Interesting. If I set phi to 2pi only, I get a slice of half a ball.

.

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- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:Niles Johnson's method seems to work.

- Code: Select all
`<< Quaternions`;`

\[Beta]0 = Quaternion[1, 0, 0, 0];

\[Beta]1 = Quaternion[0, 1, 0, 0];

\[Beta]2 = Quaternion[0, 0, 1, 0];

\[Beta]3 = Quaternion[0, 0, 0, 1];

Qcoordinates = {\[Beta]0, \[Beta]1, \[Beta]2, \[Beta]3};

trials = 4000;

n1 = ConstantArray[0, trials];

Do[q = RandomPoint[Sphere[4]];

w = q[[1]];

x = q[[2]];

y = q[[3]];

z = q[[4]];

n = {0, 2 w*y + 2 x*z, 2 y*z - 2 w*x, w^2 + z^2 - x^2 - y^2}.Qcoordinates;

n1[[j]] = {n[[2]], n[[3]], n[[4]]}, {j, trials}]

Graphics3D[Table[Point[{{0, 0, 0}, p}], {p, n1}]]

However, it won't work with our spin 1/2. Something else is needed.

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According to the "small print" in the Mathematica documentation https://reference.wolfram.com/language/ref/Sphere.html, because "4" is a positive integer, not a real, Sphere[4] is equivalent to Sphere[{0,…,0}], a unit sphere in R^4, i.e. a 3-sphere.

Your Mathematica implementation of just one of Nile Johnson's plots shows uniformly distributed points *on* S^2.

Here are the sources again: Niles Johnson's webpages https://nilesjohnson.net/hopf.html, David Lyons preprint http://csunix1.lvc.edu/~lyons/pubs/hopf_paper_preprint.pdf which finally appeared a:s David W. Lyons. An elementary introduction to the Hopf fibration. Mathematics Magazine, 76(2):87--98, 2003. Journal webpage: https://www.maa.org/press/periodicals/mathematics-magazine

- gill1109
- Mathematical Statistician
**Posts:**1653**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

Yep. We have moved on from that implementation. Looking at something different now.

.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:Yep. We have moved on from that implementation. Looking at something different now.

.

You still haven't explained why you seem to think that a uniform random sample from the parallelized 3-sphere isn't anything different from a uniform random sample from the un-parallelized 3-sphere.

Of course, there are algorithms that exploit the possibility of parallelization. For instance, run a random walk which, at each step, chooses either to step from one fibre to another or to take a step along the fibre where it is right now. It is easy to come up with schemes such that the position of the random walker after n steps is rapid, as n increases, closer and closer to a uniform distribution. After all, we are talking about a compact manifold with a smooth compact group structure.

However, I doubt that this way could give us faster algorithms than those we already have, for the same accuracy.

Here https://arxiv.org/pdf/0802.0644.pdf is an example of the kind of random walks I have in mind:

Semi-classical analysis of a random walk on a manifold

Gilles Lebeau, Laurent Michel

We prove a sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold (M,g). The proof includes a detailed study of the spectral theory of the associated operator.

Journal reference: Annals of Probability 2010, Vol. 38, No. 1, 277-315

DOI: 10.1214/09-AOP483

Last edited by gill1109 on Fri Aug 23, 2019 12:18 am, edited 1 time in total.

- gill1109
- Mathematical Statistician
**Posts:**1653**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

gill1109 wrote:FrediFizzx wrote:Yep. We have moved on from that implementation. Looking at something different now.

.

You still haven't explained why you seem to think that a uniform random sample from the parallelized 3-sphere isn't anything different from a uniform random sample from the un-parallelized 3-sphere.

Of course, there are algorithms that exploit the possibility of parallelization. For instance, run a random walk which, at each step, chooses either to step from one fibre to another or to take a step along the fibre where it is right now. It is easy to come up with schemes such that the position of the random walker after n steps is rapid, as n increases, closer and closer to a uniform distribution. After all, we are talking about a compact manifold with a smooth compact group structure.

However, I doubt that this way could give us faster algorithms than those we already have, for the same accuracy.

Points on an R^4 3-sphere won't be the same as on a quaternionic parallelized 3-sphere. But that is not the primary issue. It is how spin behaves (the action) via a parallelized 3-sphere. There is an extra parameter that varies from 0 to pi that I am trying to see if I can find it.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:gill1109 wrote:FrediFizzx wrote:Yep. We have moved on from that implementation. Looking at something different now.

