by the space between the slits or on either side of the slits? If some are blocked is there general percentage of blockage that is constant even with changes in speed and/or wavelength?

    Maybe this question will lead to a hidden variable? Please forgive me for this one, I could not pass it up.

    1. As far as I can tell, Bohr never actually said "don't ask questions". What he said was more like that “there can be no question of any unambiguous interpretation of the symbols of quantum mechanics other than that embodied in the well-known rules which allow to predict the results to be obtained by a given experimental arrangement.”

  7. The issue I have with Bohmian mechanics is that when you're familiar with Everett's relative state formulation, it feels like Bohmian mechanics just adds a bit of formalism that doesn't change the model's explanatory power (this is of course historically backwards, and Bohm should get more credit for paving the way towards the MWI). Even worse, it posits the existence of a multiverse of p-zombies.

    The Bohmian model is basically a universal pachinko machine - there's a universal wavefunction in the universal Hilbert space, and in that space there's also a single dot representing all of the particles in the world, its path determined by the wavefunction and the starting position. The wavefunction evolves deterministically with no influence from the dot. One may then conceptualize a complete description of it in spacetime, which contains information on every path the dot may take for every possible starting position, including the complete information on every observer in each of those timelines.

    One must ask then - what do you need that dot for? The one-sided causality between it and the wavefunction means the former can't be more physically real than the latter. Why should we say that we live in a universe controlled like a marionette by a vastly larger object, rather than in that object?

    1. Bravo! The best description of the relation between Bohm and Everett theories I've seen. And it also emphasises that the "quantum equilibrium hypothesis" that Sabine mentions is beside the point: if it fails, it means the ball started rolling at a relatively unlikely point, but quantum mechanics correctly describes the subsequent probability flow, which is really all that matters.

    2. The claim that BM = MWI + the location of an initial dot in Configuration Space is a Superdeterministic misinterpretation of Bohmian Mechanics. The reason it's a strawman argument is because BM's Quantum Equilibrium Hypothesis does not require spacetime to strictly adhere to the Born Rule from the moment of its inception. BM instead derives the Born Rule as a consequence of the Quantum Equilibrium Hypothesis that emerges over time, regardless of whether the universe began in that state or not. BM's non-axiomatic derivation of the Born Rule is one of the subtle ways it departs from standard QM, while nevertheless maintaining consistency with its predictive results.

      As for Superdeterminism itself, would it not require a virtually unlimited degree of numerical precision specify the exact location of that initial dot in Configuration Space? It's hard to believe all subsequent adaptive phase-locking phenomena exhibited by the unfathomable variety of feedback loops that lie at the core of all manner of emergent systems in the universe were predetermined by that sole initial condition. I think "super" falls far short of the godlike degree of determinism inherent in that conception.

    3. Lish,

      "As for Superdeterminism itself, would it not require a virtually unlimited degree of numerical precision specify the exact location of that initial dot in Configuration Space"

      No, it does not, as we have demonstrated here.

  8. Bohm's theory is convinient for quantum cosmology, since it avoids the problem of the system and the observer which are necessary in the Copenhagen interpretation so that the Copenhagen interpretation cannot be applied to the whole universe.

    As far as the Bohm´s formulation of QFT, the difficulty is that one has to work in the Schrodinger representation, so that one has to use the functionals of fields, and the mathematics of field functionals is not sufficiently developed in order to do the calculations like in QFT.

    For me, the problem with the Bohm's theory is that particles in a stationary bound state do not move, which does not sit well with the intuitive idea that particles should always move (think of an electron orbiting a proton).

  9. Bohmian mechanics - a personal journey.
    Initially fascinated by BM, I soon concluded that it's plain wrong. That is, if individual particles have definite positions, then individual runs of an experiment must conserve energy-momentum (e-m). Now, individual particles alone clearly don't do that, so a conserved quantity must involve also the wave. This is the situation in other systems involving (point) particles interacting with fields, e.g. classical electrodynamics (CED) and GR (at least this is what I thought...). But in those two examples, there exists a symbiotic relation between the particle and the field: the particle sources the field while the field guides the particle, whereas in BM there is only the latter, master-slave relation.

