Defense of de Broglie-Bohm pilot wave theory

A nice summary of typical arguments against de Broglie-Bohm theory can be found at R. F. Streater's "Lost Causes in Theoretical Physics" website. Ulrich Mohrhoff also combines the presentation of his position with an interesting rejection of pilot wave theory. The quotes from these pages haven been taken in Jan. 2008. Then, there have been two nice polemical articles in Lubos Motl's "reference frame", which I consider in a separate file. Up to now, these have been the most interesting pages I have found. If you know interesting pages critical of dBB, Nelsonian stochastics, or non-local hidden variable theories in general, please tell me about them.

Pilot wave theory violates Einstein causality

In the first place, the theory predicts effects that move faster than the speed of light, and this is due to the quantum potential, which is non-local. (Streater)

That's correct. But these effects are hidden, can be seen only indirectly, via such effects as violations of Bell's inequalities. Weak relativistic symmetry – and this is all we can test with experiments – is not destroyed. And the discussion about the violations of Bell's inequalities proves that this is a necessary feature of any realistic theory.

What is observable in pilot wave theory?

It seems quite natural to start with the following question:

If positions and velocities (and thus momenta) have precise values, then why can we not know their precisely values? (Mohrhoff)

with the seemingly natural answer that they should be observables in pilot wave theory. But that's false. Not everything, what really exists, is observable to observers. What is observable, and what remains hidden, is a question which has to be answered by the theory itself – based on the assumptions of the theory about what is real. Let's continue:

At one time Einstein insisted that theories ought to be formulated without reference to unobservable quantities. When Heisenberg later mentioned to Einstein that this maxim had guided him in his discovery of the uncertainty principle, Einstein replied something to this effect: "Even if I once said so, it is nonsense (Quatsch)." His point was that before one has a theory, one cannot know what is observable and what is not. Our situation, however, is different. We have a theory, and this tells in no uncertain terms what is observable and what is not. (Mohrhoff)

Indeed, pilot wave theory tells in no uncertain terms what is observable and what is not, as we will see. Unfortunately, the answer is not that simple, as the author seems to think. Trajectories are real. It doesn't follow that they are observable. To mingle these two notions is an error from the point of view of a realist – it is a positivistic idea. What is observable to Platon's prisoners in the cave are shadows on the wall. What is observable for us, if we live in a Bohmian world, we have to find out yet.

Let's see what we can tell, in a simple way, about measurement results in pilot wave theory. They exist. Thus, they have to be functions of real objects, which, in pilot wave theory, are the positions and the wave function. We have more or less direct access to them. This restricts them to positions. And they have to exist now. Past positions no longer exist, except as memories in our present brains, future ones do not yet exist, but the results of real experiments really exist, and exist now.

This should not be mingled with another property of the theory – that we are able to compute past positions using Bohmian equations. These trajectories are in the part of the theory which tells us what exists. Some God may be in a position to see past and future as well, together with the wavefunction of the universe, and, therefore, to compare predictions of pilot wave theory with what he sees. But we, humans, see at best what exists now. What we can do, is to use pilot wave theory to predict what we can observe now, and to compare these predictions with our observations. This, possibly, allows us to falsify some particular guesses about the trajectories, or, even more, falsify pilot wave theory as a whole.

Because this plays an important role in the following considerations, let's memorize this: The results of observations have to be functions on the configuration space Q at some fixed moment of time t.

Energy, momentum and so on ...

The definition of the measurement results does not impose any restrictions on the measurement of the current configuration q ∈ Q. But what about the other observables, like energy, momentum and so on? In pilot wave theory, these measurements can be described as well. The interesting point is that the results of these "measurements" depend not only on the state of the "measured object", but also on the state of the "measuring device". Thus, "measurement" is, in some sense, the wrong word, "result of interaction" would be more appropriate. The name used to describe such "measurements" which are, really, interactions, is "contextual".