.

You still haven't explained why you seem to think that a uniform random sample from the parallelized 3-sphere isn't anything different from a uniform random sample from the un-parallelized 3-sphere.

Of course, there are algorithms that exploit the possibility of parallelization. For instance, run a random walk which, at each step, chooses either to step from one fibre to another or to take a step along the fibre where it is right now. It is easy to come up with schemes such that the position of the random walker after n steps is rapid, as n increases, closer and closer to a uniform distribution. After all, we are talking about a compact manifold with a smooth compact group structure.

However, I doubt that this way could give us faster algorithms than those we already have, for the same accuracy.

Points on an R^4 3-sphere won't be the same as on a quaternionic parallelized 3-sphere. But that is not the primary issue. It is how spin behaves (the action) via a parallelized 3-sphere. There is an extra parameter that varies from 0 to pi that I am trying to see if I can find it.

.

Sorry Fred, uniform random points on an R^4 3-sphere are exactly the same as uniform random points on a quaternionic parallelized 3-sphere. So you can get one by making the other. But as you say, you are interested in the real and interesting issue that there are four parameters in the Hopf fibration. This is exactly the same issue that the overall phase of the wave function is superfluous.

Do you think that the fourth parameter does have physical meaning, after all? You are not the first. Today, this is what comes up top in a Google search on my computer:

https://www.scirp.org/journal/PaperInformation.aspx?PaperID=64044

"Causality of Phase of Wave Function or Can Copenhagen Interpretation of Quantum Mechanics Be Considered Complete?"

Georgiev Koprinkov, Department of Applied Physics, Technical University of Sofia, Sofia, Bulgaria.

Abstract: Theoretical and experimental evidences of a causal relation of the phase of the wave function and physical reality are presented. The Copenhagen interpretation of quantum mechanics, which gives physical meaning to the amplitude of the wave function only, cannot be considered complete on that ground. A new dynamics-statistical interpretation of quantum mechanics is proposed.

- gill1109
- Mathematical Statistician
**Posts:**1653**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

Bam!

I was looking for half a circle, 0 to pi, and there it is. I just had to normalize the output since the input is not normalized and should be. Fancy way to draw a half circle. But this certainly could be what I am looking for. It is in the 3-sphere topology.

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I was looking for half a circle, 0 to pi, and there it is. I just had to normalize the output since the input is not normalized and should be. Fancy way to draw a half circle. But this certainly could be what I am looking for. It is in the 3-sphere topology.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

Normalizing the inputs for a, b and c didn't help but it did change the output a bit.

.

- Code: Select all
`trials = 2000;`

n = ConstantArray[0, trials];

Do[basepoint = RandomPoint[Sphere[]];

a = basepoint[[1]];

b = basepoint[[2]];

c = basepoint[[3]];

al = Sqrt[0.5 (1 + c)];

be = Sqrt[0.5 (1 - c)];

ph = RandomReal[{2 \[Pi], 2 \[Pi]}];

th = ArcTan[a, b] - ph;

w = al*Cos[th];

x = -be*Cos[ph];

y = -be*Sin[ph];

z = al*Sin[th];

r = ArcCos[w]/(\[Pi] Sqrt[1 - w^2]);

(*n[[j]]=Normalize[{x*rr,y*rr,z*rr}],{j, trials}]*)

n[[j]] = {x*r, y*r, z*r}, {j, trials}]

Graphics3D[Table[Point[{{0, 0, 0}, p}], {p, n}], Axes -> True]

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- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

Here is an interesting 3D plot. Two linked Seifert surfaces?

.

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- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:Here is an interesting 3D plot. Two linked Seifert surfaces?

Or, without the noise, could be linked Hopf circles or Clifford parallels (see Fig. 1 in this paper: https://arxiv.org/pdf/0806.3078.pdf).

***

- Joy Christian
- Research Physicist
**Posts:**2161**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

Joy Christian wrote:FrediFizzx wrote:Here is an interesting 3D plot. Two linked Seifert surfaces?

Or, without the noise, could be linked Hopf circles or Clifford parallels (see Fig. 1 in this paper: https://arxiv.org/pdf/0806.3078.pdf).

