[NOTE: This piece is modified from one written in the spring of 2005, but never published because it was too demanding, so be warned.]
In 2004, two mathematics professors from Princeton University devised the simplest proof yet that the world really is unpredictable at a microscopic level.
Quantum mechanics has passed many experimental tests, but it generally predicts only the probabilities of various outcomes. Over the decades, many physicists, notably Einstein, have longed for a description that doesn't involve "throwing dice."
The Princeton "Free-Will Theorem" concludes that, if experimenters can make choices freely, then this unpredictable behavior of elementary particles is unavoidable. But other experts suspect that the result is another manifestation of the "spooky action at a distance," that dominates the quantum world.
"Physicists usually are not impressed much," admitted John Horton Conway, the inventor of the 1970 cellular-automaton game he called Life. "They actually believe quantum mechanics." Indeed, few dispute that quantum mechanics gives correct predictions. But Conway and his colleague Simon Kochen said that although their conclusions are familiar, they start with three axioms, called SPIN, TWIN, and FIN, that are much simpler than previous theorems, and avoid "counterfactual" experiments that can't be done.
The first axiom, called SPIN, is based on an unusual property of a "spin-one" elementary particle: Measuring the square of its angular momentum, or spin, as projected along three perpendicular directions will always yield two ones and one zero. This bizarre property is usually derived from quantum mechanics, but it could have been observed independently.
"We don't have to know what 'the square of the spin' means," Conway said. "It's really rather important that we don't, because the concept 'squared spin' that we're asking about doesn't exist-- that's one of the things that's proved."
If such a squared spin existed, then experimenters could, in principle, choose a direction to measure it along and know in advance whether it would be one or zero. But in a groundbreaking theorem published in 1967, Kochen and E.P. Specker showed that it is impossible to prepare a list beforehand that gives the required two ones and a zero for all possible sets of measurement directions. They concluded that there are no "hidden variables" that describe the "real" spin.
Later researchers, however, realized that such hidden variables could logically exist, but only if their values changed depending on which measurements were chosen, a property known as "contextuality."
To avoid this problem, Conway and Kochen analyzed pairs of particles with matched properties. Their second axiom, which they call TWIN, is that experimenters can make and separate such pairs. This ability is well established, and experiments on the pairs have confirmed the quantum prediction that measurements of their properties remain correlated long after they separate. (I described one recent experiment for Technology Review.)
In 1964, John Bell showed that the some measurements on the two particles can only be explained if each particle somehow continues to be affected by the other, even though they are far apart. This surprising "nonlocality" has since been confirmed in numerous experiments, which find that the correlation, averaged over many pairs, exceeds the maximum for any conceivable local theory.
To avoid the need for statistical averages, Conway and Kochen applied the Kochen-Specker theorem to a single, matched, spin-one pair. If researchers measure the same component of the squared spin for both particles, they should always find either both zeroes or both ones. (Rutgers student Douglas Hemmick also derived this result in his 1996 doctoral thesis.) A similar "Bell's theorem without inequalities" was described in 1989 by Daniel Greenberger, Michael Horne, and Anton Zeilinger, using three "spin-1/2" particles.
Conway and Kochen's third axiom, FIN, is grounded in special relativity, and says that information travels no faster than, say, the speed of light. In spite of appearances, they say, nonlocal effects do not exceed this speed limit, because they describe only coincidence between two measurements, not causation. In fact, special relativity makes it meaningless to say that either of two widely separated measurements occurred "first," so it makes no sense to talk of information passing between the two.
Combining these ingredients, Conway and Kochen imagine that the squared spin is measured in all three directions for one member of a pair. If an experimenter measures the spin of the other member along any of these directions, the result must agree with the one for the first member.
But if the experimenter is free to choose which direction to measure, then because of FIN, that information is not available to the first particle. The result of the first measurement can't depend on her choice, but since there is no way to consistently anticipate all possible measurements, Conway and Kochen conclude that no hidden variable could have predicted the outcome. The only way to avoid this unpredictability, they say, is if the experimenter wasn't really free to choose which experiment to do.
Tim Maudlin, who heard Conway present the work in a colloquium in November 2004, disputes that conclusion. A philosophy professor at Rutgers University and author of "Quantum Non-Locality & Relativity," Maudlin remarked that saying the behavior of a particle cannot be determined by information in its own past "is just what we mean by non-locality," which is already clearly established. "You've taken the contextuality and stretched it out" to include both members of the pair, he asserts.
Conway and Kochen published their Free Will Theorem in 2008, and the Princeton Alumni Weekly has posted videos of lectures by Conway. But it appears that other scientists are free to choose whether to believe it.