# Bell’s Theorem: Two Neglected Solutions

## Abstract

Bell’s theorem admits several interpretations or ‘solutions’, the standard interpretation being ‘indeterminism’, a next one ‘nonlocality’. In this article two further solutions are investigated, termed here ‘superdeterminism’ and ‘supercorrelation’. The former is especially interesting for philosophical reasons, if only because it is always rejected on the basis of extra-physical arguments. The latter, supercorrelation, will be studied here by investigating model systems that can mimic it, namely spin lattices. It is shown that in these systems the Bell inequality can be violated, even if they are local according to usual definitions. Violation of the Bell inequality is retraced to violation of ‘measurement independence’. These results emphasize the importance of studying the premises of the Bell inequality in realistic systems.

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## Notes

1. 1.

According to Bell’s original [1], a HVT is local iff (1) the force fields the theory invokes are well-localized (they drop off after a certain distance, therefore (1) can be assumed even in an experiment with static settings); and (2) it does not invoke action at-a-distance, i.e. it invokes only influences that propagate at a (sub)luminal speed, in particular between the ‘left’ and ‘right’ part of the experiment. Notice this corresponds to an extremely mild locality condition: any known physical system satisfies it.

2. 2.

(B1) takes the (almost ideal) experimental results into account. Here we do not consider certain so-called loopholes linked to the fact that real Bell experiments would not be 100 % faithful tests of the theorem (these loopholes seem to become more and more unlikely).

3. 3.

This may be shown as follows. In a deterministic HVT the physical properties σ 1 and σ 2 have a value before each instance of measurement (since σ i is determined by, i.e. a function of, λ); conversely, in a realistic theory σ 1 and σ 2 have a value even before measurement and one can, implicitly or explicitly, index that value by a hidden variable or index.

4. 4.

Here ‘nonlocal’ cannot mean ‘entangled’ because entanglement is of course not proven by Bell’s theorem. In information-theoretic texts ‘nonlocal’ is often synonymous to ‘violating the BI’, but in our discussion the term is obviously not used in that way.

5. 5.

Total determinism seems to be a popular philosophy. For what it is worth, here is the result of a little survey we did, one among about 20 physicists, experts of the foundations of quantum mechanics, one among about 20 philosophy students. In both cases, about 40 % of participants said to be in favour of determinism, 60 % in favour of indeterminism. (In a third group (30 p.), after a defense of determinism, the ratio was rather inversed.) These surveys were casual and have of course no pretension to definitiveness.

6. 6.

A well-known theory on conspiracy theories [35] proposes following ground for people believing in conspiracy. In short: extraordinary effects call for extraordinary causes. In the face of events or ‘coincidences’ that are perceived as formidable, people would have a tendency to look for formidable explanations: a conspiracy by higher powers (or simply the powerful). Now, exactly this theory [35] might apply to people calling S3 a ‘conspiracy theory’ (!): they perceive S3 as too formidable to be true, and believe that only higher powers—a conspiracy—can explain what S3 proposes. In this context, see also Spinoza [33], who denounced fallacious reasoning of a quite similar type. He analyzed in particular the case of his contemporary fellows, who, in view of the perfection of the world and the quasi-infinite potential of harmonious interaction it offers, concluded that it surely must have been made for them by a higher power. But according to Spinoza’s determinism, and modern biology, nature and mankind evolved in a lawful way so as to necessarily be in some kind of harmony—no divine plan is needed. In sum: anthropocentric reasoning is widespread and often wrong.

7. 7.

It seems noteworthy that this is an argument that even Bohr might have liked: he often stressed that a quantum system or phenomenon includes the whole measurement set-up, due to the complementary nature of such arrangements.

8. 8.

The quantum treatment shows that the J ij correspond to the exchange integrals $$\int\psi^{*}_{\mathrm{ab}}.V.\psi_{\mathrm{ba}}.\mathrm{d}^{3} \mathbf{x} _{1}.\mathrm{d}^{3} \mathbf{x} _{2}$$, with V the Coulomb potential and ψ ab(x 1,x 2)=ψ a(x 1).ψ b(x 2) (where ψ a(b)(x 1(2)) are the single electron eigenfunctions located at x 1 and x 2 respectively) ([46] Chap. 7).

9. 9.

Also, it plays no role whether these λ are classical or quantum-like, a case we will investigate elsewhere [47], as explicitly mentioned by Bell (see [2]).

10. 10.

If one wants to push the analogy with the Bell experiment further, suppose that Alice measures σ 1 and σ a, and Bob σ 2 and σ b.

11. 11.

This point will be analyzed in detail elsewhere [47]. In particular it will be shown that the existing Bell experiments, even those using dynamic analyzer settings, cannot exclude that MI is violated by a local mechanism, much like in the Ising lattices.

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## Acknowledgements

I would like to thank, for many stimulating discussions, Gilles Brassard, Mario Bunge, Emmanuel M. Dissakè, Henry E. Fischer, Yvon Gauthier, Gerhard Grössing, Andrei Khrennikov, Marian Kupczynski, Jean-Pierre Marquis and Eduardo Nahmad.

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Correspondence to Louis Vervoort.

## Appendix: Determinism in the History of Philosophy; Spinoza’s System

### Appendix: Determinism in the History of Philosophy; Spinoza’s System

In particular concerning the issue of determinism discussed in Sect. 4, there is a strong interrelation between physics and philosophy. Rejecting (total) determinism (S3) amounts, in a sense, to a dramatic discontinuity in the history of western thought (which is of course not a proof of determinism). The following is a highly condensed and selective introduction to the history of this position, an introduction unfortunately but inevitably biased by personal preference.