You might think that Bohm's story allows you to retain the classical fiction that each property of a physical system exists regardless of whether it is measured. As a matter of fact, it doesn't. On this theory, energy and momentum and spin and every particle property other than position are contextual, ... (Mohrhoff)

That's correct. But I see no problem here. Is there anything "nonclassical", not in agreement with common sense, that results of interactions can be observed? And that these results depend on the states of both parts of the interaction? Let's see:

There are, in fact, theorems in the literature to the effect that any cryptodeterministic construal of quantum mechanics will invariably have to treat certain observables as contextual. This means that the value of an unmeasured observable in general depends on which observables are measured at that time. (Mohrhoff)

This sounds, at a first reading, like something strange, noneclassical. But is it? For a contextual "observable" in pilot wave theory, the result (an interaction with the "measurement device"), depends on the particular choice of this device and, moreover, even on its actual state. Another device – another result. That's all. What makes it sounding strange is only the implicit assumption that the result of the "measurement" depends only on the state of the "measured object". But this is only the result of inappropriate naming, nothing more.

Let's note, that the scientists, using this naming conventions, have not been fools. They had good reasons for naming these interactions measurements. Namely, the probability distribution of the "measurement results" depends only on the wave function of the "observed object", as they should, if the naming convention would be ok. Only at the hidden variable level we can see the difference, the dependence on the state of the "measurement device", which makes it more appropriate to name all this an interaction.

This effect is also not strange at all. Very different measurements can lead to the same probability distribution in very classical situations. Compare, for example, distributions of the value of a fair dice x itself with the distributions of 7-x.

Importance of including measurement devices

At one point, Streater openly contradicts Bohm's equivalence results:

Secondly, the theory without contextuality cannot predict the same results as quantum mechanics, contrary to the claims made by Bohm. (Streater)

So what, once in pilot wave theory non-position measurements are contextual? It seems, that Streater thinks that pilot wave theory measurements are not contextual. They are. What is the origin of his failure? The following quote shows an important and interesting misunderstanding of the measurement concept in pilot wave theory:

It is argued that all observables ultimately come down to position measurements of the particle. (Streater)

The argumentation is quite different. Observables in pilot wave theory don't reduce to position measurements of the particle. They can be reduced to positions of the particle together with its environment. This includes, if necessary, even macroscopic measurement devices. This is quite clear from our consideration above of what is observable. The function on Q, which describes the results, is certainly not restricted to the configuration space of the particle alone.

This is important, because it allows to describe all QM measurements as measurements in pilot wave theory. To see this, we have to remember, that, in the Copenhagen interpretation, measurement results are states of macroscopic devices. We also have to use the Copenhagen interpretation to make another point: Once the result is macroscopic, the prediction no longer depends on the way it is stored at the macroscopic level. With this information, we can restrict ourself to the subclass of macroscopic QM observations, where different measurement results are stored using different macroscopic configurations. Then, the macroscopic result can be described as a function f(Q) on the macroscopic part of the configuration space of pilot wave theory.

If we restrict ourself to position measurements of a single particle alone, this argumentation fails, and we have, obviously, no way to describe the other QM observables like momentum, energy and so on.

Measurements at different times

If one thinks that measurement is restricted to position of a single particle, the natural way to measure momentum is to measure velocity, isn't it? All we need are two position measurements for the same particle. To consider this problem, let's look in more detail at measurements at different times. Streater's argument continues as follows:

It is argued that all observables ultimately come down to position measurements of the particle. It is not clear, however, whether the supporters of the theory expect all the position measurements needed to measure say momentum can be done at the same time. (Streater)

Really not clear? So let's clarify: In pilot wave theory, all measurement results have to exist at the same time t. Roughly, we can say that this means that each measurement has some well-defined moment of time t. But what about quantum measurements done at different times? Can we compare their predictions with predictions of pilot wave theory? The answer is simple: It doesn't matter much when a quantum measurement is done. The only restriction is that all measurement results should exist at the final time. This is sufficient to compare pilot wave theory and QM predictions. Fortunately, this is always possible, and we obtain an equivalence of pilot wave theory and QM predictions.

