## New GAViewer Simulation of Joy's S^3 Model

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### New GAViewer Simulation of Joy's S^3 Model

Here is the code,

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html//Joy Christian's S^3 GA Model Correlationsfunction 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+ 0.0000*e3;   return normalize(v);}   batch test(){   set_window_title("Test of Joy Christian's S^3 Local-Realistic Model");   N=50000; //number of iterations (trials)   I=e1^e2^e3;        for(nn=0;nn<N;nn=nn+1) //perform the experiment N times   {      a=getRandomUnitVector();      b=getRandomUnitVector();           lambda=getRandomLambda(); //lambda is a fair coin,                        //resulting in +1 or -1           mu=lambda * I;  //calculate the lambda dependent mu           A=(-I.a)(mu.a); //Eq.(54)of arXiv:1405.2355           B=(mu.b)(I.b);  //Eq.(55)of arXiv:1405.2355           La=A/a;           //Part of eq.(72)of arXiv:1405.2355           Lb=B/b;           //Part of eq.(72)of arXiv:1405.2355           q=0;           if(lambda==1) {q=(La Lb);} else {q=(Lb La);}           print(corrs=scalar(q), "f");  //Output the correlations           angleA=atan2(scalar(a/e1), scalar(a/e2))*180/pi;           angleB=atan2(scalar(b/e1), scalar(b/e2))*180/pi;           angle=abs(angleB - angleA);           print(angle);   //Output the angles   }   prompt();}`

And the result is for 50,000 trials with random angles at one degree resolution and plotting the data in Mathematica,

.
FrediFizzx
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### Re: New GAViewer Simulation of Joy's S^3 Model

***
Many thanks, Fred. The simulation is looking good.

Here is a link to the theory behind the S^3 model: https://arxiv.org/abs/1405.2355.

***
Joy Christian
Research Physicist

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### Re: New GAViewer Simulation of Joy's S^3 Model

Joy Christian wrote:***
Many thanks, Fred. The simulation is looking good.

Here is a link to the theory behind the S^3 model: https://arxiv.org/abs/1405.2355.

***

Your welcome. I added back in the part that shows that the cross products still vanish so we are justified in taking only the scalar part of the correlation value results.

Typical result is,

mean = 0.002779 + 0.001376*e1^e2

Showing that the cross products are vanishing.

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html//Joy Christian's S^3 GA Model Correlationsfunction 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+ 0.0000*e3;   return normalize(v);}   batch test(){   set_window_title("Test of Joy Christian's S^3 Local-Realistic Model");   N=50000; //number of iterations (trials)   I=e1^e2^e3;     s=0;   for(nn=0;nn<N;nn=nn+1) //perform the experiment N times   {          a=getRandomUnitVector();          b=getRandomUnitVector();          lambda=getRandomLambda(); //lambda is a fair coin,                        //resulting in +1 or -1          mu=lambda * I;  //calculate the lambda dependent mu          A=(-I.a)(mu.a); //Eq.(54)of arXiv:1405.2355          B=(mu.b)(I.b);  //Eq.(55)of arXiv:1405.2355          LA=A/a;         //Part of eq.(72)of arXiv:1405.2355          LB=B/b;         //Part of eq.(72)of arXiv:1405.2355          q=0;          if(lambda==1) {q=(LA LB);} else {q=(LB LA);}          s=s+q;          print(corrs=scalar(q), "f");  //Output the correlations          angleA=atan2(scalar(a/e1), scalar(a/e2))*180/pi;          angleB=atan2(scalar(b/e1), scalar(b/e2))*180/pi;          angle=abs(angleB - angleA);          print(angle);   //Output the angles      }      mean=s/N;      print(mean, "f");      prompt();}`
FrediFizzx
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
Joy Christian wrote:***
Many thanks, Fred. The simulation is looking good.

Here is a link to the theory behind the S^3 model: https://arxiv.org/abs/1405.2355.

***

Your welcome. I added back in the part that shows that the cross products still vanish so we are justified in taking only the scalar part of the correlation value results.

