## Joy Christian's program of achieving quantum correlations with Clifford algebras

In the last post I explained how the algebra of the projector operators cannot always be Boolean, otherwise the Hilbert space formalism of quantum mechanics is invalid. Today I will stay in the classical-quantum divide area and I'll talk about an invalid proposal by Joy Christian which generated a lot of debate (and acrimony). When I attended the Vaxjo conference people looking up my archive record saw that I have argued against this proposal and I was asked to explain why it is invalid.

The story begins with the EPR-B experiment and the derivation of the correlation

$$-a \cdot b$$

between Alice and Bob when Alice orients her detection device on direction a and Bob orients his on direction b. So the corelation curve is minus the cosine of the detection angles (the blue line below):

In this experiment the two spin 1/2 particles are in a singlet state:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}( |up \rangle_{left} |down \rangle_{right} - |down \rangle_{left} |up \rangle_{right} )$$

and because the observables are $$a \cdot \sigma$$ and $$b \cdot \sigma$$ the correlation is:

$$\langle \Psi | (a \cdot \sigma)\otimes(b \cdot \sigma) | \Psi \rangle$$

So how can we compute this? We use an identity:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma$$

which yields the final answer because $$\langle \Psi | \sigma| \Psi \rangle = 0$$ as the mean value for both Alice and Bob are zero for any direction because we started with a total spin zero state (a singlet state).

Now Joy noticed this identity and thought that it would be nice if he could use it in a classical setting to recover the $$-a\cdot b$$ correlation. There is a "little" problem: how to make the pesky $$i (a\times b) \sigma$$ term disappear?

So Joy came up with the following proposal: half the particle pairs obey:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma$$

and the other half obey this:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b -i (a\times b) \sigma$$

and so when averaged you get to the quantum correlation: $$-a \cdot b$$

But how can this be possible? It is all in the sign of $$\sqrt{-1}$$ Joy claimed. When complex numbers are represented in a plane, the imaginary unit corresponds to the vertical axis. So for half of the particle pairs we draw the imaginary axis bottom up, and for the other half up bottom. But do we really get the cancellation? Nope because $$a\times b$$ is a pseudo-vector which upon this reflection against the horizontal axis changes signs as well and the identity remains:

$$(a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma$$

and all of Joy's ill fated proposal is based on a "forgotten" -1 sign.

But if Joy would have presented his proposal like this, it would not have gotten very far. Instead Joy explained it all using the language of Clifford algebra which is not at all familiar to physicists. Also there was an associative faulty narrative about "topologically complete reasoning".

The main discovery of Joy was a no man's land at the intersection of math, physics, and philosophy: the mathematicians understanding Clifford algebra knew nothing of Bell, the physicists did not know how to counter Joy's philosophical narrative, and the philosophers had no clue of Clifford algebra. Add to this Joy's aggressive and patronizing defense of his proposal and you get a perfect storm of controversy.

The full story of debunking this nonsense would make for a nice soap opera. I was not the first who noticed the mathematical errors in Joy's proposal, I was the third out of four. Also I was not the first who wrote a paper about it, I was the second one out of three, but I was the first who uploaded it on the archive. There were other archive replies to Joy before me but nobody actually bothered to double check his math. The first reply by Marcin Pawlowski came very close to point out the problem but Joy's reply managed to discourage his critics into challenging his math:

"More specifically, the critics culminate their charge by declaring that, within my local realistic framework, it would be impossible to derive “a scalar in the RHS of the CHSH inequality. QED.” If this were true, then it would certainly be a genuine worry. With hindsight, however, it would have been perhaps better had I not left out as an exercise an explicit derivation of the CHSH inequality in Ref.[1]. Let me, therefore, try to rectify this pedagogical deficiency here."

And so people thought at that time that Joy is wrong, his physics and philosophical arguments were nonsense, but his math was correct and it was not a good idea to challenge him at that. But as it turned out all his math was only smoke and mirrors with more and more mathematical mistakes to cover up the prior ones, and I can write up an entire book about it.

All this controversy has hopefully came to an end with Joy resigning his FQXi membership, but he actually never accepted he was wrong and continues to this day to call his critics arguments: "strawmen arguments".

There was only one person who had more energy and spent more time than me debunking Joy's claims and this is Richard Gill

and the physics community owes him a debt of gratitude for putting this nonsense to rest. There were also two good things coming out of this challenge to Bell's theorem.

