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