Wednesday, 20 May 2015

Further Numerical Validation of Joy Christian’s Local-realistic Model based on Geometric Algebra:

This thread is a continuation of Albert Jan's excellent work using the GAViewer computer program to prove Joy Christian's classical local realistic model that contradicts Bell's theorem.  Albert Jan has kindly given me this blog space to post my version of his GAViewer script which is now tailored to Joy Christian’s much discussed (and somewhat controversial) one-page paper at  arXiv:1103.1879. For a theoretical understanding of how Joy’s local model and this computer script works, the reader may also wish to consult the appendix of Joy’s latest paper on the subject.  Otherwise feel free to ask for more explanation on this thread.  Now for the GAViewer script.

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 of Joy Christian's arXiv:1103.1879 paper");
N=20000; //number of iterations (trials)
I=e1^e2^e3;
s=0;

a=getRandomUnitVector();
b=getRandomUnitVector();
minus_cos_a_b=-1*(a.b);

for(nn=0;nn<N;nn=nn+1) //perform the experiment N times
{
lambda=getRandomLambda(); //lambda is a fair coin,
//resulting in +1 or -1
mu=lambda * I;  //calculate the lambda dependent mu
C=-I.a;  //C = {-a_j B_j}
D=I.b;   //D = {b_k B_k}
E=mu.a;  //E = {a_k B_k(L)}
F=mu.b;  //F = {b_j B_j(L)}
A=C E;  //eq. (1) of arXiv:1103.1879, A(a, L) = {-a_j B_j}{a_k B_k(L)}
B=F D;  //eq. (2) of arXiv:1103.1879, B(b, L) = {b_j B_j(L)}{b_k B_k}
q=0;
if(lambda==1) {q=((-C) A B (-D));} else {q=((-D) B A (-C));} //eq. (6)
s=s+q; //summation of all terms.
}
mean_mu_a_mu_b=s/N;
print(mean_mu_a_mu_b); //print the result
print(minus_cos_a_b);
prompt();
}
//Typical result is:
//mean_mu_a_mu_b = 0.87 + 0.00*e2^e3 + 0.00*e3^e1 + 0.00*e1^e2
//minus_cos_a_b = 0.87
//The scalar parts match and others vanish!  Proving the result is -a.b.
//Thus Dr. Christian's arXiv:1103.1879 paper is a classical local
//realistic counter-example that in fact contradicts Bell's theorem.

Here are equations (1) and (2) from the paper.

And equation (6).

What the "if else" line does for eq. (6) is to correct for GAViewer being in a fixed right handed bivector basis. And is also Joy's main physics' postulate. When lambda = 1, the particle pairs are in the right handed basis; when lambda = -1 (else), they are in the left handed basis. What do you see when looking at a left handed system from a right handed only perspective? The geometric order is reversed. Very simple.

More explanations for the GAViewer program and Joy's model can be found in previous threads on this blog.

And for further reading you may find some of the key papers by Joy at this link: http://arxiv.org/find/all/1/au:+Christian_Joy/0/1/0/all/0/1.

Best regards,

Fred Diether

PS.  Have a good day and have much fun!

Update 22-5-15:

G=a.b was taken out of the script above as it has been pointed out to me that since a and b have the same values for each iteration in the loop there is no advantage to doing that part in the process.

So the result now includes the scalar part that represents -a.b and now we have added the result for the scalar part in the next line so that one can see that they match.

Michel Fodje has translated the above GAViewer code to Python so here it is:

from __future__ import division
import numpy
from clifford import *
e0, e1, e2 = [blades['e%i'%k] for k in range(3)]
I = (e0^e1^e2)

def randVec3d(lo=0, hi=2*numpy.pi):
theta = numpy.random.uniform(lo, hi)
z = numpy.random.uniform(-1, 1)
sn = numpy.sqrt(1-z**2)
y = sn*numpy.sin(theta)
x = sn*numpy.cos(theta)
return (x^e0) + (y^e1) + (z^e2)

N = 20000
s = 0
a = randVec3d()
b = randVec3d()

for i in range(N):
L = numpy.random.choice([-1., 1.])
mu = L&I
C = (-I)*a
D = I*b
E = mu*a
F = mu*b
A = C&E
B = F&D
q = ((-C)&A&B&(-D)) if L == 1 else ((-D)&B&A&(-C))
s = s+q

print s^(1./N)
print -(a*b)

# Typical Run
#
# 0.44353 - (0.00776^e01) - (0.0068^e02) + (0.00131^e12) # Model
# 0.44353 # -a.b

1. Unfortunately your measurement outcomes are not mapped to +/-1. But congratulations on finally getting the correlation to come out right. Unfortunately it is not the correlation which experimenters find.

1. Gill still doesn't understand the difference between correlations done in experiments vs. those done in theory. A loophole free test will further validate Joy's model. Something you will never understand apparantly.

2. As anyone can see, the measurement outcomes are quite clearly mapped to +/-1 in my one-page paper, which is now unambiguously verified by Fred's and Michel's simulations. I have added a second appendix to my latest paper on the subject to make this fact even more transparent and explicit:

http://arxiv.org/abs/1501.03393

3. Also of interest in this regard is this new event-by-event simulation of my 3-sphere model for the EPR-B correlation:

http://rpubs.com/jjc/84238

2. "This script makes use of Heine's contribution of adding G=a.b" is a beautiful give-away. The computation of the correlation involves knowing in advance what the correlation should be.

