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Roderich Tumulka comments on some passages of Josef Jauch's book "Foundations of Quantum Mechanics"
This book contains a chapter titled "Hidden variables", and some things said there should be commented. Jauch writes (p. 111),
In section 7-1 we show with a thought-experiment (due to Einstein) that it is in general not possible to assign definite values to all the observable quantities of a physical system. This leaves open the question whether there might not exist in principle unobservable quantities (hidden variables) which would account for the probabilistic feature of the observed states, due to our ignorance of these variables. After a short preparatory section (7-2) we give a precise definition of hidden variables in section 7-3 and prove the theorem which shows that the existence of hidden variables of this kind is in contradiction with the empirical facts.
Jauch's book appeared in 1968, that's 16 years after Bohm's hidden variables theory. It thus appears a strange enterprise to me that Jauch tries to prove a theorem that excludes the existence of hidden variables theories satisfying a certain "precise definition"; the mere existence of Bohmian mechanics implies, of course, that Jauch's "precise definition" cannot be sensible. Let me remark also that it is not the main point about hidden variables theories to make quantum mechanics deterministic, as Jauch stresses ("which would account for the probabilistic feature of the observed states, due to our ignorance of these variables"), but rather to resolve the inconsistencies arising, among others, from the measurement problem.
Second, note that when saying "This leaves open the question...", Jauch gives the impression that it is unknown whether or not hidden variables theories exist; why doesn't he clearly say that hidden variables theories do exist, and refer to Bohm?
Third, Jauch indicates the additional variables would be "in principle unobservable quantities" because he knows every physicist would be skeptical about something "in principle unobservable". But, of course, the additional variables in Bohm's theory, namely the particle positions, are everything but unobservable! You simply put a photographic plate in the electron's way, and the dot you get is where the Bohmian trajectory hit the plate.
Jauch's no-hidden-variables proof
Now let's take a look at Jauch's proof against hidden variables. He holds that in a hidden variables theory, the experimental results come about deterministically (which is true in Bohmian mechanics) and therefore, knowledge of the hidden variables would allow us to predict the measurement result of any "observable" (which is not true in Bohmian mechanics, because two different experimental set-ups that would "measure" the same "observable" in orthodox theory might yield different results when fed with the same Bohmian state - wave function and particle positions - of the "measurement object"). Thus, according to a hidden variables theory, repeating the same experiment with the same initial state of the object will yield the same result again and again (which is true in Bohmian mechanics only if it's the same experiment), so, in hidden variables theories, states are dispersion-free in the sense that they assign a unique value to every observable (which is not true in Bohmian mechanics). But, in the chapter before, Jauch had defined that "state" means a mapping that assigns a number between zero and one to every (closed) subspace of Hilbert space, and that adds when we add orthogonal subspaces. It is Gleason's theorem that says these mappings correspond to density matrices. Now Jauch proves that, based on his definition of state, there are no dispersion-free states. That is, he only proves that there is no density matrix that will give weight either zero or one to every subspace. In formulas: what he proves is that you cannot find a trace-one operator W such that for every projection P, tr(WP) is either zero or one.
Let me put that more clearly: a hidden variables theory claims that the standard notion of "state" is wrong. It says the states are not what is usually called a state in quantum mechanics. Bohmian mechanics, e.g., says the state of a quantum system is the pair (wave function, particle positions). Standard quantum mechanics says a state is a density matrix. Now if some theory assumes all measurement results are predetermined by the initial state of the object, Jauch simply uses the double-assignment of the word "state" to secretly switch meaning: he claims he refutes the statement "all measurement results are predetermined by the initial state of the object", but his proof instead refutes the statement "all measurement results are predetermined by the initial density matrix of the system".
Jauch does not notice his fallacy. He knows there are counterexamples to his no-hidden-variables proof, but he does not understand why. He says (p. 120):
The merit of the considerations in section 7-3 lies in the fact that they show that at least one class of hidden variables, the obvious and most natural one, can thus be empirically refuted.
One more point on which I don't agree: positions are certainly the most obvious and natural sort of hidden variables.
Hidden or observable?
Another few phrases I'd like to comment on (p. 117):
What are the physical properties of systems which admit hidden variables?
Two types of answers are possible. Either there are no observable physical properties which follow from the existence of hidden variables, or there are such observable properties. In the first case, the hidden variables have no physical significance, and their use for the description of physical systems is not a question of physics and would therefore not concern us here. In the second case, one may ask whether these physical consequences are in agreement with experience or not. If they are not in agreement, then the existence of hidden variables is empirically refuted.
Does Bohmian mechanics have "observable physical properties which follow from the existence of hidden variables"? Depends on what that means. Let us first simply consider the immediate meaning: what would change in Bohmian mechanics if we simply removed the "hidden variables", i.e. all the variables beyond the wave function, i.e. the particle paths? Well, if the particles vanish, everything that's composed of particles vanishes: e.g. chairs, buildings and humans. Then space-time is empty. In particular, measurements don't have results. So we conclude that in Bohmian mechanics, there are observable physical properties which follow from the existence of hidden variables. And these consequences, e.g. the existence of buildings, are in agreement with experience.
