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 gill1109 » Thu Aug 15, 2019 10:18 am

Joy Christian wrote:
gill1109 wrote:According to the Wikipedia (and other) sources I gave you, we can use a generalization of Euler angles to higher dimensions. For S^3 there are three angles, while for S^2 there were two angles.

The claim of Wikipedia and other sources is wrong. Euler angles do not provide a singularity-free representation of S^3. Only quaternions can provide a singularity-free representation of S^3.

The issue of singularities is irrelevant. Despite Gimbal lock, there is no problem at all.

The question was: how to pick a uniformly distributed point on the quaternionic 3-sphere. The answer is well known, and it is given in the sources I gave you. Take a look at Eduarda Moura & David G. Henderson (1996), Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 978-0-13-373770-7 (Chapter 20: 3-spheres and hyperbolic 3-spaces).

Despite the fact that spherical coordinates do not, in the strict mathematical sense, define a coordinate chart on the sphere at zenith and nadir, there is no possible objection to using them to parametrize the sphere, and in particular, using them to generate a uniformly distributed point on the sphere. Obviously, one does not give each Euler angle an independent, uniform, distribution! Using the appropriate sequence of conditional distributions of each angle given the previously selected ones automatically takes care of the "singularities".

From wikipedia ("gimbal lock"): "The gimbal lock problem does not make Euler angles "invalid" (they always serve as a well-defined coordinate system), but it makes them unsuited for some practical applications." (my emphasis of the word "some")
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Re: Another simulation for QM Local functions

Postby Joy Christian » Thu Aug 15, 2019 10:59 am

gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:According to the Wikipedia (and other) sources I gave you, we can use a generalization of Euler angles to higher dimensions. For S^3 there are three angles, while for S^2 there were two angles.

The claim of Wikipedia and other sources is wrong. Euler angles do not provide a singularity-free representation of S^3. Only quaternions can provide a singularity-free representation of S^3.

The issue of singularities is irrelevant. Despite Gimbal lock, there is no problem at all.

The question was: how to pick a uniformly distributed point on the quaternionic 3-sphere. The answer is well known, and it is given in the sources I gave you. Take a look at Eduarda Moura & David G. Henderson (1996), Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 978-0-13-373770-7 (Chapter 20: 3-spheres and hyperbolic 3-spaces).

Despite the fact that spherical coordinates do not, in the strict mathematical sense, define a coordinate chart on the sphere at zenith and nadir, there is no possible objection to using them to parametrize the sphere, and in particular, using them to generate a uniformly distributed point on the sphere. Obviously, one does not give each Euler angle an independent, uniform, distribution! Using the appropriate sequence of conditional distributions of each angle given the previously selected ones automatically takes care of the "singularities".

From wikipedia ("gimbal lock"): "The gimbal lock problem does not make Euler angles "invalid" (they always serve as a well-defined coordinate system), but it makes them unsuited for some practical applications." (my emphasis of the word "some")

I disagree with this argument. Preserving the singularity-free nature of the quaternionic 3-sphere has been the key feature of my argument all along. See, for example, my 13 August 2007 reply to Philippe Grangier: https://arxiv.org/pdf/quant-ph/0703244.pdf. But you, Fred, John Reed, or anyone else are welcome to try your approach. I assure you that you will not be able to reproduce the strong correlation. And when you fail, I will claim that that is because you ignored the singularity-free nature of the quaternionic 3-sphere. :)

***
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Thu Aug 15, 2019 11:28 am

Joy Christian wrote:I disagree with this argument. Preserving the singularity-free nature of the quaternionic 3-sphere has been the key feature of my argument all along. See, for example, my 13 August 2007 reply to Philippe Grangier: https://arxiv.org/pdf/quant-ph/0703244.pdf. But you, Fred, John Reed, or anyone else are welcome to try your approach. I assure you that you will not be able to reproduce the strong correlation. And when you fail, I will claim that that is because you ignored the singularity-free nature of the quaternionic 3-sphere. :)

***

Oh, that's my queue here. :D This Mathematica simulation follows Joy's paper here mostly. I've changed the notation to match better. Also the 3-sphere complete states function should be in the first Do loop. This is how one can predict the A and B +/-1 outcomes.

EPRsims/Joy_local_CS_no0s3Dsep1.pdf
.
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Re: Another simulation for QM Local functions

Postby gill1109 » Thu Aug 15, 2019 8:39 pm

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:According to the Wikipedia (and other) sources I gave you, we can use a generalization of Euler angles to higher dimensions. For S^3 there are three angles, while for S^2 there were two angles.

The claim of Wikipedia and other sources is wrong. Euler angles do not provide a singularity-free representation of S^3. Only quaternions can provide a singularity-free representation of S^3.

The issue of singularities is irrelevant. Despite Gimbal lock, there is no problem at all.

