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.
.
FrediFizzx wrote:Yep. We have moved on from that implementation. Looking at something different now.
.
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.
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.
.
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]
FrediFizzx wrote:Here is an interesting 3D plot. Two linked Seifert surfaces?
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).
***
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.
.
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.
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 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 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.
FrediFizzx wrote:Got it! Finally!
FrediFizzx wrote:Now on to the more difficult task of getting valid parallelized 3-sphere points.
Return to Sci.Physics.Foundations
Users browsing this forum: ahrefs [Bot] and 78 guests