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Hello Everyone,

The following is my claim to collect the 10,000 Euros offered by Richard Gill. In my opinion I won the 10,000 Euros offered by him a long time ago, but he has raised some formal issues with my claim and has refused to pay up. Therefore I am now making this fresh claim, which I believe meets all of his technical requirements.

You can read the terms of his offer on this page here, and my objections to how he has been misinterpreting my proposed experiment on these pages here and here.

Without further ado, here are the documents I am submitting to meet his challenge:

(1) A simple explanation of my proposed experiment, with links to relevant papers.

(2) A proof by simulation that there indeed exist N vectors, u_k and -u_k, appearing in equation (16) of my experimental paper: http://rpubs.com/jjc/16415.

(3) An explicit calculation, using a numerical simulation, of the four specific correlations considered in his challenge: http://rpubs.com/jjc/16531. It is very important to read the commentary in this simulation to understand what is being submitted here. Here is the simulation again: http://rpubs.com/jjc/16531.

NB: There are only N directions of spin momentum (about each station) occurring in this simulation, not 2N.

(4) And finally, and perhaps most importantly from Richard's perspective, I submit two computer files, named "AliceDuv.csv" and "BobDuv.csv", as required by him.

With these submissions, I claim to have won the 10,000 euros offered by Richard Gill for producing the N angular momentum vectors specified in my experiment.



PS: The files for Alice's and Bob's spin directions are very large. Please "right click" them and do "Save Target as" to download them. Your browser may crash if you try to open them in your browser. Also, here are the text (and zipped) versions of the two files: http://www.sciphysicsforums.com/jjc/Ali ... _Files.zip (thanks, Fred).

I have received a reply from Richard. He is enjoying his vacation on Rugen (a German island), with limited access to internet. So it will take him a while to respond.

I wonder Joy, are you in it for the honour, the money or both?

gris wrote:I wonder Joy, are you in it for the honour, the money or both?

I am in it for truth and justice.

Could you print here, say, the first ten lines of each file, explain the format, and tell us the number of lines of each file?

PS It could also help if you would post the R script which creates the two files (I assume you used R to make them). That is no doubt much smaller than the two files themselves. Then I can pick up the script, run it, and verify your claim.

gill1109 wrote:PS It could also help if you would post the R script which creates the two files (I assume you used R to make them). That is no doubt much smaller than the two files themselves. Then I can pick up the script, run it, and verify your claim.

The R script is already linked above in my initial post. But here it is again: http://rpubs.com/jjc/16531 (scroll down to the bottom of the page).

The first ten directions observed by Alice and Bob are here:



From the R script linked above the meaning of these numbers should be fairly clear.

But the important thing to note here is that, since u and v specify one and the same direction of angular momentum (since u.v = 1 always), we have

A(a,u) = sign(a.u) = sign(a.v) = A(a,v),

and similarly

B(b,u) = sign(b.u) = sign(b.v) = B(b,v).

You say that the 2-vectors "u" and "v" represent the same directions. It is true that their inner products u.v are always equal to 1. Yet the lengths of the vectors u and v vary tremendously. In fact, many of these vectors are longer than 1. You only give an x and a y coordinate of each vector so we are missing the z of the Cartesian triples (x, y, z).

I am afraid I cannot process these data files. They do not unambiguously tell me one set of N directions observed by Alice and one set of N directions observed by Bob.

Code: Select all

summary(colSums(v*v))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
1.000e+00 3.000e+00 1.000e+01 3.754e+06 5.400e+01 3.568e+11
> summary(colSums(u*u))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
0.0000042 0.1765000 0.3791000 0.3797000 0.5184000 1.2180000
> summary(colSums(u*v))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
      1       1       1       1       1       1


Joy Christian wrote:

gris wrote:I wonder Joy, are you in it for the honour, the money or both?

I am in it for truth and justice.