I don't think this is going to catch on among experimenters.

1. It is proof by negation. Something else you apparently don't understand.

2. Sadly, since Gill can't figure out anything that is really pertinent to say or questions to ask that he is going to continue on his path of complete misrepresentation.

3. So now that we have proof that the model is a valid alternative explanation of quantum correlations in a classical local realistic way, the only real question that remains is; does the model hold macroscopically in a proper singlet macroscopic scenario? I think only the actual macroscopic experiment will tell us that.

4. As I mentioned, the simulation code does not generate +/- 1 valued measurement outcomes and does not compute the correlation according to the instructions to the experimenters in Christian's exploding ball paper - i.e. the conventional one from Peres' exploding ball paper.

You counter, Fred, with the argument that there is a difference between between correlations done in experiments vs. those done in theory. Here you are doing theory. I agree - this is just like the situation in conventional quantum mechanics.

To continue the analogy with conventional QM: it is dead simple to simulate a Bell-CHSH type experiment with hidden variables. Just treat the wave function as a hidden variable. Use a random generator to get the outcome of Alice's measurement, use that to calculate the collapsed wave function at Bob's place, use another random generator to get Bob's measurement outcome.

Your computer code does not simulate an experiment, it verifies a theoretical computation.

Now my question is: can you simulate a corresponding Bell-CHSH *experiment* (i.e. simulate binary outcomes, calculate the conventional experimental correlation), using the model?

Can you do it in a local-realist way, or do you need communication between the measurement settings?

1. Do you see anything non-local in the code above? Is there any "communication between the measurement settings"? Everything is local and there is no a, b connection. It's a classical local realistic process that produces the result of QM, -a.b. Thus Bell's theorem is contradicted. Period. You lost the debate. Get over it.

2. Bell's theorem is not contradicted. Bell's theorem is about experiments which generate +/-1 valued outcomes. It says that such experiments cannot be simulated by a local hidden variables theory. You agree that the GAViewer code does not simulate an experiment involving calculation of correlations from +/-1 valued outcomes.

You can't easily contradict a true theorem!

As Weatherall pointed out in his FOOP paper http://arxiv.org/abs/1212.4854 , Christian's model simply has got nothing at all to do with the whole EPR-Bell business.

I must say, these recent developments triggered by Paul Snively and Albert Jan Wonninck have been immensely clarifying. I feel we are close to "closure", as they say.

3. To continue ... regarding physical relevance, Bell's theorem says that we cannot simulate a successful loophole-free Bell-CHSH type experiment by a LHV model. This is true mathematical statement about reconciling a mathematical picture of distributed computing (a network of classical computers) with predictions about experimental observations made by another theory: quantum physics. So far, the relevant experiment has not even been done, though about five top experimental groups are actually doing one right now. In Delft, close to me in Leiden, they get one pair of measurement results every 20 seconds and are going to have to run the experiment for rather a long time, since they expect only a very small deviation from the local realism bound. They'll need weeks of data before the result is statistically significant (if the experiment comes out as they predict).

So Bell-CHSH is a mathematical theorem about some physics experiments which are actually being performed right now - this very minute, as you read these words, if you read them within a few weeks from me writing them!

I think this makes Bell *relevant*.

Now you may still not find it *interesting*. Niels Bohr would not find it interesting. He always argued that there is no point in worrying your head about what was actually going on. He would just have laughed when it turned out that doing exactly what he said you shouldn't do, leads to head-ache and weird conclusions about spooky action at a distance. Spooky action at a distance ... acting via results of measurements which are not performed.

But (if the experiment comes out as predicted) Einstein would have find it bothersome.

Anyway: we have resolved all possible contradictions and fixed an annoying sign error. Bell's theorem stands. Whether or not the metaphysical implications are interesting or not is really a matter of taste. Von Neumann (I think) always believed that if we would find a deeper theory *underlying* QM, it would be even weirder than QM. I think he is being proven right. We had better get our heads around QM and realise that its predictions are not so weird at all - we have some major prejudices about how we like to understand the world - local realism - and we have to abandon them.

4. You lost the debate no matter how much rubbish you post. Get over it.

"The important question here is: What would be the 'actual experimental outcomes' in the proposed experiment? ... what would be actually observed in the experiment are the bivectors L(a, Î») and L(b, Î») about the directions a and b, respectively. The raw scores sign ( + s^k · a) and sign ( − s^k · b) may then be calculated, but by an algorithm extraneous to the actual experiment, and long after the experiment has actually been completed. Thus they would not be an integral dynamical part of the physical experiment itself. On the other hand, thanks to the precise geometrical relation between the raw scores sign ( + s^k · a) and sign( − s^k · b) and the standard scores L(a, Î») and L(b, Î») given by the equations (7) and (8), the statistical association between the raw scores can nevertheless be inferred by calculating the covariance of the corresponding standardized variables A_a(Î») and B_b(Î») as demonstrated above, thus predicting the correlation (27)."

