A simple two-page proof of local realism

[Joy Christian]

Over on the fqxi forum http://fqxi.org/community/forum/topic/812, some of us were earlier discussing the elephant in the room which the small but real discrepancies between these various attempts to simulate Joy's model seem to have introduced.

It is abundantly clear---and it is demonstrably proven by several people beyond any shadow of doubt---that it is Gill's attempted simulation of my flawless analytical model, and only Gill's attempted simulation of my flawless analytical model, that is in error.

How do I know this? How can I be so confident? Well, just take a look at the eight elementary mathematical equations in this two-page document. One does not have to be Einstein to see the analytical validity of these equations. One does not need a simulation to prove the analytical validity of these equations. If an attempted simulation does not reproduce what is already proven analytically in these equations, then one does not need a PhD in mathematics or statistics to know who or what is in error. QED

[gris]

As some of you may know, I have written up a very simple two-page proof of Einstein's local realism: http://libertesphilosophica.info/blog/w ... 1/EPRB.pdf.

Bell's theorem supposed to have proved that it is impossible to produce the EPR-Bohm correlation purely locally, with or without determinism. But the above model proves otherwise (I thank Michel Fodje and John Reed for producing the cited simulations of the model). Further details of the proof, an extensive discussion on the origins of *ALL* quantum correlations, and the specifics of an experimental test of my hypothesis, can be found on my blog: http://libertesphilosophica.info/blog/.

It is very important to realize that the above simple proof of local realism is the end of the road for Bell's theorem. There is no way out for the supporters of Bell's theorem. Let me explain this for the benefit of those unfamiliar with the subject and those who are declining to see the logical power of elementary mathematics:


Joy Christian

The a priori question is IMO why you have to disprove anything? Since 1911 we have had the knowledge of superconductivity. The Bell theorem from later date has simply not taken into account a scenario in which you can see it as a split gearbox in which the toothed wheels are still counter-rotating. Of course they will keep on doing that if you take the parts of the gearbox further apart. Measurement will break the symmetry. That is the only common sense way of looking at it. Lex parsimony and Bayes => verbal logic as a scientific dictate. Bells theorem is spooky Vodoo at a distance and that is belief in magic. Given a common sense alternate scenario what is there to disprove? Magic is in it self immediately falsified in the light of any plausible other non magical and probable explanation. It is axiomatic. No mathematics needed.

A far better experiment to disprove the Bell theorem is to prove the mathematical possibility of balls in a super conductive box simulation, and see if you can get them to go to the order of a dynamic crystal in the center of the box. Every ball remains in its own virtual box by hitting the ball of the adjunct virtual box. You would need a super computer with a lot of very accurately modeled massless billiard balls (that accuracy should suffice I guess). That would immediately provide a fundamental answer to the observed too much order in the system/ visible universe. Pointing thus in a different direction of super conductivity at the heart of it all. A direct and much more fundamental way of dealing with the problem. Just keep at it on ever better supercomputers until you either via this or any other way can explain the too much order. If and when simple common sense can do the trick, as it thus can, then the Lex parsimony + Bayes dictate you follow that path. Mathematics on part issues is to cumbersome and not integrated. My test is integrated yet I can leave that be for I already have the product of that integration: a viable test.

Disproving Bell - if at all possible other than I just did - brings you no further.

Edit: in fact it should IMO only be possible to prove Bell to be non magical and something different than I say, such as a scenario with a sort of magnetic field > c. But to prove something like that is for the proponents of the Bell theorem. To disprove it will IMO be impossible because the proponents can always claim that your test in some way disturbed the symmetry (which then would probably be what indeed happened).

Gerhard Ris

[gill1109]

Gerhard, so your recommendation is to shut up and stop thinking and just build faster and faster supercomputers?

You say "immediately provide a fundamental answer .... Just keep at it on ever better supercomputers until you either via this or any other way can explain the too much order"

How do you know that there ever will be an answer? How do you know it won't it go on for ever? When do you predict we will be done?

I predict that it is easy to prove that the supercomputer which is needed to resolve the question your way, is bigger than the universe.

