## 3-Sphere 3D Vectors 2-Particle Simulation

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

### Re: 3-Sphere 3D Vectors 2-Particle Simulation

FrediFizzx wrote:
jreed wrote:
FrediFizzx wrote:It might actually be easier in Mathematica to just do the QM math with Pauli algebra. There is bound to be some good QM packages for Mathematica. Maybe I will look that up as I hadn't thought of it before.
.

I'm using Pauli matrices to represent the vectors in geometric algebra. The other components can be constructed from these. The Pauli matrices are already in Mathematica.

Oh, that is right. Looking forward to seeing this calculation in Mathematica.
.

Nota bene: You can divide any GA quantity by any other GA quantity (such as a bivector), but you cannot divide any quantity of any kind by a matrix, be it a Pauli matrix.

***
Joy Christian
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Joy Christian wrote:
FrediFizzx wrote:
jreed wrote:
FrediFizzx wrote:It might actually be easier in Mathematica to just do the QM math with Pauli algebra. There is bound to be some good QM packages for Mathematica. Maybe I will look that up as I hadn't thought of it before.
.

I'm using Pauli matrices to represent the vectors in geometric algebra. The other components can be constructed from these. The Pauli matrices are already in Mathematica.

Oh, that is right. Looking forward to seeing this calculation in Mathematica.
.

Nota bene: You can divide any GA quantity by any other GA quantity (such as a bivector), but you cannot divide any quantity of any kind by a matrix, be it a Pauli matrix.

***

I don't expect any division will be necessary. And you can actually do the limits in Mathematica. Maybe I will try to setup the measurement functions in Mathematica. Doing the random 3D vectors for a, b and s will be more difficult though. I think I will leave this to John to work on as my old age laziness is kickin' in. .
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Yeah, no problem doing the limits in Mathematica. .
FrediFizzx
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Joy Christian wrote:Nota bene: You can divide any GA quantity by any other GA quantity (such as a bivector), but you cannot divide any quantity of any kind by a matrix, be it a Pauli matrix.
***

This is so silly that it's surely meant to be a joke. Have you never inverted a matrix?
Heinera

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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Heinera wrote:
Joy Christian wrote:
Nota bene: You can divide any GA quantity by any other GA quantity (such as a bivector), but you cannot divide any quantity of any kind by a matrix, be it a Pauli matrix.
***

This is so silly that it's surely meant to be a joke. Have you never inverted a matrix?

It is not a joke. Matrix algebra is not a division algebra.

***
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

FrediFizzx
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

In terms of the first program Fred posted, can we agree:
1. a and b are the detector settings, and are independent of each other and everything else
2. theta is entirely determined by a and b
3. The x-axis of Fred's first posted graph is theta and the y axis is suppossed to be a calculated correlation based on the simulated trials
I think the answer is "yes, we can agree on this". Richard, do you agree? (Then I wonder how you are bucketing the 50000 theta values to create the graph, but I guess it doesn't matter.)

Then I wonder: is the correlation calculated correctly? Richard, do you think so?
Lambda appears to be necessary for this to work. That seems ok to me, if 1 above is true. Or, Richard, is the problem that the use of Lambda makes the correlation calculation invalid? If so, it still seems interesting that a single binary random variable alone, Lambda, makes the correlations match quantum theory, given that Richard's side has been arguing you would need more randomness. Has anyone else run the code and got the graph?

Let's say Richard says Lambda makes the correlation calculation invalid. Can it be interpreted in a physical manner that refutes this? What is that interpretation?

Finally, is this correlation versus theta graph what Bell's Theorem says is impossible? I know of Bell's theorem in an entirely different format. I am wondering if this is just one piece of what Joy needs to demonstrate, or if it just one of many things.

Or is the bump in the carpet somewhere else?
Lord of the Physics

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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Lord of the Physics wrote:In terms of the first program Fred posted, can we agree:
1. a and b are the detector settings, and are independent of each other and everything else
2. theta is entirely determined by a and b
3. The x-axis of Fred's first posted graph is theta and the y axis is suppossed to be a calculated correlation based on the simulated trials
I think the answer is "yes, we can agree on this". Richard, do you agree? (Then I wonder how you are bucketing the 50000 theta values to create the graph, but I guess it doesn't matter.)

Then I wonder: is the correlation calculated correctly? Richard, do you think so?
Lambda appears to be necessary for this to work. That seems ok to me, if 1 above is true. Or, Richard, is the problem that the use of Lambda makes the correlation calculation invalid? If so, it still seems interesting that a single binary random variable alone, Lambda, makes the correlations match quantum theory, given that Richard's side has been arguing you would need more randomness. Has anyone else run the code and got the graph?

Let's say Richard says Lambda makes the correlation calculation invalid. Can it be interpreted in a physical manner that refutes this? What is that interpretation?