First, Sascha Vongehr came with what he called a Quantum Randi Challenge: show you beat Bell's theorem on a computer or shut up. With programming help from Cristi Stoica I came up with this simple Java Script program which runs in any web browser which anyone can use to try to disprove Bell's theorem until they really understand why it is an impossible task.

Second, James Owen Weatherall actually manage to fulfill Joy's hope to eliminate the extra term in a mathematically valid model which was not using Clifford algebras. But then would this count as a "disproof" of Bell's theorem? NO because the actual experimental outcomes are +1 and -1 and the correlations must be computed using them and not in a space of make-believe statistics.

So even if Joy's math were valid, it would not represent in any way a "disproof" of Bell's theorem.

Now what I found completely amazing was that after resigning his FQXi membership Joy received encouragements to continue the fight. It is not clear what fight. The fight to prove +1 = -1? Didn't I say soap opera?

Post Script: In case anyone has questions regarding any past or present mathematical, physical, or philosophical claims by Joy, feel free to ask here and I will answer.
Post Post Script: in an online reply at Sci.Physics.Foundations, Joy claims his model it is not about the sign of sqrt(-1). Oh yes it is, by Hodge duality. And Joy knows that because he used to call the two Hodge duality for distinct Clifford algebras "Joy duality" which earned him high marks in his crackpot index for naming equations after yourself. The moderator of that blog, FreddyFizzx is in cahoots with Joy to promote Joy's ideas and suppress sane opposite points of view.

1. Florin:
Compared to the usual 3-4 pages long proof of Bell's result ,this way using
(a⋅σ)(b⋅σ)=−a⋅b+i(a×b)σ =-a.b is very nice and an undergrad will understand and be convinced in 5 minutes! Do you have a simple proof showing the classical red line?

1. this formula is from Asher Peres's book, page 162:
http://www.fisica.net/quantica/Peres - Quantum Theory Concepts and Methods.pdf#page=177

2. Kashyap,

The classical red line does not have a simple derivation and it involves a bit of geometry. If you have Peres' book, look at pages 160-161. I'll try to explain it here using words.

Imagine a bomb initially at rest which explodes and separates into two parts. The total angular momenta is zero and let the angular momenta after explosion be J1 and J2. Let Alice pick a measurement direction "a" and Bob pick a measurement direction "b". Each measurement direction is perpendicular with a plane (say A for Alice and B for Bob). Alice measures J1 and Bob measures J2. Alice records +1 if "J1 DOT a" is positive and -1 if "J1 DOT a" is negative. Similar for Bob. What is the classical correlation between Alice and Bob outputs of +1 and -1 if the experiment is repeated N times?

Now using the planes A and B cut a unit sphere through the center of the sphere using those planes. For easy visualization pick the angle between the vectors "a" and "b" to be say 30 degrees. Do the drawing on a piece of paper to follow along the argument. You just segmented the sphere in 4 sectors. Color the areas: black, white, black, white. You now have 2 black sectors with an angle of 30 degrees and 2 white sectors with an angle of 150 degrees. The four sectors correspond to alternative signs of the Alice-Bob outcome product and the areas is in the ratio: 30 to 150, or alpha:pi-alpha where alpha is the angle between Alice and Bob measuring directions. The alpha areas corresponds to +1 correlation, and the pi-alpha areas corresponds to the -1 correlation.

If J1 is uniformly distributed the correlation is:

[(+1)*alpha + (-1)*(pi-alpha)]/pi = -1 + 2 alpha/pi which is your red straight line.

1. That proof is good enough pedagogical. Thanks.

2. Actually this is one of those pieces of folk wisdom that turns out not to be completely correct. It's true, of course, as Bell and CHSH showed, that quantum anti-correlations (e.g. for the singlet) can be stronger than any LHV anti-correlations for various values and ranges of theta. But it's not true that the so-called "classical red line" represents the strongest possible LHV anti-correlations for all values of theta between 0 and pi/2. The best results I know of in this direction are in http://arxiv.org/abs/1307.6839 (Physical Review A 90, 062124 (2014)). They include examples of values of theta (close to pi/2) for which LHV anti-correlations are not only stronger than the "classical red line" suggests, but also stronger than the singlet anti-correlations.

3. Adrian, I recall reading your paper in the past, but I need to re-read it to refresh my memory before replying. Is this related to Caroline Thompson's chaotic balls? http://fmoldove.blogspot.com/2014/03/the-detection-loophole-in-bell-test.html

4. Not as far as I'm aware. The LHV models we consider don't use the detection loophole: they always give a definite answer.