Ah I get it: truth is you're right and justice is you get € 10000,= for winning the bet? Correct?

Joy believes he is right, and we together set up a challenge, and he can win 10 000 Euro with a winning submission before June 11. Thereafter 5 000 Euro. It is open to everyone.

Anyone can submit the two files ... For instance, those people who believe that Joy's analysis is correct can study his works and try to use his insights to beat him to it ...

gill1109 wrote:You say that the 2-vectors "u" and "v" represent the same directions. It is true that their inner products u.v are always equal to 1. Yet the lengths of the vectors u and v vary tremendously. In fact, many of these vectors are longer than 1. You only give an x and a y coordinate of each vector so we are missing the z of the Cartesian triples (x, y, z).

I am afraid I cannot process these data files. They do not unambiguously tell me one set of N directions observed by Alice and one set of N directions observed by Bob.

Code: Select all

summary(colSums(v*v))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
1.000e+00 3.000e+00 1.000e+01 3.754e+06 5.400e+01 3.568e+11
> summary(colSums(u*u))
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
0.0000042 0.1765000 0.3791000 0.3797000 0.5184000 1.2180000
> summary(colSums(u*v))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
      1       1       1       1       1       1


The directions observed by Alice and Bob are unambiguously the same. There is no mention of the magnitudes of the vectors either in your challenge or in my proposed experiment, and for good reasons. Alice and Bob do not measure magnitudes of anything. They only measure sign(a.u) = sign(a.v) and sign(b.u) = sign(b.v), respectively. I think you should read the terms of your challenge once again. Show me where in your challenge you say anything about the magnitudes of the spins.

Joy Christian wrote:Show me where in your challenge you say anything about the magnitudes of the spins.

The challenge which we drew up together talks about the directions of the spins, not about their magnitudes.

It mentions explicitly, in connection with computing the correlations, that they are represented as unit vectors in R^3.

However spherical coordinates would also be OK, you just have to tell me what representation you are using.

And obviously if we are computing correlations for measurement directions in the equatorial plane, you can drop the "z" coordinate if you specify the directions with Cartesian coordinates of unit vectors.

Your program computes a lot of 2-vectors u and v. You say that the u's and the v's represent the same directions. OK, tell me how they represent a direction, and what it is, and then we can calculate the correlations.

You claim that sign(a.u) = sign(a.v) and sign(b.u) = sign(b.v).

However this is unfortunately not true, as is easy to check:

Code: Select all

> alpha <- 0 * pi/180
> beta <- 45 * pi/180
> a <- c(cos(alpha), sin(alpha))
> b <- c(cos(beta), sin(beta))
> (E_0_45 <- mean(sign(colSums(a * u)) * -sign(colSums(b * u))))
[1] -0.70718
> alpha <- 0 * pi/180
> beta <- 45 * pi/180
> a <- c(cos(alpha), sin(alpha))
> b <- c(cos(beta), sin(beta))
> (E_0_45 <- mean(sign(colSums(a * v)) * -sign(colSums(b * v))))
[1] -0.2943

gill1109 wrote:It mentions explicitly, in connection with computing the correlations, that they are represented as unit vectors in R^3.

This is complete baloney. There is no mention of unit vectors for the spin directions. You do not say that in your challenge, and neither do I in my proposal.

Let me repeat:

The directions observed by Alice and Bob are unambiguously the same. There is no mention of the magnitudes of the vectors either in your challenge or in my proposed experiment, and for good reasons. Alice and Bob do not measure magnitudes of anything. They only measure sign(a.u) = sign(a.v) and sign(b.u) = sign(b.v), respectively. I think you should read the terms of your challenge once again. Show me where in your challenge you say anything about the magnitudes of the spins.

gill1109 wrote:You claim that sign(a.u) = sign(a.v) and sign(b.u) = sign(b.v).