Finally, is this correlation versus theta graph what Bell's Theorem says is impossible? I know of Bell's theorem in an entirely different format. I am wondering if this is just one piece of what Joy needs to demonstrate, or if it just one of many things.

Or is the bump in the carpet somewhere else?

Nobody is objecting or doubting your 1-3. They are pretty straight forward. The correlations are calculated correctly. Nobody is doubting that either since a computer is doing the calculating. However, the first code I posted is pretty irrelevant compared to the second set of code that I posted. That first set of code was just a "warm up". Lambda is not required to mathematically obtain the correct correlations for the product calculation. But lambda IS required for the correct physical picture.

Yes, this calculation of the product is what Bell's junk physic theorem says is impossible. But quantum mechanics is local with or without the hidden variable. So no more freakin' spookiness!!!!!!!!!!!!!!
.
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Nobody is objecting or doubting your 1-3. They are pretty straight forward. The correlations are calculated correctly. Nobody is doubting that either since a computer is doing the calculating. However, the first code I posted is pretty irrelevant compared to the second set of code that I posted. That first set of code was just a "warm up". Lambda is not required to mathematically obtain the correct correlations for the product calculation. But lambda IS required for the correct physical picture.

Yes, this calculation of the product is what Bell's junk physic theorem says is impossible. But quantum mechanics is local with or without the hidden variable. So no more freakin' spookiness!!!!!!!!!!!!!!
.

Looking at the second set of code and results, I see a problem I realize was now a problem for the first. Theta is the result of random variables in the simulation, and it will be rare to see the same value of theta twice in the same simulation, let alone to get enough responses for that theta to calculate a good estimate of the correlation at theta. It seems these simulations are not really simulations, but calculate the value of a derived formula at random values of theta. You do it enough times to cover an interval. But you could just go sequentally through a grid of thetas. But this would just illustrate a theoretical formula. Am I making sense to anyone? (It would be a weird formula in that you flip a coin (lambda) to decide which version of the formula to use, each time you plug into it.)
Lord of the Physics

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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Lord of the Physics wrote:
Nobody is objecting or doubting your 1-3. They are pretty straight forward. The correlations are calculated correctly. Nobody is doubting that either since a computer is doing the calculating. However, the first code I posted is pretty irrelevant compared to the second set of code that I posted. That first set of code was just a "warm up". Lambda is not required to mathematically obtain the correct correlations for the product calculation. But lambda IS required for the correct physical picture.

Yes, this calculation of the product is what Bell's junk physic theorem says is impossible. But quantum mechanics is local with or without the hidden variable. So no more freakin' spookiness!!!!!!!!!!!!!!
.

Looking at the second set of code and results, I see a problem I realize was now a problem for the first. Theta is the result of random variables in the simulation, and it will be rare to see the same value of theta twice in the same simulation, let alone to get enough responses for that theta to calculate a good estimate of the correllation at theta. It seems these simulations are not really simulations, but calculate the value of a derived formula at random values of theta. You do it enough times to cover an interval. But you could just go sequentally through a grid of thetas. But this would just illustrate a theoretical formula. Am I making sense to anyone?

You are sort of making sense. The code is an event by event calculation of the product of the measurement functions. That is all it is. Validation of the product calculation for random a, b and s vectors. We never claimed for it to be anything more than that. QM can't predict individual event by event outcomes. And we are not even going to try that.
.
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Lord of the Physics wrote: Am I making sense to anyone? (It would be a weird formula in that you flip a coin (lambda) to decide which version of the formula to use, each time you plug into it.)

Yes, you make perfect sense. This is exactly what they do.
Heinera

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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Heinera wrote:
Lord of the Physics wrote: Am I making sense to anyone? (It would be a weird formula in that you flip a coin (lambda) to decide which version of the formula to use, each time you plug into it.)

Yes, you make perfect sense. This is exactly what they do.

Lord doesn't seem to understand what lambda is. It is the left or right handedness of the singlet. If lambda = +1, then process the formula for the right handed singlet. If lambda =-1, then process the formula for the left handed singlet which is just the reverse order of the right handed formula.
.
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

FrediFizzx wrote:You are sort of making sense. The code is an event by event calculation of the product of the measurement functions. That is all it is. Validation of the product calculation for random a, b and s vectors. We never claimed for it to be anything more than that. QM can't predict individual event by event outcomes. And we are not even going to try that.
.

If your hidden variable model is correct, then it can predict event by event outcomes when the hidden variable is known. One can do an event by event simulation like this: for each theta in a grid, randomly generate the hidden variable repeatedly and calculate the outcomes (which will depend on the theta you are fixed at). Then calculate the correlations in the outcomes over the repeated trialsn for that theta. Then plot calculated correlation versus theta.
Lord of the Physics

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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Lord of the Physics wrote:
FrediFizzx wrote:You are sort of making sense. The code is an event by event calculation of the product of the measurement functions. That is all it is. Validation of the product calculation for random a, b and s vectors. We never claimed for it to be anything more than that. QM can't predict individual event by event outcomes. And we are not even going to try that.
.

