Disproof of Bell’s Theorem

Question

The half-page arxiv doc by Joy Christian of Oxford Uni, UK has the Title and Abstract:

Disproof of Bell’s Theorem

We illustrate an explicit counterexample to Bell’s theorem by constructing a pair of dichotomic variables that exactly reproduce the EPR-Bohm correlations in a manifestly local-realistic manner.

As The Bell's theorem is often cited it seems important to clarify this short doc.
Is the doc OK?

This recent paper Comments on "Disproof of Bell's theorem" by Florin Moldoveanu (July 06,2011)
can shed light in the interpretation and importance of the work of Joy and may be the case that the answers could benefit from it.

One conclusion that seems relevant, imo, is that Geometric Algebra is very promising to model the physical world.

LM post @TRF pointed to a pre-print of Gill (Simple refutation of Joy Christian's simple refutation of Bell's simple theorem)
and the Joy answer (Refutation of Richard Gill's Argument Against my Disproof of Bell's Theorem)
(thanks LM)

Answer 1

Oh crud, this! I read this paper a month ago. What Joy Christian does is to write out the Bell inequalities, and then effectively identifies quantum states with the variables in the terms of the inequality. This is silly, for the whole point is to erect the inequalities and then demonstrate how quantum states violate them. Joy identifies the quantum states with the elements in the inequality. The whole thing is what I would call tautologically false.

The only thing worse that somebody trying to disprove nonlocal properties of quantum mechanics are knuckleheads who try to show that relativity is all wrong. There seems to be an endless stream of this sort of nonsense. These types of papers are simply best ignored.

---

{\bf Addendum}

Joy Christian tacitly equates the elements in the inequality with the quantum states because there is no sign value to the outcome of a quantum measurement. There is this 2-1 relationship with a Bloch sphere and the $R^3$ sphere. The lack of this sign, say a spin up or down, involves equating the average of noncommuting variables with average of commuting variables. It is for this reason that the quantum variable effectively slip into place with the classical probability variables in the inequalities.

I do understand Bell's theorem well enough. It is a demonstration that quantum mechanics does not obey classical set theory. The corresponding case classically involve projecting onto subspaces of an entangled state $$ |\psi\rangle~=~1/\sqrt{2}(|+,-\rangle~+~|-.+\rangle) $$ for the singlet state configuration. So the Pauli matrices for the two are $\sigma_i~\tau_i$, the set of projector operators on the 1 and 2 states are employed $$ P(1)_z~=~(1/2)(1~+~\sigma_z), P(2)_z~=~(1/2)(1~+~\tau_z) $$ and for the 45 degree case $$ P(1)_{45}~=~(1/2)(1~+~(\sigma_z~+~\sigma_x)/\sqrt{2}), ~P(2)_{45}~=~(1/2)(1~+~(\tau_z~+~\tau_x)/\sqrt{2}) $$ and $$ P(1)_x~=~(1/2)(1~+~\sigma_x), P(2)_x~=~(1/2)(1~+~\tau_x). $$ The projections onto the entangled state which correspond to the classical probability rules is $$ Prob(|, /)~=~P(1)_z*P(2)_{45} $$ $$ Prob(/, \_ )~=~P(1)_{45}*P(2)_x $$ $$ Prob(|,\_ )~=~P(1)_z*P(2)_x $$ Some calculations with the matrices and the states leads to the Bell result that this does violate the inequality $Prob(|,\_)~\ge~Prob(|,/)~+~Prob(/,\_)$ expected classically.

In effect the classical inequality is derived from union and intersection rules $\cup,~\cap$, with OR and AND logical meaning, while the quantum analogue involves additions and products of operators to construct “spans” on a vector space.

There are these extraordinary claims which come about now and then. Of course 100 years ago the claims of quantum mechanics would have been seen as extraordinary. Even Einstein did not like it, though he laid down some initial groundwork on it. However, the physics of quantum mechanics and its unusual implications have a vast data base of experimental support behind them. There is no “crisis” with quantum mechanics, where the potential deviations may lie near the Planck scale. However, for most purposes these are not important. Claims that quantum physics is wrong, or that there are hidden variables or that Bell’s theorem is wrong or violated have come and gone. Much the same is likely to happen here as well.