I should stress again that our results don't contradict any of the standard results on quantum non-locality, nor (as far as I see) do they contribute to any new exploitable loopholes. They simply add to our understanding of exactly which experiments show a gap between quantum and LHV anti-correlations, and exactly how large that gap is.

5. Adrian, I re-read you paper and it is very interesting. Bell's point was about the flat correlation curve near zero vs. "the kink" from LHV. All 4 coloring models from your paper have kinks which go in the opposite direction (steeper and steeper) compared with QM which is flat due to superposition. In the post quantum domain (like PR boxes) the flatness region is extended beyond theta = 0. With detection loophole one can flatten the kink to generate -cos(theta) and can even go further and create a PR box correlation curve.

The behavior next to theta = pi/2 is most interesting.

3. At least one text on this blog is right.

1. There is only one question left unanswered: did Joy make an honest trivial mathematical mistake, or did he know from the very beginning he was wrong? In other words, is he a tragic figure, or is he like Uri Geller?

People who personally know him but did not investigate his claims in detail say that he is a very nice and sweet person, but the people who argued with him know only his abusive side.

By now Joy knows he is wrong but he continues the parade like in "the emperor is naked" story. And his "model" went through so many changes, I counted them once and I got a large number larger than 10, I think it was 18 but I could be wrong. All along he claimed that the latest model was the "original one". The last one displayed at Joy's website was actually discovered by Richard, and was extracted from an old paper by Perle on the detection loophole.

Joy also proposes an experiment to test his ideas. And this experiment is exactly the example on pages 160-161 from Peres book which I told Kashyap above.

2. I think that by now the charitable 'honest mistake' interpretation is becoming increasingly hard to sustain. I've had a look at Joy's claims recently, in particular his simulation that he claims realizes his model, and as I've pointed out here, it's really just a detection loophole model with the added twist that Joy claims that the hidden variable values leading to data rejection are unphysical---'not allowed by the metric on S^3', I think he would put it---, and thus, he effectively deletes them out of his ensemble. (I wasn't the only, nor the first one to point this out.)

The resulting ensemble then indeed violates the CHSH inequality as he claims, but the problem is that in order to compute the ensemble, i.e. the hidden variable values, explicit knowledge of the measurement directions of Alice and Bob is needed, which of course isn't available in a local model. Joy claims that these physical hidden states can be calculated without reference to those measurement directions, however upon my repeated questioning he's failed to produce any way to do so.

3. Jochen,

I spent two weeks to double check Perle's math in his paper, from which Richard extracted a particular model from it. That math computation is not trivial by any means. At the time Richard and Joy had a bet and it is simply amazing how Joy took that model and claimed it was his all along. It is not. The original Joy proposal had nothing to do with the detection loophole.

If you follow Joy's papers in historical fashion you see a comedy of mathematical errors each new one attempting to cover up the prior one. My personal favorite is this:

limit as epsilon goes to zero of (epsilon/epsilon) = 0 because the numerator goes to zero.

This was in reply to an objection by Marc Holman who noticed that the original Joy model was so rigidly deterministic that it never allowed for the +- or the -+ outcomes and only produced the ++ and -- outcomes.

4. @Luboš Motl July 19, 2015 at 1:31 AM
Sorry, this is all wrong also. Just a bunch of lies.

5. @The person posting Anonymous from UK. (can this be Joy?)

Some arguments to back your assertions would be nice.

6. http://fmoldove.blogspot.com
http://fmoldove.blogspot.co.uk

Which one is it?

7. "http://fmoldove.blogspot.com
http://fmoldove.blogspot.co.uk

Which one is it?"

You know, blogger.com has excellent reporting capabilities. It tells you the country, the browser type, the operating system, etc of all incoming traffic.

4. Joy Christian has responded at http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=183#p4944

1. I saw that and I invited him to debate here. Then someone from UK posted above: "Just a bunch of lies.". I think that was Joy.

5. Especially interesting is the second image (LaTeX display) showing two lines which appear to contradict one another. However, according to the discussion later in that thread, the contradiction is only apparent. The geometric product on the left hand side of the second equation belongs to -I and is the transpose of the one on the left hand side of the first equation, which belongs to +I.

1. I am amazed Joy still continues with his charade.

2. Yes it is extraordinary. The charade only impresses a few idiots. But it is fun watching the idiocy increase.

You wrote: Joy came up with the following proposal: half the particle pairs obey:

(a⋅σ)(b⋅σ)=−a⋅b+i(a×b)σ

and the other half obey this:

(a⋅σ)(b⋅σ)=−a⋅b−i(a×b)σ

and so when averaged you get to the quantum correlation: −a⋅b.

