A simple two-page proof of local realism

[gill1109]

... how did Christian arrive at the particular choice together with uniformly distributed over . These two ingredients determine the probability distribution of the circular caps in the rotating ball model. According to his documents EPRB.pdf and complete.pdf, the choice for the function f(.) is essentially arbitrary, it just has to be chosen to satisfy certain bounds coming from the triangle inequality for quaternions. theta_0 is supposed to represent the angle between x and g_0 where neither x nor g_0 are further specified in those documents, so why theta_0 should be taken uniform on [0, pi/2] is another mystery. If x and/or theta_0 are picked uniformly at random in S^2, the angle between the two does *not* have the uniform distribution. ... [Yet] the simulations show that these "arbitrary" choices are extraordinarily well made, though not well enough to give us the cosine correlations, exactly. ... There is no way they can be "derived" by some exact principles, since they only deliver an approximation to what we are after.

Joy does not seem about to answer my questions on how he made his particular choices for f and the distribution of theta_0. Maybe someone else will do so. Why take ? Why take uniformly distributed over ? I understand that these are *feasible* choices, but they are not the only possibilities. Other choices would not have given such good results.

[Joy Christian]

... how did Christian arrive at the particular choice together with uniformly distributed over . These two ingredients determine the probability distribution of the circular caps in the rotating ball model. According to his documents EPRB.pdf and complete.pdf, the choice for the function f(.) is essentially arbitrary, it just has to be chosen to satisfy certain bounds coming from the triangle inequality for quaternions. theta_0 is supposed to represent the angle between x and g_0 where neither x nor g_0 are further specified in those documents, so why theta_0 should be taken uniform on [0, pi/2] is another mystery. If x and/or theta_0 are picked uniformly at random in S^2, the angle between the two does *not* have the uniform distribution. ... [Yet] the simulations show that these "arbitrary" choices are extraordinarily well made, though not well enough to give us the cosine correlations, exactly. ... There is no way they can be "derived" by some exact principles, since they only deliver an approximation to what we are after.

Joy does not seem about to answer my questions on how he made his particular choices for f and the distribution of theta_0. Maybe someone else will do so. Why take ? Why take uniformly distributed over ? I understand that these are *feasible* choices, but they are not the only possibilities. Other choices would not have given such good results.

I have already answered your questions, but you---as you have done many times before---continue to ignore my answers and continue to misrepresent my model according to your own preconceptions of it. As I have already said several times, the answers to your questions can be found in these two documents: (1) and (2).

In particular, a detailed discussion of where the function comes from and why is distributed over is given in the first document. Please read section A.3.3 starting on page 253. I have repeatedly asked you to read the entire paper, but you have no intention of doing that. So I now ask that you read at least the first few paragraphs of the section A.3.3. Otherwise you would be misleading the physics community and thus blocking the progress of physics.

[gill1109]

Thank you. I read "This in turn suggests that we may treat θo ∈ [0, π/2] as an additional random parameter".

I am looking for a principled *derivation* of the new representation of hidden variables and the chosen probability distribution of the hidden variables, not for suggestions, guesses. Other guesses would have been just as well justified. These particular guesses do the job remarkably well (accuracy 0.001). Were you just extraordinarily lucky?

[Joy Christian]

Thank you. I read "This in turn suggests that we may treat θo ∈ [0, π/2] as an additional random parameter".

I am looking for a principled *derivation* of the new representation of hidden variables and the chosen probability distribution of the hidden variables, not for suggestions, guesses. Other guesses would have been just as well justified. These particular guesses do the job remarkably well (accuracy 0.001). Were you just extraordinarily lucky?

Is this how you read papers? Half a sentence here, half a sentence there, and then attack?

I have given you the principled *derivation* in the two documents above. Read the documents in full. Luck has nothing to do with the choice of . It follows from the geometry and topology of the 3-sphere, within the statistical analysis I have presented in the above documents. I would have to be extraordinarily lucky indeed to be able to reproduce, not only the EPR-Bohm correlation, but also the 3- and 4-particle GHZ correlations, and all 16 predictions of the asymmetric Hardy state, exactly. In fact, I would have to be God himself to be able to reproduce all these correlations in such exquisite detail: http://arxiv.org/abs/0904.4259 But the God here is .

[gill1109]

That was not an attack. I asked a question (have now asked it twice). You have referred me each time to specific pages (lines, even) of specific documents. Reading there, what I find is always the same, namely: the statement that a certain function can be chosen quite arbitrarily as long as it satisfies certain constaints. You appear just to pick one arbitrarily which happens to do the job. Why could it not have been another? Furthermore, no motivation at all for the probability distribution of the argument of that function (a component of your hidden variable).

Maybe you can tell me which page to read in order to find what I asked for, or you could agree that the choice is essentially arbitrary.

[Joy Christian]

That was not an attack. I asked a question (have now asked it twice). You have referred me each time to specific pages (lines, even) of specific documents. Reading there, what I find is always the same, namely: the statement that a certain function can be chosen quite arbitrarily as long as it satisfies certain constaints. You appear just to pick one arbitrarily which happens to do the job. Why could it not have been another? Furthermore, no motivation at all for the probability distribution of the argument of that function (a component of your hidden variable).

