Conway's Proof Of The Free Will Theorem
We must believe in free will. We have no choice.
-- Isaac B. Singer
BackgroundIn mid-2004, John Conway and Simon Kochen of Princeton University proved the Free-will Theorem. This theorem states "If there exist experimenters with (some) free will, then elementary particles also have (some) free will." In other words, if some experimenters are able to behave in a way that is not completely predetermined, then the behavior of elementary particles is also not a function of their prior history. This is a very strong "no hidden variable" theorem.
This result has not currently been published, however, on January 27, 2005, Dr Conway gave a public lecture at the University of Auckland on the proof of this theorem. The following is my account of his talk. As a disclaimer, please note I am not a physicist and do not understand some of the subtler nuances of quantum mechanics. I have tried to faithfully reproduce Conway's explanation and argument, however, there may still be some errors in this document. If you find such an error, do let me know so I can correct it.
In my (very weak) defense, I quote Dr Conway's retelling of Feynman's quip: If you meet someone who claims to understand quantum mechanics, the only thing you can be sure of is that you have met a liar.
Conway's talk was informative, entertaining and very accessible. The audience consisted not only of mathematicians and physicists — I recognized many computer scientists, philosophers and at least one theologian.
The Conway-Kochen proof of the Freewill Theorem relies on
three axioms they call SPIN, TWIN and FIN:
Conway stated that although he believed FIN to be true, he pointed out that, experimentally, FIN is the most contentious of the three axioms. It cannot be verified experimentally. The theories of relativity state that the speed of light c is the upper bound on the speed at which information transfer occurs. FIN does not require the theory of relativity to be correct (FIN requires any upper bound not necessarily c) although it would be sufficient. "We do not know if some unknown method allows for instantaneous transfer of information", Conway laughed, "almost by definition."
Kochen-Specker ParadoxConway then went on to describe a simple version of the Kochen-Specker paradox. This paradox is a consequence of the SPIN axiom.
Suppose a particle has already decided its spin in every
direction. When an experimenter measures its spin in some
direction the particle simply "answers" with the value of
spin it has predetermined. Conway showed that this is not
possible because there is no way to assign 0s and 1s to
all directions that we can measure a particle from and
still be consistent with the SPIN axiom. In his talk,
Conway showed that even if an experimenter was limited to
merely 33 directions, there was no way for a particle to
predetermine the square of its spins in all 33 directions
and still be consistent with SPIN.
Imagine a cube that snuggly surrounds a sphere. On each face of the cube, we inscribe a circle and inside each circle we draw a square that touches the circle at the squares four corners. We divide each such square into four smaller squares and mark the following points on the cube.
We get 33 points (9 points per face x 3 faces + 1 point
per edge x 6 edges) on the cube in this way. These
represent 33 directions of measuring a particle.
Lets try to assign a possible set of values of 0 and 1 to these 33 points consistent with SPIN. First note that we only need 33 points and not 66 because whether you measure a particle from one direction or from exactly the opposite direction, we get the same value. This is because we are measuring not spin, but spin squared (so 12 = -12 and 02 = 02).
If we continue in this fashion we deduce the spin that
would be measured on a particle if it is to satisfy
SPIN. However, having successfully deduced the spin for 32
of the directions (marked either green for 0 or red for 1
in the above diagrams), we find these 32 measurements and
SPIN force the final measurement (marked yellow) of spin
to be to be both 0 and 1 which is impossible!|
There is no way to assign a spin value to each of the 33 directions that we have chosen to measure this particle in a manner consistent with SPIN. There is a nice Python script that allows you to interactively try out this experiment available here. This script was used to generate the pictures above.
What does this mean? It means one of two things:
"Once you step into a river, you cannot step into that river a second time because in some sense its now a different river."
EPR ParadoxNext Conway talked about the EPR Paradox and the TWIN axiom. In 1935, Albert Einstein, Boris Podolsky and Nathan Rosen proposed a thought experiment hoping to demonstrate what they felt was a short-coming of quantum mechanics. This thought experiment is called the EPR-paradox.
Imagine two particles that are allowed to interact initially. We measure their total angular momentum or spin we will get a value between -2 and 2. Once in a while when we are lucky we get a total angular momentum of 0. This means we have a pair of particles whose individual spins sum to zero — we may have a particle with a spin of 1 and another with -1, or we may have two particles with 0 spin. Notice if we square the particle spins, we get identical values for each pair. What this says is that if two particles total angular momentum is zero, then the square of the spin of one particle measured from some direction is identical to the square of the spin of the other, irrespective of how far apart these particles may drift before they are measured.
The paradox that Einstein and his colleagues desired to demonstrate was that making a measurement in one part of a quantum system can have an instantaneous effect on measurements made elsewhere in the system.
Conway-Kochen ProofFinally Dr Conway had set up sufficient background to deliver his proof.
Suppose we have two particles Phome and Paway. Suppose further that these particles together have a zero angular momentum. We allow Paway to drift away from Phome so that the two particles become separated by some large distance. Now we would like to measure Phome from some three orthogonal directions namely x, y and z. Also we would like to measure Paway from a direction ω where ω is the same as one of x, y or z.
Lets measurement of the particle Phome and Paway be represented as two functions fhome and faway. The output of this function may depend on:
Consider the two functions:
| Once the particles are separated by distance ,
it takes a time T for information from one particle
to reach another. The information shown as
IA in the diagram is available only to
Phome and not to Paway.
If we do our experiment within time T then
Paway has no access to
IA so IA cannot possibly
affect the output of faway. Similarly,
Phome cannot access IB
so IB cannot affect the output of
fhome. So we can remove these variables
from our measurement function:
Conway thus concluded that if the experimented had sufficient freewill to decide the directions in which he would measure the particle then the particle too must have the freewill to decide on the value of its spin in those directions such that it can be consistent with the 101-property. In concluding Dr Conway said that he believed he did have freewill. Holding up a piece of chalk, he said he felt he could choose whether or not he would drop it or continue to hold it. His theorem he said leads him to accept that the universe is teeming with freewill. He also said that while he did not have any proof for it, he believed that the cumulative freewill of particles is the source of his freewill as a person.
QuestionsWhen the floor was opened for questions, one member of the audience questioned Dr Conway's use of the term "Free Will". She asked whether Dr Conway was "confusing randomness and free will".
In a passionate reply, Dr Conway said that what he had shown, with mathematical precision, that if a given property was exhibited by an experimenter than that same property was exhibited by particles. He had been careful when constructing his theorem to use the same term "free will" in the antecedent and consequent of his theorem. He said he did not really care what people chose to call it. Some people choose to call it "free will" only when there is some judgment involved. He said he felt that "free will" was freer if it was unhampered by judgment - that it was almost a whim. "If you don't like the term Free Will, call it Free Whim - this is the Free Whim Theorem".
|I would like to thank Morris W. Hirsch for correcting a typographical and presentation error in this document.|
Thank you to Frank Foreman for sending me the following picture.