But this is not what Jauch meant. What Jauch has in mind are the additional variables Einstein talks of, and that these are "in principle unobservable quantities". I don't know how he comes to this opinion, as this is hardly what Einstein says, but apparently that's what he has in mind. And indeed, it would be a strange idea to introduce in principle unobservable quantities. However, in Bohmian mechanics, particle positions are absolutely observable. So again we conclude that in Bohmian mechanics, there are observable physical properties which follow from the existence of hidden variables. And these consequences, e.g. the dots on the photographic plate, are in agreement with experience.
There is still another way how to read Jauch's question about the "physical properties of systems which admit hidden variables", namely: for any given hidden variables theory (e.g. Bohmian mechanics), is there an experimental test against standard quantum theory? Before I answer this question for Bohmian mechanics, let me say something about the status or the role of Bohmian mechanics, as I see it.
In the 18th century, the common view about the differential calculus was that it is about infinitely small quantities. But this leads to difficulties, to inconsistencies, and after all, it is simply unclear whether infinitely small quantities exist at all. The solution was the epsilon-delta technique which explains why algorithms like "to find the maximum of a function, compute its derivative according to derivation rules, and equate it to zero" work. This explanation (the espilon-delta one) is completely clear and consistent, but it breaks with beliefs like the existence of infinitely small quantities. For practical work, you only need to know the algorithms and differentiation rules, and for memorizing these rules, the idea of infinitely small quantities might be helpful. But if you want to know what this really means and why all this really works, you have to apply epsilon and delta.
The analogy is: calculus algorithms and derivation rules - quantum formalism; story of infinitesimals - orthodox quantum theory; rigorous epsilon-delta definitions - Bohmian mechanics. In quantum mechanics, there is a formalism that tells you how to compute the probabilities of the possible results of an experiment from the spectral decomposition of a self-adjoint operator and a density matrix. But as soon as you believe that density matrices are just what is out there in the world, and that experiments measure an "observable", you run into difficulties and inconsistencies like the measurement problem. It is Bohmian mechanics that explains the formalism in a consistent and clear way. It is Bohmian mechanics you have to apply when you want to know what the quantum formalism really means and why it really works. That's how I see the status and the role of Bohmian mechanics. It would be a strange question whether one can test the epsilon-delta theory against the infinitely-small-quantities theory. In much the same way, it is a strange question whether one can test Bohmian mechanics against orthodox QM in experiment, as Bohmian mechanics is what comes out if you try to make QM consistent.
Nevertheless, I will try and answer the question whether Bohmian mechanics and orthodox quantum mechanics make different predictions. First, the usual formalism (due to von Neumann) can be derived from Bohmian mechanics. But note that this formalism does not really answer all the experimental questions, because it assumes that a "measurement" is done at a certain point in time. Some measurements, however, are not done at a certain point in time. Think of a time-of-arrival measurement: you wait until the detector clicks, and then look at your stopwatch. The statistics of such an experiment cannot be derived from von Neumann's formalism (because the formalism only considers measurements that are done at some freely chosen point in time). Of course, orthodox quantum theory knows how to deal with the statistics of time-of-arrival measurements: positive-operator-valued (POV) measures on the line replace projection-valued (PV) measures on the line. This can also be deduced from Bohmian mechanics, so again the predicted statistics coincide with those of orthodox quantum mechanics. But to be fair, note that whereas the POVs were deduced in Bohmian mechanics, they are based on a new postulate in orthodox QM. There are other cases where we wait for the particle to hit a detector rather than measure its position at a certain time: scattering experiments. Thus, Born's algorithm for computing the scattering cross section (which agrees perfectly with experiment) cannot be deduced from von Neumann's formalism. It can be deduced from Bohmian mechanics, but in orthodox quantum mechanics, it is another postulate based only on heuristic grounds. Both theories agree again, but this is an unfair competition. One more point should be made: in some sense, the formalism does indeed determine the probability distribution for every experiment, even those when we don't choose a certain time when to "measure". We might solve the Schrödinger equation for the "measurement" apparatus, and not invoke the collapse postulate until reading off the result from the apparatus at some very late time. (It can be shown that this procedure gives the same probabilities at least for von-Neumann-type measurements; of course, this procedure must give the true probabilities in every case, or Schrödinger's equation is refuted.) Since it is just the same Schrödinger equation that one has to solve in Bohmian mechanics, and since the Bohmian particle positions are $|\psi|^2$ distributed, this procedure gives the same probabilities as Bohmian mechanics. In this sense, Bohmian mechanics and orthodox QM always make the same empirical predictions.
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