## Monday, 30 March 2015

### Numerical validation of vanishing of the second term in the proof of Joy Christian: The prequel

My previous post, and the code there, successfully showed that in the formula 19 from the article from Joy Christian (1) the second term in the second line vanishes (the integral having the outer product).
In the discussion that followed it became clear that the step from line 1 to line 2 is not obvious. This step is explained earlier by Joy in the article in formula 17, where µ can be either I or -I.

However, when using -I in GAViewer, 17 does not hold. The reason for this seems to be that in GA and GAViewer the I has a fixed handedness. So more in detail, a positive value of I always corresponds with a right handed coordinate system, and a negative I with a left handed coordinate system.

The situation can somewhat be compared with a (fixed) transparent clock. As in GA one can look from both sides to get the desired result, in this case the time. But on one side one has to calculate the dots from the top clockwise to the hand to get the correct time, on the other side counter-clockwise.
So if nature is able to randomly change the front and the back of the clock, as in Joy's model, the persons looking at it have to compensate their calculation of the time accordingly.

So the code below checks for the handedness specified in lambda, and then adapts the calculation. It shows again that the second term vanishes, and the expectation value E(a,b) equals -a.b for different randomly chosen unit vectors for a and b. A typical result in GAViewer from the program below

Addendum: Joy added appendix A to Macroscopic Observability of Spinorial Sign Changes: A Reply to Gill,  http://arxiv.org/abs/1501.03393, in which the findings from this simulation are explained.

function getRandomLambda()
{
if( rand()>0.5)
{
return 1;
}
else
{
return -1;
}
}

function getRandomUnitVector() //uniform random unit vector:
//http://mathworld.wolfram.com/SpherePointPicking.html
{
v=randGaussStd()*e1+randGaussStd()*e2+randGaussStd()*e3;
return normalize(v);
}

batch test()
{
set_window_title("Test Joy Christian");
N=10000;
I=e1^e2^e3;
s=0;
aa=getRandomUnitVector();
bb=getRandomUnitVector();

minus_cos_a_b=-1*(aa.bb);

for(nn=0;nn<N;nn=nn+1) //perorm the experiment N times
{
lambda=getRandomLambda(); //lambda is a fair coin, resulting in +1 or -1

v=I.aa ;
w=I.bb;
q=0;
if(lambda==1)
{
q=v w;
}
else
{
q=w v;
}

s=s+q; //summation of second term. "." is inner product,
//"op(aa,bb)" is outer product

}
mean_mu_a_mu_b=s/N;
print(mean_mu_a_mu_b); //print the result
print(minus_cos_a_b);
prompt();

}

1. Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables, Joy Christian, 2010, http://arxiv.org/pdf/quant-ph/0703179v3.pdf

1. And I will add that when we do Heine's maneuver of adding aa.bb to the average we get,

mean_mu_a_mu_b = 0.01*e2^e3 + -0.00*e3^e1 + 0.00*e1^e2

Everything vanishes as it should. So this a simple and elegant way of compensating for GAViewer being in a fixed right hand bivector basis.

1. I have updated my latest arXiv refutation of Gil’s vacuous claims. I have added an appendix discussing the reasoning behind Albert Jan’s code, which at last puts an end to the nonsensical claims by Gill and other naysayers:

http://arxiv.org/abs/1501.03393 .

2. Here is a link to a .g text file with code above with a small addition for testing Heine's maneuver.

http://1drv.ms/1HfBmzm

1. This new program confirms the identity

(v w + w v)/2 = -aa.bb

Is this Joy's theory?

2. This new GAViewer script confirms what you have to do to see Joy's model (which is clearly stated above in Albert Jan's description) from a right handed only perspective.

3. The translation of my model to the GAViewer with a fixed "+I" is:

(-I.a)(-I.b) ---> (I.b)(I.a) ,

as explained by Albert Jan and Fred.