Please add to your code the calculation of the "raw scores" and please also calculate their correlation in the conventional way. Only if the result is the same as (27) have you shown that you have "a classical local realistic counter-example that in fact contradicts Bell's theorem".

You've got about half way. You did part of the theory. Now you have to relate it to experiment.

6. Valid simulations of Joy's model have already been done. It is not our fault that you don't understand how the complete states theory connects to S^3 topology. But even if you did understand it, you would never accept it as you would then have pay Joy the bet you lost.

Joy has profusely explained and justified the "raw scores" and standard scores in his papers. Sorry that you don't understand it.

5. Let me just mention regarding computer simulations: http://arxiv.org/abs/1505.04431 is a new paper on the Pearle model, with some new results; and connected to that, https://pubpeer.com/publications/06814F3D1ACB675517256B37BB4EF1 refers to Christian's paper with his version of Pearle, http://arxiv.org/abs/1405.2355

1. You still don't have a clue about http://arxiv.org/abs/1405.2355, do you?

6. This comment has been removed by the author.

7. Having rewritten equation (6) of http://arxiv.org/pdf/1103.1879v1.pdf, with the sequence of multiplication reversed when lambda = -1, you'll have to work back and rewrite equation (5) as well. That will be tricky, I think. I suppose it can be done but it means that you will have to redefine correlation. After all, in the original definition, LHS of (5), there is a *fixed order* (first a then b), but you want to reverse the order when lambda = -1. If you manage to come up with something, you will have given to the world yet another new correlation ... the Diether-Christian-Pearson correlation!

Has Joy Christian authorized this major rewrite of his theory?

1. LOL! We have already done that and it is very easy to do with GAViewer. You could probably even do it yourself if you understood Geometric Algebra. Most people have no problem going from eq. (5) to eq. (6). Your right handed only brain is still having trouble seeing what a left handed system looks like.

2. The problem is how to go from the new version of (6) to the old left hand side of (5). Equation (5) starts with the definition of the Christian-Pearson correlation E(a, b). It involves a summation of products, then division by "n", then a limit for n going to infinity, then a division by theoretical standard errors. However in equation (6) the division by theoretical standard errors is done *inside* the summation over i from 1 to n, and you divide from the left or the right depending on the sign of lambda^i.

This is a case of "out of the frying pan, into the fire".

3. As Franz Heymann would say, "Rubbish!". R.I.P. There is no difference between eqs. (5) and (6) which one can verify using GAViewer.

After that, add the observed outcomes +/-1 to the code and show that the relative frequency of coincidences (Alice's outcome = Bob's outcome) minus the relative frequency of anti-coincidences is also equal to the theoretical correlation defined in (5).

Only when this is done, is your simulation an "event-based simulation" and have you disproved Bell's theorem. Right now there are no events. No measurement putcomes. No detector clicks!

5. More rubbish. The GAViewer code for eq. (5) can be accomplished very easy from what I have presented above. I guess you are not capable of doing it yourself. Figures.

As I have said before; simulations for the model have already been done but you won't accept them because then you would have to pay Joy the bet you lost. It is trivial that Bell's theorem has been contradicted since the desired result of -a.b has been obtained. You lost the debate. GET OVER IT!

6. (5) is easy to do in GAViewer. Unfortunately, the result is that Christian's (5) does not equal Diether's (6).

Diether's code proves that Christian's one page paper is hopelessly riddled with errors.

7. PS in (5), there is a displayed fraction: a limit of an average of products, divided by another product. In GAViewer there are no geometric algebra fractions. One has to choose whether to divide on the left or to divide on the right. One can see in Christian's claimed chain of equalities (5) = (6) = (7) that he interprets something like x / (yz) as y^{-1} x z^{-1}.

Maybe Diether has an alternative. Probably the simplest is to redefine E(a, b) = Fred's (6) = (7).

8. LOL! Gill is still struggling with trying to understand geometric algebra. Anyone that understands GA has no problem going from eq. (5) to (6). In fact, just using my A thru F, one can use eq. (5) to get the result of -a.b. Try it. Complete the equation below. I will tell you if you are right or wrong.

if(lambda == 1) {q = ???..........................................;} //eq. (5)

9. Equation (5) starts with a definition.

E(a, b) =

{lim_{n to infty} n^{-1} sum_{i=1}^n A(a, lambda^i) B(b, lambda^i})}

divided by

{ {- sum_j a_j beta_j} times {sum_l b_k beta_k} }

So you propose to put the "if" statement inside the inner summation in the numerator? This is where we appear to need the right-hand versus left-hand basis trick, since we are multiplying the bivectorial outcomes A and B in the i'th trial and you propose to use the Diether-product = GAViewer's AB if lambda == +1, GAViewer's BA if lambda == -1.

Define A^i = A(a, lambda^i) and B^i = B(b, lambda^i}}. If you go back to equations (1) and (2), you will find that A^i B^i = B^i A^i = -1 whether lambda^i is +1 or -1, so in fact we do not need the Diether product here. In fact according to the definition (first equality of (5)), E(a, b) = - 1 divided by { {- sum_j a_j beta_j} times {sum_l b_k beta_k} }.

Yes this is all very easy to implement in GAViewer. Unfortunately it GAViewer tells us that (5) is not equal to Diether's (6).