Maybe you can tell me which page to read in order to find what I asked for, or you could agree that the choice is essentially arbitrary.

OK. Let us look at this in two different ways. First, I have given detailed analysis in the longer paper (60 pages) of how one can go from my analytical model (which was summarized in my one-page paper) to what you call 2.0 representation. In that paper I also have a long footnote on pages 242 and 243, which provides the essential reason for the function . But you are not satisfied with the derivation or the explanation given in section A.3.3. You see as arbitrarily chosen.

I disagree to some extent, but not entirely. After all, is the initial or complete state of the physical system, which deterministically determines the outcomes of measurements. If you choose a different initial state L in the function A(a, L) of Bell, then you would get a different correlation. Why should that be surprising? That is exactly what we would expect. It all depends on which system is under consideration. If instead of a rotationally invariant singlet state we consider some other state (say, the Hardy state, for example), then the corresponding local-realistic initial state would be quite different from , perhaps involving a function different from . That is why it is called the initial state. In a full, dynamical theory would correspond to an initial value in a dynamical equation, like in Schrodinger's equation. Thus there is nothing mysterious about certain amount of arbitrariness in the function . It just reflects a choice of a physical system. In my 3-sphere model this choice, for the rotationally invariant singlet state, pops out naturally, as can be seen from my discussions in the paper I mentioned.

[Joy Christian]

I have posted the following comment on the Welcome page of my blog:

I am grateful to one of my former detractors, Richard Gill, for reproducing yet another simulation of my 3-sphere model for the EPR-Bohm correlation. It is good to see my local model being confirmed in so many different simulations, especially because many critics of my work had previously misinterpreted my one-page paper and claimed that my model predicts constant correlation, namely -1, for all values of the experimental parameters a and b. The recent plethora of simulations confirming my local model clearly show that the critics (which included Richard Gill, Scott Aaronson, James Weatherall, Adrian Kent, Lucien Hardy, an exquisitely qualified panellist recruited by FQXi to evaluate my work, a distinguished board member of Physical Review D, numerous referees and editors of many established journals, as well as some lesser known critics) were grossly mistaken. So by taking a U-turn Richard Gill has now done a great service to physics by reproducing some of the simulations of my analytical local model in a different programming language.

[gill1109]

Well, my interpretation of the code I have published is that it gives a simulation of two different flatland models: one is based on S^1 (subset of R^2) and the other is based on S^2 (subset of R^3).

One can interpret both of these models as "detection loophole" models if one takes the outcomes "no outcome" as corresponding to pairs of particles (or states) which did exist but did not get succesfully measured.

One can interpret both of these models as "conspiracy loophole" models if one takes the population of states (pairs of particles) as corresponding to the subset for which two acceptance criteria were both satisfied, depending on the two settings a, b.

Geometrically they can both be seen as variants of Caroline Thompson's flatland spinning ball models - she had the detection loophole interpretation in mind. She had in mind to replace the "sharp" circular disk caps of fixed radius by some kind of fuzzy caps, and here that is implemented by picking the common radius of the four caps at random.

Both of them take some formulas from Joy's second edition ("Christian 2.0") works: in particular, the choice f(theta_0) = 1/2 sin^2(theta_0) with theta_0 uniformly distributed on [0, pi/2].

If anyone wants to interpret these models as S^3 models making use of what I like to think of as "The Math of Joy" (no offence intended - but for a flatlander like me Joy's maths is incomprehensible as mathematics, but does make a lot of poetic sense), they are welcome to do so. The code stands. It is clear what classical probability model it represents. The simulation is just a numerical computation of certain integrals using the Monte Carlo technique.

[Joy Christian]

If anyone wants to interpret these models as S^4 models making use of what I like to think of as "The Math of Joy" (no offence intended - but for a flatlander like me Joy's maths is incomprehensible as mathematics, but does make a lot of poetic sense), they are welcome to do so. The code stands. It is clear what classical probability model it represents. The simulation is just a numerical computation of certain integrals using the Monte Carlo technique.

To begin with, my model is based on the hypothesis that we live in S^3, not R^3 (and most certainly not S^4). "The Joy of Math" is a good way to present my ideas. After all, I have only used high-school trigonometry in the current representation of my 3-sphere model: http://libertesphilosophica.info/blog/w ... 1/EPRB.pdf.

Speaking of poetry, how is your poetry "disjunction replaced by conjunction" going? I mean, you do believe that it is logically correct to replace disjunctions like

AB or AB'

with conjunctions like

AB and AB'

in your arguments, don't you? You replace disjunctions by conjunctions and conclude that there is voodoo in the world, don't you?

[gill1109]

Sorry, S^4 instead of S^3 was a misprint.

I don't conclude there is voodoo in the world. I conclude that there is irreducible randomness in the world.

(Nor, IMHO, do I replace conjunctions by disjunctions!).

But you are welcome to your own interpretations.

[Joy Christian]

I don't conclude there is voodoo in the world. I conclude that there is irreducible randomness in the world.

"irreducible randomness" is the same voodoo by a different name.