4. And for a simple demonstration using GAViewer,

>> product=w v
product = 0.38 + 0.57*e2^e3 + -0.36*e3^e1 + -0.63*e1^e2
>> t=crosspr(aa,bb)
t = 0.57*e1 + -0.36*e2 + -0.63*e3
>> result=-aa.bb+I.t
result = 0.38 + 0.57*e2^e3 + -0.36*e3^e1 + -0.63*e1^e2

product = result so reversing the order does give us the proper identity when lambda toggles to -1. It's a slam dunk folks. The model works as advertised.

5. 6. Got a flash of deja vu. Seems like this was all done analytically on the FQXi blog 2 or three years ago. But now we have proof via a computer program that Joy's model is correct.

E(a, b) = -a.b

7. As Heinera remarks, Jan Albert's code confirms some well known identities from geometric algebra. They show that Christian needs to rewrite his math.

And after Christian has done that, he can try again to explain why his complete model is a local hidden variables model. The couple of lines which Albert Jan has kindly repaired for Christian are not the complete story. Now the GAViewer project must be extended to include what leads up to formula (19). Where is are the measurement outcomes A(a, lambda) and B(b, lambda)?

This was a good start. However, I prophesy that it will not help. Bell's theorem has not been disproved yet. In fact, Bell's theorem is true. So Christian's project is bound to collapse, sooner or later. The bump under the carpet can be moved about, but it cannot be made to vanish altogether. It has now got shifted from formula (19) to one of the surrounding formulas.

8. Bell's theorem was disproved in 2007:

http://libertesphilosophica.info/blog/

3. This comment has been removed by the author.

4. Yes, this is fantastic! It proves that Christian's mathematics is stuffed full of errors. In order to make the derivation correct, one needs ad hoc local fixes. The theory needs complete rewriting.

The next stage for Albert Jan is to add in the prequel to the prequel. Where are the binary measurement outcomes? The formula Albert Jan presently starts with is already derived from earlier assumptions.

This brings us to Christian's one page paper http://arxiv.org/pdf/1103.1879v1.pdf, where the complete model is described, starting with binary outcomes A(a, lambda) and B(b, lambda). Please take a look at http://rpubs.com/gill1109/onion where I have coded the complete model in R. It is easy to do the same in GAViewer.

1. Little yappin' doggie is in denial. GAViewer has just proven that Joy's model does give the result -a.b. Nothing else needs to be done. Learn enough GA so you might understand what has just happened here. Physics is not going to be the same.

2. Gill claims to have coded my model in R.

Any moron can write a code that does not work.

3. I have coded the model of the one page paper in R. Completely, from beginning to end. I hope that Albert Jan will try in GAViewer.

GAViewer has helped us find mistakes in formula (19). They need to be fixed. Now we need to extend the GAViewer program to include the steps leading up to (19). We don't have a counterexample to Bell's theorem yet because we don't yet have binary outcomes A(a, lambda) and B(b, lambda) and we haven't been shown that the resulting hidden variables model is a *local* hidden variables model.

So: please continue the good work!

4. Bell's theorem has been dead since 2007:

http://libertesphilosophica.info/blog/

5. Actually, I was wrong when I said there is nothing more to do. The macroscopic experiment needs to be done to see if the model holds macroscopically.

1. To begin with, simulate the macroscopic experiment. Now that Albert Jan has fixed a bug in one line of the theory, show us that there are no remaining bugs.

2. Gill's right handed brain does not connect with his left handed brain. LOL! Woof, woof! Elephants have arrived at their destination for now.

3. Elephants have still got a long, long way to go. Bell's theorem says they will never reach their destination. So come on dear elephants, now that Albert Jan has fixed one bug, use his help to simulate your macroscopic experiment. Simulate a local hidden variables loophole free successful Bell-CHSH experiment. I wager 5000 Euro you can't do it.

4. Bug???? The dingo ate your baby!

5. You think so, Fred? OK, simulate the complete model now. Not just one line of calculation in the middle. Prove by a computer simulation that the macroscopic simulation has a chance of success. You want to have it done, don't you?