Again, we have to include macroscopic measurement devices into the considerations. But once this is done, it is no problem at all to store the results. We simply include classical storage devices into the configuration. Thus, for any QM set of measurements, done at any set of measurement times ti<tmax, we can easily modify the macroscopic configuration in such a way that the measurement results are a function f(Q) at time tmax. This can be done without changing any QM predictions.

After this modification, we already have a situation which is covered by Bohm's equivalence theorem. This theorem is not questioned by Streater:

He [Bohm] shows that for any time t, his theory predicts the same distribution for the position X(t) of a particle at time t as does the quantum theory of Schrodinger, and also any function of X(t) is also distributed as in quantum theory. (Streater)

Thus, once we can easily modify any quantum measurement so that it fits into this particular form, we obtain complete equivalence of QM and pilot wave theory predictions.

Again, the inclusion of the macroscopic part, in this case of macroscopic storage for earlier measurements, is essential for the conclusion. Note that this depends in an essential way on the internal consistency of quantum theory itself: The predictions do not change if we include the measurement device, even a macroscopic one, into the quantum part, and leave the classical measurement to a final moment.

Now, the pilot wave theory-QM-equivalence is proven, even for QM measurements at different times. Nonetheless, let's illustrate what happens. To describe a measurement of positions of one particle q at two different times t0, t1, we need at least two positions of something at the final time t1. We can use q(t1) itself as one value. The other value should be a different one, say, q'(t1). And q' has somehow interacted with q at the time of the first measurement t0. But there was also some unknown initial value q'(t0-ε) before this first measurement. And, whatever the interaction between the two particles, it is impossible to get rid completely of the dependence of the two final results q(t1),q'(t1) on the initial value q'(t0-ε).

In quantum theory, the positions of the particle at different times do not commute, whereas Bohm's theory predicts their joint distribution. This is an indication that there are observables whose distribution is not correctly given by Bohm's theory. Indeed there are. This easy fact is denied by most of the authors in Duerr et al. ...

Bohmists take this as the definition of reality: [Duerr et al, bottom of p 35]. For them, energy, momentum and spin are not elements of reality. However, the claim that position is an element of reality (in Einstein's sense) at all times, true of Bohmian mechanics, is not true in quantum mechanics, contrary to their belief. In QM, positions at different times do not commute, so there is no representation of them by random variables on a common sample space: some correlations referring to positions at different times, fail to satisfy Bell's inequalities, which they would do if there were such a representation. (Streater)

"Denied" and "contrary to their belief"? I have not checked, but at least I don't believe that in QM position operators at different times commute. Is there a contradiction? No. What is measured if we measure, in the QM meaning of the word, positions of the same particle at different times, is something very different from the positions of the same particle at different times. At least one, the earlier, quantum measurement result is, from pilot wave point of view, only the stored value of some earlier interaction of a test particle with the "measured" particle.

Thus, Bell's inequality holds for the true positions, but not for the measured positions. No contradiction.

Strange Bohmian trajectories

As a natural consequence of these considerations, trajectories may differ from the results of "measurements of trajectories" in the sense of quantum theory. Knowing this, we are not confused reading, for example, the following:

The classical velocity of the Bohmian particle is not related to the measured values of the velocity in quantum theory. In any real ψ, the Bohmian velocity is zero, whereas the measurements (as predicted by quantum theory) is a random variable with distribution given by |φ|^2, where φ the Fourier transform of ψ. The ground state of the hydrogen atom is real for all time, and so in Bohmian physics, nothing moves. (Streater)

Ok, we already know this as a general fact. Not much need for particular illustrations. But, of course, if you like more of this, no problem:

Y. Aharonov and L. Vaidman (pp 141-154) in "About position measurements", give many examples ..., similar to those in Englert, B.-G., Scully, M. O., Sussmann, G., and Walthier, H. : Surrealistic Bohm Trajectories, Zeitsschrift fur Naturforschung, 47a, 1175-1186, 1992. Ahahonov and Vaidman admit: "We worked hard, but in vain, searching for an error in our and Englert et al arguments"[sic]. They conclude "The proponents of the Bohm theory do not see the phenomena we describe here as difficulties of the theory", and quote a riposte of the Bohmists [Duerr, D., Fusseder, W., Goldstein, S., Zanghi, N., Comment on Surrealistic Bohm Trajectories, Z. Fuer Natur, 48a, 1261-1262, 1993. (Streater)

I don't see any problem too. No question – pilot wave theory makes strange, often counterintuitive, predictions about the trajectories. It makes the same strange predictions as quantum theory, thus, it should not surprise us much if the trajectories look surrealistic. Who cares once we cannot measure them anyway?

The role of Einstein causality in Bell's theorem

One of the main points of these pages in general is to find support for a hidden preferred frame. Once Einstein causality is used in Bell's proof, we can save realism and hidden variables, if we reject Einstein causality for the hidden variables. But is this sufficient? What about proofs of Bell's inequality, which don't use Einstein causality? Here, Streater uses one such theorem without mentioning Einstein causality:

Thus, the probability theory is classical, ie obeys the axioms of Kolmogorov. By Bell's theorem, the theory predicts the Bell inequalities for certain pairs of compatible variables. (Streater)

Streater's argumentation seems to suggest that Einstein causality is not the point. To prove Bell's inequality, we don't need it. Criticising "Quantum Theory from Quantum Gravity", by Markopoulou and Smolin, Streater writes:

They try to avoid Bell's theorem by saying "The non-local hidden variables required to satisfy the conditions of Bell's theorem are the links ... in the graph". The authors seem to be trying to construct a non-local quantum dynamics. There is no need to, since relativistic quantum mechanics is, or should be, local. The authors final theory is not full quantum mechanics. It is a Nelson theory in which there is a wave function satisfying a version of Schrodinger's equation. The mod-squared wave function is the probability density that the particle is present at the time in question. However, as in Bohmian theory, the velocity is not an observable incompatible with the position, but is the average of Nelson's forward and backward velocities. (Streater)

If correct, this would mean giving up Einstein locality for the hidden variables would not save them. Fortunately, that's not the case, as we have already seen. True velocity should not be mingled with measured velocity, they are different. The argumentation is the same as for measurements at different times (no wonder, the natural way to measure velocities is to measure positions at different times). "Measuring velocities" in pilot wave theory measures something completely different, depending on the state of the measurement device. It is a "contextual measurement".

Nonetheless, what's the point of introducing Einstein causality? It allows to exclude simple contextual explanations. A theorem, which requires non-contextuality as an assumption, is something very weak. If translated into common language: "We cannot explain the results of the conference without assuming that the speaches of some participants have been influenced by some speaches of other participants." Would such a theorem surprise you? Not much (except, maybe, for a Breshnev-time communist party conference). But this is, essentially, the type of impossibility theorems, which rely on non-contextuality.

This situation changes drastically, if we use, as in Bell's theorem, Einstein causality. (If the speaches are held in different places, far away from each other, so that no radio translation of one speach could have reached any of the other speakers, the theorem we have mentioned is already surprising.) Now we no longer need any assumptions about non-contextuality. What we need, except Einstein causality, is a weak notion of realism (Kolmogorovian probability theory). The influence, which we have to exclude, is no longer the influence of the measurement instrument (which contains quantum parts) on the result of the observation, but that of the human (macroscopic) decision what to measure, which is made far away from the "measurement" itself..