But what Joy has been saying for about a month now is that all this time (8 years?) we have been misreading his formulas. Actually he is saying that half the particle pairs obey:

(a⋅σ)(b⋅σ)=−a⋅b+i(a×b)σ

and the other half obey this:

(b⋅σ)(a⋅σ)=−a⋅b−i(a×b)σ

and so when averaged you get to the quantum correlation: −a⋅b

The mathematical claim is now correct!

The problem now shifts to the question, which nobody seems to think about, why the hell are we averaging products of bivectors, anyway? That is because of a crazy proposal to replace the ordinary correlation between +1 and -1 valued observations with a Pearson correlation for (noncommutative) bivectorial observations. Which are not the observations made by experimentalists. They just get to count detector clicks. The physicist's correlation between +/-1 valued observations is just the fraction of equal observations minus the fraction of unequal observations. It's all about counts of coincidences and anti-coincidences.

The math of Joy's model is totally irrelevant, because the "model" itself is not a model of anything in reality at all. It's just some kind of wet dream.

3. The problem is that you cannot average the two. If you can add the two equations then you are also allowed to subtract them. And what do you get?
$$0=2i(a\times b)\sigma$$
which means axb = 0 and you get nonsense.

4. Florin, when you substract *your* two equations you get nonsense. When you substract *my* two equations you get sense. (a⋅σ)(b⋅σ) is not equal to (b⋅σ)(a⋅σ). The geometric product is not commutative.

What nobody noticed for about 8 years is that when Joy writes his two equations, which then seem identical to yours, he actually means my two equations.

He only realised this himself, about two months ago. See the discussion at http://vixra.org/abs/1504.0102

I wrote (two months ago): "The algebraic problem identified in my viXra paper has at last been fixed, by use of two geometric products: a left hand geometric product and a right had geometric product, depending on the sign of the hidden variable. Thanks to the combined efforts of Heine Rasmussen (who first suggested this solution), Albert Jan Wonninck (whose GAViewer code made it painfully obvious that something needed to be done), Fred Diether (who further polished Wonninck's code)."

Joy initially disagreed with this version of the history, and Heine disowned it, but Fred corrected his master, and wrote: "Well to be fair, it was Heine's addition of a.b that somewhat lead us and Albert Jan to finally realize that looking at a left handed system from a right handed only perspective that the geometric order would be reversed. The solution is very simple and elegant and right arm!"

After that, Joy acquiesced. And this is now the party line.

5. I gave up following the latest twist and turns of what Joy claimed it was his original model all along.

(b⋅σ)(a⋅σ)=−a⋅b−i(a×b)σ is the latest incarnation of the so many different variations of the attempts to arrive at -a.b, but I only consider the very first Joy proposal. If Joy will ever admit that was mistaken, then I can entertain considering a new proposal.

6. Jochen wrote "I think that by now the charitable 'honest mistake' interpretation is becoming increasingly hard to sustain". However I think that the problem is that we have been *over*-estimating Joy's *mathematical* intelligence all these long years. I think that his *mathematical* intelligence is close to rock-bottom zero. So to answer the question "is he a tragic figure, or is he like Uri Geller?" I would say: this is beyond tragic, this is quite pathetic. The really interesting thing is why he was taken so seriously for so long.

7. Have a look at the “Retraction Watch” blog.

Entry: “Physics journal retracts paper without alerting author”

Retraction of the paper: “Local causality in a Friedmann–Robertson–Walker spacetime"

http://retractionwatch.com/2016/09/30/physicist-threatens-legal-action-after-journal-mysteriously-removed-study/#comment-1242479

8. Florin:
I would like to understand whether the equations (67) - (75) in Joy Christian’s paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” make any sense at all. I don't understand how the mathematical limes operation are carried out.

9. When following Joy Christian’s answers to concrete questions at the “Retraction Watch” blog (entry: “Physics journal retracts paper without alerting author”), one observes a simple pattern. In most cases, he never responds in a plain way; he often merely tries to divert attention from the questions by concealment tactics pointing to a new set of problems. That is easily comprehensible.

I think that Harry G. Frankfurt’s essay "On Bullshit" (2005) describes such a - so to speak - cat-and-mouse game concisely:

“Frankfurt determines that the bullshit is speech intended to persuade (a.k.a. rhetoric), without regard for truth. The liar cares about the truth and attempts to hide it; the bullshitter doesn't care if what they say is true or false, but rather only cares whether or not their listener is persuaded”.
(from https://en.wikipedia.org/wiki/On_Bullshit)