by gill1109 » Thu Aug 22, 2019 10:02 pm
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.
.
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
[quote="FrediFizzx"]Niles Johnson's method seems to work.
[code]
<< 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}]]
[/code]
[img]http://www.sciphysicsforums.com/spfbb1/EPRsims/nilesj.jpg[/img]
However, it won't work with our spin 1/2. Something else is needed.
.[/quote]
According to the "small print" in the Mathematica documentation [url]https://reference.wolfram.com/language/ref/Sphere.html[/url], 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 [url]https://nilesjohnson.net/hopf.html[/url], David Lyons preprint [url]http://csunix1.lvc.edu/~lyons/pubs/hopf_paper_preprint.pdf[/url] which finally appeared a:s David W. Lyons. An elementary introduction to the Hopf fibration. Mathematics Magazine, 76(2):87--98, 2003. Journal webpage: [url]https://www.maa.org/press/periodicals/mathematics-magazine[/url]