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 » Fri Aug 09, 2019 10:53 am

jreed wrote:Please post the code. I want to take a look at it.

Here is a PDF,

EPRsims/Quat_local_CS_no0s3Dsep3.pdf

And here is a link for the notebook file for download,

EPRsims/Quat_local_CS_no0s3Dsep3.nb

Doing the quaternions slows the simulation down to a crawl. It takes forever to get through the first 3 Do loops. There is probably a more efficient way to do it no doubt.
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Sat Aug 10, 2019 11:54 am

I found a different way of doing uniform unit 3D vectors in Mathematica. Mathematica has a built-in function for x, y, and z points on a sphere,

Code: Select all
RandomPoint[Sphere[]]
Norm[%]
{0.892316,-0.254865,-0.37258}
1.

Normalize@RandomVariate[NormalDistribution[],3]
Norm[%]
{-0.191735,0.981317,-0.0159656}
1.

RandomPoint[Sphere[4]]
Norm[%]
{0.130034,0.629256,-0.692968,0.326991}
1.

The first one is maybe a little bit faster than the second way that I have been using. And we can see they are all normalized to 1. Now..., is the last one points on a 3-sphere?
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Sat Aug 10, 2019 12:11 pm

FrediFizzx wrote:I found a different way of doing uniform unit 3D vectors in Mathematica. Mathematica has a built-in function for x, y, and z points on a sphere,

Code: Select all
RandomPoint[Sphere[]]
Norm[%]
{0.892316,-0.254865,-0.37258}
1.

Normalize@RandomVariate[NormalDistribution[],3]
Norm[%]
{-0.191735,0.981317,-0.0159656}
1.

RandomPoint[Sphere[4]]
Norm[%]
{0.130034,0.629256,-0.692968,0.326991}
1.

The first one is maybe a little bit faster than the second way that I have been using. And we can see they are all normalized to 1. Now..., is the last one points on a 3-sphere?
.

Nope. The finer points in Mathematica says it is R^4.
.
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Sat Aug 10, 2019 1:58 pm

I take it back. That is the coordinates of the points of a unit 3-sphere. But which ones are the x, y and z coordinates? The first set of three or the last set of three?
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Re: Another simulation for QM Local functions

Postby Joy Christian » Sat Aug 10, 2019 2:09 pm

FrediFizzx wrote:
I take it back. That is the coordinates of the points of a unit 3-sphere. But which ones are the x, y and z coordinates? The first set of three or the last set of three?
.

Any three of the four coordinates can be taken as x, y, and z. Normalized vectors in R^4 can certainly give a 3-sphere, but that would not be a quaternionic (or parallelizable) 3-sphere. Therefore the correlation among its points will not be as strong as -a.b. It can only give a linear correlation. What is needed are uniformly distributed points on a quaternionic 3-sphere.

***
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Sat Aug 10, 2019 2:52 pm

Joy Christian wrote:
FrediFizzx wrote:
I take it back. That is the coordinates of the points of a unit 3-sphere. But which ones are the x, y and z coordinates? The first set of three or the last set of three?
.

Any three of the four coordinates can be taken as x, y, and z. Normalized vectors in R^4 can certainly give a 3-sphere, but that would not be a quaternionic (or parallelizable) 3-sphere. Therefore the correlation among its points will not be as strong as -a.b. It can only give a linear correlation. What is needed are uniformly distributed points on a quaternionic 3-sphere.

***

Yep, but no one shows how to do that. :D Not sure it is possible.
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Re: Another simulation for QM Local functions

Postby gill1109 » Wed Aug 14, 2019 12:59 am

Joy Christian wrote:
FrediFizzx wrote:I take it back. That is the coordinates of the points of a unit 3-sphere. But which ones are the x, y and z coordinates? The first set of three or the last set of three?
.

Any three of the four coordinates can be taken as x, y, and z. Normalized vectors in R^4 can certainly give a 3-sphere, but that would not be a quaternionic (or parallelizable) 3-sphere. Therefore the correlation among its points will not be as strong as -a.b. It can only give a linear correlation. What is needed are uniformly distributed points on a quaternionic 3-sphere.

Maybe I can help? What exactly do you mean by "quaternionic 3-sphere"? Then we can discuss what we should mean by "uniform distribution".

This should mean that you define a continuous group of transformations of the sphere. The 3-sphere is compact, so there exists a unique finite Haar measure which can (therefore) be normalised to a unique invariant probability measure. Given any parametrization of the sphere we can now calculate how to pick a random uniform point on it (we just figure out the joint probability distribution of the parameters).

So the starting point is to define the space of quaternions, then we need a metric on the space. Now we can define the sphere of radius 1. We had a metric, so we have a topology on the space of quaternions, and this gives us a topology on the sphere. Now we need a set of transformations of the sphere. The rest is calculus. Or differential geometry, if you prefer to use that language. We need to verify that the set of transformations has the required properties. We need a parametrization of the sphere. Then we are ready to calculate.