10. More nonsense, rubbish and misrepresentation by Gill. He can't face the fact that he lost the debate.

11. Dear Fred clearly does not know how to fix his GAViewer code.

And that is plain for everyone to see :-)

12. It is really quite unbelievable that Gill couldn't do something that is so simple that even a monkey could probably do.

if(lambda==1) {q=(A B)/(C D);} else {q=(B A)/(D C);} //eq. (5)

And the result is,

mean_mu_a_mu_b = 0.87 + -0.00*e2^e3 + 0.00*e3^e1 + 0.00*e1^e2
minus_cos_a_b = 0.87

Proof that even eq. (5) gives the result of -a.b. You lost the debate; get over it!

8. Hint: appendix B of Christian's recent http://arxiv.org/pdf/1501.03393v4.pdf should tell you how to relate the experimenter's measured correlation based on the +/-1 valued observations sign(s^k . a) and - sign(s^k . b) to the theoretical correlation based on the Clifford algebra valued (grade 2) L(a, lambda^k) and L(b, lambda^k). Equations (7) and (8) tell you how to find s^k; see also (B2) and (B3).

9. Christian http://arxiv.org/pdf/1501.03393v4.pdf: "In practice the above dynamical variables s^k are supposed to be measured in the proposed experiment by directly observing the polar vectors s^k dual to the random bivectors L(s, Î»k). It is only after all the runs of the experiment are
completed and the vectors s^k are fully recorded, the traditional dynamical variables sign ( + s^k · a) and sign ( − s^k · b) supposed by Bell are to be calculated, by an algorithm extraneous to the actual experiment. These calculations may be done, for example, years after the experiment has been completed. Thus they are not supposed to be an integral part of the physical experiment itself, and this fact must be accounted for in the ensuing statistical analysis of data."

Thus all the experiments done to date, which did violate CHSH etc, were actually analysed wrongly? We have to throw away Aspect et al, Weihs et al, Giustina et al, Christensen et al.... ? Or is it just that quantum experiments should be analysed in the classical way, but non-quantum experiments in the traditional way?

1. Typo: "but non-quantum experiments in the new way".

2. More nonsense, rubbish and misrepresentations!!!

10. I think we all agree that http://arxiv.org/pdf/1501.03393v4.pdf is well worth reading. I think we have now reached closure.

11. I have examined your GAViewer script and I am sorry to tell you that it fails to meet the locality requirement of a true EPR simulation. In the calculation of the variable g in the following...

if(lambda==1) {q=((-C) A B (-D));} else {q=((-D) B A (-C));}

you form the geometric product of (-C) and (-D). This is a distinctly "non-local" operation! C is the dual of vector a and D is the dual of vector b. Multiplying one bivector by the other uses information from both sides of the experimental apparatus at the same time. That's "non-local"!

Also, try printing your final results with floating point precision using the following modification

print(mean_mu_a_mu_b, "f");
print(minus_cos_a_b, "f");

You will find that the correlation is uncannily accurate, to as many decimal places as you like. This remains the case even when the iteration count is reduced to a single sample!

1. Basically what C and D are doing is dividing out the detectors. You have to read Joy's papers to understand what is going on.

Thanks for the tip on "f".

Yeah, I like it better when a and b are random in the loop then you add a.b to the sum to see that -a.b is completely cancelled out in addition to the other terms going to zero.

2. Basically, the theory says "the correlation = - a . b because the correlation = - a . b". The program verifies that when GAViewer calculates - a . b twice, it gets the same answer each time.

3. Still, this is quite some improvement from the first version of the theory back in 2007 or whenever. It didn't even get - a . b = - a . b then. So there's been progress of a kind.

4. More nonsense, rubbish and misrepresentations.

12. Fred has proposed that (5) should be implemented in GAViewer as

if(lambda==1) {q=(A B)/(C D);} else {q=(B A)/(D C);} //eq. (5)

which he says is so simple, that even a monkey could do it. But I have a problem here, because lambda depends on the iteration counter i.

Equation (5) reads E(a, b) = middle expression = right hand expression

So is Fred evaluating the middle expression or the right hand expression?

Please show us how you make GAViewer compute the middle expression, now that GAViewer's geometric product has to be transposed for every iteration in which lambda^i = -1

(5) E(a, b) = middle expression = right hand expression

should be interpreted in a different way, where also the equality signs needs, in a sense, to be transposed?

Let me use the symbol := to stand for "is defined as". Then perhaps you are saying that

(5) E(a, b) = middle expression = right hand expression

is actually shorthand for two separate definitions

E(a, b) := right hand expression

middle expression := right hand expression

1. Does anyone understand the above gibberish by Gill? Someone should get the poor guy some professional help. Thank you.

2. Oh yes, I understand it. It is a cry of desperation and despair.

13. Mr. Gill makes plenty of sense, it is your defense that contains the gibberish. When he states "The program verifies that when GAViewer calculates -a.b twice, it gets the same answer each time." he is describing the overall situation quite succinctly.

To simulate an EPR experiment, you need to make a one calculation using elements only from Alice's side of the apparatus and a separate calculation using elements only from Bob's side. For example....

q1=(-C) A;
q2=B (-D);

Now, if you were to display the interim values of q1 and q2 you would see that they are both bivectors and not even scalar. But, in a real EPR experiment the results from either side of the apparatus are binary values.