I have spent far more time on this than I wanted to.

Answer 2

My reading of Joy's paper —just as it is, without having carefully read the arXiv paper I cited, nor all of Joy's responses to critics that I also mentioned— is, so far: the left and right hand sides of eq(1) and eq(2), without the central interpolations, state that $A(\mathbf{a},\lambda)=\lambda$ and $B(\mathbf{b},\lambda)=-\lambda$, where $\lambda$ takes the values $\pm 1$. $A(\mathbf{a},\lambda)$ and $B(\mathbf{b},\lambda)$ are independent of $\mathbf{a}$ and $\mathbf{b}$, respectively, hence the expected value of the product is $-1$.

The central interpolations introduce nine algebraic objects, each of which is the basis of and satisfies the algebraic relations of a quaternion algebra, $\beta_i$ and $\beta_{i'}(\lambda)$, with $\lambda=\pm 1$. For $\lambda=+1$, the $\beta_{i'}(+1)$ satisfy the same algebra as the $\beta_i$; for $\lambda=-1$, $-\beta_{i'}(-1)$ satisfy the same quaternion algebra, with the sign change to be noted. To fix the algebraic structure further, which is absolutely necessary so we know how to handle products like $\beta_i\beta_{i'}(+1)$, Joy states that $\beta_{i}(\lambda)=\lambda\beta_i$, so we are in fact dealing with a purely quaternion algebra, of real dimension 4. The whole of the prelude to eq(5-7) could be stated using only $\beta_i$; for me the $\beta_i(\lambda)$ just obscures things. I would like to see a mathematical justification for introducing the $\beta_i(\lambda)$ instead of just using $\lambda\beta_i$.

The notation of eq(5-7) is problematic because it seems to play fast and loose with the non-commutative structure of the quaternions. One cannot in general write $\frac{p}{q}$ for two quaternions $p$ and $q$, because in general $pq^{-1}$ is different from $q^{-1}p$. Since eq(5-7) obtains a different result from the result that I get in my first paragraph, I'd want to see the whole thing rewritten using inverses so that the order of the multiplications is kept under control. Unless there is a potent reason for using the $\beta_{i'}(\lambda)$ notation, I'd like to see everything written out using only the $\beta_i$. If the answer is still $-\mathbf{a.b}$ I'd want to check that it does not make any unwarranted reversal of the quaternions $a_i\beta_i$ and $b_i\beta_i$, even one of which would be exactly enough to get the result $-\mathbf{a.b}$ instead of $-1$.

I currently cannot see any way to justify the jump from the left hand expression of eq(6) to the right hand expression. Perhaps someone can show me how to get from one to the other.

If my discussion above is OK, this leaves questions about Joy's earlier papers. My impression is that Joy tried to make the argument of his earlier papers as succinct as possible. He may have made a mistake in doing so, in which case, if he claims the earlier papers do not make any mistake, then they have to be considered on their own merits. On the other hand, before I would consider checking that I think I would want to see Joy withdraw or replace this paper on the arXiv with something that at least made play at addressing my discussion here.

Finally, I look forward to comments.

Answer 3

Helder,

A lot of comments surround this question and we are awaiting some further responses from the Author of the papers. However this is my understanding of the conclusion of these papers:

Every theory - even classical physics - violates the Bell Inequalities

So in a sense there is no dispute with the Bell calculation as a demonstrable result of Quantum Mechanics. All of that and the Aspect experiments are fine and to be expected.

The problem, it is claimed, lies in a mathematical (topological) assumption at the root of the Bell calculation about the geometric nature of classical spin. Once this is taken account of then one can show that a corresponding classical system would have similar properties. Indeed Joy has even proposed a classical experiment involving an exploding ball to test this result (I cannot find the link right now.)

There are mathematical technical counter challenges in the Papers quoted by Peter Morgan, and there is the question as to what this all would really mean for the other Classical-Quantum questions that exist (determinism, non-locality, etc) and I am not too clear about that as yet. I would hope that further answers could clarify these aspects. Otherwise after I study these other papers I might be able to finalise an overall opinion.