If your hidden variable model is correct, then it can predict event by event outcomes when the hidden variable is known. One can do an event by event simulation like this: for each theta in a grid, randomly generate the hidden variable repeatedly and calculate the outcomes (which will depend on the theta you are fixed at). Then calculate the correlations in the outcomes over the repeated trialsn for that theta. Then plot calculated correlation versus theta.

If you can figure out how QM can predict individual outcomes event by event then let us know. This is a mapping to GA from a QM local model. The best QM can do is predict A = +/-1 and B = +/-1 50-50.
.
FrediFizzx
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

FrediFizzx wrote:If you can figure out how QM can predict individual outcomes event by event then let us know. This is a mapping to GA from a QM local model. The best QM can do is predict A = +/-1 and B = +/-1 50-50.
.

I contend that to simulate a refutation of Bell, you randomly generate a bunch of initial stuff. Then independently after that, you randomly generate two detector settings independent of each other. And after that the exact outcomes are completely determined. You seem to say that there is still some randomness left after the detector setting are chosen. Ok, then generate random variables independent of everything else at that point to get an outcome. I guess you are allowed to let the extra random stuff at detector A depend on thr detector A setting, but not the setting at B (and vice versa).
Lord of the Physics

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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

Lord of the Physics wrote:
FrediFizzx wrote:If you can figure out how QM can predict individual outcomes event by event then let us know. This is a mapping to GA from a QM local model. The best QM can do is predict A = +/-1 and B = +/-1 50-50.
.

I contend that to simulate a refutation of Bell, you randomly generate a bunch of initial stuff. Then independently after that, you randomly generate two detector settings independent of each other. And after that the exact outcomes are completely determined. You seem to say that there is still some randomness left after the detector setting are chosen. Ok, then generate random variables independent of everything else at that point to get an outcome. I guess you are allowed to let the extra random stuff at detector A depend on thr detector A setting, but not the setting at B (and vice versa).

We don't need to simulate a refutation of Bell. Bell's theorem can be disproven without any simulation. This is proof that QM is in fact local for the EPR-Bohm scenario.
.
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

FrediFizzx wrote:Yeah, no problem doing the limits in Mathematica. .

But Mathematica did not evaluate the result. It just copied the formula you gave it. You still need to supply a value of s (and of a) if you want to get an answer.
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

gill1109 wrote:
FrediFizzx wrote:Yeah, no problem doing the limits in Mathematica. .

But Mathematica did not evaluate the result. It just copied the formula you gave it. You still need to supply a value of s (and of a) if you want to get an answer.

I can't believe you are having this much trouble with the limits. Of course it would have to have a value of s and a to get +a or -a. It is a symbolic result. But we use,
Code: Select all
`if(sgn(a.s)==1) {S1=Da;} else {S1=-Da;}  //polarizer takes S1 to +/-Daif(sgn(b.s)==1) {S2=Db;} else {S2=-Db;}  //polarizer takes S2 to +/-DbWhich in the QM local model would be because s = s1 = s2,if(sgn(a.s)==1) {s1=a;} else {s1=-a;}  //polarizer takes s1 to +/-aif(sgn(b.s)==1) {s2=b;} else {s2=-b;}  //polarizer takes s2 to +/-b`

Hopefully those functions are more understandable for you. They do the same thing as the limits.
.
FrediFizzx
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

FrediFizzx wrote:
gill1109 wrote:
FrediFizzx wrote:Yeah, no problem doing the limits in Mathematica. .

But Mathematica did not evaluate the result. It just copied the formula you gave it. You still need to supply a value of s (and of a) if you want to get an answer.

I can't believe you are having this much trouble with the limits. Of course it would have to have a value of s and a to get +a or -a. It is a symbolic result. But we use,
[code]
if(sgn(a.s)==1) {S1=Da;} else {S1=-Da;} //polarizer takes S1 to +/-Da
if(sgn(b.s)==1) {S2=Db;} else {S2=-Db;} //polarizer takes S2 to +/-Db

Which in the QM local model would be because s = s1 = s2,

if(sgn(a.s)==1) {s1=a;} else {s1=-a;} //polarizer takes s1 to +/-a
if(sgn(b.s)==1) {s2=b;} else {s2=-b;} //polarizer takes s2 to +/-b
[/code
Hopefully those functions are more understandable for you. They do the same thing as the limits.
.