    So I set out to complement the guiding equation with a `sourcing equation' so a Lagrangian can be written-->conservation laws. This innocuous looking task turned out a lot harder than I expected, and seemed to work only in a fully relativistic theory (the result is the "central ECD system" in

    There was, however, a price to be paid for this completion of BM. Now, that each particle comes with its own wave, there is no longer a single `master wave' encoding the statistical results of an experiment (even for a single particle; with multiple particles the situation was worse). But then, a single ensemble system is now a particle-wave pair, and there doesn't seem to be any natural measure on such a complex set, so why not QM I reasoned.

    It took me a while to grasp it, but lying in front of me was CED of interacting charges - free of its infamous self-force problem, which has never been properly solved before (see confession of Jackson 3rd ed. p.745). And when I asked: what is the simplest statistical description of such a consistent CED, the answer was straightforwardly QM (, just as anticipated.

    I have since been exploring that grand-child of Bohm which I named Extended Charge Dynamics, and recently showed that the missing matter problem may very well be just another, hitherto ignored facet of CED

    In that long journey, done mostly outside academia, I had engaged with no less than four communities, which seemed rather oblivious to each other. And it was unbelievably frustrating to get them to take a more panoramic view of physics, reveling, to my judgment, the root cause for the long stagnation in the foundations of physics: a wrong turn at the beginning of the twentieth century with regard to the self-force problem of CED.

  10. Regardless of the validity of Bohm’s interpretation of QM, it does provide a connection to modern nonlinear dynamics, aka, chaos theory. As you point out, the actual trajectory followed by the electron in the double slit experiment is sensitively dependent on its initial position prior to passing through the slit(s). It’s an interesting way to think about what happens.

  11. The Wheeler-DeWitt equation

    G_{ijkl}δ^2Ψ[g]/δg_{ij}δg_{kl} + R^(3)Ψ[g]= 0

    for the wave functional in a polar form Ψ[g] = Ae^{-S[g]} will result in a minisuperspace version of the quantum potential. It has never been clear to me this does much for understanding quantum gravity. Maybe with my idea of building a path integral something could emerge. However, the path integral I derived was no different from standard path integrals anyway.

  12. Isn't there also a rather more philosophical problem with Bohm's theory? Its guiding wave affects particles but is not itself affected by particles. Action without reaction is not something the rest of physics allows.

    1. The presumption that physical particles should exert an effect on the BM guiding wave is a prevalent misconception that underlies the mistaken claim that BM requires faster-than-light communication between entangled particles. If that were so, it would imply that the BM guiding wave is itself a relativistic field that propagates through physical space (e.g. like electromagnetic waves). The reason none of these effects occur is because the guiding wave is manifested by the evolution of the quantum wavefunction in non-relativistic Configuration Space (i.e. where the wavefunction itself is defined). The physical particles we observe are projections of quantum events from complex-valued, multi-dimensional Configuration Space into real-valued, physical 3D space. Since there is no projection of physical particle properties back into Configuration Space, particle interactions have no effect on the pilot waves that guide them.

  13. The Base for the Ultimate Theory of Everything (TOE) is the entangled multiverse:
    Cramer's TI is the same as CP ( Charge Parity) symmetric Multiverse entanglement. (clocks are running only backwards over there)
    TI is: Transactional Interpretation of John Cramer.
    Libet's measurement results (RPI and RPII) are the measurement proof example of TI

  14. Bohm's theory is convinient for quantum cosmology, since it avoids the problem of the system and the observer which are necessary in the Copenhagen interpretation. Hence the Copenhagen interpretation cannot be applied to the whole universe while the Bohm interpretation can be applied.

    As far as the Bohm´s formulation of QFT, the difficulty is that one has to work in the Schrodinger representation, so that one has to use the functionals of fields, and the mathematics of field functionals is not sufficiently developed in order to do the calculations like in QFT.

    For me, the problem with the Bohm's theory is that particles in a stationary bound state do not move, which does not sit well with the intuitive idea that particles should always move (think of an electron orbiting a proton).