So, you must first convert each bivector to plus or minus one before they can be multiplied together (correlated). Assuming you had a function like this, called "binary", you could then perform the geometric product without the conditional logic. For example...

q=binary(q1) binary(q2);

It is up to you to decide how to implement this function!

As a bit of background:

I have read most of Joy Christians papers going back many years now and I am familiar with their criticisms. I've read EPR papers from many of the original experimentalists; Aspect, Weihs, etc. I am quite familiar with GAViewer the GASandbox, having used them many times; plus I am a professional programmer. Lastly I've been investigating entanglement using computer simulations as a hobby for years. So you can assume I have the requisite knowledge to comment on subject intelligently.

I will be happy to evaluate your reasoned argument and/or updated script in response. Any ad hominem attack will be considered unscientific.

1. You have my respect. Anyone who understands Gill's gibberish has my respect.

There is no need for Fred or Albert Jan to revise or update their scripts. Their codes are just fine. They confirm the calculations in my one-page paper.

2. gauser317 seems to think this is an EPRB simulation using +/- 1 outcomes. It is not. Again, for the benefit of any lurkers it is validation of a theoretical calculation of Joy's model. And to address the other concern, all one has to do is add two line and change one.

G=A (-C);
H=B (-D);
q=0;
if(lambda==1) {q=(G H);} else {q=(H G);}

And the result with more significant figures is,

mean_mu_a_mu_b = -0.661198 + 0.000705*e2^e3 + -0.004928*e3^e1 + -0.003485*e1^e2
minus_cos_a_b = -0.661198

We get the proper scalar result with the other terms vanishing.

3. The point of my original post was to highlight the fact that their code is not fine!

Carl Sagan, popularized the saying "extraordinary claims require extraordinary evidence." You claim that you have disproved Bell's theorem and can replicated the results of an EPR experiment using a computer simulation! The onus is on you to provide some extraordinary evidence to support this.

Bell's Theorem concerns the correlation of measurements that resulted in binary outcomes. The first stipulation in his famous paper "On the Einstein Podolsky Rosen Paradox" states...

A(a,lambda) = +/- 1, B(b,lambda) = +/- 1

But the program as written makes no attempt to verify that this is in fact the case. I and others claim it cannot. It is you who must prove otherwise.

If the program works as you say it does then Fred or Jan should have no trouble splitting the calculation into two parts and displaying the interim results to demonstrate as much.

Nothing short of that will provide the extraordinary evidence necessary to support your claim.

4. I can tell from your comments that you don't understand Joy's model very well. Perhaps you should be asking questions about what you don't understand? Did you miss the picture of eq. (1) and (2) above? Do you need help calculating that in geometric algebra?

5. Well, there are several parts to my program. Albert Jan and Fred have verified one significant part of the program. This was important because Gill and others have been falsely accusing me of making an error in my GA calculations, even though it takes only 20 minutes of analytical calculation to recognize that there is no such error. But now we have an explicit simulation that verifies that there is indeed no such error in my papers.

Then there is a separate issue of event-by-event simulations of the model, with +1 or -1 outcomes. For that there are several possibilities. One of them is to verify equation (B10) of my latest reply to Gill:

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

We have made no progress so far in that direction.

Finally there is a completely different way of simulating the model, and you may want to check out this latest simulation to see how that is done:

http://rpubs.com/jjc/84238 .

6. If, as you claim, the program is a faithful representation of the mathematics described in your one-page paper then your paper does not provide a counterexample to Bell's Theorem.

In particular, you offer up an expectation value in equation 5 that does not conform to the requirements as set up by Mr. Bell. The probability distributions covered by his theorem are consistent with equation 2 from his famous paper. In that equation all functions to the right of the integral sign produce, as you put it, dichotomic values. Yet you introduce an equation (5) that contains a "non-local" bivector product in the denominator. This product, in the general case, produces a rotor and not a scalar. Since the right hand side of the equation is not a scalar it is hard to see how it might even be considered a probability.

That being said, how can you claim that your correlation function operates in the same mathematical domain in which Bell's Theorem holds?

The above mentioned rotor remains in all the equations on lines 5 and 6 of your paper and is also contained in the variable "g" in the script. But a simple examination of the variable "g" shows that it is in fact a rotor which is, as we've been saying, not a dichotomic result!

To illustrate this point, after running your script, use the View menu in GAViewer to select object "g". You will see that it is a rotor! That's not just my description, that's how Fontijne, Dorst, and Bouma describe the object too. And when all is said and done 'mean_mu_a_mu_b" is still a rotor! Do the same for "C" and "D" and you will see that they are both described as bivectors, again not scalars as required by Mr. Bell.

7. There is no "g" in the script. Are you talking about "q"? It is pretty apparent that you don't understand what is going on with the geometric algebra related to Joy's model. Sure "q" shows as a rotor in GAViewer because with only 10K trials the terms other than the scalar have not completely vanished. But as N goes to infinity they will vanish just leaving the scalar.

I showed you above how to get the result of -a.b without any a, b connection. So as I said before, you should be asking questions about what you don't understand.