***

I can rotate that 3D plot around in Mathematica. It definitely has the Seifert twist in it and they are linked. But dang, now I forgot how it did it.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:Joy Christian wrote:FrediFizzx wrote:Here is an interesting 3D plot. Two linked Seifert surfaces?

...

Or, without the noise, could be linked Hopf circles or Clifford parallels (see Fig. 1 in this paper: https://arxiv.org/pdf/0806.3078.pdf).

I can rotate that 3D plot around in Mathematica. It definitely has the Seifert twist in it and they are linked. But dang, now I forgot how it did it.

.

Cool pictures! I think it is not noise that you see, because there is none in that code, but, in each case, a small bunch of adjacent circles, forming a circular strip.

Joy wrote in the cited paper by himself "The central message of Refs. [1] and [2] and the above discussion is that EPR-Bohm correlations have nothing to do with entanglement or non-locality per se, but are a vestige of geometry and topology of the physical space." I agree that EPR-Bohm correlations have everything to do with the geometry and topology of physical space. In my opinion, the geometry and topology of Euclidean space is, at a fundamental level, the cause of "quantum entanglement" and "quantum non-locality". (I have thought so, for a very long time).

I put those two expressions "quantum entanglement" and "quantum non-locality" in quotation marks because they are technical terms in mathematical physics with clear and generally agreed mathematical definitions.

References:

[1] J. Christian, Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables: arXiv:quant-ph/0703179.

[2] J. Christian, Disproof of Bell’s Theorem: Further Consolidations: arXiv:0707.1333; See also arXiv:0904.4259.

- gill1109
- Mathematical Statistician
**Posts:**1653**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

gill1109 wrote:Cool pictures! I think it is not noise that you see, because there is none in that code, but, in each case, a small bunch of adjacent circles, forming a circular strip.

Joy wrote in the cited paper by himself "The central message of Refs. [1] and [2] and the above discussion is that EPR-Bohm correlations have nothing to do with entanglement or non-locality per se, but are a vestige of geometry and topology of the physical space." I agree that EPR-Bohm correlations have everything to do with the geometry and topology of physical space. In my opinion, the geometry and topology of Euclidean space is, at a fundamental level, the cause of "quantum entanglement" and "quantum non-locality". (I have thought so, for a very long time).

I put those two expressions "quantum entanglement" and "quantum non-locality" in quotation marks because they are technical terms in mathematical physics with clear and generally agreed mathematical definitions.

References:

[1] J. Christian, Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables: arXiv:quant-ph/0703179.

[2] J. Christian, Disproof of Bell’s Theorem: Further Consolidations: arXiv:0707.1333; See also arXiv:0904.4259.

Thanks. I remember how I did it now. I set a, b and c all to the same random spherical point in Cartesian coordinates and out pops linked Hopf bands. If I go to spherical coordinates, I only get one Hopf band.

Strange. Of course since alpha and beta are defined they way they are, c has to be the radius in spherical coordinates. Oh..., that is two linked Hopf fibers though. I'm slowly getting a handle on how this 3-sphere code works.

- Code: Select all
`trials = 10000;`

n = ConstantArray[0, trials];

Do[basepoint = ToSphericalCoordinates[RandomPoint[Ball[]]];

a = basepoint[[1]];

b = basepoint[[1]];

c = basepoint[[1]];

al = Sqrt[0.5 (1 + c)];

be = Sqrt[0.5 (1 - c)];

ph = RandomReal[{0, 2 \[Pi]}];

th = ArcTan[-a, b] - ph;

w = al*Cos[th];

x = -be*Cos[ph];

y = -be*Sin[ph];

z = al*Sin[th];

r = ArcCos[w]/(\[Pi] Sqrt[1 - w^2]);

(*r=ArcCos[w];*)

(*n[[j]]=Normalize[{x*rr,y*rr,z*rr}],{j, trials}]*)

n[[j]] = {x*r, y*r, z*r},

{j, trials}]

Graphics3D[Table[Point[{{0, 0, 0}, p}], {p, n}], Axes -> True]

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

I finally got a piece of a circle without having to normalize the outputs. I'm still looking for that half circle we need.

Well, I have got part if it so the rest of it must be there somewhere.

Well, I have got part if it so the rest of it must be there somewhere.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:I finally got a piece of a circle without having to normalize the outputs. I'm still looking for that half circle we need.

...

Well, I have got part if it so the rest of it must be there somewhere.

