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

Numerical validation of vanishing of the second term in the proof of Joy Christian

In the 2010 publication (1) Joy uses the vanishing of the second term a formula in his proof, which can be seen here:

Florin Moldoveanu has argued that this was not correct because of an illegal use of a minus (2). This argument has later been used by other opponents:

The issue can now easily be verified by using a GA based programming tool GAViewer (3). This tool interprets e1, e2, e3 etc. as the unit vectors that forms the orthogonal base of space. Furthermore it can naturally calculate with geometric objects like bivectors and trivectors, which are used in Joy Christians framework.

The e1, e2 and e3 base, and a bivector in gaviewer.

The code I use is quite easy and listed below. It can be checked by saving it in a text file with the extension '.g', open it with the GAViewer and type 'test()' in the console window. The calculated term is

After a few seconds the result will be something like 

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

, a bivector that has a weight of almost 0, and thereby verifies Joy's assumtion.



Code of the .g test file:
 

function crosspr(a, b)
{
    c=a^b;
    I=e1^e2^e3;
    return c/I;
}

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;               //number of iterations
    I=e1^e2^e3;

    s=0;

    aa=getRandomUnitVector();
    bb=getRandomUnitVector();
    t=crosspr(aa,bb);      //"crosspr(aa,bb)" is cross product
 
    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
        q=mu.t;            //"." is inner product
        s=s+q;             //summation of second term. 
              
    }
    res=s/N;               //calculate average
    print(res);            //print the result

    prompt();

}

Addendum: The code above doesn’t yet cover the full relevant analysis of Joy in (1), say equation 17 to 19, as some of the commentators correctly indicated. So there is still more to be done to disproof Florin Moldoveanu.

1. Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables, Joy Christian, 2010, http://arxiv.org/pdf/quant-ph/0703179v3.pdf
2. Disproof of Joy Christian’s “Disproof of Bell’s theorem”, Florin Moldoveanu , 2011 , http://arxiv.org/abs/1109.0535v1
3. GAViewer: http://geometricalgebra.org/gaviewer_download.html

Recent activity

I like to mention some recent activity on the subject of this blog. First I like to refer to a (type A) simulation from Michel Fodje in python (1), which very neatly shows how CSHS violation could be obtained in practice while having detectors that operate on 100%. Just like in real experiments one has to match the detector countings afterwards using some timestamp algorithm. Michel shows that just a fraction of  0.01% of single particles leaving the emitter and obscuring the countings can result in a CHSH of 2.828, which is equal to the QM value.

 

Current discussions on Joy Christians work can be found on the Blog of Paul Snively (2). Paul is intrigued by the theory from Joy, and suggests to validate the formulas Joy uses in 'CAS': software that can resolve (simplify) formulas that contains variables and symbols. Richard Gill, one of the most active opponents of Joy, is recently blocked on the Blog. Paul Snively somewhat motivates this in one of the discussion posts - and of course he owns the blog - but I would say one normally is not expelled from a discussion for the given reasons. 
Anyway, the discussions are interesting.


1) https://github.com/minkwe/epr-clocked
2)  Blog of Paul Snively http://psnively.github.io/blog/2015/01/22/Fallacy