6. Maybe the dingo ate your baby? Woof, woof!

6. And for your viewing pleasure, here is a link to a picture from GAViewer of the model for one instance of vectors a and b.

https://rzz4qa.sn2.livefilestore.com/y2phZMXx5kXsMQgKmI_ojumOn2lO3uU5nQBaO1yq2_Fqq7wOQuTGs1LZlIf8MYwlkQoUfMJ9fQNDWxGPvtUN1IQtS2OZIMQUfCu80pBq4GIe0wKEilttBwuNmVEgxRwBX1gYFrpnI84OizEq5wqrwsiIw/joy.png?psid=1

Hope this works.

7. So: Albert Jan has identified a bug in one line of the theory. Now we need to find out if the rest is still OK.

1. The only bug around is a bug in your right handed brain that prevents you from seeing the left handed part in a right hand perspective.

8. Now we wait for a simulation of the macroscopic experiment. Albert Jan can try to win the 5000 Euro I offered. (Joy and Fred each already sent in submissions, which failed. The rules are that any single person may only make one attempt).

1. The path to truth is thinner than a razor's edge!

9. Albert Jan: could you now take a look at the one page paper? I'd like to see that in GAViewer too. In that paper we start with the construction of A(a, lambda) and B(b, lambda). You know, the ones which according to (1) and (2) are equal and opposite and either equal -1 or +1. Next, notice in formula (5) the unconventional definition of "correlation". Then we have formulas (5) to (6) which need to be fixed with the same trick as you have used in this thread.

10. Albert Jan did the most important part already. The average of (I.a)(I.b) = -a.b. The rest is merely window dressing. And since all of it has been explained to you in extreme detail many times, you will most likely never be able to understand it anyways.

1. I'm afraid, Fred, that the rest is not window-dressing.

Just try yourself now to apply Albert Jan's trick to the one page paper. It doesn't work there. The problem is that there is some distance to cover between Christian's definitions of the measurement functions A( , ) and B( , ) to where he comes up with his formula (19). You can't just change the definition of bivector algebra product in just one formula and leave all the rest the same. You either need to use a consistent definition all the way through the paper or explain why you change definition from time to time. This is not a conjuring trick (pulling a rabbit - a . b out of a hat). At least, it's supposed not to be a conjuring trick. It's supposed to be physics.

But it is now exposed as a conjuring trick. Nothing more.

2. We don't expect you to ever understand.

11. Here is how it goes in R: http://rpubs.com/gill1109/onion2 I apply Albert-Jan's patch to the bug in the central formula in the one-page paper. It doesn't work there. The problem is that there are more bivectorial products in the derivation of E(a, b) when we start at the beginning, with the definition of the measurement functions A and B. You can't just redefine "times" as it suits you, when it suits you.

1. Right hand brain can't see what the left hand is. Too bad for you.

12. Something that will be very cool is to write a script to animate this for viewing in GAViewer. I will probably be working on that but maybe someone else can do it faster.

13. The problem with the one page paper is the transition from (6) to (7). The multiplication table we need is beta_j(lambda^i)beta_k(lambda^i) = - delta_jk - lambda^i epsilon_jkl beta_l(lambda^i). The left hand side of (7) thus contains two parts: the first part reduces to -a_j b_j while the second part should actually contain, inside the limit and summation, lambda^i epsilon_jkl beta_l(lambda^i). Because of a notational mixup, Christian write just beta_l instead of beta_l(lambda^i). Since beta(lambda) = lambda beta he "loses" in this wat one lambda from what should have been lambda^2 thus creating out of nowhere, a random sign. This way the "a x b" part is made to vanish since about half of the n contributions to a x b have a minus sign and in the limit, cancel against the other half.

So it is indeed rather cool to work through the one page paper with GAViewer or with R+onion. You come face up against a little slip-up caused by notational ambiguity: Christian uses the same notation, beta_j, to stand for a specific generators of his geometric algebra, and for generic generators of a geometric algebra. He starts with a specific collection beta_1, beta_2, beta_3. But later he uses the same symbols beta_1, beta_2, beta_3 to stand both for +beta_1, +beta_2, +beta_3 and for -beta_1, -beta_2, -beta_3

It is a typical beginner's error (a non-mathematician beginner's error: lack of mathematical discipline).

14. The only problem with the one page paper is that you will never understand it. But that is your problem and we don't care about your problem with it. Maybe the dingo ate your baby.