  15. I’m surprised that you claim that in QM a particle can be in two places at once. You say:

    * “particle described by a wave-function that says the particle is in two places at once”

    * “in Bohmian mechanics, the particle is really always in only one place, which is not the case in quantum mechanics”

    But QM doesn’t say that at all. All QM says is that the wave-function can be non-zero in many places. QM never says that the wavefunction *is* the particle, nor that the particle exists in multiple places. That would be an interpretation, not part of QM itself.

    1. I am using the standard way of verbalizing the concept of a wave-function. If you think you can do it better, you are very welcome to set up your own YouTube channel.

    2. Hmm, I think it’s more standard to talk of the wave function as a “calculational tool” or “the state, from which we can calculate probabilities.”

      Anyway, I didn’t mean to reproach. Given your interest in foundational issues (such as your Superdeterminism paper, which I liked), I just thought you’d phrase it differently, is all.

      I do have a video channel, but think I cannot link (spam filter?). See the site physicsisnotweird (dot com): the first three videos are on foundational questions, future dependence, and the block universe. (I made them before papers by you and Wharton, so the terminology differs.)

  16. Phase -

    What if the particle actually always IS there, but, either, it's polarity fluctuates, making the equipment either not attuned or not able to measure it ; or it is 'out of phase' and simply not observable without learning how to calibrate instruments to detect wave functions in different phases?(In Sci-Fi, 'phase shifting', lol. But, obviously, there are real examples of this work being done.)

    Then, let's assume all particles are in this constant state of polar reversals, perhaps randomly and affectually, as dominoes, or perhaps according to some as yet undiscovered cosmological governance. A mass body moving through the field would presumably be charged - or +, and guided by the interactive forces with the particles. In this case it would follow that the polarity shifting is indeed governed by a mathematical principle, and a geometrically symmetrical one, as would be evidenced by observing inertia in a vacuum. Further to that I would boldly predict this symmetry would be 'perfection', symmetrical on an infinite number of planes, maybe a sphere.

    Also, I've been thinking, since the recent news of an additional state of matter being identified. Why do we make these, what are imho, assumptive classifications in the beginning? I have to start working with the assumption that there is only matter, it's 'state' is irrelevant. The factors used in equations, i.e. electron speed, atomic field size, melt point, etc., are particle specific and labelling their 'state' is meaningless, except as an English word. We should look at matter as being in a single state, a manifestation of these, particle properties. So, instead of solid, liquid, gas, you have high speed, 'free of force effect particles' on one end of the spectrum, and motionless,'fully bound by force' on the other. ... Idk, but, why would heat weaken atomic binding force? Where's the math for that? Why do they speed up, is heat a barrage of wave functions? .... If heat is concentrated wave function, then, perhaps, this is similarly the case with measuring instruments, but rather than the heat-activity quotient, it is a wave-polarization relationship that affects the particle being observed.

    ".. the loud sound it seemed to fade, came back like a slow voice on a wave of phase." - Starman, David Bowie

    (...the radio faded out, and we heard another voice. It came on a different wave of phase) ... Bowie credited dreams for telling him what to write. The song also says he is sitting there, but doesn't say hi. It doesn't specify, but I assume he can't be seen cz he is out of phase. d:o}

    Peace, physics nerds. Y'all can thank me later. .....with money;)

  17. One thing I found interesting reading your article about Bohmian mechanics is a connection I noticed to what another physicist said regarding point particles (unfortunately I can't remember where I read it or who said it). He wrote that he regarded point particles as "a disturbance in the field" rather than being continual existing points of energy. That seemed to fit nicely with Bohm's guiding field explanation.

  18. Originally the idea of a pilot wave was raised by de Broglie. It was discussed and refuted at the Solveig conference in 1927. John Bell has investigated that discussion and says (in his book “Speakable and Unspeakable in QM”) that this decision was not the result of physical arguments but the result of the personality of Heisenberg; who used the occasion to slam de Broglie.