Typical result is,

mean = 0.502779 + 0.001376*e1^e2

Showing that the cross products are vanishing.

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html//Joy Christian's S^3 GA Model Correlationsfunction 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+ 0.0000*e3;   return normalize(v);}   batch test(){   set_window_title("Test of Joy Christian's S^3 Local-Realistic Model");   N=50000; //number of iterations (trials)   I=e1^e2^e3;     s=0;   for(nn=0;nn<N;nn=nn+1) //perform the experiment N times   {          a=getRandomUnitVector();          b=getRandomUnitVector();          lambda=getRandomLambda(); //lambda is a fair coin,                        //resulting in +1 or -1          mu=lambda * I;  //calculate the lambda dependent mu          A=(-I.a)(mu.a); //Eq.(54)of arXiv:1405.2355          B=(mu.b)(I.b);  //Eq.(55)of arXiv:1405.2355          LA=A/a;         //Part of eq.(72)of arXiv:1405.2355          LB=B/b;         //Part of eq.(72)of arXiv:1405.2355          q=0;          if(lambda==1) {q=(LA LB);} else {q=(LB LA);}          s=s+q          print(corrs=scalar(q), "f");  //Output the correlations          angleA=atan2(scalar(a/e1), scalar(a/e2))*180/pi;          angleB=atan2(scalar(b/e1), scalar(b/e2))*180/pi;          angle=abs(angleB - angleA);          print(angle);   //Output the angles      }      mean=s/N;      print(mean, "f");      prompt();}`

Fred,

Does it not remain possible to obtain the EPRB result

E(AB) = - cos(θ) (1)

from a countable infinity of tests (if you wish) in which no two cross-products are the same?

That is: is it not possible to hold the angle between the detectors constant (ie, = θ over a countable infinity of N tests, if you wish) while the cross-product of the detector settings (say a_i and b_i) with i = 1, 2, …, N) never repeats over N tests?

In other words, is it not the case that valid tests can be constructed in which the cross-products accumulate and never vanish?

Thanks; Gordon

(me declaring my view that 3-space is an adequate venue for any Bell-test)
Gordon Watson

Posts: 255
Joined: Wed Apr 30, 2014 4:39 am

### Re: New GAViewer Simulation of Joy's S^3 Model

Gordon Watson wrote:Does it not remain possible to obtain the EPRB result

E(AB) = - cos(θ) (1)

from a countable infinity of tests (if you wish) in which no two cross-products are the same?

That is: is it not possible to hold the angle between the detectors constant (ie, = θ over a countable infinity of N tests, if you wish) while the cross-product of the detector settings (say a_i and b_i) with i = 1, 2, …, N) never repeats over N tests?

In other words, is it not the case that valid tests can be constructed in which the cross-products accumulate and never vanish?

Thanks; Gordon

(me declaring my view that 3-space is an adequate venue for any Bell-test)

I don't think so. The cross product is a x b = c. The average of c will vanish at infinity.
FrediFizzx
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Posts: 1833
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Location: N. California, USA

### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
Gordon Watson wrote:Does it not remain possible to obtain the EPRB result

E(AB) = - cos(θ) (1)

from a countable infinity of tests (if you wish) in which no two cross-products are the same?

That is: is it not possible to hold the angle between the detectors constant (ie, = θ over a countable infinity of N tests, if you wish) while the cross-product of the detector settings (say a_i and b_i) with i = 1, 2, …, N) never repeats over N tests?

In other words, is it not the case that valid tests can be constructed in which the cross-products accumulate and never vanish?

Thanks; Gordon

(me declaring my view that 3-space is an adequate venue for any Bell-test)

I don't think so. The cross product is a x b = c. The average of c will vanish at infinity.

Let (a_i) x (b_i) = c_i. Let Σ(c_i) = Nc via accumulation. Then Σ(c_i)/N = c ≠ 0.
.
Gordon Watson

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Joined: Wed Apr 30, 2014 4:39 am

### Re: New GAViewer Simulation of Joy's S^3 Model

Gordon Watson wrote:Let (a_i) x (b_i) = c_i. Let Σ(c_i) = Nc via accumulation. Then Σ(c_i)/N = c ≠ 0.
.