The question was: how to pick a uniformly distributed point on the quaternionic 3-sphere. The answer is well known, and it is given in the sources I gave you. Take a look at Eduarda Moura & David G. Henderson (1996), Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 978-0-13-373770-7 (Chapter 20: 3-spheres and hyperbolic 3-spaces).

Despite the fact that spherical coordinates do not, in the strict mathematical sense, define a coordinate chart on the sphere at zenith and nadir, there is no possible objection to using them to parametrize the sphere, and in particular, using them to generate a uniformly distributed point on the sphere. Obviously, one does not give each Euler angle an independent, uniform, distribution! Using the appropriate sequence of conditional distributions of each angle given the previously selected ones automatically takes care of the "singularities".

From wikipedia ("gimbal lock"): "The gimbal lock problem does not make Euler angles "invalid" (they always serve as a well-defined coordinate system), but it makes them unsuited for some practical applications." (my emphasis of the word "some")

I disagree with this argument. Preserving the singularity-free nature of the quaternionic 3-sphere has been the key feature of my argument all along. See, for example, my 13 August 2007 reply to Philippe Grangier: https://arxiv.org/pdf/quant-ph/0703244.pdf. But you, Fred, John Reed, or anyone else are welcome to try your approach. I assure you that you will not be able to reproduce the strong correlation. And when you fail, I will claim that that is because you ignored the singularity-free nature of the quaternionic 3-sphere. :)

How wonderful that we have met with a contradiction. Now we have some hope of making progress. [not quite Niels Bohr].

I assure you, we will *exactly* reproduce the strong correlation - because all we are doing is using a different representation of exactly the same mathematical structure as the standard structure of conventional quantum mechanics! Recall, this is also the line of Jay Yablon - he suggests that your (Joy's) approach can also be rewritten in terms of Pauli matrices; and that if you claim that your GA model is local realistic, then the same holds for the standard QM computations, and the only question is really, what is your definition of "local" and "realistic".

I tend to agree with Jay here, in that all we have is the standard calculation, and all we have are arguments about formalising the intuitive concepts "local" and "real". Back to EPR and the Einstein-Bohr debate. Remember, those two would never ever have come to an agreement, however many decades they kept on arguing, though of course, many bystanders did come to conclusions. But not all the same one. So I predict that Jay will succeed in convincing himself that QM as we already have it is local and realistic, but that he won't convince any new people to come over to his side! But we will see. I have made predictions in the past, which were (also?) wrong.

Anyway, we see that we are just arguing about words. The mathematics is immutable. (As long as there are no mathematical errors. That's another matter).
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Re: Another simulation for QM Local functions

Postby Joy Christian » Thu Aug 15, 2019 9:29 pm

gill1109 wrote:I assure you, we will *exactly* reproduce the strong correlation ...

Ok, so let's get cracking. Fred already has a working code in Mathematica. Can that be adapted to your approach? Or can your approach be adapted to Fred's code?

***
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Thu Aug 15, 2019 9:55 pm

Joy Christian wrote:
gill1109 wrote:I assure you, we will *exactly* reproduce the strong correlation ...

Ok, so let's get cracking. Fred already has a working code in Mathematica. Can that be adapted to your approach? Or can your approach be adapted to Fred's code?

***

Image

Image produced with +/-1's from A and B outcomes at one degree resolution. Yes, I would say it works pretty good to simulate how local QM and for that matter any of Joy's local models predicts the strong correlations. 3-sphere topology is the way. What else could it be?
.
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Re: Another simulation for QM Local functions

Postby gill1109 » Fri Aug 16, 2019 3:39 am

Joy Christian wrote:Ok, so let's get cracking. Fred already has a working code in Mathematica. Can that be adapted to your approach? Or can your approach be adapted to Fred's code?

Here is Python code for generating uniformly distributed points on the 3-sphere
Code: Select all
import numpy as np
N = 600
dim = 4
norm = np.random.normal
normal_deviates = norm(size=(dim, N))
radius = np.sqrt((normal_deviates**2).sum(axis=0))
points = normal_deviates/radius

You generate 4-variate standard Gaussians and normalise the length to 1.

Here is an old reference to some stone-age methods:

W.P. Petersen and A. Bernasconi (1997), Uniform sampling from an n-sphere: Isotropic method, Technical Report, TR-97-06, Swiss Centre for Scientific Computing.
ftp://ftp.math.ethz.ch/hg/scsc/pub/papers/TR/TR-97-06/TR-97-06.2of3.ps
This paper actually derives analytically the successive conditional distributions of the spherical coordinates. They are not so weird at all. These probability distributions could be used directly, you just need the usual probability integral transform method, but the authors, writing as I said in the stone age, come up with approximate rejection methods which save you from computing arc cosines and things like that.

They were interested in fast methods for computers with less memory or disk storage and much less speed than present-day cheap smart phones and want to avoid using mathematical functions which would need to be computed on the fly.