    It is surprising that de Broglie and Bohm obviously did not investigate the internal structure of elementary particles and particularly did not look for a *cause* of the pilot wave. At Bohm’s time there was already some knowledge about it which he could have used. When de Broglie predicted the interference of electrons he must have assumed an internal oscillation in the particle. Later Dirac and Schrödinger found that this internal oscillation is going on at the speed of light (which was earlier already assumed by Lorentz). David Hestenes later took it as the origin of the spin.
    The electron, to take this example, has internal charges, which on the one hand means the electrical charge which we know of the electron. And in addition charges which keep the internal structure bound. As these charges orbit, they cause an alternating external field which in motion causes waves. So the particle itself is the cause of the surrounding field and the position of the particle in the field is always well defined. No requirement here for non-locality.

    This field causes an interference pattern at the double slit which guides the particle precisely to the detector. The individual path of the particle in a single case is then not only defined by its initial position but also by its actual phase when it arrives at the slits.

    Such particle model explains in addition a lot of properties like mass (very precisely), constancy of spin, frequency to energy relation, and Pauli principle, which are stated but not explained by Copenhagen QM. So, to be able to explain these phenomena (besides others) is a great benefit of this approach following the pilot wave idea, in comparison to the Copenhagen QM.

    The problem of entanglement, however, is not solved by such model. But it is as well not solved by the Copenhagen QM, where it is merely stated to exist.

    1. That's the way to go. For the entanglement it seems to be nececcary to find out also external structure, not only internal - how internal and external together conserves antipodes over spatiality. Temporal variation is ruled by speed constant c.

      Of course the next conclusion is related to the definitions of naming selectable things: parity; where come left and right, how are ruled opposite directions and opposite charges. To me it revealed that the larger structure have to be the key - in coherent quanta group only antipodes make difference.

  19. Sabine:
    "Bohm had pretty much the whole community against him. ... And in the end, they just ignored him."

    Max Planck, Werner Heisenberg, Niels Bohr, Louis de Broglie, Paul Dirac, and Erwin Schrödinger were all members of the Pontifical Academy of Sciences, the scientific academy of the Vatican City. Latest member is physics Nobel prize winner Reinhard Genzel for his studies on black holes. Co-winner Roger Penrose has been a guest of the Specola Vaticana as featured speaker on a workshop on Singularities and Black Holes, held to celebrate the work of Georges Lemaître, the inventor of the Big Bang.

    The reason for Bohm's ex-communication most likely can be found in Vatican City.

    1. @Gerd: you should suggest this to Dan Brown for the plot of his next novel ;) Maybe you can even get some royalties.

    2. @opamanfred:
      “Science could not exist without faith! Science needs faith that with it we can find truth, and that this truth is worth finding” commented Guy Consolmagno, Director of the Vatican Observatory, on the award of the Nobel Prize in Physics 2020 (vaticannews).

      Fact is that science needs no faith at all.

      If you now think of conspiracy theories ("Dan Brown") you should read something about the German "Reichskonkordat", negiotiated soon after the Enabling Act of 1933 between the Vatican and the German Nazis, which is still in force today.

    3. For those searching for a sociological motive for the rejection of Bohm theory, a more productive avenue is to note Bohm's early communist leanings, which eventually forced him to leave his home country after being hounded by the McCarthyites. Many of his cold-warrior contemporaries saw Bohm theory as guided by Marxist materialism, hence for instance Heisenberg's description of the theory as "ideological". I don't know what Schrödinger's reaction was, but Schrödinger was neither a cold warrior nor a part of either the Copenhagen group or the Copenhagen consensus as Sabine suggests in the article. On the other hand, Schrödinger's attempts to reform/re-interpret QM were in a different direction from Bohm's.

  20. We humans are our own entangled reflections from far away, in the mirror symmetric Multiverse. The Base for the Ultimate Theory of Everything (TOE) is the reflective entangled multiverse supported by: John Cramer's TI is the same as CP (Charge Parity) symmetric multiverse entanglement.

