Conway's Proof Of The Free Will Theorem

[bascule]

http://www.cs.auckland.ac.nz/~jas/one/freewill-theorem.html

 

Conway's Proof Of The Free Will Theorem

 

Background

 

In 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.

 

Assumptions

 

 

The Conway-Kochen proof of the Freewill Theorem relies on three axioms they call SPIN, TWIN and FIN:

 

1. SPIN
   Particles have the 101-property. This means whenever you measure the squared spin of a spin-1 particle in any three mutually perpendicular directions, the measurements will be two 1s and a 0 in some order.
    
   
2. FIN
   There is a finite upper bound to the speed at which information can be transmitted.
    
   
3. TWIN
   If two particles together have a total angular momentum of 0, then if one particle has an angular momentum of s, the others must necessarily have an angular momentum of -s.
   

 

Conway expanded on each of these axioms during his talk. He insisted that given his proof, if you disagreed with his conclusion, you must necessarily also disagree with one of these axioms. These are axioms and so they are stated without proof, however, the two axioms SPIN and TWIN can be experimentally tested and verified. Moreover some of these experiments have actually been performed and they support SPIN and TWIN.

 

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 Paradox

 

Conway then went on to describe a simple version of the Kochen-Specker paradox. This paradox is a consequence of the SPIN axiom.

 

 

When measuring the spin of a spin-1 particle along a direction, the values that are possible are:

 

• parallel to the direction (+1)
  
• perpendicular to the direction (0)
  
• anti-parallel to the direction (-1)
  

 

As stated earlier, the SPIN axiom talks about the square of the spin and thus the values are limited to 1, 0 and 1.

 

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).

 

 

Without loss of generality let us assume we get a measurement of 0 when looking towards the particle from the center of one of the cube faces. Then by the 101 principle we know that the measurement from the direction of the remaining faces must be 1 because those faces are orthogonal.

 

 

In fact every point on the plane through the middle of the cube shown is orthogonal to our initial direction and hence must give a squared spin of 1.

 

 

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:

 

1. Each measurement of a particle is not independent but rather depended on context. In other words, the order in which you make measurements matters and the value of a particles spin in a given direction depends on the history of measurements of that particle in other directions. The measurements are not commutable.
   
2. Alternatively, the particle does not decide what the value of its spin is in any direction until the experimenter actually makes a measurement!
   

 

This result is known as Kochen-Specker paradox and was discovered in 1967 by Simon Kochen and Ernst Specker. Conway said that while this was an interesting result, the hypothesis that measurements are commutable is untestable.

 

"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 Paradox

 

Next 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 Proof

 

Finally Dr Conway had set up sufficient background to deliver his proof.

 

(Please see the original link; I'm too lazy to reformat this for the forums!)

[[Tycho?]]

I dont especially believe in free will, I think we act the way we do simply because of how our neurons fire in a given situation. If that exact situation could be repeated exactly, I would expect the exact same response. But whatever.

[Sisyphus]

I don't see how he's equating randomness with free will, nor do I see how predetermination precludes free will. If you are able to make a choice, that is an exercise in will. Whether or not that choice is entirely determined by prior history doesn't seem at all relevant. Random=free? I guess you can define it that way, but what possible significance could that have?

[bascule]

I don't see how he's equating randomness with free will, nor do I see how predetermination precludes free will. If you are able to make a choice, that is an exercise in will. Whether or not that choice is entirely determined by prior history doesn't seem at all relevant. Random=free? I guess you can define it that way, but what possible significance could that have?

 

I'm a compatibilist as well (I believe free will can be an inherently deterministic process). However, this proof is a scientifically-derived argument against dualism (i.e. disproof of the soul?)

[Severian]

Isn't his axiom 'FIN' in contradiction with the statement "making a measurement in one part of a quantum system can have an instantaneous effect on measurements made elsewhere in the system"? So 'FIN' and 'TWIN' are incompatible in the sense that he uses them.

 

In other words, he is not allowing for the non-locality of Quantum mechanics. In fact, I would have said that his 'proof' is in fact a proof of non-locality, via reductio ad absurdum.

[jcarlson]

Isn't his axiom 'FIN' in contradiction with the statement "making a measurement in one part of a quantum system can have an instantaneous effect on measurements made elsewhere in the system"? So 'FIN' and 'TWIN' are incompatible in the sense that he uses them.

 

In other words' date=' he is not allowing for the non-locality of Quantum mechanics. In fact, I would have said that his 'proof' is in fact a proof of non-locality, via reductio ad absurdum.[/quote']

 

Actually, I believe this apparent contradiction is what his theory rests on. In the link it explains how, if the expiriment is conducted within a time T, it would be impossible for the two particles to somehow communicate, but since their total angular momentum is zero, and they will always have opposite spins, the functions that determine the behavior cannot rely on ANY information either of the particles receive during time T, but only on the direction from which the experimenter observes them.

[Severian]

In the link it explains how, if the expiriment is conducted within a time T, it would be impossible for the two particles to somehow communicate, but since their total angular momentum is zero, and they will always have opposite spins

 

But the non-locality of QM means that they DO communicate within a time T. The collapse of the wavefunction is instantaneous and global, so both electrons have access to the function information at time T.