[gris]

No, Richard, it is like solving a crime scene. We miss a lot of data that is obvious. So first we observe all the data we do have in its essence and then we use the instrument between our ears: i.e. we think. Formulating the right questions. In my mind it subsequently popped up that it might be a dynamic crystal. This would indeed fundamentally explain the to much order we observe in our universe. After that it is thus indeed doggedly following that lead (AND every other testable lead especially if it like this idea potentially can integrate it all in a TOE like this idea.)

I don't know how much computing power is needed to simulate say 1000 x 1000 x 1000 mass-less rigid billiard balls (this amount of balls and amount of accuracy should do the trick) each given the room of say a billiard table square. All with the same speed in random directions. In a large box with superconductive walls. Sufficient accuracy is crucial. That doesn't require a computer the size of the universe. It requires tinkering, if you don't have the computing power then less balls might do the trick. And indeed you follow the lead like good police work solving a cold case when new DNA methods come along.

It might be possible to simulate balls a different way that requires less computing power, and you can slow the simulation down. Anyway it is potentially testable, and it would in one go solve a most fundamental question. Edit: come to think of it, it might suffice only to have points representing the center of each ball and have the computer do vector analysis every-time these points are at a certain distance from each other.

Your questions pose the same problem with what you are doing. We don't know. Yet we can do an educated guess via probabilistic reasoning. I believe in an absolute truth that we will never absolutely prove. Yet I'm convinced that by going at it in the right rational way we will get every close to this truth. Like Bayes in fact has done, for neurology more and more points in the direction that it is this algorithm in all our brains. Bayes didn't know that.

BTW what would you say to a prosecutor who tries to solve a crime scene by only looking at part of the evidence and using mathematics in stead? That in effect is what you are doing.

[Heinera]

Over on the fqxi forum http://fqxi.org/community/forum/topic/812, some of us were earlier discussing the elephant in the room which the small but real discrepancies between these various attempts to simulate Joy's model seem to have introduced.

It is abundantly clear---and it is demonstrably proven by several people beyond any shadow of doubt---that it is Gill's attempted simulation of my flawless analytical model, and only Gill's attempted simulation of my flawless analytical model, that is in error.

How do I know this? How can I be so confident? Well, just take a look at the eight elementary mathematical equations in this two-page document. One does not have to be Einstein to see the analytical validity of these equations. One does not need a simulation to prove the analytical validity of these equations. If an attempted simulation does not reproduce what is already proven analytically in these equations, then one does not need a PhD in mathematics or statistics to know who or what is in error. QED

I have looked into the minkwe-model from an analytical, and not numerical perspective. It is clear that it will not reproduce the cosine correlation; actually, it is possible (and not very hard) to show by analytical means that on the interval [Pi/3, 2*Pi/3] the minkwe-correlation is in fact a linear (or more correctly, affine) function of the form

C*(Pi/2 – theta),

where theta is the angle between detectors and C is a constant parameter that must be determined numerically, and is close to 3/Pi. I use Richard’s sign convention here. Also, as far as I can see this result agrees very well with Richard’s numerical simulation in R. At the borders of the interval [Pi/3, 2*Pi/3], the function changes in a continuous way into another function with (probably) no closed form expression, so it must be computed numerically. I didn’t look much further into this part, as it is clear to me that overall, the function cannot be the cosine.

Obviously, symmetric results hold for the interval [Pi/3, 2*Pi/3] + Pi.

[gill1109]

what would you say to a prosecutor who tries to solve a crime scene by only looking at part of the evidence and using mathematics in stead? That in effect is what you are doing.

I am looking at all of the existing evidence and mathematics is providing part of it. Using the computing machine between our ears ... we can generate very powerful evidence much much more cheaply than bothering the taxpayers of the world to build the biggest supercomputer ever, which probably will only ever get around to telling us that the answer might be something like 42, but the problem of posing the question properly requires an even bigger supercomputer.