  21. The Theory of Everything needs a Multiverse ! even Bohm single photon dual slit experiment.
    We humans, even photons are entangled reflections in the mirror multiverse.!
    The Base for the Ultimate Theory of Everything (TOE) is the entangled multiverse:
    1: Cramer's TI is the same as CP ( Charge Parity) symmetric Multiverse entanglement. (clocks are running only backwards over there)
    TI is: Transactional Interpretation of John Cramer.
    2: Libet's measurement results (RPI and RPII) are the measurement proof example of TI.
    3: Bohmian Double Slit Interpretation by Dual Entangled Universes, and the Benjamin Libet experiment.

  22. so Bohmian mechanics is like what you do to Maxwell's equations when you derive the physical optics formulation (before you start applying approximations to it) ?

  23. Thank you Dr.Sabine,
    I was one of the requesters of this topic.

  24. One question about this sentence:
    "Bohmian mechanics says that the reason we can only make probabilistic predictions in quantum mechanics is just that we did not exactly know where the particle initially was."

    Is this "exactly" here still subject to the indetermination principle? In other words, was Bohm assuming that also the indetermination threshold could be bypassed until enough precision aobout initial condition could be reached?
    Or, viceversa, he assumed that it is possible to make a non probabilistic prediction based on more accurate initial conditions but still in compliance with Heisenberg principle?

    1. Heisenberg's Uncertainty Principle says that we cannot determine the exact position of the particle. But in Bohmian mechanics, the particle actually has an exact position which it is not possible to determine, and the fact you can't determine it is the source of the probabilities inherent in quantum mechanics.

  25. How would particles be "guided by a field which in return depends on the particle"? And how could someone go about proving Bohm's interpretation?

    Isn't Bohm claiming that quantum mechanics is more or less same as Newtonian mechanics? Is so, why does the water wave analogy fail? If not, what's different?

    What if instead our notions of 'wave' and 'particle' are fundamentally wrong?

    1. wrote: "What if instead our notions of 'wave' and 'particle' are fundamentally wrong?"

      Right. We haven't arrived at a clear picture of what "quantum objects" are. De Broglie's beautiful idea that they have both wave and particle properties has not really come to fruition in Bohmian mechanics. What I especially dislike about the dBB theory is that interference patterns formed by electrons and photons actually require separate explanations! (Photons do not have a position operator.) It's like a twentieth century apartment fitted with rococo furniture.

    "Isn't Bohm claiming that quantum mechanics is more or less same as Newtonian mechanics? Is so, why does the water wave analogy fail? If not, what's different?"

    Water waves require a medium which would correspond to the disproved luminiferous aether.
    For Newtonian mechanics/quantum mechanics rather think of a mobile gun (speed v) driving on a circular path which fires an infinite number of bullets ("particles" with speed c) at the same time in all directions ("wave"). How fast (v/c) must the gun drive to hit at least one bullet from the side?

    1. Lost in math? Problems like that can be easily solved using Euler's complex e-function, where you can arbitrarily mix Cartesian coordinates with cylindrical coordinates (see my last post in Quantum mechanics #5). If you define t=0 as the time when be bullets are fired, you can immediately write down the position of the bullets as
      b(t)=R + ivt + cte^(iφ) for all angles φ, where one angle represents one individual bullet, and the position of the gun as
      g(t)=R*e^(iωt), ω=v/R,
      which are vector equations.
      The results are solutions to b(t)=g(t).

      It is no secret to solve equations like that. The biggest mystery however is, why such simple basic equations do not appear in quantum mechanics.

    2. Gerd Termathe: I get the impression you have missed the whole point of the "Lost In Math" title.

      The issue isn't that ANY of the math is difficult.

      The issue is that physicists are seeking "beauty" in their math and trying to drive the science by aesthetic considerations that have no scientific justification.

      Then you show you too are "Lost in Math": You say, ironically, "The biggest mystery however is, why such simple basic equations do not appear in quantum mechanics."

      Why is that a mystery? By what scientific justification should quantum mechanics equations be simple? Simplicity is just an aesthetic quality, when scientifically all that matters is whether the equations capture the observed behavior, in the wild or in controlled experiments.

      It seems to me you've missed the whole point.

  27. Dr. Hossenfelder,

    I believe one of the reasons Bohm privileged non-relativistic QM in his approach is because it is easy to provide an account of the measurement process in terms of NR-QM, which is what he did in his 1952 paper. I was wondering if you (or anyone else who reads this blog) could refer me to a work that describes the measurement process in detail using quantum fields?