- gill1109
- Mathematical Statistician
**Posts:**1653**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

gill1109 wrote:FrediFizzx wrote:I finally got a piece of a circle without having to normalize the outputs. I'm still looking for that half circle we need.

...

Well, I have got part if it so the rest of it must be there somewhere.

The a, b, c and phi inputs are not supposed to be random thus the "noise". But we can see when they are random, we get the Hopf bands with linked fibers from the basic fiber parameter formula.

So, I need to setup some animation controls in Mathematica to feed those 4 inputs in a controlled way. Ugh!

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

It is OK to normalize the outputs since the inputs are not supposed to be random anyways. So we get our half circle we are looking for by setting phi to pi,

And here is the other half when setting phi to 2pi.

So I think we are good. 0 to pi comes out naturally in the 3-sphere formula.

.

And here is the other half when setting phi to 2pi.

So I think we are good. 0 to pi comes out naturally in the 3-sphere formula.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

I finally almost obtained the half circle without normalizing the outputs from Niles Johnson's formula for the Hopf fibration.

I say "almost" because it is not a true half circle as its height is very close to 1 but its width is about 0.57 so it is slightly oblong. But we are going from 0 to pi. Here is a PDF of the code with some annotations and the outputs.

EPRsims/nilesj7.pdf

You can see a funny thing happens to theta. It toggles randomly between -5pi/4 and -9pi/4. ??? Phi is set just below 2pi to keep y from going to zero.

.

I say "almost" because it is not a true half circle as its height is very close to 1 but its width is about 0.57 so it is slightly oblong. But we are going from 0 to pi. Here is a PDF of the code with some annotations and the outputs.

EPRsims/nilesj7.pdf

You can see a funny thing happens to theta. It toggles randomly between -5pi/4 and -9pi/4. ??? Phi is set just below 2pi to keep y from going to zero.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

Got it! Finally!

I finally realized that Niles Johnson's 3-sphere mapping formula had the x, y and z outputs times r to obtain the Hopf fiber links that you saw in previous postings. For the purpose of getting the half circle we just want the x, y and z outputs. Then I made a minor adjustment to theta subtracting 45 degrees. Then I converted x and z to polar coordinates to get theta for the complete states function's s0 and used it for the simulation producing,

So this gives a somewhat better justification for s0 going from 0 to pi via Niles Johnson's 3-sphere mapping formula. The 3-sphere topology is definitely not trivial for the EPR-Bohm scenario. Here are PDF's of the Mathematica code for the half circle plot and for the simulation.

EPRsims/nilesj8.pdf

EPRsims/Joy_local_CS_no0s3Ds0.pdf

Now on to the more difficult task of getting valid parallelized 3-sphere points.

.

I finally realized that Niles Johnson's 3-sphere mapping formula had the x, y and z outputs times r to obtain the Hopf fiber links that you saw in previous postings. For the purpose of getting the half circle we just want the x, y and z outputs. Then I made a minor adjustment to theta subtracting 45 degrees. Then I converted x and z to polar coordinates to get theta for the complete states function's s0 and used it for the simulation producing,

So this gives a somewhat better justification for s0 going from 0 to pi via Niles Johnson's 3-sphere mapping formula. The 3-sphere topology is definitely not trivial for the EPR-Bohm scenario. Here are PDF's of the Mathematica code for the half circle plot and for the simulation.

EPRsims/nilesj8.pdf

EPRsims/Joy_local_CS_no0s3Ds0.pdf

Now on to the more difficult task of getting valid parallelized 3-sphere points.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1715**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:Got it! Finally!

Excellent!

FrediFizzx wrote:Now on to the more difficult task of getting valid parallelized 3-sphere points.

Pick uniform random points on the plain vanilla 3-sphere. They *are* valid uniformly distributed parallelized 3-sphere points. How do you want to parametrize them? Is not good enough? Do you prefer spherical coordinates ?

The 3-sphere *is* the set of unit quaternions. The product of two unit quaternions is a unit quaternions. Hence the unit quaternions form a group acting by multiplication on the left, and also by multiplication on the right. We can define the uniform distribution as being the unique probability measure on S^3 which is invariant under these group actions.

You can look at S^3 from the point of view of the Hopf fibration but this does not lead to any new, different, notion of uniform distribution.

https://en.wikipedia.org/wiki/Hopf_fibration

- gill1109
- Mathematical Statistician
**Posts:**1653**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

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