15. No, who cares, anyway. Maybe -1 = +1. And maybe physicists will adopt Christian's novel definition of correlation. But the one page paper is great fun and easy to code in GAViewer. Try it, Fred! Show us the results!

1. Just keep on spammin' this blog with your misrepresentations. For sure the dingo ate your baby and you are finished.

16. Here is a careful implementation of the one page paper in GAViewer: http://www.math.leidenuniv.nl/~gill/test4.g

Notice that I set the random seed at the beginning of the test function:
randSeed(1234);
Comment out this line if you want a different pair of measurement directions with each new test.

I have commented the code, including references to each formula in the one page paper.

If you think it is incorrect, please tell me what you think needs to be changed.

Instructions:
- fire up GAViewer
- input this code through the File -> Open menu
- type "test();" and enter in the console window

You'll see something like:

>> Parsing '/Users/richard/Desktop/test4.g'
>> test();
aa = -0.40*e1 + -0.58*e2 + -0.71*e3
bb = -0.61*e1 + 0.38*e2 + 0.69*e3
Eab = -0.47 + -0.13*e2^e3 + 0.72*e3^e1 + -0.51*e1^e2
dotprod = 0.47
crossprod = -0.13*e1 + 0.72*e2 + -0.51*e3

17. Albert-Jan: you have now fixed the transition from (17) to (19) in Christian's http://arxiv.org/pdf/quant-ph/0703179v3.pdf

Now you need to start at (8), definition of measurement functions A and B. From there we go to (7). Now you need to find a bridge from (7) to (19). Christian writes in the neighbourhood of (19): "Similarly, the joint expectation value of the two outcomes in the manifestly local form (7) works out to be ... where we have used the results (17) and (18)".

However, (17) and (18) are not enough to get from (7) to (19).

So either you can "believe" the last part of the derivation (the bit you have fixed), in which case we do not have a complete local hidden variables model yet, or you start at (8) and compute (7) by simulation of (8). If you get the same answer as Christian gets in (19), i.e., proof by simulation, then we can try to find the missing math to complete his derivation, ie turn the proof into a mathematical proof.

However my guess is that you won't succeed!

1. This comment has been removed by the author.

2. Sorry, in http://arxiv.org/pdf/quant-ph/0703179v3.pdf, A and B are defined in (16). We don't have quite so far to go ...

I was mislead by Christian's remark that he uses (7) in getting to (19). But I guess that the remark is meant to apply to (16): we see the same "manifestly local form".

So it should be easier to find out if the model can be entirely rescued, yes or no!

3. Just keep on spammin' away!

4. 18. Hmm, formula (16) of http://arxiv.org/pdf/quant-ph/0703179v3.pdf is rather problematic. It reads:

A_n(mu) = B_n(mu) = mu.n =/~ +/- 1 in S^2

From Christian's definitions, and see (15), mu.n is a unit bivector so it does *correspond* to a unit vector, thus an element of S^2. But what does the "approximately equal to" or perhaps "equivalent to" symbol actually mean? Are we supposed to associate elements of S^2 with +1 or -1 depending on whether they are in the upper or the lower hemisphere? In that case we are back with the "exploding ball" model which Christian earlier rejected, formulas (8), (9) and (10).

I come to the conclusion that the paper http://arxiv.org/pdf/quant-ph/0703179v3.pdf does not contain a local hidden variables model at all: it merely contains some wishful thinking.

The one page paper is a great improvement: everything is there, in full explicit detail.

1. Truly the mark of a desperate man that can't admit defeat.

2. Excellent, Fred; so you do know exactly what "~/=" is supposed to mean in (16) of http://arxiv.org/pdf/quant-ph/0703179v3.pdf ??? So you will be able to help out Albert-Jan, in translating the whole model into GAViewer language ... ?

I have my own impression who is getting desperate here ... GAViewer certainly has put the cat among the pigeons. Or: is sorting out the men from the boys.

3. Well, if you truly want to understand Joy's model, I think an extreme attitude adjustment will be necessary on your part. I don't think anyone is really interested in helping you to understand it right now. Although, most likely you are not capable of understanding it. Sorry.

