bells-inequalitydeterminismquantum mechanicsquantum-interpretations
It seems to be common consensus that the world is non-deterministic and this is proved by Bell's theorem.
But even though Bell's experiments proved that the theory of quantum mechanics work, How does it prove the non-existent of local hidden variables?
Isn't it possible that there are hidden variables at work, and the results that were derived from these hidden variables coincide with the predictions of quantum mechanics?
The issue is locality of what? Which quantities are assumed to be local?
If you say the reults of all experiments, hypothetical as well as actual, and say that these must be assignable definite values, then this is in conflict with quantum mechanics. But this is the assumption that is called "hidden variables", the reason is that this is the assumption Einstein made in the EPR paper. This is the assumption that if you could perform an experiment to determine a quantity, then it is legitimate to give this quantity a definite value, even if you didn't do the experiment. The hidden variables are just the names of the extra quantities that determine the outcome of those hypothetical experiments.
The results of the quantum experiments themselves are only obviously nonlocal if their results are assumed to be definite things, even in those cases where you didn't actually go on to perform the experiment. This is not accepted in the standard interpretations, although for slightly different reasons in different philosophies.
Suppose you have three polarization settings, A,B,C such that the measurement of A,B,C is always the same between the two distant spins, A and B are 99% correlated, B and C are 99% correlated, and A and C are 96% correlated. This is the starkest violation of Bell's inequality that I know. In this case, you can see the violation without a calculation--- if A and B are the same except for 1 time in a hundred, and B and C are the same except for one time in a hundred, how can A and C be different more than 2 times in a hundred? Obviously they can't, and that's Bell's inequality. But obviously they are, in quantum mechanics, because the probability difference between polarization settings goes as the square of 1 minus the (small) angle squared, so that doubling the (small) angle increases the discrepancy in the measurements by a factor of 4, not by a factor of 2.
In order to argue that this is a locality violation, when you are measuring A on particle 1 and B on particle 2, you have to assume that were you to measure A on particle 2, the result would have been the same. The only locality violations are regarding the answers to these hypothetical questions. But you aren't measuring A on particle 2, so that this is a counterfactual statement. You don't have to believe that counterfactual statements of this sort are meaningful. If you believe that the results of experiments come out of thin air, that they don't come from the laws of physics, but of an irreducible interaction between the particles and the measuring device, which is called an observation.
This is Bohr's language, so I am just stating Bohr's position that the results of experiments come out of thin air. They aren't determined by anything previous. So it makes no sense to say "were you to measure A, the result would have been such and so", because you aren't measuring A. The assumption of "counterfactual definiteness" is the central thing, that it makes sense to talk about the answers to experiments you haven't done. I don't like Bohr's way of stating it, I prefer Everett's, but the two answers are exactly the same when push comes to shove.
The assumption of hidden variables is only used to allow you to talk about what the outcome of the A measurement on particle 1 would have been for those instances where you are doing a B measurement on particle 1. You can determine what they answer would have been using the hidden variables, assuming that the hidden variables determine the outcome of the experiments. You don't have to use the hidden variables if you are willing to assume that it is ok to talk about the value you would have gotten in experiments you haven't done.
Since it is only through considering couterfactual experiments that you get a contradiction, if you don't have counterfactuals, you don't have contradictions. The contradiction is that there is no way to assign consistent definite results to all the possible counterfactuals. You could just refuse to assign answers to hypothetical measurements. Equivalently, you could assign amplitudes with those answers, not definite results. Its only when you try to assign definite results that you get a contradiction.
This argument of Bohr's is very difficult to understand, and, like everything else, it is entirely cleared up by switching to Everett's many-worlds philosophy. This explains exactly what is going on, in a detailed mechanistic way that is equivalent to Bohr except for philosophical mumbo-jumbo. Since it is equivalent to Bohr, modulo philosophy, I don't see any point in retaining Bohr's harder to understand language, except to the extent that it is interesting that you can state things this way too.
In the many-worlds point of view, the A measurement splits the observer, and it splits the observer in a different direction than the B measurement does, it entangles the observer with a different property of the particle. The relative statistics that the two observers see are only determined when the two observers come to talk, and then which copy of the observer over here meets which copy of the other observer over there is always a matter of how they are tilted relative to each other in wavefunction space. The tilt between the copies of the observers is entirely determined by which quantity they chose to measure. It is now obvious that there is no reason that Bell's inequality has to hold in this sort of thing, because the results not only of counterfactual experiments are undertermined, but even the results of actual experiments are undetermined! There is still another copy of you which got the opposite results!
Bohr's philosophy is obtained by focusing on one branch, and rejecting the other counterfactual branches as non-existing.
I suppose you could see even the Everett version as violating locality, because the world-label (and the wavefunction) are global constructs. But I don't think this is the best practice. If the world is quantum mechanical, those experiments which split observers will always be indefinite.
Since many-worlds requires some slightly nonobvious philosophy, it is best to make the argument in a purely classical universe, where there are no philosophical headaches. You can violate Bell's inequality inside a semi-local classical theory, with world-splitting. The theory will be a classical analog to the Many-Worlds quantum theory, but no philosophical headaches, because the splitting will be built into the theory, and not emerge from philosophical contortions.
The theory will be local in that it will obey the principle:
It will not be local in this sense:
These variables describe the analog of the entanglement between A and B.