The (sum of c_i)/N will certainly be 0 in the limit as N --> infinity. Are we doing sums or multiplication? Nc is multiplication.
.
FrediFizzx
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
Gordon Watson wrote:Let (a_i) x (b_i) = c_i. Let Σ(c_i) = Nc via accumulation. Then Σ(c_i)/N = c ≠ 0.
.

The (sum of c_i)/N will certainly be 0 in the limit as N --> infinity. Are we doing sums or multiplication? Nc is multiplication.
.

Fred,

In deriving the expectation E(AB) under the constrained scenario, let the sum Σ of the N vectors c_i be the vector C (which is not zero). Then C/N = c = average vector contribution per test; which is not zero, even as N --> infinity.
.
Gordon Watson

Posts: 255
Joined: Wed Apr 30, 2014 4:39 am

### Re: New GAViewer Simulation of Joy's S^3 Model

Gordon Watson wrote:
FrediFizzx wrote:
Gordon Watson wrote:Let (a_i) x (b_i) = c_i. Let Σ(c_i) = Nc via accumulation. Then Σ(c_i)/N = c ≠ 0.
.

The (sum of c_i)/N will certainly be 0 in the limit as N --> infinity. Are we doing sums or multiplication? Nc is multiplication.
.

Fred,

In deriving the expectation E(AB) under the constrained scenario, let the sum Σ of the N vectors c_i be the vector C (which is not zero). Then C/N = c = average vector contribution per test; which is not zero, even as N --> infinity.
.

Unless C is infinity, then anything X/N as N --> infinity will be zero.
.
FrediFizzx
Independent Physics Researcher

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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
Gordon Watson wrote:
FrediFizzx wrote:
Gordon Watson wrote:Let (a_i) x (b_i) = c_i. Let Σ(c_i) = Nc via accumulation. Then Σ(c_i)/N = c ≠ 0.
.

The (sum of c_i)/N will certainly be 0 in the limit as N --> infinity. Are we doing sums or multiplication? Nc is multiplication.
.

Fred,

In deriving the expectation E(AB) under the constrained scenario, let the sum Σ of the N vectors c_i be the vector C (which is not zero). Then C/N = c = average vector contribution per test; which is not zero, even as N --> infinity.
.

Unless C is infinity, then anything X/N as N --> infinity will be zero.
.

But under the proposed constraint, C --> infinity as N --> infinity.
.
Gordon Watson

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Joined: Wed Apr 30, 2014 4:39 am

### Re: New GAViewer Simulation of Joy's S^3 Model

Gordon Watson wrote:But under the proposed constraint, C --> infinity as N --> infinity.
.

But that is not what is happening since half of the "c's" will cancel out the other half in the summation so C doesn't go to infinity. If not zero, it will be a very tiny number.
.
FrediFizzx
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
Gordon Watson wrote:But under the proposed constraint, C --> infinity as N --> infinity.
.

But that is not what is happening since half of the "c's" will cancel out the other half in the summation so C doesn't go to infinity. If not zero, it will be a very tiny number.
.

Fred, Gordon is not talking about my model. In my model the bivector term vanishes exactly in the N --> infinity limit, which means integrating rather than summing.

***
Joy Christian
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### Re: New GAViewer Simulation of Joy's S^3 Model

Here is the GAViewer code for the CHSH version of the GA S^3 model with test angles in degrees of a1= 0, a2 = 90, b1 = 45, and b2 = -45. With an equivalent notation change from the previous code.

