Another simulation for QM Local functions

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

Re: Another simulation for QM Local functions

Postby FrediFizzx » Sat Aug 17, 2019 2:28 pm

local wrote:As far as I can see, the Christian model basically relies upon data rejection ("detection loophole") but justifies the rejection by claiming that the rejected detections correspond to "states" that cannot occur in 3-sphere topology. So my questions are:

1. Is it asserted that the world itself has 3-sphere topology, or just that emitted photon pairs have 3-sphere topology?

2. What are these "states" referred to when models are labeled as (for example) "complete states", etc. Quantum mechanics has only the singlet state for EPRB (leaving aside nonmaximal states).

3. Has it been shown that this 3-sphere topology leads to nonexistence of just those "states" whose rejection produces the quantum correlation? Where is this demonstration given in a form digestible by mere humans?

Thank you for any light you can shed on these questions!

The Christian model does NOT rely on data rejection if the states don't exist in the first place!
1. We are mainly talking about EPR-Bohm so it is fermions with spin 1/2 not photons.
2. We don't understand the question.
3. The simulations speak for themselves. How else could Nature do the quantum correlations without "spooky action at a distance" which is complete nonsense? This is how. Action in Nature is local.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby local » Sat Aug 17, 2019 4:47 pm

FrediFizzx wrote: The Christian model does NOT rely on data rejection if the states don't exist in the first place!

I asked what these states are and you said you do not understand the question. I would like to ask again what are these states that you are talking about and how are they excluded from existing? Thank you for your reply and for being patient as I try to understand the Christian model.
local
 
Posts: 295
Joined: Mon Aug 05, 2019 1:19 pm

Re: Another simulation for QM Local functions

Postby Joy Christian » Sat Aug 17, 2019 5:21 pm

local wrote:
FrediFizzx wrote: The Christian model does NOT rely on data rejection if the states don't exist in the first place!

I asked what these states are and you said you do not understand the question. I would like to ask again what are these states that you are talking about and how are they excluded from existing? Thank you for your reply and for being patient as I try to understand the Christian model.

The complete state is defined by the pair of vectors e_o and s_o. See Fig. 1, page 4, of this paper: https://arxiv.org/pdf/1405.2355.pdf. See also the comments just below eq. (40), page 7.

***
Joy Christian
Research Physicist
 
Posts: 2793
Joined: Wed Feb 05, 2014 4:49 am
Location: Oxford, United Kingdom

Re: Another simulation for QM Local functions

Postby FrediFizzx » Sat Aug 17, 2019 7:30 pm

local wrote:
FrediFizzx wrote: The Christian model does NOT rely on data rejection if the states don't exist in the first place!

I asked what these states are and you said you do not understand the question. I would like to ask again what are these states that you are talking about and how are they excluded from existing? Thank you for your reply and for being patient as I try to understand the Christian model.

We are talking about singlet states. What else is there for states for the EPR-Bohm scenario? :roll:

Spin 1/2 is not like the ordinary spin of say a spinning top. Its spin vector e0 will not just point in any 3D direction in the same was as the spin vector of a top could.

Joy gave you an answer to your question about "excluded" in his paper linked above.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby gill1109 » Sun Aug 18, 2019 12:32 am

FrediFizzx wrote:
Joy Christian wrote:
FrediFizzx wrote:
Joy Christian wrote:***
That's is 2-sphere in the picture, not a 3-sphere. How can that possibly help with the problem?

***

Look at the Mathematica code. It's 3-sphere points projected onto the 2-sphere.
.

I understand that. But the points are uniform on the 2-sphere, not on the 3-sphere; and it is the 3-sphere that is supposed to be modeling the physical space.

***

It is not possible to do a 3D graphic of the 3-sphere directly. This is the x, y, and z coordinates of the 3-sphere projected onto the 2-sphere.
.

What you can do, of course, is to consider the four Euclidean coordinates (x, y, z, t) as three spatial coordinates and one time coordinate. So you can visualise the 3-sphere in a movie. However, it will be hard to judge if the movie is showing us what we hope to see.