Answer 4

Mr. Crowell has yet again demonstrated his total lack of understanding of Bell’s theorem, as well as my argument against it. To begin with, Bell’s theorem has nothing to do with quantum mechanics per se, or classical set theory for that matter. It is a theorem about any possible future theory of physics---without prejudice or preconception displayed by Mr. Crowell concerning what that theory could be---and involves only some very basic assumptions about the completeness of that theory, and whether or not it could be a locally causal theory. My papers too have nothing to do with quantum mechanics per se (so please stop misrepresenting them without ever having read them). They have to do with how Bell’s argument against a certain class of possible locally causal theories is simply wrong, and more importantly how we can better understand the origin and strength of the observed correlations in nature. Mr. Crowell, you DO NOT understand Bell’s theorem, or my work in particular. Your opinion about my work is based on prejudice and ignorance.

Now in response to the original question raised by Helder Velez let me describe the essence of my argument in the short paper. Given the background described above, the basis of Bell’s theorem is the claim that no local and realistic model can reproduce the experimental data observed in the EPR-Bohm type bipartite experiments. Therefore, all one has to do to refute Bell’s theorem is to produce such a local-realistic model. In the above paper I show that half-a-page is all it takes to produce such a model. No elaborate arguments are needed, since Bell’s argument explicitly rests on the impossibility of such a local model. Thus, a correct response to Helder Velez’s question should have been an attempt by the participants to assess whether or not I have produced the local model I claim I have produced. But that is not what we are getting so far.

Answer 5

The short answer to the OP is "no". Since then there has been little interest in the Christian model.

However, there have been some recent internet forum discussions on implementing Christian's model using software for geometric algebra. Moreover, one paper got published in Springer's journal IJTP: the International Journal of Theoretical Physics. This led me to look again at some of Christian's papers and look again at the possible use of geometric algebra in quantum theory: for a preliminary report see http://vixra.org/abs/1504.0102

It seems that many early critics found geometric algebra too abstract and unfamiliar in order to actually work through the maths in Christian's papers. However, the most simple geometric algebra used in these works is actually very easy to understand and moreover already very familiar to those working in quantum information theory: it is nothing more or less than the algebra of 2x2 complex matrices over the reals. Thus we have matrix multiplication, matrix addition, and scalar multiplication by reals. The latter making this algebra an eight-dimensional vector space.

Once one knows what mathematical universe Christian is operating in, it is very easy indeed to work through this mathematics ... and run into both conceptual errors and algebraic errors.

Already, the pioneers of geometric algebra (Hestenes, Doran, Lasenby ...) had worked out in detail how to rewrite the usual mathematics of quantum information theory in the language of geometric algebra. There were a lot of papers in the late 90's and early 20's ... but it did not catch on. For a single qubit, nothing new happens, and the geometric picture of rotations of the Block sphere is already very familiar. For several entangled qubits, the dimensions of the usual Hilbert space model and the obvious corresponding tensor product Clifford algebra space do not match - the latter is too large. An ad hoc fix has to be made in order to recover the "usual" objects inside the Clifford algebra objects. It appears that no new insights were generated, so all we had was simply yet another parametrisation and yet another collection of computational tricks.

Answer 6

After reading relevant papers and comments I have impression that it is sometimes using the same terms by different people for different things. In his paper Bell described concrete values, particular physical experiments, but not some abstract theoretical formula and he discussed that the results of the experiments may not be explained by classical correlations.

Joy Christian wrote: “Central to Bell’s theorem [1] is the claim that no local and realistic model can reproduce the correlations observed in the EPR-Bohm experiments.”

I doubt, Bell achievement could be formulated in such a simple way: let us recall discussions that even the usual quantum mechanics without reduction is local theory (from some works of Deutsch et al or maybe even directly from definitions). It is possible to consider standard expression for correlations in quantum mechanics as a formal postulate and it is precisely reproduced in all experiments. From mathematical point of view this expression might be equivalent with formula of Christian (if later is correct, after all). The problem is not to write an abstract expression for correlation, but to explain, how it appears in real experiment.

The subtlety is that reduction or some other way of transition from quantum to classical world used in description of experiments produce problems and misconceptions with interpretation of non-local effects and it is just a reason of late understanding of significance of Bell work and subject of many discussions on foundations of quantum mechanics.