The Free Will Theorem of Conway and Kochen
John Conway and Simon Kochen assume three axioms, which they call "SPIN", "TWIN" and "FIN". The spin and twin axioms can be established by entanglement experiments. Fin is a consequence of relativity theory.
1. SPIN: Measuring the square of the component of spin of certain elementary particles of spin one, taken in three orthogonal directions, results in a permutation of (1,1,0). 2. TWIN: It is possible to "entangle" two elementary particles, and separate them by a significant distance, so that they give the same answers to corresponding questions. The squared spin results are the same if measured in parallel directions. If the first experimenter A (on Earth) performs a triple experiment for the frame (x, y, z), producing the result x → j, y → k, z → l while the second experimenter B (on Mars, at least 5 light minutes away) measures a single spin in direction w, then if w is one of x, y, z, its result is that w → j, k, or l, respectively. 3. FIN: There is a finite upper bound to the speed with which information can be effectively transmitted. Conway and Kochen say this is a consequence of "effective causality." [But the collapse of the probability amplitude wave function is instantaneous and not so limited. ]The formal statement of the Free Will Theorem is then
If the choice of directions in which to perform spin 1 experiments is not a function of the information accessible to the experimenters, then the responses of the particles are equally not functions of the information accessible to them.Conway and Kochen say:
The theorem states that, given the axioms, if the two experimenters in question are free to make choices about what measurements to take, then the results of the measurements cannot be Why do we call this result the Free Will theorem? It is usually tacitly assumed that experimenters have sufficient free will to choose the settings of their apparatus in a way that is not determined by past history. We make this assumption explicit precisely because our theorem deduces from it the more surprising fact that the particles’ responses are also not determined by past history. Thus the theorem asserts that if experimenters have a certain property, then spin 1 particles have exactly the same property. Since this property for experimenters is an instance of what is usually called “free will,” we find it appropriate to use the same term also for particles. determined by anything previous to the experiments. The idea of a "free choice" of the experimenter goes back to the response of Niels Bohr to Albert Einstein, Podolsky, and Rosen and their EPR paradox. EPR argued that entangled particles could be regarded as separate systems, and since they could choose which type of measurement to make on the first system, it would make an instantaneous difference in the state and properties of the second system, however far away, violating special relativity.
We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two different wave functions to the same reality (the second system after the interaction with the first).Bohr replied:
As pointed out by the named authors, we are therefore faced at this stage with a completely free choice whether we want to determine the one or the other of the latter quantities by a process which does not directly interfere with the particle concerned. ...we are, in the "freedom of choice" offered..., just concerned with a discrimination between different experimental procedures which allow of the unambiguous use of complementary classical concepts.In his long 1938 essay on "The Causality Problem in Atomic Physics" Bohr again emphasizes the "free choice" of an experimental procedure in his solution to the EPR paradox.
the paradox finds its complete solution within the frame of the quantum mechanical formalism, according to which no well defined use of the concept of "state" can be made as referring to the object separate from the body with which it has been in contact, until the external conditions involved in the definition of this concept are unambiguously fixed by a further suitable control of the auxiliary body. Instead of disclosing any incompleteness of the formalism, the argument outlined entails in fact an unambiguous prescription as to how this formalism is rationally applied under all conceivable manipulations of the measuring instruments. The complete freedom of the procedure in experiments common to all investigations of physical phenomena, is in itself of course contained in our free choice of the experimental arrangement, which again is only dictated by the particular kind of phenomena we wish to investigate.In all recent EPR experiments to test Bell's Inequalities, "free choices" of the experimenters are needed when they select the angle of polarization. Note that what determines the second experimenter's results is these tests is simply the first experimenter's measurement, which instantaneously collapses the superposition of two-particle states into a particular state that is now a separable product of independent particle states. Since the free will theorem applies to any arbitrary physical theory consistent with the axioms, it would not even be possible to place the information into the universe's past in an ad hoc way. The argument proceeds from the Kochen-Specker theorem, which shows that the result of any individual measurement of spin was not fixed (pre-determined) independently of the choice of measurements. Conway and Kochen describe new bits of information coming into existence in the universe, and we agree that information is the key to understanding both EPR entanglement experiments and human free will. They say
...there will be a time t0 after x, y, z are chosen with the property that for each time t < t0 no such bit is available, but for every t > t0 some such bit is available. But in this case the universe has taken a free decision at time t0, because the information about it after t0 is, by definition, not a function of the information available before t0!Their anthropomorphization of the universe as "taking a free decision" is too simplistic, but it is essential to solutions of the problem of measurement to recognize that the "cut" between the quantum world and the classical world is the moment when new information enters the universe irreversibly. In "The Strong Free Will Theorem," Conway and Kochen replace the FIN axiom with a new axiom called MIN, which asserts only that two experimenters separated in a space-like way can make choices of measurements independently of each other. In particular, they are not asserting that all information must travel finitely fast; only the particular information about choices of measurements made by the two experimenters. In an article on the free will theorem dedicated to John Wheeler, Conway and Kochen write:
One advantage of the Free Will Theorem is that by making explicit the necessary Free Will assumption, it replaces all these dubious ideas by a simple consequence, FIN, of relativity. A greater one is that it applies directly to the real world rather than just to theories. It is this that prevents the existence of local mechanisms for reduction. The world it presents us with is a fascinating one, in which fundamental particles are continually making their own decisions. No theory can predict exactly what these particles will do in the future for the very good reason that they may not yet have decided what this will be! Most of their decisions, of course, will not greatly affect things – we can describe them as mere ineffectual flutterings, which on a large scale almost cancel each other out, and so can be ignored. The authors strongly believe, however, that there is a way our brains prevent some of this cancellation, so allowing us to integrate what remains and producing our own free will. The mere existence of free will already has consequences for the philosophy of general relativity. That theory has been thought by some to show that “the flow of time” is an illusion. We quote only one of many distinguished authors to that effect: “The objective world simply is, it does not happen”(Hermann Weyl). It is remarkable that this common opinion, often referred to as the “block universe” view, has come about merely as a consequence of the usual way of modeling the mathematics of general relativity as a theory about the curvature of an eternally existing arena of space-time. In the light of the Free Will theorem this view is mistaken, since the future of the universe is not determined. Theodore Roosevelt’s decision to build the Panama Canal shows that free will moves mountains, which implies, by general relativity, that even the curvature of space is not determined. The stage is still being built while the show goes on. Einstein could not bring himself to believe that “God plays dice with the world,” but perhaps we could reconcile him to the idea that “God lets the world run free.”Although Conway and Kochen do not claim to have proven free will in humans, they assert that should such a freedom exist, then the same freedom must apply to the elementary particles. (Recall that Arthur Stanley Eddington was mistakenly charged with the idea that human free will was the same idea as that electrons are "free.") What Conway and Kochen are really describing is the indeterminism that quantum mechanics has introduced into the world. While indeterminism is a necessary precondition for human freedom, it is insufficient by itself to provide free will. Another way of looking at their work is to say that Conway and Kochen are trying to close a "loophole" in the Bell inequality tests. We might call this loophole the "determinism loophole" or better the "pre-determinism loophole." If determinism is true, then all the experimental tests might have been pre-determined to show that quantum mechanics is correct, and indeterminism exists, where the real underlying nature of the universe is deterministic. This is beyond belief, but not beyond the hope and dreams of many thinkers, especially mathematical physicists, who hope to show that information is conserved, and that Einstein, Schrödinger, de Broglie, Planck, and friends were right, Heisenberg and Bohr were wrong.