To do that, I would proceed as follows: (1) show that the marginal distribution of t is what it should be. Draw a histogram and superimpose on it the theoretical probability density. (2) for each of the small bins of t-values, you have a sample of (x, y, z) conditional on t approx equal to the bin centre. Those (x, y, z) should be points approximately uniformly distributed on the 2-sphere of radius sqrt(1 - t^2).

In other words, decompose the joint probability distribution of (x, y, z, t) as a marginal distribution of t and a collection of conditional distributions of (x, y, z) given t.

That is essentially the same as stopping the movie at a discrete sequence of time points. But that is not enough. One must not forget to check that time runs at the correct rate. Hence also check the marginal distribution of t.
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

Re: Another simulation for QM Local functions

Postby FrediFizzx » Sun Aug 18, 2019 8:53 am

gill1109 wrote:What you can do, of course, is to consider the four Euclidean coordinates (x, y, z, t) as three spatial coordinates and one time coordinate. So you can visualise the 3-sphere in a movie. However, it will be hard to judge if the movie is showing us what we hope to see.

To do that, I would proceed as follows: (1) show that the marginal distribution of t is what it should be. Draw a histogram and superimpose on it the theoretical probability density. (2) for each of the small bins of t-values, you have a sample of (x, y, z) conditional on t approx equal to the bin centre. Those (x, y, z) should be points approximately uniformly distributed on the 2-sphere of radius sqrt(1 - t^2).

In other words, decompose the joint probability distribution of (x, y, z, t) as a marginal distribution of t and a collection of conditional distributions of (x, y, z) given t.

That is essentially the same as stopping the movie at a discrete sequence of time points. But that is not enough. One must not forget to check that time runs at the correct rate. Hence also check the marginal distribution of t.


Video of Hopf fibration for the 3-sphere.

https://nilesjohnson.net/hopf.html

This is what we are dealing with. :D
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby FrediFizzx » Sun Aug 18, 2019 10:40 am

Here is something new about 3-spheres that I didn't know about. Hmm...

https://en.wikipedia.org/wiki/Seifert_surface
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby Lord of the Physics » Sun Aug 18, 2019 7:21 pm

FrediFizzx wrote:
gill1109 wrote:What you can do, of course, is to consider the four Euclidean coordinates (x, y, z, t) as three spatial coordinates and one time coordinate. So you can visualise the 3-sphere in a movie. However, it will be hard to judge if the movie is showing us what we hope to see.

To do that, I would proceed as follows: (1) show that the marginal distribution of t is what it should be. Draw a histogram and superimpose on it the theoretical probability density. (2) for each of the small bins of t-values, you have a sample of (x, y, z) conditional on t approx equal to the bin centre. Those (x, y, z) should be points approximately uniformly distributed on the 2-sphere of radius sqrt(1 - t^2).

In other words, decompose the joint probability distribution of (x, y, z, t) as a marginal distribution of t and a collection of conditional distributions of (x, y, z) given t.

That is essentially the same as stopping the movie at a discrete sequence of time points. But that is not enough. One must not forget to check that time runs at the correct rate. Hence also check the marginal distribution of t.


Video of Hopf fibration for the 3-sphere.

https://nilesjohnson.net/hopf.html

This is what we are dealing with. :D
.


Good fibrations! Explicitly *dancing in my underwear*!
Lord of the Physics
 
Posts: 7
Joined: Wed Jul 10, 2019 10:32 am

Re: Another simulation for QM Local functions

Postby FrediFizzx » Sun Aug 18, 2019 7:33 pm

Lord of the Physics wrote:
FrediFizzx wrote:
Video of Hopf fibration for the 3-sphere.

https://nilesjohnson.net/hopf.html

This is what we are dealing with. :D
.

Good fibrations! Explicitly *dancing in my underwear*!

:D Definitely non-trivial for spin 1/2.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby gill1109 » Mon Aug 19, 2019 4:22 am

FrediFizzx wrote:
gill1109 wrote:What you can do, of course, is to consider the four Euclidean coordinates (x, y, z, t) as three spatial coordinates and one time coordinate. So you can visualise the 3-sphere in a movie. However, it will be hard to judge if the movie is showing us what we hope to see.