As a consequence, a hidden variable theory needs contextuality as well as non-locality. Pilot wave theory has both properties. The non-locality of pilot wave theory is important to see the error in the following argument:

The famous EPR experiment, in which the momentum of a particle is 100% anti-correlated with another at a far distance, shows that by measuring the momentum of the far particle we can find that of the first. Naturally, EPR thought that this does not change the first particle in any way. So they concluded that momentum should be classed as an element of reality [in any good theory]. Bohm concocted a similar EPR pair, in which the spins are 100% anti-correlated. Thus we might conclude that Bohmists regard both the momentum of a particle, and its spin, as elements of reality. Not a bit of it! (Streater)

Have you found the error? Pilot wave theory is nonlocal, thus, not a "good theory" according to Einstein causality. (Streater himself has used this against pilot wave theory.) The guiding equation contains an explicit dependence on the position of the far away particle as well as the situation around it. So Bohmists have no reason at all to believe that the first measurement does not influence the state of the second particle. Instead, if interested, they could compute the trajectories and look, explicitly, how the far away momentum measurement influences the state of the particle here. Thus, the application of the EPR criterion of reality does not tell us anything about momentum and spin as elements of reality: It claims "If, without in any way disturbing a system, we can predict with certainty ... the value of a physical quantity, than there exists an element of physical reality corresponding to this physical quantity." But in the pilot wave world the momentum measurement of one particle does disturb the far away particle.

Impossibility theorems

Unfortunately, Streater is not alone making such claims:

This subject was assessed by the NSF of the USA as follows [Cushing, J. T., review of Bohm, D., and Hiley, B., The Undivided Universe, Foundations of Physics, 25, 507, 1995.] "...The causal interpretation [of Bohm] is inconsistent with experiments which test Bell's inequalities. Consequently...funding...a research programme in this area would be unwise". (Streater)

A really strange situation. We have a simple theory – pilot wave theory. A mathematically quite simple theory, with a simple proof that the predictions coinside with QM predictions. But some people reject it based on some general impossibility theorems. Instead of caring about the question what has happened. It seems, something has to be wrong – or the impossibility theorem, or the equivalence proof.

The solution of the puzzle, in case of the known impossibility theorems, is simple. The impossibility theorems have to make assumptions about the theories. And pilot wave theory simply fails to fulfill these assumptions. In case of Bell's inequality, this assumption is Einstein-locality. In case of Kochen-Specker and others, non-contextuality.

Once this is clarified, the impossibility theorems strenghten the case for pilot wave theory. Why? Very simple – you may not like some properties of pilot wave theory, like non-locality and contextuality. But, if there is an impossibility theorem, which proves that you cannot do better (at least without giving up other assumptions of the theorems, especially realism), this is no longer a failure, but a necessary property of the theory. Non-locality and contextuality of realistic theories are necessary properties of realistic theories, they are consequences of what we observe in the quantum world.

Do we need a wave function reduction?

The theory is not consistent in its interpretation. It is set up as a probability theory; but a measurement of the position of the particle does not result in a conditioning of the probability distribution of the position. This is at the insistence of J. S. Bell, who championed the Bohm theory when it was more or less discredited. Bell said "No-one can understand this theory until he is willing to see ψ as a real objective field rather than just a probability amplitude" ... The usual rules of probability would lead to the replacement of psi by a conditional probability, after a measurement has given us more info as to the whereabouts of the particle. (Streater)

The truth is that pilot wave theory is not set up as a probability theory. The equations, as for the wave function, as for the particle, are deterministic equations for objectively existing things. Results about the invariance of some probability distributions are derived later.

Moreover, the reduction of the predicted probabilities can be easily done without any reduction of the wave function itself. The solution is simple. The description of the measurement process has to cover the measured object as well as the measurement device. It predicts their joint probability distribution. Everybody is free to use this joint probability distribution to obtain conditional probability distributions for the observed object taken alone for various values of the measurement result.