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html//CHSH evaluation of Joy Christian's S^3 Modelbatch test(){     set_window_title("Test of Joy Christian's CHSH derivation");     N=5000; //number of iterations (trials)     I=e1^e2^e3;     s=0;     a1=sin(0)*e1 + cos(0)*e2 + 0.000*e3;     b1=sin(pi/4)*e1 + cos(pi/4)*e2 + 0.000*e3;     a2=sin(pi/2)*e1 + cos(pi/2)*e2 + 0.000*e3;     b2=sin(7*pi/4)*e1 + cos(7*pi/4)*e2 + 0.000*e3;     for(nn=0;nn<N;nn=nn+1) //perform the experiment N times     {          if(rand()>0.5) {lambda= 1;} else {lambda = -1;}          mu=lambda*I;  //calculate the lambda dependent mu          A1=(-I.a1)(mu.a1);          B1=(mu.b1)(I.b1);          A2=(-I.a2)(mu.a2);          B2=(mu.b2)(I.b2);          q=0;          if(lambda==1) {q= (A1 B1)/(a1 b1) + (A1 B2)/(a1 b2) +(A2 B1)/(a2 b1)-(A2 B2)/(a2 b2);}           else {q= (B1 A1)/(b1 a1)+ (B2 A1)/(b2 a1)+(B1 A2)/(b1 a2)-(B2 A2)/(b2 a2);}          s=s+q;     }     Joy_CHSH=abs(s/N);     print(Joy_CHSH, "f");      prompt();}`

And the result is again,

Joy_CHSH = 2.828427

Note that the bivector "residue" has completely canceled out.
.
FrediFizzx
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:Here is the GAViewer code for the CHSH version of the GA S^3 model with test angles in degrees of a1= 0, a2 = 90, b1 = 45, and b2 = -45. With an equivalent notation change from the previous code.

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html//CHSH evaluation of Joy Christian's S^3 Modelbatch test(){     set_window_title("Test of Joy Christian's CHSH derivation");     N=5000; //number of iterations (trials)     I=e1^e2^e3;     s=0;     a1=sin(0)*e1 + cos(0)*e2 + 0.000*e3;     b1=sin(pi/4)*e1 + cos(pi/4)*e2 + 0.000*e3;     a2=sin(pi/2)*e1 + cos(pi/2)*e2 + 0.000*e3;     b2=sin(7*pi/4)*e1 + cos(7*pi/4)*e2 + 0.000*e3;     for(nn=0;nn<N;nn=nn+1) //perform the experiment N times     {          if(rand()>0.5) {lambda= 1;} else {lambda = -1;}          mu=lambda*I;  //calculate the lambda dependent mu          A1=(-I.a1)(mu.a1);          B1=(mu.b1)(I.b1);          A2=(-I.a2)(mu.a2);          B2=(mu.b2)(I.b2);          q=0;          if(lambda==1) {q= (A1 B1)/(a1 b1) + (A1 B2)/(a1 b2) +(A2 B1)/(a2 b1)-(A2 B2)/(a2 b2);}           else {q= (B1 A1)/(b1 a1)+ (B2 A1)/(b2 a1)+(B1 A2)/(b1 a2)-(B2 A2)/(b2 a2);}          s=s+q;     }     Joy_CHSH=abs(s/N);     print(Joy_CHSH, "f");      prompt();}`

And the result is again,

Joy_CHSH = 2.828427

Note that the bivector "residue" has completely canceled out.
.

However, please note that this does not constitute a violation of Bell-CHSH since we know for a fact that a1 and a2 or b1 and b2 can't happen at the same time in an EPR-Bohm scenario. This illustrates the same trick that QM uses to show a "violation". What this does illustrate is that a local-realistic model gets the same prediction as QM for a CHSH-like string of expectation terms. And that is the important point that only matters here.
.
FrediFizzx
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Posts: 1833
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
FrediFizzx wrote:Here is the GAViewer code for the CHSH version of the GA S^3 model with test angles in degrees of a1= 0, a2 = 90, b1 = 45, and b2 = -45. With an equivalent notation change from the previous code.