In fact, it is well known that the quaternions form a four-dimensional real vector space, and there is a very sensible way to define a (real) norm on the space making it a normed vector space of dimension 4. This Euclidean norm is our metric. The set of transformations could be just the right multiplications by a quaternion. The uniform distribution is normalized right Haar measure. It turns out that the quaternionic 3-sphere is just the ordinary sphere of radius 1 in 4-dimensional (real) Euclidean space, our old friend , and the transformations are the ordinary "rotations". I think we already know how to pick uniformly distributed points on .

See for instance the question and answers at:
https://stackoverflow.com/questions/15880367/python-uniform-distribution-of-points-on-4-dimensional-sphere
The questioner is muddled about terminology. The repliers sort him out and give some solutions.

See also
https://en.wikipedia.org/wiki/N-sphere#Uniformly_at_random_on_the_(n_%E2%88%92_1)-sphere
https://en.wikipedia.org/wiki/3-sphere
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Re: Another simulation for QM Local functions

Postby Joy Christian » Wed Aug 14, 2019 1:42 am

gill1109 wrote:
Joy Christian wrote:
FrediFizzx wrote:I take it back. That is the coordinates of the points of a unit 3-sphere. But which ones are the x, y and z coordinates? The first set of three or the last set of three?
.

Any three of the four coordinates can be taken as x, y, and z. Normalized vectors in R^4 can certainly give a 3-sphere, but that would not be a quaternionic (or parallelizable) 3-sphere. Therefore the correlation among its points will not be as strong as -a.b. It can only give a linear correlation. What is needed are uniformly distributed points on a quaternionic 3-sphere.

Maybe I can help? What exactly do you mean by "quaternionic 3-sphere"? Then we can discuss what we should mean by "uniform distribution".

S^3, or quaternionic 3-sphere, is defined precisely, for example, in eq. (1) of this paper: https://arxiv.org/pdf/1211.0784.pdf. See also eqs. (2) to (5) for further details.

***
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Re: Another simulation for QM Local functions

Postby gill1109 » Wed Aug 14, 2019 6:01 am

That "precise" answer is not so precise. You send me back to a thick book by Doran and Lasenby, and to a paper by David Hestenes which will take me some time to get my hands on. And I don't even know if your two references use the same notations and definitions:

[4] C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003)
[5] D. Hestenes, Am. J. Phys. 71, 104 (2003)

You also refer me to your own paper's equations (2) to (5) where there is a lot of notation which isn't explained.

Perhaps you can just tell me whether or not my guess is correct, that the quaternionic 3-sphere, for our present purposes, is the "ordinary" three-sphere - the surface of the unit ball in four-dimensional real Euclidean space. Left multiplication by a particular quaternion corresponds to a particular rotation of the ball, i.e. an element of SO(4). The problem of drawing a uniformly distributed point on the quaternionic 3-sphere is therefore solved in the internet references which I gave you.

Alternatively, please give me references including page numbers/formula numbers etc.

Maybe it's just time for us all to move on to fields anew? I've apologised to you profusely for my fanatic attacks on your work. (I do still stand by the scientific content of those attacks). It looks to me that everyone is getting a little bored ... do we really need a symposium on this ancient stuff? I doubt that yet another symposium is going to definitively resolve the EPR paradox and the problem of the interpretation of quantum mechanics. (Many people think it is all solved, yet they have different solutions. Jay thinks he can come up with a new solution. From personal experience, I doubt that he will succeed, though of course, it is always possible that someone fresh to the field comes up with truly fresh insight. I get to talk with several "young hopefuls" every year - usually retired gentlemen who somehow regret not having followed up their youthful interest in physics with an academic career in physics. I think it is both wonderful and laudable).
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Re: Another simulation for QM Local functions

Postby Joy Christian » Wed Aug 14, 2019 7:37 am

gill1109 wrote:Perhaps you can just tell me whether or not my guess is correct, that the quaternionic 3-sphere, for our present purposes, is the "ordinary" three-sphere - the surface of the unit ball in four-dimensional real Euclidean space. Left multiplication by a particular quaternion corresponds to a particular rotation of the ball, i.e. an element of SO(4). The problem of drawing a uniformly distributed point on the quaternionic 3-sphere is therefore solved in the internet references which I gave you.

The quaternionic 3-sphere is not the same as "ordinary" three-sphere - the surface of the unit ball in four-dimensional real Euclidean space. Far from it. The latter is sometimes called a "round sphere" because it is round. The quaternionic sphere, on the other hand, is not round. It cannot be round, because quaternions obey strange multiplication rules. A product of two quaternions points to a point on the "sphere" that can be reached only by a quaternionic multiplication, not by scalar multiplication of 4D vectors. The two spheres thus differ in topology.

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Re: Another simulation for QM Local functions

Postby gill1109 » Wed Aug 14, 2019 8:01 am

Joy Christian wrote:
gill1109 wrote:Perhaps you can just tell me whether or not my guess is correct, that the quaternionic 3-sphere, for our present purposes, is the "ordinary" three-sphere - the surface of the unit ball in four-dimensional real Euclidean space. Left multiplication by a particular quaternion corresponds to a particular rotation of the ball, i.e. an element of SO(4). The problem of drawing a uniformly distributed point on the quaternionic 3-sphere is therefore solved in the internet references which I gave you.