[gill1109]

I have looked into the minkwe-model from an analytical, and not numerical perspective. It is clear that it will not reproduce the cosine correlation; actually, it is possible (and not very hard) to show that on the interval [Pi/3, 2*Pi/3] the minkwe-correlation is in fact a linear (or more correctly, affine) function of the form

C*(Pi/2 – theta),

where theta is the angle between detectors and C is a parameter that must be determined numerically, and is close to 3/Pi. I use Richard’s sign convention here. Also, as far as I can see this result agrees very well with Richard’s numerical simulation in R. At the borders of the interval [Pi/3, 2*Pi/3], the function changes in a continuous way into another function with (probably) no closed form expression, so it must be computed numerically. I didn’t look much further into this part, as it is clear to me that overall, the function cannot be the cosine.

Obviously, symmetric results hold for the interval [Pi/3, 2*Pi/3] + Pi.

Wonderful Heinera, hereby you also answered a question put by Fred Diether. No need to do even bigger simulations. We know the answer now.

In the meantime I also discussed the Chantal-Sabsay simulation with Daniel Sabsay, http://cyberneticmoments.com. He seemed to agree with my criticisms and said he would improve his Javascript accordingly. In particular he admitted that his cosmetic rescaling of the vertical axis was not really kosher. I hope that Joy will get around to putting a more honest simulation and more honest graphic of its output on his webpages.

[Joy Christian]

Here, finally, is my version of Richard Gill's S^2 version of Michel Fodje's classic simulation:

http://rpubs.com/chenopodium/joychristian (with thanks to Chantal Roth).

Do scroll all the way down on the page to check out the second plot. What do you see?

[gill1109]

I see that the numbers "2", "1/2" have been treated as free parameters and tweaked to get us really close to the cosine! Congratulations! A victory for the programming language R, I would say, and proof of Chantal and my respective skills.

Caroline Thompson would be really proud to see this. She knew that her chaotic spinning ball model could be tweaked, by introducing "unsharp" membership of the circular caps, to get as close as you like (for any practical purposes) to the cosine. And at last, this has been done, to a fantastic degree of accuracy.

http://freespace.virgin.net/ch.thompson1/Papers/The%20Record/TheRecord.htm

[gill1109]

It's like the arms race ... we are all going to go bankrupt buying bigger and bigger faster and faster computers ...

http://rpubs.com/gill1109/ChaoticUnsharpBall

This is just the latest Christian-Roth co-production but with higher resolution (sample size 10 million). There is something unhealthy going on at around 60 degrees.

[Joy Christian]

So let us put some perspective on my version of Richard Gill's S^2 version of Michel Fodje's original simulation of my 3-sphere, or SU(2), model for the EPR-Bohm correlation.

As we can see from the details discussed in the above paper, my analytical model predicts the (negative) cosine correlation exactly. What is more, in Bell's local-realistic framework (discussed in his 1964 paper) we are completely free to choose whatever initial or complete state of the system we like. In the present representation of the 3-sphere this initial state is represented by the pair (e_o, theta_o), which are simply four numbers. These numbers depend on the system under consideration. They depend on the symmetry of the physical situation. For example, for the GHZ or Hardy case the initial state would be nothing like these four numbers.

Now have a look at the derivation of this state within my 3-sphere model. As we can see from just above the box of eq.(10), I had made a simple choice for the initial state by choosing the function f(theta_o) defined in eq.(7). This choice simply specifies the magnitude of the sum of the two initial quaternions, p_o and q_o, and thereby also specifies the initial state (e_o, theta_o). In Michel Fodje's simulation the choice I made for f(theta_o) seemed necessary and sufficient to produce the correct correlation. It now appears that when one zooms-in with greater precision, the initial state is in fact what has now been chosen in the latest simulation.

I am grateful to Richard Gill for insisting on greater precision for the simulation, which helped me discover a more accurate choice for the initial state (e_o, theta_o) in the EPR-Bohm case. I am also grateful to Chantal Roth for encouraging me to learn R and thus investigate Richard Gill's simulation myself. The result of my investigations is not devoid of beauty.

[gill1109]

I re-ran the latest simulation http://rpubs.com/chenopodium/joychristian at larger sample size still (now 10^7) and ask you to take a look at a different spot on the curve ... say at around 60 degrees.

http://rpubs.com/gill1109/ChaoticUnsharpBall

Getting closer but not quite there yet.