The usual rules of probability would lead to the replacement of ψ by a conditional probability, after a measurement has given us more info as to the whereabouts of the particle. However, this conditioning is exactly the collapse of the wave packet, and this is a dirty word among Bohmists: to allow it here might weaken the case against quantum mechanics. (Streater)

Is it? There is, of course, no need to consider all the time the wave function of the whole universe. (It is this wave function of the universe which is considered to be a really existing thing.) There is no reason to object at all against a replacement of this unknown and much too complex wave function by an effective wave function. It gives an equivalent description of the part of the universe we want to study.

But this reduced, effective wave function is no longer considered as an independent element of reality. It is only an effective description of the influence of the really existing wave function of the universe, as well as the configuration of the remaining part of the universe, on our part. The mathematics of this reduction is quite simple: Put the configuration of the part of the universe we want to ignore into the wave function of the universe, and you have the conditional wave function of the part you want to consider. If this part is approximately independent from the remaining universe, then this conditional wave function follows a reduced Schrödinger equation.

In this understanding, the reduction of the wave function is related with a changing the part of the universe we want to describe, a reduction of the considered degrees of freedom. The same thing done by considering conditional probabilities.

The interpretation itself is quite consistent. Reality consists of the state q of the universe, and the wave function of the universe. The effective wave function is, instead, nothing real, but a mathematical device for computing an approximation. It depends on the arbitrary, subjective choice of the scientist, who decides, which part of the universe is interesting to him, and what he likes to ignore.

You think that such a difference of the status of the wave function of the universe and the effective wave function of some part of it is artificial and strange? Me too. There is a lot of room for improvement of pilot wave theory.

What practicing physicist would really want to bother with a hidden variable theory of the Bohmian type,...? Its defining feature appears to be no more than this: more difficult mathematics than the standard theory, while at the same time making no further predictions that can actually be tested. Hardly a bargain. (Fuchs, arXiv:quant-ph/0105039, p.240)

This is something I don't understand. Fuchs is, judging from his papers, a physicist able to work with quite complex mathematics, in particular with the Schrödinger equation. The only new mathematics in pilot wave theory – the guiding equation – is, mathematically, a much simpler equation: an evolution equation for one variable q(t), when the Schrödinger equation is a partial differential equation for a function ψ(q,t). Naming this "more difficult mathematics" sounds strange.

Of course, every more fundamental theory has to add something new. In modern physics, with its highly mathematized character, we cannot avoid to add new mathematics. What we have to look at is not if there is new mathematics (if a theoretical physicist does not like new mathematics, he has the wrong job), but if these new mathematics allow to derive some of the old mathematics, therefore explaning them. Despite adding new mathematics, the axioms of the new theory may contain less, or easier, mathematics, because the axioms of the old theory may be derived.

This is the case in pilot wave theory. One part of the mathematics remains unchanged – the Schrödinger equation. We add a much simpler equation for the configuration. But we can derive (thus, remove from the definition of the theory) all the measurement theory – a theory which requires much more complicate mathematics than the Schrödinger equation, namely the general theory of operators in Hilbert spaces. Thus, we add easy mathematics and can derive complex mathematics.

Moreover, the equation we add is not only easy mathematically, it is simple in any other understanding of the word as well: conceptually simple, in agreement with our intuitions. The thing we can derive now – quantum measurement theory – has none of these features. It is not only mathematically complex, but unintuitive and strange as well.

Of course, if one works every day with operators, one becomes used to them. The time when we have learned all this, and wondered about the strangeness of a theory which needs such strange and complex things as operators on Hilbert spaces in it's definition, have been almost forgotten. Once we get used to it, this complexity no longer counts. On the contrary, learning something new becomes more and more difficult with age. Thus, our intuitions about the mathematical simplicity are clearly distorted in favour of what we have learned in our youth, and what we are using every day. Without such a heavy bias, claims about mathematical complexity of pilot wave theories would be unexplainable.