The quaternionic 3-sphere is not the same as "ordinary" three-sphere - the surface of the unit ball in four-dimensional real Euclidean space. Far from it. The latter is sometimes called a "round sphere" because it is round. The quaternionic sphere, on the other hand, is not round. It cannot be round, because quaternions obey strange multiplication rules. A product of two quaternions points to a point on the "sphere" that can be reached only by a quaternionic multiplication, not by scalar multiplication of 4D vectors. The two spheres thus differ in topology.

Who is talking about scalar multiplication of 4D vectors here? I am not talking about scalar multiplication of Euclidean vectors. What I am saying is that multiplication by one unit quaternion from the left, as a transformation of the set of all unit quaternions, is isomorphic to rotation of the sphere S^3. And successive multiplications is isomorphic to successive rotations.

Notice that the product of two unit quaternions is another unit quaternion.

With all due respect, Joy, I think you need to take a refresher course on the topics of division algebras and Clifford algebras and geometric algebra. Please take note of Hurwitz' theorem. https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)

The quaternionic sphere is very round indeed! The two spheres have the same topology once they have been brought into the right one-to-one correspondence.
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Re: Another simulation for QM Local functions

Postby Joy Christian » Wed Aug 14, 2019 8:15 am

gill1109 wrote:What I am saying is that multiplication by one unit quaternion from the left, as a transformation of the set of all unit quaternions, is isomorphic to rotation of the sphere S^3. And successive multiplications is isomorphic to successive rotations.

I agree with this.

gill1109 wrote:Notice that the product of two unit quaternions is another unit quaternion.

I agree with this too.

gill1109 wrote:The quaternionic sphere is very round indeed! The two spheres have the same topology once they have been brought into the right one-to-one correspondence.

I don't agree with this. If this were true, then we don't need Pauli matrices in quantum mechanics. We could just represent spins using Euler angles. But we know that that does not work.

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Re: Another simulation for QM Local functions

Postby FrediFizzx » Wed Aug 14, 2019 9:46 am

Joy Christian wrote:I don't agree with this. If this were true, then we don't need Pauli matrices in quantum mechanics. We could just represent spins using Euler angles. But we know that that does not work.

***

It is actually quite strange. As far as we can tell, electrons are round with respect to electric charge but they are not round with respect to magnetic charge. And I would guess that since spin is related to magnetic charge, they are not round with respect to spin 1/2.
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Re: Another simulation for QM Local functions

Postby local » Wed Aug 14, 2019 12:33 pm

FrediFizzx wrote:electrons ... are not round with respect to magnetic charge

Please, I do not understand. What does this mean and what is the evidence? Lorentz invariance is false?
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Wed Aug 14, 2019 12:42 pm

local wrote:
FrediFizzx wrote:electrons ... are not round with respect to magnetic charge

Please, I do not understand. What does this mean and what is the evidence? Lorentz invariance is false?

Electrons have a magnetic dipole moment whereas they don't have an electric dipole moment that we know of.

https://en.wikipedia.org/wiki/Electron_magnetic_moment
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Re: Another simulation for QM Local functions

Postby local » Wed Aug 14, 2019 2:03 pm

So earth is not round because it has a North and South pole (never mind effects at the equator due to rotation)? We can't have round magnets?
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Re: Another simulation for QM Local functions

Postby FrediFizzx » Wed Aug 14, 2019 3:10 pm

local wrote:So earth is not round because it has a North and South pole (never mind effects at the equator due to rotation)? We can't have round magnets?

:D The concept that electrons are not round with respect to spin 1/2 has nothing to do with that. I suppose you can have round magnets but never thought about it.
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Re: Another simulation for QM Local functions

Postby gill1109 » Wed Aug 14, 2019 10:24 pm

Joy Christian wrote:
gill1109 wrote:What I am saying is that multiplication by one unit quaternion from the left, as a transformation of the set of all unit quaternions, is isomorphic to rotation of the sphere S^3. And successive multiplications is isomorphic to successive rotations.

I agree with this.

gill1109 wrote:Notice that the product of two unit quaternions is another unit quaternion.

I agree with this too.

gill1109 wrote:The quaternionic sphere is very round indeed! The two spheres have the same topology once they have been brought into the right one-to-one correspondence.

I don't agree with this. If this were true, then we don't need Pauli matrices in quantum mechanics. We could just represent spins using Euler angles. But we know that that does not work.

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.
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Re: Another simulation for QM Local functions

Postby Joy Christian » Wed Aug 14, 2019 10:45 pm

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 sigularity-free representation of S^3.

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Re: Another simulation for QM Local functions

Postby jreed » Thu Aug 15, 2019 9:56 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 sigularity-free representation of S^3.

***

That's correct. Look up Gimbal Lock. You'll find that this occurs when Euler angles are used for geometrical representations and represents a singularity. Quaternions avoid this problem.
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