    Thank you

  28. It occurred to me that Bohm's is not an interpretation but a theory. And same goes for "Many Worlds". Both of these postulate a new element: "pilot wave" and "other worlds" respectively. In contrast, the Copenhagen interpretation just talks about the waveform collapse which is not a new element. True?

    On another note, thanks for the feedback, especially on the wave-particle view.

    1. Don't worry about the mythical collapse of the wave function. One could call it a spurious feature added by the Copenhagen interpretation -- the quantum formalism works just fine without it. We use wave funtions (or traces of operators, in the Heisenberg picture) to compute probabilities and expectation values. That's it. Wave function collapse occurs only in the minds of people who erroneously think of "the wave function" as representing an "individual" quantum systm.

    2. If the wave-function cannot represent individual quantum systems, then quantum mechanics is not a fundamental theory. I have explained this earlier.

    3. Sabine,

      I've been following your blog for about a year now, but I still fail to understand why statistical theories seem to be second-rate to you. Do you believe that the universe "is" deterministic? Do you think that only a theory that (in principle) can predict the exact decay time of a neutron deserves to be called fundamental? That's a tall order! Quantum theory is not likely to be replaced by something more "fundamental" soon.

    4. I have no idea what makes you think I consider statistical theories "second-rate". I say they are not fundamental. That's a technical term that says nothing about how relevant I consider such theories to be, which in any case I think complete besides the point.

      No, I do not think that a fundamental theory needs to be deterministic. Again, I do not know what makes you think so, as I am positively sure I have never voiced such an opinion, not here and not elsewhere.

    5. I'm still puzzled by your statement "If the wave function cannot represent individual quantum systems, then quantum mechanics is not a fundamental theory." It may well take me another year to understand your (technical) definition of the term "fundamental". But if your verdict is based on the "measurement problem", I'd like to caution that such an assessment should not rest on a flawed interpretation or misunderstanding,

      Hoping for an imminent solution of the measurement problem and wishing you all the best,


    6. Werner,

      I am not "interpreting" anything. You do. I am merely looking at the math. The only way quantum mechanics can be internally consistent is if it is not fundamental.

    7. David Bohm was monist and didn`t see wave and particle as separate things.Both the Copenhagen Interpretation and Many World theory have free will as axiom and are not scientific.

    8. Since you disagree so strongly with each other, and I am still not clear (others too?), is this question worthy of its own blog post?

    9. I have written several blogposts and papers about it and if that didn't make it clear I don't think repeating it will.

    10. Of course there's no point in making the exact same blog posts again. But there's still progress to be made. The discussion must go on, and it should get around the stagnation point. The measurement problem isn't solved yet!

      In the case of photons there is no wave function after the polarization has been measured. The photon is gone. What is relevant, also for electrons, is the wave function before the measurement. The wave function after the measurement (whether collapsed or not, or non-existent) is irrelevant.
      And isn't it highly questionable that measurements are supposed to happen in an instant? (In less than a Planck time?) In the Heisenberg picture measurement can be treated ss a process, and the evolution considered as a whole. If wave function collapse is real, where is its place in the Heisenberg picture? (Do I need to remind you that in the Heisenberg picture state vectors remain constant?)

      Yes, look at the math, and find out if wave function collapse features in actual calculations.

    11. Without the measurement update (= wavefunction collapse), quantum mechanics does not describe observations correctly. There is a reason this axiom is in the theory. It is beyond me that we even need to discuss this.

    12. Trapped in axioms ...
      Obviously you have deeply absorbed the quantum formalism, or should I say quantum dogmatics? I've always been more interested in the physics, rather than formalism. None of my quantum theory textbooks mentions axioms. Can you direct me to the official, "canonical" set?

    13. Needless to say, there are infinitely many logically equivalent sets of axioms for quantum mechanics. Any one of those will do.

    14. Werner: Axiomatic just means it is self-evidently true. Most axioms are implicit. Is mass quantifiable? We think that is self-evidently true, but we don't list it as an axiom of physics.