8. Correction; that should be mean_mu_a_mu_b instead of "q" above.

9. Perhaps reading the Appendix B of this paper might help:

http://arxiv.org/abs/1501.03393

10. Yes, I meant to write "q" not "g" above, please replace one with the other.

Now, when you say "because with only 10K trials the terms other than the scalar have not completely vanished." that admits the flaw directly. There can be no non-scalar terms in the final term at all and still qualify as a "real" result, no matter how small they are! Not after 10K iterations not after even just one.

Bell's Theorem simply does not pertain when your calculation employs imaginary numbers (bivectors).

11. The flaw is indeed there, but only in your crude understanding of my model. The bivector term vanishes EXACTLY in Eq. (19) displayed by Albert Jan from my first paper in his original post. The simulation of Eq.(19) necessarily requires replacing the Riemann integral by a sum over large N. The correlation predicted by my model are therefore EXACTLY equal to -a.b.

12. The bivectors may vanish in Eq. (19) but they most certainly do not vanish before you take their geometric product in either your one-page paper or in the script.

Unfortunately, for the quantum experimentalist the bivectors must vanish because his "observables" are always just real-valued. Consequently, Bell's Theorem is only applicable in a single dimensional algebra where this reduction to scalar values has already occurred. In your "counterexample" this reduction never occurs!

Using Fred's code from above one could write for example...

G=scalar(A (-C));
H=scalar(B (-D));
if(lambda==1) {q=(G H);} else {q=(H G);}

or maybe...

G=norm(A (-C));
H=norm(B (-D));
if(lambda==1) {q=(G H);} else {q=(H G);}

Or any other scalar-valued operator or function. Whatever the means, all terms must be reduce to grade_0 before taking the product of G and H. Anything short of that and your still operating in the full 3D vector space.

I will be happy to help Fred and Albert with the GAViewer programming if they have any difficulties accomplishing this.

13. As I said before, you should be asking questions; not making statements. You are totally lost in how geometric algebra works with Joy's model. We would be happy to help you sort it out if you have a question about it.

14. Fred, if you think that you can wave off this criticism by simply leveling personal attacks at me then you are sadly mistaken.

I believe that I have explained my position pretty clearly by now. In addition I have illustrated just how the issue can be easily resolved by adding only a few lines of program code.

Here's a simple question.

Does this mean that you are unwilling to insert this or similar code in your program?

Because if that is the case, then you are also unwilling to support your contention that it is I and not you who fails to understand how geometric algebra works!

15. You are the one that started with the personal attacks by implying that we don't know what we are doing. Get lost and don't come back until you have a pertinent question to ask. The code we have done is perfect for Joy's model.

16. Once again you try to wave off my criticism rather than address it directly. Most unprofessional!

17. O.K. Let me be a peace maker here. If you think that you can write a better code than what Albert Jan and Fred have produced (and by that I mean more faithful to what is seen in the experiments as well as to my model), then please post your code somewhere on the Internet and let us all see it. I have already suggested that you should be looking at equation (B10) of my latest paper I mentioned above.

18. Equation (B10) of http://arxiv.org/abs/1501.03393 does give a link to binary outcomes. To see them, we apparently need to calculate the vectors s^k. Christian writes " The raw scores sign ( + s^k · a) and sign ( − s^k · b) may then be calculated, but by an algorithm extraneous to the actual experiment, and long after the experiment has actually been completed."

The paper does not contain, as far as I know, such an algorithm, and it does not tell us where we can find one.

19. PS: Concerning this "missing think", Joy wrote, and I was really impressed by his openness and honesty: "We have made no progress so far in that direction."

Yes, the GAViewer does now reproduce the one page paper, in the sense that we now know how the formulas in the one page paper should be interpreted. The one page paper does not contain the link between the elements of a Clifford algebra and the raw scores. As Christian has admitted, this problem has not been quite sorted out yet. The simulations of Pearle's model are perhaps an attempt in this direction.

So the obvious thing to do is to rewrite Pearle in GAViewer and then try to mesh the two together. This means: come up with the algorithm to compute vectors s^k from from the geometric algebra objects which are in the present GAViewer code. I see a problem here concerning the "rejection" part in the Pearle algorithm, where a "pre-ensemble" is thinned down to generate a "true ensemble". It is not difficult to check that, for each different pair of settings a and b, one gets a different "true ensemble". But Christian's papers say that this is not true. I conclude that the Pearle simulation is *not* a simulation of Christian's model. It does not enable us to compute *one* ensemble of vectors s^k and *then* compute raw scores, for whichever settings a and b you might be interested in.

20. PPS sorry for all the gibberish https://en.wikipedia.org/wiki/Gibberish

14. Joy, the question here in not "is the program a faithful representation of the paper?" Nobody really doubts that.

The real question is "can you apply equation (B10) of http://arxiv.org/abs/1501.03393 in the EPR context?"

I and others believe that you cannot. The program helps to highlight the reason why i.e. it's not real-valued.

I am simply trying to illustrate that point through the algorithm in an effort to get you to defend the extraordinary claim you make in the title and abstract of your one-page paper, arXiv:1103.1879.

1. "Can you apply equation (B10) of http://arxiv.org/abs/1501.03393 in the EPR context?"

