I've been pondering realism for the past few months. In particular I've been thinking about Bell's theorem, and FQXi member Joy Christian's claimed disproof of it. I recently posted a paper on this (http://arxiv.org/abs/1107.1007) and I'd like to discuss the topic here.
You can watch Joy Christian describing his research at FQXi's 2009 conference in the Azores below:
[youtube: J2Sc0ZvNMe4, 560, 340]
(After reading my post, you should also read Joy Christian's response: here.)
Some background: John Bell coined the word "beable" to capture the essence of any realistic theory of nature. It is made of two parts: "Be" as in Hamlet's tortured question, "To Be or Not To Be?", and "able", an ending shared with "Observ-able". Beables were supposed to be the building blocks of the observable. If beables exist, however, then they are at odds with quantum mechanics because quantum mechanics on predicts only probabilistic outcomes and also it claims to be complete. (There is a fundamental theorem by Gleason which states that the only prescription of obtaining predictions in quantum mechanics is what quantum mechanics is using. What this means is that quantum mechanics is unique and cannot be improved upon any further.)
For the strong believer in realism there was only a way out: quantum mechanics is incomplete. This is the conclusion reached by Einstein, Podolski, and Rosen in now their famous EPR paper. If quantum mechanics is incomplete, then there must be some hidden variable, some "beable", theory which captures the essence of nature. These beables cannot be directly measured in any experiments carried out by us, as clumsy macroscopical beasts in the flimsy world of atomic domain. The experimenter must be a hopeless "bull in a china shop," trampling delicate beables when making any measurement.
For a long time after the EPR paper, neither the realists, nor the supporters and practitioners of quantum mechanics could convince the other that their viewpoint was right. Then Bohm found a hidden variable theory, which claimed to recover all predictions of quantum mechanics, and yet it was a realistic theory. That may sound too good to be true, but the price Bohm paid for this was introducing instantaneous action at a distance. Later on, more serious problems were discovered in his model, like its inability to have a realistic description for spin. (For me what I find disturbing in the Bohmian model is how the hydrogen atom is described: two opposite electric charges standing still at a fixed distance from each other.)
However, after Bohm introduced his model, John Bell studied it in detail and played with other hidden variables models, attempting to find a conclusive proof in favor of realism. Specifically, John Bell was studying the correlations of two physical systems separated at a large enough distances that relativity forbids communication between them during the measurement. What he found in the end (and not without irony) was a result now called Bell's theorem, which was a turning point in the demise of all hidden variable theories.
So why is Bell's theorem special? After all, we see correlations in our daily lives and no quantum mechanics is required. For example, if I have one glove in each pocket, and I extract a left glove from my right pocket, I know with certainty without looking that the right glove is left in my left pocket. But here is the catch: if quantum mechanics is right, there are residual effects of its inherent randomness which would result in additional correlations that cannot be explained by any hidden variable theory. We may be clumsy macroscopic objects, but John Bell found a way to circumvent this limitation. And now the discussion between realists and quantum mechanics can come down from the rarefied air of philosophical discussions into experimental verification. The experiment was performed by Aspect and nature ruled in favor of quantum mechanics. Quantum systems exhibit this funny behavior called entanglement where the best knowledge of the whole system does not contain the best knowledge of the subsystems. Hidden variable theories are dead; case closed and let's move on. Or so it seemed.
But the story does not end there. Along comes FQXi's Joy Christian, who recently made a very bold and provocative statement: Bell's theorem is wrong.
Christian argues that Bell proved his theorem by making a "topological error" and by demanding that beables must commute with each other. To see he argument, take rotations in 3-dimensions, for example. It can qualify for the beable status because rotations are trivial macroscopic classical operations. Yet, if you perform two rotations around different axes on a body, you'll find that the end result depends on the order you carry out the rotations. (If you don't believe it, there's a cute demonstration here using a rotating panda.)
Rotations, then, are non-commutative. Therefore it looks like one of the assumptions in John Bell's theorem is artificial and not physically justified. So what happens if this assumption of commutativity is dropped? Here is the remarkable thing: Joy Christian managed to prove in several important cases that entanglement is an illusion due to our commutative prejudice, and complete separable descriptions--that is descriptions of the two physically separated subsystems independent of each other--are possible.
It was at this point that I became very interested in his result, and recently I had completed an analysis of it attempting to put the result in the proper context (http://arxiv.org/abs/1107.1007). Let me try to explain Joy Christian's results and why, in my opinion, the claim that Bell's theorem is wrong in not fully justified.
Key-Ring Model of Hopf Fibration
The way Joy Christian proceeds is by counting all elements of reality in a given experimental setting, meaning that he counts both factuals and counterfactuals. This is not what usually is understood in the context of Bell's theorem, which is supposed to explain correlations between actual experimental results, but let's follow along. The space of all actuals and counterfactuals forms a topological space. For example, in Bohm's variant of the EPR experiment, the experimenter measures spin on one direction, and the outcome can be plus or minus one. The topology of the actual results is a disconnected set of two points (plus and minus one), while the topology of all actuals and counterfactuals is an ordinary two-dimensional sphere corresponding to measuring spin on all possible directions in space. This "S2" sphere is embedded into an "S3" sphere where the so-called Hopf fibration construction is possible.