Actually, consulting general encyclopedia of philosophy would suffice to see that determinism and free will belong to the most hotly debated topics of philosophy. Virtually all well-known philosophers—and countless scholars from other fields—have written about the topic [24, 50]. The debate is millennia old: among the first known western philosophers who defended determinism were Leucippus and his pupil Democritus (5th cent. BC), who stated that everything happens out of necessity, not chance. Democritus, often called the ‘Father of Science’, is most famous for having elaborated a detailed and incredibly modern-looking atomic theory. It is, in this context, a fascinating question to inquire on what basis Democritus could conjecture so precociously the existence of ultimate and indivisible constituents of matter, governed by laws. According to Sextus Empiricus he did so on the basis of empirical observations, such as the fact that certain substances dissolve in water in constant ratios, that certain physical and biological components degrade but also regenerate, etc.; but also of theoretical principles—namely the principle of determinism (everything has a cause) and the related idea of ‘nihil ex nihilo’ (nothing comes from nothing). The latter idea has been retraced to Parmenides (6th cent. BC), but might be much older, since it was generally accepted in Greek antiquity. It is not exaggerated to say, we believe, that these ideas are among the very few founding postulates of science and philosophy.

Since the ancient Greeks, some of the philosophers who defended determinism (understood: total determinism) were Avicenna, Spinoza, Leibniz, Locke, Kant, Schopenhauer, Laplace, Russell, Einstein, S. Hawking—to name a few. Needless to say, the views of these philosophers on determinism may differ in certain respects; but the key ingredient clearly remains. As already mentioned, the majority of philosophers who believed in determinism did not believe in ‘free will’ in the usual sense (but not all of them, see [50]). Aristotle was one of the early advocates of indeterminism (the action of irreducible chance or randomness). Leibniz’ name is forever linked to his celebrated ‘principle of sufficient reason’, stipulating that everything must have a reason. Kant famously elected the thesis that all events have a cause one of his ‘synthetic a priori principles’.

In the above list I believe Baruch Spinoza (1632–1677) deserves a special mention. I find Spinoza’s defense of determinism particularly attractive and powerful, since he puts determinism at the very basis of a systematic theory of the world and of human action [33, 34]. (Needless to say, strong philosophical theses can in general not be proven, but they can acquire cogency if they are part of full-blown theories that explain things. The more the theory explains, the more convincing the founding premises are.) In Spinoza’s principal work, the Ethics, constructed as a deductive system based on axioms, determinism is omnipresent from the start [33, 34]. Moreover, it is simple and radical. One typical example is Spinoza’s Proposition 29 of the Ethics, Part 1: “Nothing is fortuitous in Nature; everything is determined by the necessity of Nature to exist and produce effects in a given manner.” In other places Spinoza illustrates this thesis by stating that every individual action of any human being is as determined, as necessary to happen, as it is necessary that the sum of the angles of a triangle is 180. In sum, within Spinoza’s philosophy human free will is an illusion, or rather, should be redefined (which however does not bring Spinoza to fatalism, but to a wonderful pro-active ethical theory).

Even this very selective review will illustrate that ‘measurement independence’, a necessary assumption of all Bell theorems, cannot be considered trivial, as is so often done. In many well-known, popular, and solid ontologies, such as Spinoza’s, it does not hold. As already said, in one sense the indeterminism of S1 represents a formidable discontinuity in the history of science. S1 implies that a property σ acquires during measurement a certain value (+1 or −1) based on no ‘reason’ (cause) whatsoever; and that no theory ever will be able to provide such a reason (i.e. new physical parameters of which σ will appear to be a function, or that determine the probability P(σ)). But the history of scientific discovery is the history of finding explanations of phenomena that are only random at first sight. S1 says: we can stop our search for explanations here, forever. And yet, S1 may be the right interpretation. The moral is inevitable: a good dose of agnosticism seems in place.

As a last remark, notice that for a true determinist also supercorrelation (S4) can be considered compatible with the principle of determinism, the hypothesis that everything has a cause. Indeed, suppose that a theory would exist agreeing with supercorrelation. If that theory would predict some probabilities then these may of course be supposed to result from hidden causes not part of the theory. Probability theory does not prohibit such an assumption. Indeed, one of its fathers, Laplace, believed that any probability is only a tool we need because of our ignorance of hidden causes. And the examples in physics in which probabilistic behavior can very well be retraced to deterministic laws, are countless.

Thus full determinism (S3) remains possible at least as a philosophy. S4 is (more easily) subject to scrutiny as a part of a physical theory. And indeed, some will find that S3 or S4 explains the ‘connectedness’ of things in a less mysterious way than S1, the Copenhagen interpretation, which essentially just accepts it.

From the point of view of philosophy, one further remarkable point is that S3 offers a solid scientific basis for theories such as Spinoza’s; a possible link with oriental philosophies will not have escaped from the attention of experts. We refer to Spinoza’s work [33] to remind the reader that there a subtle view on free will and human interaction is exposed. On a very personal note, it may therefore be utterly relevant for the formidable problems western society faces: the latter is based on a belief in the virtually unrestricted freedom of the individual. Which may be too simple a picture.

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Vervoort, L. Bell’s Theorem: Two Neglected Solutions. Found Phys 43, 769–791 (2013). https://doi.org/10.1007/s10701-013-9715-7

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### Keywords

• Bell’s theorem
• Superdeterminism
• Supercorrelation
• Measurement independence
• Correlated spin systems
• Ising lattices