T'hooft says something intriguing about hidden-variables and Bell's Theorem, and I haven't yet figured out whether it is profound or not.
I'm stripping the stuff about cellular automata out, and getting to what I think is the heart of t'hooft's explanation of how they evade Bell's theorem.
If you're trying to mimic the predictions of quantum mechanics for the EPR experiment using classical means, you need to have a setup for a "game" of the following type:
There are three "players", call them Alice, Bob and Charlie. Play consists of a number of rounds, and each round has a number of turns:
- Turn 1: Charlie creates two messages and puts them in sealed envelopes. Call them M_A and M_B. Meanwhile, Alice chooses a direction \vec{a} and Bob chooses a direction \vec{b}. Nobody gets to see what anyone else is doing during this round. (So that we can get good statistics, let's assume that \vec{a} and \vec{b} are chosen from finite sets of possibilities.)
- Turn 2: Charlie sends M_A to Alice and M_B to Bob.
- Turn 3: Alice uses some fixed algorithm F_A(\vec{a}, M_A) to compute a result R_A which is either +1 or -1. Bob uses another fixed algorithm F_B(\vec{b}, M_B) to compute R_B (again either +1 or -1).
After playing many, many rounds, for different choices of M_A, M_B, \vec{a} and \vec{b}, we compute probability functions:
P_1(\vec{a}) = the fraction of rounds in which Alice chooses \vec{a} and R_A = +1.
P_2(\vec{b}) = the fraction of rounds in which Bob chooses \vec{a} and R_B = +1.
P_3(\vec{a}, \vec{b}) = the fraction of rounds for which Alice choose \vec{a} and Bob chooses \vec{b} and R_A = R_B = +1
The challenge for such a classical explanation of EPR is to make the predictions match those of quantum mechanics:
P_1(\vec{a}) = P_2(\vec{b}) = 1/2
P_3(\vec{a}, \vec{b}) = 1/2\ cos^2(\theta/2) where \theta is the angle between \vec{a} and \vec{b}.
The loophole that t'hooft suggests is that the choices made by the players, \vec{a}, \vec{b}, M_A, M_B are NOT independent. They are correlated.
They certainly can be correlated. Even though people may think that they are "freely choosing" something, in fact, they could be basing their choice on some fact about the world. Even though they may think of themselves as making unpredictable choices, there's actually a deterministic algorithm at work. Since the three players all (presumably) share a common past history (they had to have gotten together at some point in the past in order to agree to play the game), it's certain that the states of their brains are correlated by past interactions. It's then possible that this correlation would give rise to a correlation in their seemingly random choices.
T'hooft's idea is certainly a logical possibility, but it seems wildly implausible to me. Even if the players' choices are determined by their pasts, the exact algorithm for making a choice may be incredibly complex. Maybe Alice is using the digits of pi. Maybe Bob is basing his choice on the latest soccer scores from the World Cup. I can see how correlated initial states of the players might cause correlations in the outcomes, so there is no necessary reason for Bell's inequality to hold. But the weird thing is that the statistical predictions of QM for EPR are completely insensitive to the details of how the choices are made. It's hard for me to see how it is possible to make a "superdeterministic" theory that explains EPR in any way short of Charlie precisely simulating the decision-making processes of Alice and Bob, and then making his choice with their eventual choices for \vec{a} and \vec{b} in mind.
