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arXiv:0905.4641

What Does the Free Will Theorem Actually Prove?

Sheldon Goldstein, Daniel V. Tausk, Roderich Tumulka, Nino Zanghı̀

Published May 28, 2009• 42 citations
arXivHTMLPDF

Abstract

Conway and Kochen have presented a "free will theorem" (Notices of the AMS 56, pgs. 226-232 (2009)) which they claim shows that "if indeed we humans have free will, then [so do] elementary particles." In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition "MIN" motivated by relativistic symmetry. We point out that for stochastic models this conclusion is not correct, while for deterministic models it is not new.

DOI
10.48550/arxiv.0905.4641

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Conway and Kochen have presented a "free will theorem" (Notices of the AMS 56, pgs. 226-232 (2009)) which they claim shows that "if indeed we humans have free will, then [so do] elementary particles." In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition "MIN" motivated by relativistic symmetry. We point out that for stochastic models this conclusion is not correct, while for deterministic models it is not new.
I begin with description of quantum entanglement and its relevance to the EPR-type experiments, which played a decisive role in rejecting determinism and hidden-variable theories. Thereby I prove the closely-related Kochen-Specker paradox, on which also the main object of my paper, the recently published Conway and Kochen’s Free Will Theorem, is founded. It is being shown there, on the basis of three physical axioms, following from the special theory of relativity and quantum mechanics, that if the choice for a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. Therefor the response of the particles can be seen as free. In conclusion I mention some critical remarks on the theorem, expressed in subsequent responses to the original article.
Introduction The two theories that revolutionized physics in the twentieth century, relativity and quantum mechanics, are full of predictions that defy common sense. Recently, we used three such paradoxical ideas to prove “The Free Will Theorem” (strengthened here), which is the culmination of a series of theorems about quantum mechanics that began in the 1960s. It asserts, roughly, that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity. More precisely, if the experimenter can freely choose the directions in which to orient his apparatus in a certain measurement, then the particle's response (to be pedantic—the universe's response near the particle) is not determined by the entire previous history of the universe. Our argument combines the well-known consequence of relativity theory, that the time order of spacelike separated events is not absolute, with the EPR paradox discovered by Einstein, Podolsky, and Rosen in 1935 and the Kochen–Specker Paradox of 1967 (see [5]). We follow Bohm in using a spin version of EPR and Peres in using his set of thirty-three directions, rather than the original configuration used by Kochen and Specker. More contentiously, the argument also involves the notion of free will, but we postpone further discussion of this to the last section of the chapter.
If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly certain preliminary goal, and the choice is only the mean, by which it can be achieved or not by the one who determines the goal. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior.The choice is usually linked to very complicated systems such as human brain or society and even often associated with consciousness. In its background, the material world is deterministic and absolutely devoid of choice. However, quantum mechanics introduces the choice in the fundament of physical world, in the only way, in which it can exist: All exists in the “phase transition” of the present between the uncertain future and the well-ordered past. Thus the present is forced to choose in order to be able to transform the coherent state of future into the well-ordering of past. The concept of choice as if suggests that there is one who chooses. However quantum mechanics involves a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature.There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors, Conway and Kochen, and according to them: “Do we really have free will, or, as a few determined folk maintain, is it all an illusion? We don’t know, but will prove in this paper that if indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity” “The import of the free will theorem is that it is not only current quantum theory, but the world itself that is non-deterministic, so that no future theory can return us to a clockwork universe”.Those theorems can be considered as a continuation of the so-called theorems about the absence of “hidden variables” in quantum mechanics.
The title refers to the Free Will Theorem by Conway and Kochen whose flashy formulation is: if experimenters possess free will, then so do particles. In more modest terms, the theorem says that individual pairs of spacelike separated particles cannot be described by deterministic systems provided their mixture is the same for all choices of measurement settings. We reformulate and generalize the Free Will Theorem theorem in terms of systems of random variables, and show that the proof is based on two observations: (1) some compound systems are contextual (non-local), and (2) any deterministic system with spacelike separated components is non-signaling. The contradiction between the two is obtained by showing that a mixture of non-signaling deterministic systems, if they exist, is always noncontextual. The "experimenters' free will" (independence) assumption is not needed for the proof: it is made redundant by the assumption (1) above, critical for the proof. We next argue that the reason why an individual pair of particles is not described by a deterministic system is more elementary than in the Free Will Theorem. A system, contextual or not and deterministic or not, includes several choices of settings, each of which can be factually used without changing the system. An individual pair of particles can only afford a single realization of random variables for a single choice of settings. With this conceptualization, the "free will of experimenters" cannot be even meaningfully formulated, and the choice between the determinism and "free will of particles" becomes arbitrary and inconsequential.