      I haven't read your textbooks, but I'd speculate they probably don't mention axioms because they rely on repeatable experiments to build up the quantum model, grounded in undisputed truths (axioms) about how the real world works. If those are mentioned, it is as undisputed facts of nature, mathematics and logic.

      If you want to understand the axioms, study the foundations of math with set theory, and the foundations of calculus, which physics takes as given.

    15. Dr. A.M. Castaldo wrote: "Axiomatic just means it is self-evidently true."

      Thanks. I've heard about Euclid's and Peano's achievements. Axiomatization can be useful in mature fields with clear-cut concepts. But in quantum theory it was fuelled by the hope of making a vague term like "measurement" precise by embedding it in a set of axioms. I can see now why Sabine cannot call quantum theory fundamental if it contains an undefinable primitive concept like measurement. Axiomatization has made Euclidian geometry a gem. It may have added rigidity to quantum theory, but not much clarity. The result is not like a crystal, but more like a coprolith. (Please pardon the expression, Prof. Edwards.)

      I laughed out loud about your innocent use of the word "self-evident". What is considered self-evident has changed over the times. ("The earth is flat", "The sun revolves around the earth".) For Maxwell it was self-evident that there is an aether. How could light waves propagate without a medium? Had he chosen to axiomatize electrodynamics, we might still be talking of stresses in the aether instead of electric fields.

      Bohr's insistence that quantum theory must be formulated using classical concepts ("particles", "fields") has stifled progress for almost a century. We have forgone the search for concepts that are more appropriate to the theory. The measurement problem is built into the axioms that Sabine has chosen to put her faith in. She will probably be pondering the measurement problem for the rest of her life.

    16. Werner: It was not naive, it was simplistic, for somebody asking simplistic questions.

      My apologies for failing to detect your sarcasm, you sounded like a petulant high school student.

      It was never self-evident that there was an aether; as you indicate, that was a conclusion; akin to belief in God because "what else could it be?" All your other examples are the same; jumping to conclusions on the basis of "What else could it be?"

      Those are not self-evident, they are fallacious logic.

    17. Quantum interpretations are the imposition of axioms onto QM. They are in a sense analogous to how alternative axioms to Euclid's 5th axiom of geometry give rise to various geometric models. Some interpretations appear to be falsifiable. The results at San Grasso indicated gravitation plays no role in wave function collapse. [ ] The constraints are pretty tight and it seems to put the ideas of Penrose and Pullin et al in doubt. I was at one time in favor of these ideas. However, it appears these things may just be plain wrong. I have in this case found how it is an experiment can really change how you think about things.

    18. Lawrence Crowell wrote: "an experiment can really change how you think about things"

      That's how it should be. After all, physics is an empirical science. But wave function collapse is different. There aren't even theoretical arguments that it must happen. The "collapse postulate" is put forward by people who confuse state preparation and measurement. (In state preparation you just create a subensemble to get a pure state.) It's a useless axiom hat plays no role in actual calculations. MWI and Bohmian mechanics can do without it, and do so without discernible consequences.
      The Heisenberg picture provides the strongest argument that wave function collapse is fictitious. Sabine may have a blind spot here, but you must surely be aware that the Heisenberg and Schrödinger pictures are equivalent.

      Of course there's a strong psychological force to believe in wave function collapse. Many seem to think that, according to quantum theory, an electron "is" a wave function. Because the electron is not destroyed, there must be a new wave function (so the reasoning goes) after the measurement. But this new wave function isn't used in calculations. The wave function is a mere tool, a piece of mathematical apparatus that must not be identified with an individual particle.

    19. You are offering an epistemic perspective, which is no more verifiable than an ontological perspective. That there is decoherence and the loss of quantum phase, say that quantum phase is taken up by a needle state or environment, is not that controversial. This leads to a set of diagonal elements in the density matrix that are probabilities. The off diagonal terms corresponding to superpositions or entanglements is lost to some large reservior of states. What this does not tell us is how the configuration of the system is realized by one of the states with a probablity. It is reduced to a pure classical case of "reading the dice," but where we have no theory to tell us how that happens.