Yes.

"I and others believe that you cannot. The program helps to highlight the reason why i.e. it's not real-valued."

You and others need new reading glasses and better programing skills.

There is not a single extraordinary claim in any of my papers. The demonstrations and derivations in all of my papers are utterly ordinary and quite elementary.

Bell and his followers, however, have been making extraordinary claims for the past 50 years. They have been claiming, for example, that there is voodoo in the world. Worse still, their claims are based on elementary topological mistakes and embarrassing logical fallacies, as explained here:

http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=115&start=30#p3977 .

2. Interesting. On 8 June 2015 at 13:47, Joy Christian wrote "Then there is a separate issue of event-by-event simulations of the model, with +1 or -1 outcomes. For that there are several possibilities. One of them is to verify equation (B10) of [...] http://arxiv.org/abs/1501.03393 . We have made no progress so far in that direction."

3. Gill is English by birth, by the way, as far as I know. It is therefore truly astonishing that he can neither read nor understand simple English. Really astonishing.

4. To reiterate, the extraordinary claim is right there in the title of your paper "Disproof of Bell's Theorem."

Bell's Theorem shows that for the class of expectation values that are defined by equation 2 of his classic paper it is not possible to reproduce the quantum mechanical expectation value in his equation 3. A disproof must show that at least one member of the class can.

Now it could be argued that your formulation has reproduced the quantum mechanical results, but you fail to convince me or others that you have started with an equation (5 in your paper) that is of the class that is equivalent to Bell's equation 2.

When confronted with the obvious inconsistency you continuously offer the same three-part defense.

First, without elaboration, declare that you are the one who is correct.

Second, bait the critic by directing an unwarranted ad hominem attack at him or her.

Finally, try to redirect the discussion away from the present conflict, often into the mare's nest of your other papers.

Anything but provide the questioner with an educated response.

Yet my question remains, how can your equation 5 which is formulated in a 3 dimensional vector space be considered a member of the class of expectation values as defined in Bell's equation 2. Bell's equation is limited to real numbers?

5. This comment has been removed by a blog administrator.

6. You have no interest in being educated.

7. This comment has been removed by a blog administrator.

8. Fred and Joy, please no name calling on the blog

9. Fair enough, Albert Jan.

But it is not hard to see what the true intentions are of this blogger:

https://www.blogger.com/profile/02322665002841448855

15. Here is your answer, gauser317: given by Joy Christian, 8 June 2015 at 13:47

"Then there is a separate issue of event-by-event simulations of the model, with +1 or -1 outcomes. For that there are several possibilities. One of them is to verify equation (B10) of [...] http://arxiv.org/abs/1501.03393. We have made no progress so far in that direction."

I think this is why Fred and Joy can only hurl abuse at anyone who asks a sensible question about the "model". They are stuck.

1. So? Wachya gonna do about that, Mr. Gill? Resume your day job of writing malicious letters about me, like a disgruntled old granny, to the President of my College in Oxford?

2. Lets move on.

The cognitive dissonance here seems to derive from your belief that appendix B of http://arxiv.org/abs/1501.03393 establishes a rigorous proof of equation B10.

You seem firm in your conviction that the case has been made and you go on to reach the inevitable corollary, that Bell's Theorem must be incorrect.

But the further application of your method has lead to absurd results that I and others have objected to.

My English language reading of equation B10 is that the real-valued expectation values formed on the left side of your equation are "always" equal to the quantum mechanical expectation values formed on the right hand side.

Bell's Theorem makes the counter argument that these two values can "never" be the same.

Now, as a counter example to your equation B10, consider the case when the hidden variable is always +1. By my reading, the summation in the left hand side would amount to -1. However the dot product on the right hand side is only -1 in the particular case that b=-a.

I must be missing something.

On a side note, I would like to thank Albert Jan for keeping this thread open and relevant.

3. If you have objections to my work, then write them up and publish them at least on the arXiv. I am done with you here.

4. Having been subjected to your intellectually dishonest debating technique and curt attitude for a full week now, I think I will wait until one of these papers is published in a peer-reviewed journal.

As far as my intentions are concerned, I am only seeking the scientific truth here. I harbor no animus toward you or any of the parties involved in this debate.

Years ago, when I first noticed your papers, I believed that the claims therein might be true. As a amateur physicist I wanted the results to be true. But as a programmer, having a long experience with the EPR paradox, I know that this is never enough.

I honestly thought that my criticism would be enlightening but I fear now that I have only caused you anguish. I hope you will believe me when I say this was never my intent.

Live long and prosper.

But of course you never had any intention to read my papers.

6. Yeah, talk about dishonest debating. Still also waiting to see some code from qauser317.

7. You assume too much my friend.

I am well aware that the springer-link article passed peer-review and was published. I think that's great! I have no issue with your findings there.

To save time, I will even concede that everything contained in that paper is established mathematics.

My trouble centers around Appendix B in your Reply to Gill. This is where you directed me, of course I read it, and I've been cogently commenting on it's contents all day. I believe it is here that you may have extend your accepted findings a little bit to far.

If you are at all interested in engaging me further on this subject then my next question to you would be...

