by gill1109 » Fri Dec 18, 2020 5:24 am
This is the complete paper by Bell to which Fred referred. For reasons of internet typography I replaced lambda with mu, rho with pi, an overline with an underline, and dropped Bell's hats on bold a and bold b. This was written by John Bell some time after his paper Bell (1964) appeared, but before the Bell papers using CHSH started coming out. "His theorem" is therefore his 1964 theorem, not his later, stronger, theorems.
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The editor has asked me to reply to a paper, by G. Lochak [1], refuting a theorem of mine on hidden variables. If I understand correctly, Lochak finds that I failed somehow to allow for the effect on these variables of the measuring equipment. I will try to explain why I do not agree. The opportunity will also be taken here to comment on another refutation [2], by L. de la Peña, A. M. Cetto and T. A. Brody, and on another [3] by L. de Broglie. Yet another refutation of the same theorem, by J. Bub[4], has already been refuted by S. Freedman and E. P. Wigner [5].
Let us recall a typical context to which the theorem is relevant. A ‘pair of spin 1/2 particles’ is produced in a space-time region 3 and activates counting systems, preceded by Stern–Gerlach magnets, in space-time regions 1 and 2. The system at 1 is such that one of two counters (‘up’ or ‘down’) registers each time the experiment is done; correspondingly we label the result there by A ( = + 1 or – 1). Likewise the system at 2 is such that one of two counters registers each time the experiment is done, giving B ( = + 1 or – 1). We are interested in correlations between the counts in 1 and 2, and define a correlation function AB which is the average of the product of A and B over many repetitions of the experiment.
Now it would certainly be better to give a purely operational, technological, macroscopic, description of the equipment involved. This would avoid completely any use of words like ‘particle’ and ‘spin’, and so avoid the possibility that someone feels obliged to form a personal microscopic picture of what is going on. But it would take quite long to give such a purely technological specification. So, please accept that the words ‘particle’ and ‘spin’ are used here only as part of a conventional shorthand, to invoke without lengthy explicit description the kind of experimental equipment involved, and with no commitment whatever to any picture of what, if anything, really causes the counters to count.
Suppose that part of the specification of the equipment is by two unit vectors a and b (e.g., the directions of certain magnetic fields at 1 and 2). Then according to ordinary quantum mechanics situations exist for which
AB = –a.b (1)
to good accuracy.
Actually it is this last statement which is challenged by de Broglie. Although his paper is called ‘Sur la réfutation du théorème de Bell’, it is not in fact concerned with any reasoning of mine. He is of the opinion that the correlation function (1) simply cannot occur for macroscopic separations, either in nature or in ordinary quantum mechanics: ‘Nous échappons complétement à cette objection puisque, pour nous, les mesures du spin sur des électrons éloignés ne sont pas corrélées’. As regards ordinary quantum mechanics, de Broglie disagrees here with most students of the subject, and I am unable to follow his reasons for doing so. As regards nature, he seems to disagree also with experiment [6].
Now we investigate the hypothesis that the final state of the system, in particular A and B, would be fully determined by the equations of some theory if the initial conditions were fully specified. So to parameters like a and b, subject to experimental manipulation, we add a list of hypothetical ‘hidden’ parameters µ. We can take these µ to be the initial values (say just after the action of the source) of some corresponding dynamical variables. We have no interest in what subsequently happens to these variables except in so far as they enter into the measurement results A and B. But in so far as they do enter into A and B we allow fully for the effect of the measuring equipment by allowing A and B to depend not only on the initial values µ of the hidden parameters but also on the parameters a and b, specifying the measuring devices:
A(a, b, µ), B(a, b, µ). (2)
We have no need to enquire into the precise nature of this dependence on a and b, nor into how it comes about, whether by the effect of the measuring equipment on the hidden variables of which the µ are the initial values, or otherwise.
Can one find some functions (2) and some probability distribution π(µ) which reproduces the correlation (1)? Yes, many, but now we add the hypothesis of locality, that the setting b of a particular instrument has no effect on what happens, A, in a remote region, and likewise that a has no effect on B:
A(a, µ), B(b, µ). (3)
With these local forms, it is not possible to find functions A and B and a probability distribution π which give the correlation (1). This is the theorem. The proof will not be repeated here.
Lochak illustrates the way in which the output of a single instrument A depends on its setting a, as allowed for in (3), in the hidden parameter theory of de Broglie. I think this is very instructive. But more instructive for the present purpose is the case of two instruments and two particles. Then one finds that in de Broglie’s theory the dependence is not of the local form (3) but of the nonlocal form (2). I have made this point on several occasions, in two of the three papers referred to by Lochak and elsewhere[7]. It may be that Lochak has in mind some other extension of de Broglie’s theory, to the more-than-one-particle system, than the straightforward generalization from 3 to 3N dimensions that I considered. But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says.
The objection of de la Peña, Cetto, and Brody is based on a misinterpretation of the demonstration of the theorem. In the course of it reference is made to
A(a', µ), B(b', µ)
as well as to
A(a, µ), B(b, µ).
These authors say “Clearly, since A, A', B, B' are all evaluated for the same µ, they must refer to four measurements carried out on the same electron–positron pair. We can suppose, for instance, that A' is obtained after A, and B' after B”. But by no means. We are not at all concerned with sequences of measurements on a given particle, or of pairs of measurements on a given pair of particles. We are concerned with experiments in which for each pair the ‘spin’ of each particle is measured once only. The quantities
A(a', µ), B(b', µ)
are just the same functions
A(a, µ), B(b, µ)
with different arguments.
References
[1] G. Lochak, Fundamenta Scientiae (Universite de Strasbourg, 1975), No 38, reprinted in Epistemological Letters, p. 41, September 1975.
[2] L. de la Pena, A. M. Cetto and T. A. Brody, Nuovo Cimento Letters 5, 177 (1972).
[3] L. de Broglie, CR Acad. Sci. Paris 278, B721 (1974).
[4] J. Bub, Found. Phys. 3, 29 (1973).
[5] S. Freedman and E. Wigner, Found. Phys. 3, 457 (1973).
[6] S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938 (1972). A brief account is given by M. Paty, Epistemological Letters, p. 31, September 1975.
[7] J. S. Bell, On the Hypothesis that the Schrödinger Equation is Exact, CERN Preprint TH. 1424 (1971).