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html//CHSH evaluation of Joy Christian's S^3 Modelbatch test(){     set_window_title("Test of Joy Christian's CHSH derivation");     N=5000; //number of iterations (trials)     I=e1^e2^e3;     s=0;     a1=sin(0)*e1 + cos(0)*e2 + 0.000*e3;     b1=sin(pi/4)*e1 + cos(pi/4)*e2 + 0.000*e3;     a2=sin(pi/2)*e1 + cos(pi/2)*e2 + 0.000*e3;     b2=sin(7*pi/4)*e1 + cos(7*pi/4)*e2 + 0.000*e3;     for(nn=0;nn<N;nn=nn+1) //perform the experiment N times     {          if(rand()>0.5) {lambda= 1;} else {lambda = -1;}          mu=lambda*I;  //calculate the lambda dependent mu          A1=(-I.a1)(mu.a1);          B1=(mu.b1)(I.b1);          A2=(-I.a2)(mu.a2);          B2=(mu.b2)(I.b2);          q=0;          if(lambda==1) {q= (A1 B1)/(a1 b1) + (A1 B2)/(a1 b2) +(A2 B1)/(a2 b1)-(A2 B2)/(a2 b2);}           else {q= (B1 A1)/(b1 a1)+ (B2 A1)/(b2 a1)+(B1 A2)/(b1 a2)-(B2 A2)/(b2 a2);}          s=s+q;     }     Joy_CHSH=abs(s/N);     print(Joy_CHSH, "f");      prompt();}`

And the result is again,

Joy_CHSH = 2.828427

Note that the bivector "residue" has completely canceled out.
.

However, please note that this does not constitute a violation of Bell-CHSH since we know for a fact that a1 and a2 or b1 and b2 can't happen at the same time in an EPR-Bohm scenario. This illustrates the same trick that QM uses to show a "violation". What this does illustrate is that a local-realistic model gets the same prediction as QM for a CHSH-like string of expectation terms. And that is the important point that only matters here.
.

Fred, given that we can match particle-pairs by coincidence-matching, why is "at the same time" relevant to the settings in your testing scenario?

In particular, if Joy's model integrates over the whole space of the HVs, you can simply set up 4 benches side-by-side and run all tests "at the same time" (near-enough), each test integrating over the whole HV space in each case. The 4 bench settings will be (a1-b1), (a1-b2), (a2-b1), (a2-b2).

Or am I missing something?
.
Gordon Watson

Posts: 255
Joined: Wed Apr 30, 2014 4:39 am

### Re: New GAViewer Simulation of Joy's S^3 Model

Gordon Watson wrote:Fred, given that we can match particle-pairs by coincidence-matching, why is "at the same time" relevant to the settings in your testing scenario?

In particular, if Joy's model integrates over the whole space of the HVs, you can simply set up 4 benches side-by-side and run all tests "at the same time" (near-enough), each test integrating over the whole HV space in each case. The 4 bench settings will be (a1-b1), (a1-b2), (a2-b1), (a2-b2).

Or am I missing something?
.

Yes... you are missing the difference between dependent expectation terms and independent terms. With the former, the bound is |2|. With the later the absolute bound is |4| and the QM bound is 2*sqrt(2). In the code above, A1 from the first term is the same A1 in the second term, etc. per particle pair. In your scenario of doing 4 separate experiments, A1 in the first term is not necessarily the same as the A1 in the second term, etc. for the B's. So it will have completely independent expectation terms with an absolute bound of |4|.

But in the code above, a1, a2, b1, and b2 are all happening at the same time which is impossible for an EPR-Bohm scenario.

But the bottom line is that Joy's model gets the same prediction as QM in the same "tricky" way for a CHSH-like string of terms. If your local-realistic model simply has a prediction of -a.b, then that is all you really need.
.
FrediFizzx
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
But in the code above, a1, a2, b1, and b2 are all happening at the same time which is impossible for an EPR-Bohm scenario.

But the bottom line is that Joy's model gets the same prediction as QM in the same "tricky" way for a CHSH-like string of terms. If your local-realistic model simply has a prediction of -a.b, then that is all you really need.
.

The underlying reason for the strong correlations and bounds of 2\/2 on CHSH is the same for both my model and quantum mechanics; namely, the non-commutative Clifford algebra of orthogonal directions in the 3D space. Consequently, the bivectors I.a1, I.a2, I.b1 and I.b2 do not commute, unlike the vectors a1, a2, b1 and b2.