To do that, I would proceed as follows: (1) show that the marginal distribution of t is what it should be. Draw a histogram and superimpose on it the theoretical probability density. (2) for each of the small bins of t-values, you have a sample of (x, y, z) conditional on t approx equal to the bin centre. Those (x, y, z) should be points approximately uniformly distributed on the 2-sphere of radius sqrt(1 - t^2).

In other words, decompose the joint probability distribution of (x, y, z, t) as a marginal distribution of t and a collection of conditional distributions of (x, y, z) given t.

That is essentially the same as stopping the movie at a discrete sequence of time points. But that is not enough. One must not forget to check that time runs at the correct rate. Hence also check the marginal distribution of t.


Video of Hopf fibration for the 3-sphere.

https://nilesjohnson.net/hopf.html

This is what we are dealing with. :D
.

That is what we are dealing with when we superimpose the parallelization on top of the already well-defined 3-sphere. However, the question of what is a uniform distribution on the 3-sphere, and how to sample from a uniform distribution on the 3-sphere, is not affected by the parallelization. Or to be more careful: one can easily sample from the uniform distribution on the parallelized 3-sphere while not taking any notice at all of the parallelization. However, I admit that there are, no doubt, ways to sample which do use the parallelization. For instance, the limiting distribution of the ordinary symmetric random walk on a *naturally discretized parallelized 3-sphere* will be the uniform distribution on the 3-sphere. Maybe I will program this for you in the coming days... first I have to decide if there is a natural way to define discretizations while preserving all the symmetries (invariances). Uniform distributions are defined once we have a family of transformations, under which probability mass should be invariant! ie we need *symmetries*.
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

Re: Another simulation for QM Local functions

Postby FrediFizzx » Mon Aug 19, 2019 9:54 am

It would be good if there were another way to get the points of a parallelized 3-sphere. I may have to install Sage on my computer and play around with Niles Johnson's program. Perhaps it can be adapted for this purpose.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby gill1109 » Mon Aug 19, 2019 12:53 pm

FrediFizzx wrote:It would be good if there were another way to get the points of a parallelized 3-sphere. I may have to install Sage on my computer and play around with Niles Johnson's program. Perhaps it can be adapted for this purpose.
.

Fred: the points of the parallelized 3-sphere are the points of the ordinary common or garden 3-sphere. And the uniform distribution on either is the same distribution on both of them.

Or do you use a different definition of "uniform distribution" from the rest of the world? If so, please tell us what is your definition.
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

Re: Another simulation for QM Local functions

Postby FrediFizzx » Mon Aug 19, 2019 3:47 pm

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}]]

Image

However, it won't work with our spin 1/2. Something else is needed.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby FrediFizzx » Mon Aug 19, 2019 4:41 pm

Yeah, something not right as it works without even involving the quaternions.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby FrediFizzx » Mon Aug 19, 2019 5:43 pm

FrediFizzx wrote:Yeah, something not right as it works without even involving the quaternions.
.

Oh, the quaternions were involved in getting this formula,

n = {0, 2 w*y + 2 x*z, 2 y*z - 2 w*x, w^2 + z^2 - x^2 - y^2}
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby gill1109 » Tue Aug 20, 2019 12:56 am

FrediFizzx wrote:
FrediFizzx wrote:Yeah, something not right as it works without even involving the quaternions.
.

Oh, the quaternions were involved in getting this formula,

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

Can you write your Mathematica code as ordinary mathematics? I don't understand it.

Mathematica itself can convert its native code language into regular human-readable typeset (LaTeX) mathematics, right?
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

Re: Another simulation for QM Local functions

Postby gill1109 » Tue Aug 20, 2019 3:33 am

Here's the most direct way to pick a uniformly distributed point on S^3 (whether parallelized or not).