19. I think I have decoded the symbol "~/=" in (16) of http://arxiv.org/pdf/quant-ph/0703179v3.pdf. Let's look at Alice's measurement outcome, A_a(mu), where a is an (ordinary) unit vector in R^3, thus an element of S^2. According to the text at formula (14), Bell's hidden variable lambda is replaced by mu = +/-I, where I = e_x ^ e_y ^ e_z. A_a(mu) = +/- I a, B_b(mu) = +/- I b. These are unit length bivectors and the set of unit length bivectors is indeed isomorphic to S^2. Given a and b, each is dichotomous, so we can associate the outcomes +/- I a and +/- I b each with +/- 1. We just need to decide what the mappings are from S^2 to {-1, +1}. Now since the distribution of mu is assumed to be isotropic, all this comes down to the two possible signs having equal probability, and therefore the model has exactly two possible "realisations": A_a(mu) = - B_b(mu) = +/-1 each with probability 1/2; and A_a(mu) = B_b(mu) = +/-1 each with probability 1/2.

According to the first choice the correlation , E(a, b) is identically equal to -1; according to the second choice, E(a, b) is identically equal to +1.

Christian's mistake is to try to calculate the correlation by averaging the product of the two bivectors. Thus he takes the step from (16) to (17). But the two bivectors are just intermediate physical variables which determine within each measurement device whether the measurement outcome is actually -1 or +1. And it is the actual measurement outcomes which need to be correlated.

1. Joy has no mistake. You are the only one mistaken. Too bad, so sad.

2. So where's your GAViewer implementation of Joy's model? Of any of Joy's models, Fred?

3. There will be an answer, let it be. Let it be.

20. Albert-Jan has a new blog item on geometric algebra. http://challengingbell.blogspot.nl/2015/04/joy-christian-for-dummies-like-me.html

Not only is GAViewer a fine tool, the accompanying book by Dorst, Fontijne and Mann is excellent. I found a pdf of the first edition on internet.

21. One can buy the revised first edition at http://www.buecher.de/ for 73 Euro

1. There will be an answer, let it be. Let it be.

22. Google "geometric algebra for computer science pdf" to find a free copy of the first edition (unrevised) at http://www.itpa.lt/~acus/Knygos/Clifford_algebra_books/p_Leo_Dorst,_Daniel_Fontijne,_Stephen_Mann%5D_Geometr%28BookFi.org%29.pdf

23. In fact this Lithuanian particle physicist Artūras Acus http://www.tfai.vu.lt/index.php?siteaction=personnel.view&id=191&menu=about has a big collection of books on geometric algebra ...

1. That is good. The big question is will you ever understand GA?

Happy Easter everyone!

2. Christian's claim back in 2007: "It is shown that Bell's theorem fails for the Clifford algebra valued local realistic variables. This is made evident by exactly reproducing quantum mechanical expectation value for the EPR-Bohm type spin correlations observable by means of a local, deterministic, Clifford algebra valued variable, without necessitating either remote contextuality or backward causation. Since Clifford product of multivector variables is non-commutative in general, the spin correlations derived within our locally causal model violate the CHSH inequality just as strongly as their quantum mechanical counterparts."

3. Learn GA, then someone might want to have a discussion with you. But probably not even if you did understand it.

4. I wrote a "quick-start" introduction to geometric algebra with analysis of two variants of Christian's model: http://www.math.leidenuniv.nl/~gill/GA.pdf

5. ROTFLMAO! It is pretty pathetic when a dude doesn't even know he has lost the debate. As I said before; you will never understand it no matter how hard you try.

"A man's got to know his limitations". --Dirty Harry

6. The above crackpot paper by Gill was rejected by arXiv moderators as a personal attack on me devoid of any scientific value. In fact it is of *negative* scientific value because it is riddled with schoolboy mathematical and conceptual errors. It is extraordinary that after eight years of efforts Gill has yet to learn the first thing about geometric algebra, let alone about my work based on this mathematical tool. It is heartening to know that the arXiv moderators spotted the crackpot-ness of the paper and rejected it.

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