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)

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    $\begingroup$ as a general rule, better to cite the abstract page of the paper on arXiv. Amongst other things, that gives a chance to see whether the author has filled in a journal reference, and also to download, for example, postscript instead of PDF. $\endgroup$ Commented Mar 24, 2011 at 15:34
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    $\begingroup$ There are various analyses of Joy Christian's work, such as arxiv.org/abs/0707.2223v3. He cites a few himself in his "reply to critics". Quite a few people in the field know him personally, which may be part of why he gets attention when others don't. People have had trouble finding a clear way to show him why his construction is not interesting, wrong, or whatever. Note that his construction uses quaternions in an essential way, eq (3) in the paper you cite. I remember seeing it said, amongst other things, that his model is contextual, but I couldn't find where I saw that said. $\endgroup$ Commented Mar 24, 2011 at 16:04
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    $\begingroup$ Dear Helder, the "disproof" paper is complete rubbish. Bell's theorem is a theorem that means a very particular thing and that is trivial to prove, see e.g. pages 7-8 of karlin.mff.cuni.cz/~motl/entan-interpret.pdf - and if it may be proved, it cannot be disproved. In equations 1,2,3, the author wants to establish some non-commuting observables in his "local realist" theory. But that's not possible and the meaningless formalism is just masking it. In equation 5, he shows he has no clue what $\lambda$ means - he has several $\lambda^i$ variables and sums over the index. Just BS. $\endgroup$ Commented Mar 25, 2011 at 14:05
  • $\begingroup$ Comments on "Disproof of Bell's theorem" by Florin Moldoveanu (July 06,2011) $\endgroup$ Commented Jul 12, 2011 at 14:02
  • $\begingroup$ Clifford algebras are used "in the context of Bell's theorem" since Tsirelson work 1993 and so it is not "a new idea". $\endgroup$ Commented Jul 12, 2011 at 19:07

6 Answers 6


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.

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    $\begingroup$ I did not notice that in the paper he is only citing himself. Anyway I'm convinced that a half-page of math would be insufficient. Thanks L.B.Crowell (Your book is very nice :) $\endgroup$ Commented Mar 24, 2011 at 14:33
  • $\begingroup$ @Helder, the other papers spell out more of the argument. $\endgroup$ Commented Mar 24, 2011 at 14:50
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    $\begingroup$ I have seen many comments on my work on Bell’s theorem during the past four years, but I haven’t seen one as ill-informed, prejudicial, closed-minded, and silly as this. It makes it plane that Mr. Crowell has absolutely no understanding of Bell’s theorem, let alone my own argument against it. The readers of this blog are urged to make up their own minds after reading what I have to say in that half-a-page. On the other hand, I appreciate Peter Morgan’s comments, but my latest paper is simply an explicit counterexample to Bell’s theorem, whose demonstration presupposes contextuality in any case $\endgroup$
    – user2745
    Commented Mar 24, 2011 at 16:23
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    $\begingroup$ Sorry Lawrence, I downvoted this partly because of your comment "as quickly as possible". The "Oh crud" is most of the rest of my reason. Your Answer also seemed to me too rushed, but I almost never downvote because "not useful" is so strong. I admit a personal aspect, which you can see openly elsewhere here, that I knew Joy, though not well, for about 6 years, until about 7 years ago. I presume he would downvote your Answer if he could. If Joy decides to play the SE "game", he's smart enough and knows enough to accrue reputation. $\endgroup$ Commented Mar 24, 2011 at 16:58
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    $\begingroup$ I think this is becoming an unmanageable conversation. Lawrence, I'm not sure from the above that you have understood the full array of assumptions that are required to derive Bell inequalities. Perhaps try, out of an infinity of Bell citations, my "Bell inequalities for random fields", J. Phys. A 39 (2006) 7441-7455; if you don't like the QFT context of that paper, or think it's too long or not clear enough to bother with, perhaps the Valdenebro A G 2002 Eur. J. Phys. 23 569 citation I give there, which I like. Mine is not great paper, sadly. $\endgroup$ Commented Mar 24, 2011 at 20:23

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.