Pick independently of one another three uniformly distributed random variables u, v, w (note: each one is uniform on a different interval),
then solve for phi_1, phi_2 and phi_3:

phi_1 in [0, pi] is the solution of u = phi_1 - 1/2 sin(2 phi_1) = u where u ~ unif(0, pi)
phi_2 in [0, pi] equals acos(v) where v ~ unif(-1, 1)
phi_3 in [0, 2 pi] equals w where w ~ unif(0, 2 pi)

Finally, set

x = cos(phi_1)
y = sin(phi_1) cos(phi_2)
z = sin(phi_1) sin(phi_2) cos(phi_3)
t = sin(phi_1) sin(phi_2) sin(phi_3)
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

Re: Another simulation for QM Local functions

Postby gill1109 » Tue Aug 20, 2019 7:45 am

Here's a picture which explains why the generation of phi_1 makes sense:
Image
Here's the R code of this image:
Code: Select all
png("Phi1 of S3.png")
plot(x, (x - sin(2 * x)/2), type = "l", axes = FALSE,
main = "Spherical coordinate phi_1 of uniform random point on S^3",
sub = "y = (x - sin(2 x))/2",
xlab ="x-axis: a uniform random angle between 0 and pi",
ylab = "y-axis: phi_1, an angle between 0 and pi")
abline(h = c(0, pi))
abline(v = c(0, pi))
graphics.off()

I first thought that I could convert this into something simpler involving just polynomials and roots. But no, I don't think it can be done. But still, finding, for given y between 0 and pi, the unique a root x in (0, pi) of y = x - sin(2 * x)/2 is computationally trivial and fast.
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

Re: Another simulation for QM Local functions

Postby FrediFizzx » Tue Aug 20, 2019 8:58 am

gill1109 wrote:
FrediFizzx wrote:
FrediFizzx wrote:Yeah, something not right as it works without even involving the quaternions.
.

Oh, the quaternions were involved in getting this formula,

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

Can you write your Mathematica code as ordinary mathematics? I don't understand it.

Mathematica itself can convert its native code language into regular human-readable typeset (LaTeX) mathematics, right?

I'm just doing Niles Johnson's eta(q) formula here,

https://nilesjohnson.net/hopf-production.html

The Mathematica code is pretty straight forward mathematics. What don't you understand? eta(q) (n in my code) is just translating the points on a 4 dimensional Euclidean sphere to that of a 2-sphere.

Niles' formula seems to work but it is not really what we want. And the quaternions in my code are not necessary as they were used by Johnson to get the eta(q) formula.
.
FrediFizzx
Independent Physics Researcher
 
Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

Re: Another simulation for QM Local functions

Postby gill1109 » Tue Aug 20, 2019 9:15 am

FrediFizzx wrote:
gill1109 wrote:
FrediFizzx wrote:
FrediFizzx wrote:Yeah, something not right as it works without even involving the quaternions.
.

Oh, the quaternions were involved in getting this formula,

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

Can you write your Mathematica code as ordinary mathematics? I don't understand it.

Mathematica itself can convert its native code language into regular human-readable typeset (LaTeX) mathematics, right?

I'm just doing Niles Johnson's eta(q) formula here,

https://nilesjohnson.net/hopf-production.html

The Mathematica code is pretty straight forward mathematics. What don't you understand? eta(q) (n in my code) is just translating the points on a 4 dimensional Euclidean sphere to that of a 2-sphere.

Niles' formula seems to work but it is not really what we want. And the quaternions in my code are not necessary as they were used by Johnson to get the eta(q) formula.
.

I have no idea at all what

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

means. I have no idea what lots of the assignments mean.

What does this mean: Qcoordinates;
n1[[j]] = {n[[2]], n[[3]], n[[4]]}, {j, trials}]
gill1109
Mathematical Statistician
 
Posts: 2812
Joined: Tue Feb 04, 2014 10:39 pm
Location: Leiden

PreviousNext

Return to Sci.Physics.Foundations

Who is online

Users browsing this forum: ahrefs [Bot] and 81 guests

cron
CodeCogs - An Open Source Scientific Library