***
Joy Christian
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### Re: New GAViewer Simulation of Joy's S^3 Model

FrediFizzx wrote:
Gordon Watson wrote:Fred, given that we can match particle-pairs by coincidence-matching, why is "at the same time" relevant to the settings in your testing scenario?

In particular, if Joy's model integrates over the whole space of the HVs, you can simply set up 4 benches side-by-side and run all tests "at the same time" (near-enough), each test integrating over the whole HV space in each case. The 4 bench settings will be (a1-b1), (a1-b2), (a2-b1), (a2-b2).

Or am I missing something?
.

Yes... you are missing the difference between dependent expectation terms and independent terms. With the former, the bound is |2|. With the later the absolute bound is |4| and the QM bound is 2*sqrt(2). In the code above, A1 from the first term is the same A1 in the second term, etc. per particle pair. In your scenario of doing 4 separate experiments, A1 in the first term is not necessarily the same as the A1 in the second term, etc. for the B's. So it will have completely independent expectation terms with an absolute bound of |4|.

But in the code above, a1, a2, b1, and b2 are all happening at the same time which is impossible for an EPR-Bohm scenario.

But the bottom line is that Joy's model gets the same prediction as QM in the same "tricky" way for a CHSH-like string of terms. If your local-realistic model simply has a prediction of -a.b, then that is all you really need.
.

Thanks Fred, it's that "trickiness" in the terminology that I seek to avoid.

In response to your, "If your local-realistic model simply has a prediction of -a.b, then that is all you really need." I agree, and it does!

PS and NB: 2√2 comes directly off my computer keyboard via ... 2 "option v" 2.

Thanks.
Gordon Watson

Posts: 255
Joined: Wed Apr 30, 2014 4:39 am

### Re: New GAViewer Simulation of Joy's S^3 Model

Joy Christian wrote:
FrediFizzx wrote:
But in the code above, a1, a2, b1, and b2 are all happening at the same time which is impossible for an EPR-Bohm scenario.

But the bottom line is that Joy's model gets the same prediction as QM in the same "tricky" way for a CHSH-like string of terms. If your local-realistic model simply has a prediction of -a.b, then that is all you really need.
.

The underlying reason for the strong correlations and bounds of 2\/2 on CHSH is the same for both my model and quantum mechanics; namely, the non-commutative Clifford algebra of orthogonal directions in the 3D space. Consequently, the bivectors I.a1, I.a2, I.b1 and I.b2 do not commute, unlike the vectors a1, a2, b1 and b2.

***

Fred, I should add here that what appears in the denominators in the code are actually bivectors I.a1, I.a2, I.b1 and I.b2, not vectors a1, a2, b1 and b2, with I^2 = -1.

***
Joy Christian
Research Physicist

Posts: 2260
Joined: Wed Feb 05, 2014 4:49 am
Location: Oxford, United Kingdom

### Re: New GAViewer Simulation of Joy's S^3 Model

Joy Christian wrote:
Joy Christian wrote:
FrediFizzx wrote:
But in the code above, a1, a2, b1, and b2 are all happening at the same time which is impossible for an EPR-Bohm scenario.

But the bottom line is that Joy's model gets the same prediction as QM in the same "tricky" way for a CHSH-like string of terms. If your local-realistic model simply has a prediction of -a.b, then that is all you really need.
.

The underlying reason for the strong correlations and bounds of 2\/2 on CHSH is the same for both my model and quantum mechanics; namely, the non-commutative Clifford algebra of orthogonal directions in the 3D space. Consequently, the bivectors I.a1, I.a2, I.b1 and I.b2 do not commute, unlike the vectors a1, a2, b1 and b2.

***

Fred, I should add here that what appears in the denominators in the code are actually bivectors I.a1, I.a2, I.b1 and I.b2, not vectors a1, a2, b1 and b2, with I^2 = -1.
***

Ah... that would explain why I got -2.828427 without the absolute value taken. It should be -(a1 b1), etc.
.
FrediFizzx
Independent Physics Researcher

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Location: N. California, USA