However this is unfortunately not true, as is easy to check:

Code: Select all

> alpha <- 0 * pi/180
> beta <- 45 * pi/180
> a <- c(cos(alpha), sin(alpha))
> b <- c(cos(beta), sin(beta))
> (E_0_45 <- mean(sign(colSums(a * u)) * -sign(colSums(b * u))))
[1] -0.70718
> alpha <- 0 * pi/180
> beta <- 45 * pi/180
> a <- c(cos(alpha), sin(alpha))
> b <- c(cos(beta), sin(beta))
> (E_0_45 <- mean(sign(colSums(a * v)) * -sign(colSums(b * v))))
[1] -0.2943

This too is complete baloney. The correct calculations are given in my simulation, which anyone can reproduce for themselves: http://rpubs.com/jjc/16531.

Let me reproduce what I have explained in the commentary of the simulation:

## Note that, despite appearances, u and v specify EXACTLY THE SAME DIRECTION
## of the spin momentum of the bomb fragment (and likewise the vectors -u and
## -v). To prove this, we simply calculate their dot product u.v. This can be
## easily done on paper, but we can also let our program calculate it for us:

(u.v <- mean(colSums(u * v)))

## [1] 1

## This proves that u and v are parallel to each other: Since u.v = 1, the
## angle between u and v is ALWAYS ZERO. Thus u and v represent ONE AND THE
## SAME DIRECTION of the spin momentum. Consequently, Alice and Bob would be
## oblivious to whether they have observed a 'u' or 'v' direction of the spin
## momentum. In other words, although the topology of the 3-sphere imparts
## some difference between u and v, as far as Alice and Bob are concerned
## there are only N directions of spin momentum in this simulation, not 2N.

## If you are puzzled by why u and v are the same and yet look so different,
## then you have never watched a rotation through a mirror: u and v are just
## mirror images of each other, with u.v = 1 indicating a 'zero' rotation.
## This becomes clearer if we recall that the relative angle between any two
## rotation axes in the physical space is half the rotation angle of the spin
## about that axis: psi_uv = 2 eta_uv. Ultimately this has to do with the
## double covering of the rotation group SO(3) within the 'spin' group SU(2).
## For further details of how rotations are composed in the physical space,
## please consult my 'experimental' paper: http://arxiv.org/abs/1211.0784.

Richard you are a man of honour. Now Joy what is your answer to my last posting?

Once again Richard Gill is bobbing and weaving and finding ridiculous excuses.

The matter is extremely elementary. If u and v are two vectors, then they specify one and the same direction if u.v = 1. Any school kid knows that.

I learnt at school that u.v = length(u) times length(v) times cosine angle between u and v.

Here are the values of a.u and a.v for the first u, v in the files supplied by JJC:

Code: Select all

> sum(a*v[ ,1])
[1] 0.7358574
> sum(a*u[ ,1])
[1] 0.6794795

They seem to be different.

Moreover, the word "unit vector" is certainly present in the challenge. But if you prefer spherical coordinates, that is fine by me. It's the direction that counts, only the direction. I generously do not mind too much how you specify directions in R^3. But you are obliged to give me an unambiguous representation of one direction for Alice and one direction for Bob, for each of the N runs. Do you wish to take this to the adjudicators? You only have one chance at adjudication...

gill1109 wrote:Here are the values of a.u and a.v for the first u, v in the files supplied by JJC:

Code: Select all

> sum(a*v[ ,1])
[1] 0.7358574
> sum(a*u[ ,1])
[1] 0.6794795

They seem to be different.

I do not understand what this calculation is. What are you calculating here? You cannot just do any random calculation like this. You have to adhere to my model. It is part of the challenge, whether you like it or not.

gill1109 wrote:Do you wish to take this to the adjudicators? You only have one chance at adjudication...

No.

You have obfuscated my simple calculation sufficiently to confuse everyone. The adjudicators will be no exception. I will try to clear up the seeming ambiguity in the two directions and repost the two files.

Continue to enjoy your vacation.