Assume a classical Newtonian world, particles interacting by a retarded potential (so that you have relativistic locality), but with a secret world integer W. The world variable is just a stupid parameter that tells you which world you are on.
Two particles only interact with each other when they have the exact same value of the variable "W" (so that W is not a position, exactly, but a sheet-number), and initially, there is one copy of the particles on world 0.
The particles can, in addition to the Newtonian stuff, also do a "fork", which adds a new world to the list of all existing worlds (a new integer label), and copies all the particles nonlocally at their current positions into the new world, and the particle which is forked is altered in a subtle way--- it has a coupling to a forking-detector material (this can all be done explicitly in a computer model).
The copying of particles to the new world doesn't have to happen right away--- since the new world is identical to one of the old worlds, you can use the old world variables right up to the moment of first altered interaction, which propagates outward from the fork position at the speed of light.
Further, the forks have internal variables, the phases. Any two forks have a relative phase, which is a variable on the unit circle. When you fork A and fork B at angle $\theta$, you make worlds in the proportion: $cos^2(\theta)$ AB's and A'B' , $sin^2(\theta)$ AB' and A'B's. In the preceding, A and A' are the two splits descended from A, and B and B' are the two splits descended from B, and the relative angle only tells you how correlated the two splits are with each other, by arranging more copies at the point of collision (when an A split meets a B split) in the proportion determined by their relative angle.
This world-splitting classical model reproduces the quantum situation. This is not a surprise, because it is just made up to do so.
The point of this is that the quantum mechanics is only as nonlocal as this classical model. It is not clear whether the splitting and the angles are to be considered nonlocal, since they are only nonlocal in the sense of extra data, not nonlocal in the sense of influence. All the influences travel strictly at less than the speed of light.
The idea of my latest paper is simple. I experienced in other blogs that most people refuse to go with me all the way. I'll give my argument step by step and you may choose where you want to step out.
Consider superstring theory, in its original, completely quantized version. Many people believe it might have something to do with the world we live in. It has interesting low energy modes that show some resemblance with what happens in the Standard Model: fundamental fields for particles with spin 0, 1/2 and 1, as well as gravitons for the gravitational field, as gravitinos. The theory is not universally accepted, but it is an interesting model with many features that look like our world. Certainly not obviously wrong, and certainly very much quantum. There is a Hilbert space of states. I only use it as a model to illustrate my ideas. But step out here if you want.
The transverse coordinates of the string form a simple integrable quantum field theory on the string world sheet. This integrable system has left-movers and right-movers, forming quantum states, the string excitations. Now, I discovered a unitary transformation that transforms the basis of this Hilbert space into another basis. In QM, we do this all the time, but what is special in the new basis is that it is spanned completely by a set of left-movers and right-movers that are integer-valued, in units whose fundamental length is $2 \pi \sqrt{\alpha^\prime}$. Thus, we have operators taking integer values, and they are all commuting. What's more, they commute at all times. The evolution operator here translates the left-movers to the left and the right-movers to the right. Intuitively, you might find that the result is not so crazy: these integers are of course related to particle occupation numbers in quantum theory. I still have Hilbert space, but it is controlled by integers. If you don't like this result, please step out.
Do something similar to the fermions in the superstring theory. They can be transformed into Boolean variables using a Jordan-Wigner transformation. The superstring theory of course has supersymmetry on the world sheet. That does not disappear, but does become less conspicuous. Also the fermions are transversal. The Boolean variables also commute at all times. Next stop.
Realize that, if Nature starts in an eigen state of these discrete operators, it will continue to be in such an eigenstate. There is a super-selection rule: our world can't hop to another mode of eigen states, let alone go into a superposition of different modes. Thus, if at the beginning of the universe, we were in an eigenstate, we are still in such an eigenstate now. Step out if you want.
I can add string interactions. My favorite one is that strings exchange their legs if they have a target point in common. This is deterministic, so the above still applies. This is a stop where you may get out.
Rotations and Lorentz transformations. To understand these, we need to know the longitudinal coordinates. The original, completely quantized superstring tells you what to do: the longitudinal coordinates are fixed by solving the gauge constraints (both for the coordinates and the fermions). The superstring has only real-number operators, of course non-commuting. This step tells us that only 10 dimensions work, and fixes the intercept a. Don't like it? Please step out.
What I have here is a Lorentz invariant theory equivalent to the model generated by the original superstring theory, but acting like a cellular automaton. It IS a cellular automaton. Any passengers left?
Best Answer
Bell's theorems indeed rule out simple theories where hidden variables obey local equations. However, no matter how you reason, it's always at some point where you need another assumption. In its simplest form, it is the assumption that two observers, Bob an Alice, have the "free will" to choose along which axis they will measure the spin of a particle (photon, electron, or something else). Well, one could object that in a deterministic theory they have no such free will; their decisions were made in the far past.
But that does not invalidate Bell, because now you can say: Bell's theorem would imply that entangled photons emitted by a physical source are correlated non locally in an unnatural way with the nerves in Bob's and Alice's brains long before they made their decisions. That's called "conspiracy". So now the assumption is: there can't be conspiracy. Can't there? Spacelike non-local correlations in physical states are common in the physical world. In fact, in quantum field theory it's the propagators of all physical particles that describe correlations, and they do not vanish far outside the light cone. But the kind of conspiracy quantum systems seem to display (when described in terms of "hidden variables") looks disgusting to many researchers. So it is usually dismissed. Is "disgusting" a sound mathematical argument? You decide ...