Can you supply me with a reference that explains how the standard deviation of a bivector product is equal to the first bivector times the standard deviation of the second as utilized in equations (B7) and (B8) of appendix B?

8. The published paper is the reference you seek. But it is behind the paywall. However, its preprint version is available here:

http://arxiv.org/abs/1211.0784

The relevant derivation is given in great detail in the main text (cf. Eqs. 105 to 110) as well as in the first appendix of the preprint.

9. Joy, I appreciate your informative reply it is both considerate and constructive. Please allow me a few days to review this additional material before I respond.

Fred, not so much.

10. I've had a chance to review the material you suggested. Your derivation of the standard deviation in the second appendix is exceedingly detailed and appears quite sound. I applaud your efforts in explaining this interesting extension to some of the better known statistical methods.

That said, I still cannot reconcile the two sides of equations 102 and 103.

Although both sides return +1 or -1 results, the conditions under which these occur seem to be quite different to me. While the right-hand side is dependent (as noted) only upon lambda the left-hand side seems to be dependent upon both lambda and the direction of vector a (or b). To me, the congruence operator seems to be misapplied in this instance.

Can you offer an explanation to this conundrum?

11. Equations (102) and (103) of that paper are the same as equations (7) and (8) of this paper:

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

The LHS and RHS of these equations define one and the same number, +1 or -1, in two different ways. Think about what the sign-function actually does to the dot products s.a and s.b, and you will see what I mean. It may also help to recall that vector s is dual to the bivector I.s [which is what L(s, lambda) is)], and the sign-functions pick only those s that are either parallel to a or anti-parallel to a. If this fails to convince you, then you can simply take equations (7) and (8) as identities that define the quantities involved in these equations.

12. Equations (7) and (8) demonstrate the issue even more clearly.

Take equation (7).

On the right, the bivector product is only a scalar at the limit (as indicated) when s is "equal to" a. Otherwise there remains a bivector term than cannot be reconciled with the real-valued sign function on the left.

But on the left, the s vector must be allowed to range over 3-D space to produce any -1 values. If s is "equal" to a then the LHS produces +1 all the time.

The equality can only be guaranteed in the case that s = +/- a.

I hope you can see my problem.

13. When cos(s, a) is between 0 and pi/2, the sign-function on the LHS produces +1, and when cos(s, a) is between pi/2 and pi, the sign-function on the LHS produces -1. On the RHS the same +/- function is performed by the orientation lambda. There is no problem.

14. Except for the fact that the RHS only produces +/- in the limited case of s=a.

15. So does the LHS.

Draw a picture of (s dot a) and try to work out what the sign-function does to (s dot a) to produce the numbers +/-1.

I also recommend the paragraph around eq. (17) of this paper:

http://arxiv.org/abs/0806.3078 .

16. Actually, I can do better than that.

Here's a GAViewer script that calculates the sign of the dot product of two vectors. It also multiplies two of these sign functions together just like you do in the LHS of equation (B10) in "Reply to Gill."

function getRandomVector()
{
v = randGaussStd()*e1 +
randGaussStd()*e2 +
randGaussStd()*e3;
return(v);
}

function getRandomUnitVector()
{
v = getRandomVector();
return normalize(v);
}

function sign(x)
{
if(x < 0) return(-1);
return(1);
}

batch test()
{
set_window_title("Test of Joy Christian's Correlation Function");
t = 0;
n = 20000;

a=getRandomUnitVector();
b=getRandomUnitVector();

for(i=0; i<n; i=i+1)
{
s = getRandomVector();
A = sign(s.a);
B = sign(-s.b);
t = t + A*B;
}
Sign_Sign = t/n;
print(Sign_Sign, "f");
Minus_Dot = -a.b;
print(Minus_Dot, "f");
}

You can cut-and-paste this code into a .g file and run it under GAViewer. Ask Albert Jan or Fred to try it if you have any trouble.

This code replicates the results from your proposed experiment in http://arxiv.org/abs/1211.0784. But, as running this program will show, the LHS of equation (B10) in http://arxiv.org/abs/1501.03393 will not produce the results you show on the RHS.

Run this program as many times as you like it will not produce the quantum mechanical correlation. The results speak for themselves!

Please show me how to modify this program to reproduce the results of equation (B10) in http://arxiv.org/abs/1501.03393.

1. Perhaps Albert Jan and / or Fred will try out your code. But I take your word that your code does not work. Perhaps it can be fixed. Programming is not my forte, so I will leave the task for someone more knowledgeable. Eq. (B10) holds analytically, so it should be possible to verify it numerically.

2. I can already tell you the above code is not right for your model. It doesn't have the left handed part from the right hand only perspective. Please try again gauser317.

3. My bad; it is different for the "s" vectors. I don't think GAViewer is right for that part.

17. The above code is exactly what the model described in http://arxiv.org/abs/1211.0784 calls for.

The "right hand only perspective" has already been provided by your script. I recapitulate those results with the variable called Minus_Dot.

Unfortunately, your script fails to provide the "left handed part" which I have thoughtfully volunteered. What is missing of course is any way to reconcile your results (RHS) with mine (LHS). You need to provide that part or stop saying that you have!

You can try to ignore the issue with another cheap excuse or you can address the problem and come up with a solution.

18. Hello Gauser317,