(The reason "S3" is used is because "S3" is parallelizable. To see what this means let's start with a regular 3-dimensional sphere S2 and use the so-called "hairy ball theorem". Imagine adding hair at any point on this sphere. Then try to comb the hair continuously with no singularities anywhere. This cannot be done: there will always be at least one singularity (in an actual head called a cowlick). S3 is different because this combing can always be performed. The "combed hair" in S3 can represent for example a vector field arising from tracing a particular dynamical evolution of the system. When you can always comb without singularity a manifold, that manifold is called parallelizable.)
So here is Joy Christian's prescription: start with an experimental setting, count all potential experimental outcomes, and then embed this result in a parallelizable manifold. Call the resulting topological space Omega and this represents elements of reality or beables. The particular parallelizable manifolds Joy Christian used were spheres. There are only 4 spheres which are parallelizable: S0, S1, S3, and S7 corresponding to the four normed division algebras: real numbers, complex numbers, quaternions, and octonions. There are other parallelizable manifolds, but the spheres enjoy an additional property critical for disproving Bell's theorem: in any normed division algebra, the norm of the product is the product of the norms. This core property is the key of proving entanglement wrong because one can "disentangle" or separate the whole into parts neatly. This method was applied for the original spin one half EPR-Bohm experiment, for GHZ states, and for Hardy-type theorems. So it seems that Bell's theorem was "disproved" by counterexample.
The original reaction of the physics community was to search for mathematical errors in Joy Christian's arguments. Joy Christian uses geometric algebra for his results, and many original criticisms were plagued by mistakes in using this unusual mathematical formalism.
So is Joy Christian right or wrong? Both actually. His counterexamples are valid entangle-free realistic models. This is were he is right. But one still cannot exceed Bell's classical correlations using only local resources. Rotations as beables are not "actionable beables" and any mechanical device realizing them in forms of gears like the ones used in clocks, for example, cannot exceed Bell's correlations for spatially separated subsystems because relativity forbids rigid bodies. In fact there are two theorems by Clifton (arXiv:quant-ph/9711009v1) demanding commutativity for beables in two very critical cases: relativistic field theories with bounded energies, and for what I would call "non-contextual ontologies". Joy Christian counterexamples are in fact contextual "environmental" hidden variable theories. What do we mean by that? Broadly speaking hidden variable theories falls into two main classes: contextual and non-contextual, while the non-contextual ones sub-separate into two more subclasses: "algebraic" and "environmental". It takes too long to explain in detail here, so I will take a shortcut instead.
Let's take a look at the space Omega. The topology of this space depends critically on which experiments are about to be performed. For the same initial preparation of the system, select different measurements and you may get different separable beables and hidden variables. The ontology of the realistic model is contextual (dependent) on the experimental setting. Contextual realism is almost a contradiction of terms. All interpretations of quantum mechanics aimed at explaining it in usual classical terms suffer from a problem or another: Bohm's theory demands a faster than the speed of light "quantum potential", transactional interpretation uses signals from the future, etc. Joy Christian's local realistic model demands a contextual ontology which is at least just as strange as quantum mechanics itself. Common sense everyday activities have no place for contextual ontology--althoughpoliticians love contextual ontology as they usually claim: "I was not flip-flopping, I was merely cited out of context", to an average person this is a sign of dishonesty and not a valid explanation. (Who knows, maybe if Nixon had known about quantum mechanics, he would have said: but this is how quantum mechanics and nature works, I am not a crook.)
Joy Christian's examples cannot be non-contextual hidden variable theories as they are ruled out by the Kochen-Specker theorem. They cannot be algebraic hidden variables theories because all predictions of quantum mechanics are recovered in the particular experimental setting. The only possibility left are environmental contextual hidden variable theories. And indeed, one can find a particular example where this becomes explicit: an EPR-Bohm experiment with spin one particles (the Heywood-Redhead experiment). The beauty of this experiment is two-fold. First, for spin one, we can use the Kochen-Specker theorem and prove that all hidden variable theories in this case are contextual. Second, Howell and collaborators (arXiv:quant-ph/0105132v2) showed the equivalence of this experiment with two spin one-half EPR-Bohm settings where we can apply directly Joy Christian's analysis. All that is left to do is to marry the two lines of argument and show explicitly that indeed Joy Christian's model for spin one-half is a contextual environmental hidden variable model.
So now is time to tally up the pluses and minuses of Joy Christian's claims.
On the plus side:
Joy Christian's results are important because they show that entangled particles can be disentangled sometimes in a particular mathematical formalism. Entanglement is no longer universal as it depends on the mathematical formalism used. Bell's theorem alone cannot rule out all contextual hidden variable theories obeying local realism. Joy Christian's result does chip away some of the importance and glamour from Bell's theorem, as it cannot be regarded anymore as the silver bullet which killed all hidden variable theories.
On the minus side:
Bell's limit from Bell's theorem still stands and local realistic models of quantum mechanics are still not possible for ALL experimental contexts. Bell's theorem is not "disproved" mathematically because Joy Christian starts with different mathematical assumptions. Bell's theorem importance is not "disproved" because Bell's limits continue to stand for all local operations and classical communication processes. Bell's and Clifton's theorems together now do the job of what people originally attributed to Bell's theorem alone. Hidden variable theories remain a dead end in physics.
I'm grateful to Joy Christian for posting his response to my thoughts: "Quantum Music from a Classical Sphere".