Here's yet another reference, not quite so old, with more methods still:

Harman, R. & Lacko, V. On decompositional algorithms for uniform sampling from n-spheres and n-balls Journal of Multivariate Analysis, 2010
https://www.sciencedirect.com/science/article/pii/S0047259X10001211
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Fri Aug 16, 2019 8:22 am

gill1109 wrote:Here is Python code for generating uniformly distributed points on the 3-sphere
Code: Select all
import numpy as np
N = 600
dim = 4
norm = np.random.normal
normal_deviates = norm(size=(dim, N))
radius = np.sqrt((normal_deviates**2).sum(axis=0))
points = normal_deviates/radius

You generate 4-variate standard Gaussians and normalise the length to 1.

Mathematica has a neat function to do that in one line,

RandomPoint[Sphere[4]]

Code: Select all
RandomPoint[Sphere[4]]
Norm[%]
Out[]={0.704782, 0.170765, 0.33992, -0.598812}
Out[]=1


And we see that the norm is 1. But that is not for a parallelized 3-sphere. I've been searching for a different way of getting points for the parallelized 3-sphere, but no luck so far.
.
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Re: Another simulation for QM Local functions

Postby Heinera » Fri Aug 16, 2019 12:24 pm

FrediFizzx wrote:And we see that the norm is 1. But that is not for a parallelized 3-sphere. I've been searching for a different way of getting points for the parallelized 3-sphere, but no luck so far.
.

And what is the exact mathematical definition of a parallelized 3-sphere, that makes it different from an "ordinary" 3-sphere?
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Fri Aug 16, 2019 2:42 pm

Heinera wrote:
FrediFizzx wrote:And we see that the norm is 1. But that is not for a parallelized 3-sphere. I've been searching for a different way of getting points for the parallelized 3-sphere, but no luck so far.
.

And what is the exact mathematical definition of a parallelized 3-sphere, that makes it different from an "ordinary" 3-sphere?

Well, we have one way of doing it.
Code: Select all
trialcs=10,000;
e0 = ConstantArray[0, trialcs];
z = ConstantArray[0, trialcs];
Do[e0[[j]] = RandomPoint[Sphere[]];(*3D uniform random unit vectors*)
 s0 = RandomReal[{0, \[Pi]}];
 z[[j]] = (2/Sqrt[1 + (3 s0)/\[Pi]] - 1);
 n = RandomPoint[Sphere[]];
 If[Abs[n.e0[[j]]] > z[[j]], n1 = n, n1 = "notvalid"],
 {j, trialcs}]

It would be good if there was another way.
.
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Re: Another simulation for QM Local functions

Postby gill1109 » Sat Aug 17, 2019 2:42 am

Heinera wrote:
FrediFizzx wrote:And we see that the norm is 1. But that is not for a parallelized 3-sphere. I've been searching for a different way of getting points for the parallelized 3-sphere, but no luck so far.
.

And what is the exact mathematical definition of a parallelized 3-sphere, that makes it different from an "ordinary" 3-sphere?

There is a nice description here: https://en.wikipedia.org/wiki/Parallelizable_manifold

One starts with something nice and smooth. For instance, in our case, the ordinary 3-sphere. You can think of it as the surface of the four-dimensional unit ball in four-dimensional Euclidean space. Now we ask ourselves, can we determine three orthogonal tangent vectors at each point of the surface, so that as we move smoothly over the surface, the set of three tangent vectors changes *smoothly* too.

You can do this for the 3-sphere and the 7-sphere. For the 3-sphere, the trick is to make a one-to-one correspondence between the 3-sphere and the set of unit quaternions. You can also do it for the 7-sphere - you use the octonions.

Notice that 3 and 7 are 1 less than 4 and 8. This whole game is connected to the theorem that the only normed division algebras exist in dimensions 1, 2, 4 and 8 and can be identified respectively with the real numbers, the complex numbers, the quaternions, and the octonions.

S^0, S^1, S^3 and S^7 are parallelizable, and each in exactly one way.

S^2 is not parallizable - that's a corollary of the hairy ball theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem

What does it mean to pick a point uniformly at random on a manifold, such as the 3-sphere? As I wrote before, *uniformity* is defined relative to a collection of transformations, such that the probability of some particular region on the manifold does not change when one applies any of the transformations to the region. Well, there is an obvious way to apply this concept to the set of unit quaternions, since multiplying by another unit quaternion is a 1-1 smooth invertible transformation of that set to itself. But all this is, is the natural generalisation of rotations from three to four dimensional space! That's why people doing virtual reality like to use geometric algebra (ie, quaternions), to do geometric operations on geometric objects in 3d space.

So it is very very well known how to generate uniformly distributed points on the parallelized 3-sphere, and there are hundreds of ways to do it, some of which I posted about, yesterday.

There is a lot of subtlety in these definitions and facts. What *degree* of smoothness are we talking about? See for instance https://en.wikipedia.org/wiki/Exotic_sphere
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