    20. > "an epistemic perspective [...] is no more verifiable than an ontological perspective"

      All physical theories must of course be verifiable, i.e. agree with observations. But the ontological perspective is in fact a mistaken, misleading misconception of quantum theory that obscures its statistical character and has caused endless confusion, as these discussions amply demonstrate. (In the Heisenberg picture one never deals with a single state vector, but always takes traces of operators, i.e. explicitly considers ensembles.)

      > "a pure classical case of "reading the dice" [...] where we have no theory to tell us how that happens"

      There's the rub. You seem to think that a theory of "how that happens" ought to exist. I don't. Life is short, and I wouldn't want to waste my time searching for a fundamental theory that in actual applications would produce the statistical rules we already have. The real problem with quantum theory is that we have no clear picture of what actually happens. We have foisted the classical idea of "particles" on the microworld and are puzzled about the weird nature of "quantum objects". Indeed, I think the very notion of quantum "object" is mistaken.

  29. Sabine wrote: "I am merely looking at the math."

    Of course there's a clash between unitary evolution and the collapse postulate. We disagree about the significance of the collapse postulate.

  30. @Dr. A.M. Castaldo (6:13 AM, October 28, 2020)
    "I get the impression you have missed the whole point of the "Lost In Math" title."

    The title here is David Bohm’s Pilot Wave Interpretation of Quantum Mechanics.

    "... scientifically all that matters is whether the equations capture the observed behavior, in the wild or in controlled experiments."

    So let's see, if the equations match the observed behavior.

    Starting from to the gun/bullet example we replace the gun by and an electron and the bullets by the field of the electron emitted at t=0. In addition we add another electron on the opposite side for symmetry, the nucleus in the center, and arrive at the known stable Helium configuration.

    Circular motion requires that the net force acting on each electron must point to the center of rotation, where acting means the point in time when the field arrives.

    We have R+ivt = R*e^(iα)*sec(α) with α=arctan(vt/R)=arctan(ωt) for the center of the field and R*e^(iωt) for the position of the electron.

    This gives for the possible angles the discrete solutions ωt=arctan(ωt), starting with 4.493409..., 7.725252..., 10.904122..., and so on.

    Based on these Eigenvalues you can do the rest of the calculations, which finally result in the observed discrete frequencies and also in Bohm's wave-guided particles.

    1. For each of these Eigenvalues we get two solutions for v/c
      v*t/c*t = v/c = v*t/(|sec(vt)| +- 1)

      For the first one (4.493409) we get
      v/c = 1.2470... and 0.8019... , one being the inverse of the other.

      The next ones are each closer to 1, resp.

  31. Sabine wrote: "...depending on the exact initial position of the particle, the guiding field tells the particle to go either one way or another. But the guiding field has a lot of valleys where particles could be going. So what happens with the empty valleys if you make a measurement? In principle, these empty valleys continue to exist. David Deutsch has claimed this means “pilot-wave theories are parallel-universes theories in a state of chronic denial.”

    These "empty valleys" are superposed solutions of the quantum wavefunction that do not correspond to any observed particle. According to parallel-universe theories, these solutions represent multiple mutually-inaccessible worlds in which corresponding particles were in fact found. Bohmian Mechanics instead resolves this multiplicity of worlds by claiming that only one of these solutions manifests observable effects in the real world - the one in which the particle is actually found.

    So what happens to the "empty valleys"? All such solutions are produced by the evolution of the quantum wavefunction in complex-valued, multi-dimensional Configuration Space, rather than in real-valued relativistic physical space. According to Bohmian Mechanics, guiding waves are not fields that propagate through physical space (i.e. like electromagnetic waves), they are confined to Configuration Space. What does manifest in physical space are the observable particles whose motion is guided by Bohmian Pilot Waves. It is only the waves that correspond to actual particles that make observable effects on the physical world.

    This is also how BM explains why the multiple worlds of MWI do not actually exist - because no such multiplicity of corresponding particles has ever been observed to pop into existence, and without a particle to guide, the "empty" Pilot Waves have no means of manifesting any effects on the physical world.



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