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Hidden Variables in Quantum Theory
An excerpt from Bohm's article "Hidden Variables in Quantum Theory" in D. R. Bates' 1962 book, ... in the field of physics, when it was discovered that spores and smoke particles suffer a random movement obeying certain statistical laws (the Brownian motion) it was supposed that this was due to impacts from myriads of molecules, obeying deeper individual laws.
This is just like the work of Immanuel Kant, in his
The statistical laws were then seen to be consistent with the possibility of deeper individual laws, for, as in the case of insurance statistics, the over-all behaviour of an individual Brownian particle would be determined by a very large number of essentially independent factors. Or to put the case more generally, Idea for a Universal History with a Cosmopolitan Intent, which led to the ideas of Adolph Quételet and Henry Thomas Buckle
lawlessness of individual behaviour in the context of a given statistical law is, in general, consistent with the notion of more detailed individual laws applying in a broader context.
In view of the above discussion, it seems evident that, at least on the face of the question, we ought to be free to consider the hypothesis that results of individual quantum mechanical measurements arc determined by a multitude of new kinds of factors, outside the context of what can enter into the quantum theory. These factors would be represented mathematically by a further set of variables, describing the states of new kinds of entities existing in a deeper subquantum mechanical level and obeying qualitatively new types of individual laws. Such entities and their laws would then constitute a new side of nature, a side that is, for the present, “hidden.” But then, the atoms, first postulated to explain Brownian motion and large-scale regularities, were also originally “hidden” in a similar way, and were revealed in detail only later by new kinds of experiments (e.g. Geiger counters, cloud chambers, etc.) that are sensitive to the properties of individual atoms. Similarly, one may suppose that the variables describing the subquantum mechanical entities will be revealed in detail when we will have discovered still other kinds of experiments, which may be as different from those of the current type as the latter are from experiments that are able to reveal the laws of the large-scale level (e.g., measurements of temperature, pressure, etc.). At this point, it must be stated that, as is well known, the majority of modern theoretical physicists () have come to reject any suggestion of the type described above. They do this mainly on the basis of the conclusion that the statistical laws of the quantum theory are incompatible with the possibility of deeper individual laws. In other words, while they would in general admit that some kinds of statistical laws are consistent with the assumption of further individual laws operating in a broader context, they believe that the quantum mechanics could never satisfactorily be regarded as a law of this kind. The statistical features of the quantum theory are thus regarded as representing a kind of irreducible lawlessness of individual phenomena in the quantum domain. All individual laws (e.g., classical mechanics) are then regarded as limiting cases of the probability laws of the quantum theory, approximately valid for systems involving large numbers of molecules.
We shall now consider the main arguments on which are based the conclusion that quantum mechanical indeterminism represents a kind of irreducible lawlessness.
We begin with a discussion of Heisenberg’s indeterminacy principle. He showed that even if one supposes that the physically significant variables actually existed with sharply defined values (as is demanded by classical mechanics) then we could never measure all of them simultaneously. For the interaction between the observing apparatus and what is observed always involves an exchange of one or more indivisible and uncontrollably fluctuating quanta. For example, if one tries to measure the coordinate, x, and the associated momentum, p, of a particle, then the particle is disturbed in such a way that the maximum accuracy for the simultaneous determination of both is given by the well-known relation It is well known that in such an experiment a statistical, interference pattern is still obtained, even if we pass the particles through the apparatus at intervals so far apart that each particle essentially enters in separately and independently of all the others. But, if the whole ensemble of such particles were to split into subensembles, each corresponding to the electron striking the grating at a definite value of x, then the statistical behaviour of every sub-ensemble would be represented by a state corresponding to a delta function of the point in question. As a result, a single sub-ensemble could have no interference that would represent the contributions from different parts of the grating. Because the electrons enter separately and independently no interference between sub-ensembles corresponding to different positions will be possible either. In this way we show that the notion of hidden variables is not compatible with the interference properties of matter, which are both experimentally observed and necessary consequences of the quantum theory. Von Neumann generalized the above argument and made it more precise; but he came to essentially the same result. In other words, he concluded that nothing (not even hypothetical and “metaphysical” hidden variables) can be consistently supposed to determine beforehand the results of an individual measurement in more detail than is possible according to the quantum theory.
Niels Bohr showed in his 1927 Como Lecture that the uncertainty principle is only a consequence of the limited resolving power of our instruments and not a mechanical disturbance, which greatly embarrassed Heisenberg.
The third important argument against hidden variables is closely connected with the analysis of the paradox of Einstein el al* This paradox arose out of the point of view, originally rather widespread, of regarding the indeterminacy principle as nothing more than an expression of the fact that there is a minimum unpredictable and uncontrollable disturbance in every measurement process. Einstein, Rosen, and Podolsky then suggested a hypothetical experiment, from which one could see the untenability of the above interpretation of Heisenberg’s principle.
We shall give here a simplified form of this experiment. Consider a molecule of zero total spin, consisting of two atoms of spin, ℏ/2. Let this atom be disintegrated by a method not influencing the spin of either atom. The total spin then remains zero, even while the atoms are flying apart and have ceased to interact appreciably.
Since total spin is conserved as zero while the particles separate, each of the spins individually remains a constant of the motion. These are "hidden constants," not hidden variables, but they explain the perfect anti-correlation of distant measurements.
Now, if any component of the spin of one of the atoms (say. A) is measured, then because the total spin is zero, we can immediately conclude that this component of the spin of the other atom (B) is precisely opposite. Thus, by measuring any component of the spin of the
atom, A, we can obtain this component of the spin of atom B, without interacting with atom B in any way.
If this were a classical system, no difficulties of interpretation would occur; as each component of the spin of each atom is always well-defined and always remains opposite in value to the same component of the spin of the opposite atom. Thus the two spins are correlated; and this permits us to know the spin of particle B when we measure that of A. But now, in the quantum theory, we have the additional fact that only one component of the spin can be sharply defined at one time, while the other two are then subject to random fluctuations. If we wish to interpret the fluctuations as nothing but the result of disturbances due to the measuring apparatus, we can do this for atom A, which is directly observed. But how does atom B, which interacts in no way either with atom A or with the observing apparatus, “know” in what direction it ought to allow its spin to fluctuate at random? The problem is made even more difficult if we consider that while the atoms are still in flight, we are free to reorient the observing apparatus arbitrarily, and in this way to measure the spin of atom A in some other direction. This change is somehow transmitted immediately to atom B, which responds accordingly. Thus, we are led to contradict one of the basic principles of the theory of relativity, which states that no physical influences can be propagated faster than light. The behaviour described above not only shows the untenability of the notion that the indeterminacy principle represents in essence only the effects of a disturbance due to the measuring apparatus; it also presents us with certain real difficulties, if we wish to understand the quantum mechanical behaviour of matter in terms of the notion of a deeper level of individual law operating in the context of a set of hidden variables. Normal | Teacher | Scholar |