Amusingly, there is another interpretation of the simulation model: this is Caroline Thompson's chaotic spinning ball model with a kind of unsharp observation of the circular caps - the circular caps don't have a fixed radius, but a random radius S, and presently it has the following probability distribution: S = (sin(Theta)^1.32)/3.16 where Theta ~ Unif(0, pi/2).

http://freespace.virgin.net/ch.thompson ... Record.htm

Conjecture 1: there does exist a radius distribution such that the cosine is reproduce exactly.

Conjecture 2: there are no constants a, b such that S = (sin(Theta)^a)/b where Theta ~ Unif(0, pi/2) reproduces the cosine exactly.

I do agree that these investigations are not devoid of beauty, and they are not devoid of mathematical interest either.

[Joy Christian]

"So let us put some perspective on my version of Richard Gill's S^2 version of Michel Fodje's original simulation of my 3-sphere, or SU(2), model for the EPR-Bohm correlation.

As we can see from the details discussed in the above paper, my analytical model predicts the (negative) cosine correlation exactly."

This implies that there does exist an initial state (e_o, theta_o) such that the cosine correlation is reproduced exactly.

[Joy Christian]

I re-ran the latest simulation http://rpubs.com/chenopodium/joychristian at larger sample size still (now 10^7) and ask you to take a look at a different spot on the curve ... say at around 60 degrees.

http://rpubs.com/gill1109/ChaoticUnsharpBall

Getting closer but not quite there yet.

The Monte Carlo accuracy of Richard Gill's simulation is about 0.0001.

The slight deviation in my plot he is now making fuss about is much smaller than 0.0001.

[gill1109]

The distance between the two curves is about 0.01 and indeed, the accuracy of the simulation is about 0.0001

I conjecture that you'll never get there exactly by further tweaking your two parameters but it can be done by tweaking infinitely many.

This is what you have at the moment:



M <- 10^7

Code: Select all

theta <- runif(M, 0, pi/2)
s <- (sin(theta)^1.32)/3.16
R <- acos(s)*180/pi
hist(R, breaks=1000,xlim=c(70,90), freq=FALSE, main = "The cap radius distribution", xlab = "Angular radius (degrees)")


I re-ran the latest simulation http://rpubs.com/chenopodium/joychristian at larger sample size still (now 10^7) and ask you to take a look at a different spot on the curve ... say at around 60 degrees.

http://rpubs.com/gill1109/ChaoticUnsharpBall

Getting closer but not quite there yet.

The Monte Carlo accuracy of Richard Gill's simulation is about 0.0001.

The slight deviation in my plot he is now making fuss about is much smaller than 0.0001.

[gill1109]

This is what Christian calls "the slight deviation ... much smaller than 0.0001"

[FrediFizzx]

This is how you do it.



Those small deviations around the peaks will disappear in the limit when the number of trials goes to infinity and the degree increments are infinitesimal.

[gill1109]

You are a bit behind the game, Fred. And you are writing the same post in two threads. I've replied to this propaganda poster at the appropriate place http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=11 Computer Simulation of EPR Scenarios

[Joy Christian]

OK, we are done. I have revised my simulation: http://rpubs.com/chenopodium/13653.

Let me remind again that the initial state of the system is still (e_o, theta_o), as derived in http://libertesphilosophica.info/blog/w ... mplete.pdf, but the choice of the initial function f(theta_o) is now different:

f(theta_o) = (1/2.47) sin(theta_o)^{1.61}.

It is also important to note that the Monte Carlo accuracy of the simulation is about 0.0001, but any remaining wrinkles in the correlation function are much smaller than 0.0001.

[gill1109]

OK, we are done. I have revised my simulation: http://rpubs.com/chenopodium/13653.

Let me remind again that the initial state of the system is still (e_o, theta_o), as derived in http://libertesphilosophica.info/blog/w ... mplete.pdf, but the choice of the initial function f(theta_o) is now different:

f(theta_o) = (1/2.47) sin(theta_o)^{1.61}.

It is also important to note that the Monte Carlo accuracy of the simulation is about 0.0001, but any remaining wrinkles in the correlation function are much smaller than 0.0001.

The difference between simulated correlation and cosine is about 0.002. Rather bigger than than the Monte Carlo error (= the statistical error) of about 0.0001. See the bottom picture.