I understand your code very well indeed. Why don't you write math formulas, or Mathematica expressions, which correspond to your code?
Using the symbol "colon equals" for "left hand side is assigned the result of evaluating the right hand side", you have:

s1:= sign(a.s)a
s2:= sign(b.s)b
What is s? I guess a uniformly distributed unit vector in R^3
And then what are the formulas for the two binary measurement outcomes?
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### Re: 3-Sphere 3D Vectors 2-Particle Simulation

gill1109 wrote:
FrediFizzx wrote: ...
I can't believe you are having this much trouble with the limits. Of course it would have to have a value of s and a to get +a or -a. It is a symbolic result. But we use,
Code: Select all
`if(sgn(a.s)==1) {S1=Da;} else {S1=-Da;}  //polarizer takes S1 to +/-Daif(sgn(b.s)==1) {S2=Db;} else {S2=-Db;}  //polarizer takes S2 to +/-DbWhich in the QM local model would be because s = s1 = s2,if(sgn(a.s)==1) {s1=a;} else {s1=-a;}  //polarizer takes s1 to +/-aif(sgn(b.s)==1) {s2=b;} else {s2=-b;}  //polarizer takes s2 to +/-b`

Hopefully those functions are more understandable for you. They do the same thing as the limits.
.

I understand your code very well indeed. Why don't you write math formulas, or Mathematica expressions, which correspond to your code?
Using the symbol "colon equals" for "left hand side is assigned the result of evaluating the right hand side", you have:

s1:= sign(a.s)a
s2:= sign(b.s)b
What is s? I guess a uniformly distributed unit vector in R^3
And then what are the formulas for the two binary measurement outcomes?

Good that you understand the code. The limits reflect the actual physics better since those sign functions don't happen until the particles hit the polarizers and are the easiest to implement in the measurement functions so we will be sticking with them. I guess you haven't even looked at the code yet otherwise you would know what s and the measurement functions are. Here it is again.

Code: Select all
`//Adaptation of Albert Jan Wonnink's original code based on GAViewer for Joy Christian's S^3 Model of the 2-particle //correlation.  This is a mapping of the quantum mechanics project to GA.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; //3D Vectors   return normalize(v);}function sgn(y){     if(y < 0) {return(-1);} else {return(1);}}   batch test(){   set_window_title("3D Test of QM mapped to GA S^3 Model for the 2-particle correlation");   default_model(p3ga);   N=5000;                               //number of iterations (trials)   I=e1^e2^e3;   ss=0;   t=0;   u=0;   for(nn=0;nn<N;nn=nn+1)                  //perform the experiment N times   {          a=getRandomUnitVector();          Da=I a;          b=getRandomUnitVector();          Db=I b;          s=getRandomUnitVector();          //S1=I s;               //bivector for particle spins          //S2=S1;               //-S1 + S2 = 0 for singlet zero spin          if(sgn(a.s)==1) {S1=Da;} else {S1=-Da;}  //polarizer takes S1 to +/-Da          if(sgn(b.s)==1) {S2=Db;} else {S2=-Db;}  //polarizer takes S2 to +/-Db          lambda=getRandomLambda();   //lambda is a fair coin, giving the +1 or -1 choice           A=(Da*lambda*(-S1));      //Measurement function limit is function above          B=(lambda*S2*Db);      //Measurement function limit is function above          q=0;          //if(lambda==1) {q=A B;} else {q=B A;}          //Above is the usual form of the product calculation but we will expand it.          //if(lambda==1) {q=(Da*lambda*(-S1)) (lambda*S2*Db);} else {q=(lambda*S2*Db) (Da*lambda*(-S1));}          //Next we will use the fact that S2 = S1 and make some replacements to conform to eq.(16)          //and it is easy to see that lambda cancels out.  We are now ready to perform the correlations.          if(lambda==1) {q=(Da (-S2))(S2 Db);} else {q=(Db S1)((-S1) Da);}          ss=ss+q;          p_a=atan2(scalar(Da/(e3^e1)), scalar(Da/(e2^e3)));  //Get angle for a vector in x-y plane          p_b=atan2(scalar(Db/(e3^e1)), scalar(Db/(e2^e3)));  //Get angle for b vector in x-y plane          neg_adotb=-(a.b);          print(neg_adotb, "f");             //Outputs -a.b event by event          if(p_a*p_b>0) {theta=acos(a.b)*180/pi;} else {theta=-acos(a.b)*180/pi+360;}          print(theta, "f");                 //Output the angles event by event          print(correlation=scalar(q), "f"); //Output the correlations event by event          t=t+A;          u=u+B;      }      mean=ss/N;      print(mean, "f");    //shows the vanishing of the non-scalar part      aveA=t/N;      print(aveA, "f");    //verifies that individual average < A > = 0      aveB=u/N;      print(aveB, "f");    //verifies that individual average < B > = 0      prompt();}`
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