  • $\begingroup$ Peter, this deserves a point +1 at least because you are showing that the argument is a little bit condensed. I have been searching the other papers for (5)-(7) but no luck yet. The point about p/q commutativity is important too. The lambda definition could be explained more too, although a kind of explanation of $\beta(\lambda)$ could be inferred from other papers. $\endgroup$ Commented Mar 25, 2011 at 15:13
  • $\begingroup$ @Roy, my impression is more that the argument can be more condensed. Just before eq(4) [corrected, I originally wrote eq(5)], Joy writes that $\beta_i(\lambda)=\lambda\beta_i$. Why can't he use that equality from the beginning? Doing so makes the argument look rather(!) different, eq(1) and eq(2) become trivial. Enough so that I feel I might be reading it all wrong. $\endgroup$ Commented Mar 25, 2011 at 18:18


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.

  • $\begingroup$ I ain't been able to make this Question work well. It's a bit out of control. I'm very conscious that my comments are responding to Lawrence, not to the OP. Joy used to have a community in Oxford, and if he still does he may blow this off. FWIW, I thunks that I agrees with your italicized para, +1, but the assumptions are very tricky, and can be put in many different ways, some that are almost alien to my Bell paper in JPhysA. I think topological is not an issue, insofar as continuity as such plays no part in Bell inequalities. I thunks there are many issues. If you get clear, ... $\endgroup$ Commented Mar 24, 2011 at 23:04
  • $\begingroup$ @Peter, I have seen your comments to Lawrence, and have no idea whether Joy will return to this site. You could post an answer directly for the OP expressing your current views on what works and what seems questionable in these papers. Or maybe post another question as this is technically only about this latest small paper. $\endgroup$ Commented Mar 25, 2011 at 9:15
  • $\begingroup$ I put up what I think is an Answer, which is as close a reading of the math as I could manage. Thanks for your patience. $\endgroup$ Commented Mar 25, 2011 at 13:09
  • $\begingroup$ Two particle were born with spin and anti-spin and then they move far away. When we measure one we know the state of the other without comunication. Why are we saying and measure, on x,y,z axes, and, as Einstein did, expect to obtain less than 25% ? I think that light only respects his referential: head-on and anti head-on. This gives always 50%. When I read the Lubos lecture on entanglement and see angles of 120 degrees between detectors I see a probable complication. Do you think that Joy is doing a paradigm shift or is only math? $\endgroup$ Commented Mar 29, 2011 at 15:46

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.

  • $\begingroup$ @Joy This may be moved by the moderators as well. The comments are restricted to 600 characters, which keeps everyone from being too verbose in comments. That's good for me! The moderators idea that you should leave comments was too hasty because you can't leave comments except on your own Questions and Answers until you have 50 Reps. That was a large part of why I upvoted your Answer. I'm afraid that the Answer as you have initially written it here is not something that will convince anyone, but you can edit it, and it can be as long as you like, I think. $\endgroup$ Commented Mar 24, 2011 at 19:00
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    $\begingroup$ Dear @Joy, don't take @Lawrence's comments personally. This site has all sorts of people with all sorts of opinions. The best way to get your point across, IMHO, is a point-by-point rebuttal of any criticisms in a, hopefully, neutral language. I have a feeling you have to rebut a LOT of criticism, so it must get really tiring. But then again, I'm sure you knew what you were getting into when you decided to write papers with such titles as "disproofs of bell's theorem" ;) $\endgroup$
    – user346
    Commented Mar 24, 2011 at 19:11
  • $\begingroup$ @Joy, Welcome! I think that your comment became this answer moved here by the moderators (the comment on LC is still present). The answer could be reworded (EDIT) to introduce your main points and address some common misconceptions about your work. One is really addressing the questioner here, rather than other Answerers. $\endgroup$ Commented Mar 24, 2011 at 19:24
  • $\begingroup$ Bell’s theorem sets up inequalities using classical set theory and illustrates how quantum mechanics violates them. Quantum mechanics is a relationship of states based on projections and the span of vector subspaces, not on unions and intersections. That really is at the core what this tells us. In some ways it is a rather trivial result. Oh well, I will let this stand at this point. OTOH, where is Lubos Motl when you really need him! $\endgroup$ Commented Mar 24, 2011 at 19:31
  • $\begingroup$ @Joy: I'd forgotten about the fact Peter mentioned, that you can't leave comments (except on your own questions or answers) with less than 50 reputation. On the other hand, what you've written in this post isn't an answer. So: I'll leave this here for now and upvote it so that you have the 50 reputation required to leave a comment. At the end of the list of comments on Lawrence's answer, you should now see a text box where you can enter your own comments. $\endgroup$
    – David Z
    Commented Mar 24, 2011 at 20:43

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.


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.


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