This is my last post on key results that were presented during the New Directions in the Foundations of Physics conference in Washington DC. (You can see my other posts about building a black hole in the lab, how quantum mechanics isn't as strange as it could be, and quantum effects and time travel.) Now I want to talk about advances by FQXi's Roderich Tumulka in creating a relativistic GRW quantum collapse theory and the Free Will Theorem controversy.
To recap, the Free Will theorem, proposed by John Conway and Simon Kochen, claims that no deterministic interpretation of quantum mechanics is compatible with the notion that humans have free will. So why does that hit up against Tumulka's model?
Tumulka presented what looked like a consistent relativistic version of the GRW approach to quantum mechanics--that is, an approach in which quantum collapse happens spontaneously, rather than as a result of a measurement. (However, this result contained no interaction, and this is where the hard part lies in light of Haag's theorem of the impossibility to have a strictly Lorenz invariant vacuum or a Hilbert space in the interaction picture in field theory.) GRW-type theories are stochastic; for an individual particle wavefunction, there is a probability of collapse, but no certainty. The clash with the Conway-Kochen Free Will Theorem arises because Conway and Kochen assert that any GRW-type theory is also proved wrong by their theorem. Randomness, they argue, is no better than determinism.
In light of the talk I re-read the original papers on the archive, and I watched a six part presentation by Conway recorded last year. It is really hard to say who is right and who is wrong--the arguments are very subtle on both sides. But I think I understand clearly enough Conway's position on randomness to be able to attempt to present it here.
Here is what Conway has in mind: He starts with the "second running" argument for causality. You go to the movies and watch a very exciting story. You do not know the outcome and to you it looks like the characters have free will. Then the next day you take a friend with you to see the same movie. To him, the characters have free will, just as it did for you yesterday, but by now for you they no longer look like that because you know the outcome. This argument shows that it is logically impossible to argue against determinism with a believer in determinism; you would get no logical contradiction. But logical consistency does not guarantee agreement with reality.
A second Conway argument is based on how backgammon tournaments--with multiple pairs of players all conducting games at the same time--are conducted. The random rolling of the dice is done ahead of time and everyone plays the same rolls in parallel to ensure fairness among all players. It does not make any difference to the outcome of the backgammon games if the players roll the dice as they go, or if the random sequence is pre-generated.
To this, Tumulka had two counter-claims: (1) The Free Will Theorem is nothing new, and it was in fact proven by Bell in an equivalent form; and (2) This theory does not apply to stochastic theories.
In his talk, Tumulka specifically focused on the critical statement of the Free Will paper in which Conway and Kochen assert that randomness does not help. In Tumulka's theory, collapse involves "random flashes." Conway insists that just as the outcome of the backgammon games is not changed, whether the dice are rolled during the matches or ahead of time, it makes no difference to GRW predictions if the random flashes are given before the Big Bang versus being "computed" when needed, "on the fly," by nature. This is just as bad as being determined.
Now back to Tumulka's position. Why is "rolling of the dice" before the Big Bang so bad? Because of the meaning of non-locality. Conway and Kochen assume that the random quantum mechanical behavior can only depend on the past causal cone because one cannot signal faster than the speed of light. Is this bad? Yes, because Tumulka argues that quantum mechanics (and his relativistic GRW theory) accesses non-local information beyond the past causal cone while obeying the no-signaling condition. Moreover, the right way to understand Bell's result is not in the popular understanding of choosing between locality and hidden variables, but as an outright rejection of locality. Without the incorrect confusion between no-signaling and non-locality, one of the three original assumptions of the Free Will Theory (the FIN, or its later modification to MIN) is wrong, and this opens a loophole allowing the existence of no-signaling, non-local stochastic quantum mechanical theories.
So who is ultimately right hinges on rejecting locality outright from Bell's inequalities. I feel that the question is not settled one way or another. Take the Aharonov-Bohm effect. It surely looks non-local, until you notice that the vector potential in electromagnetism is the connection in its corresponding gauge theory and it does obey the micro-causality condition in field theory. Or take Bell's inequality. Joy Christian managed to produce a local explanation at the expense of using geometric algebra instead of real numbers. While his interpretation of measurement is in minority among physicists, his result can be understood as a local theory in the sense of Conway and Kochen.
So here is the challenge: Does anyone have any conclusive proof that Bell's inequality rejects locality? I would surely like to hear about it and debate it.
Corrected Post Script
After reading all relevant archive papers on this subject, the core papers are as follows: arXiv:0905.4641v1 for another presentation of Tumulka's arguments, arXiv:1002.1392 for a nice solution of the puzzle of the dispute by Nicolas Gisis and arXiv:1006.2485 (and references inside) by Antoine Suarez for some relevant comparison with his "before-before" model.
The moral of the story is that the origin of the dispute stems from a different expectation about the degree to which a relativistic theory should be covariant. (Tumulka's "flash ontology" requires a less stringent covariant condition.) Therefore Conway's claim that FWT rules out stochastic models is false.
Suarez's „before-before" model shows something more: SPIN, TWIN, and MIN does not prove the free will theorem because he managed to produce a counter-example (albeit logically inconsistent and ruled out by experiments). Instead, the FWT should be derived from full quantum formalism and the MIN axiom for deterministic models. Suarez's toy model obeys SPIN and TWIN under the standard EPR experiment, but it is incompatible with the C* algebra (and in particular with Hardy's 5 axioms) because it predicts violations of entanglement under certain conditions.