(Nor, IMHO, do I replace conjunctions by disjunctions!).

Yes, you do.

But you are welcome to your own interpretations.

As you are to yours.

[gill1109]

I just published another R simulation comparing two "interpretations" of Joy's model (what I call the S^1 and S^2, or if you prefer the R^2 and R^3, models), and the predictions of Joy's model and of QM (the cosine curve).

Sample size is up to 100 million and I just compare the simulated correlations for one pair of measurement directions: the angle between the two directions is 105 degrees. (Of course, 75 would have been just as good, since the simulation models have all the expected symmetries "built in").

The S^1 interpretation is due to Minkwe (Michel Fodje). The S^2 interpretation is my own - but it brings Minkwe's formula's closer to Joy's (and the results are indeed better, though not perfect).

http://rpubs.com/gill1109/13366

Let's forget about interpretations. Let's discuss the maths. Why are we not (exactly) getting the cosine?

[Joy Christian]

I just published another R simulation comparing two "interpretations" of Joy's model (what I call the S^1 and S^2, or if you prefer the R^2 and R^3, models), and the predictions of QM (the cosine curve). Sample size is up to 100 million and I just compare the simulated correlations for one pair of measurement directions: the angle between the two directions is 105 degrees. (Of course, 75 would have been just as good), since the simulation models have all the expected symmetries "built in".

The S^1 interpretation is due to Minkwe (Michel Fodje). The S^2 interpretation is my own - but it brings Minkwe's formula's closer to Joy's (and the results are indeed better, though not perfect).

http://rpubs.com/gill1109/13366

Let's forget about interpretations. Let's discuss the maths. Why are we not (exactly) getting the cosine?

Now we are talking. Leave the interpretation to me. I have been trained to do that by the world's foremost expert on the subject---namely, Professor Abner Shimony.

Now, speaking of maths, let us leave my analytical model aside for the moment. There too, I believe, I have an upper hand. But it is indeed curious that you are seeing slight discrepancies in the correlations. I have a number of ideas why this could be happening. To begin with, the correct statistical analysis should be applied to the graded basis using Clifford algebra, as I have done in my one-page paper. You are using the standard flatland statistics based on non-graded basis, so naturally you are not going to see an exact match. So from my perspective all simulations are flawed from the start. The correct simulation should be intrinsically based on Clifford algebra elements, applying statistical analysis to both the scalars as well as the bivectors in the parameterization of the 3-sphere.

But there could be even simpler reasons for the discrepancies. One possible reason is that my analytical model includes the states in the analysis when |cos(a - e)| = 1/2 sin^2(t), whereas Michel's simulation excludes the states from the analysis when |cos(a - e)| = 1/2 sin^2(t). So there is a minute statistical discrepancy here between my analytical model and Michel's simulation. I am not qualified to judge how this discrepancy may be playing out in various simulations.

Then there is a possibility that the phase shift between Alice's particles and Bob's particles is not exactly but . I believe such a small shift in the phase difference would make a significant (if not dramatic) difference in the correlation.

So there. These are my initial suspicions.

[Joy Christian]

Let's forget about interpretations. Let's discuss the maths. Why are we not (exactly) getting the cosine?

Michel's simulation assumes a phase shift of between the observations of A and B.

Here is a good approximation of the number :



How is defined in your program?

[gill1109]

In my programs I simulated the "perfect correlation" case, not the perfect anti-correlation case. So there is no phase shift between the observations of A and B, neither in the 2-D nor in the 3-D version. That's why my plots show a cosine, and not a negative cosine.

But pi does turn up quite often in my R scripts. I guess my R's pi it is about as accurate as Minkwe's Python's pi. R on my computer represents real numbers in 64 bits numbers: IEC 60559 floating-point (double precision) arithmetic. The smallest real number epsilon such that 1 + epsilon is larger than 1 is 2.220446e-16.

[Joy Christian]

In my programs I simulated the "perfect correlation" case, not the perfect anti-correlation case. So there is no phase shift between the observations of A and B, neither in the 2-D nor in the 3-D version. That's why my plots show a cosine, and not a negative cosine.

But pi does turn up quite often in my R scripts. I guess my R's pi it is about as accurate as Minkwe's Python's pi. R on my computer represents real numbers in 64 bits numbers: IEC 60559 floating-point (double precision) arithmetic. The smallest real number epsilon such that 1 + epsilon is larger than 1 is 2.220446e-16.

What about:

"But there could be even simpler reasons for the discrepancies. One possible reason is that my analytical model includes the states in the analysis when |cos(a - e)| = 1/2 sin^2(t), whereas Michel's simulation excludes the states from the analysis when |cos(a - e)| = 1/2 sin^2(t). So there is a minute statistical discrepancy here between my analytical model and Michel's simulation. I am not qualified to judge how this discrepancy may be playing out in various simulations."

[gill1109]

In my programs I simulated the "perfect correlation" case, not the perfect anti-correlation case. So there is no phase shift between the observations of A and B, neither in the 2-D nor in the 3-D version. That's why my plots show a cosine, and not a negative cosine.

But pi does turn up quite often in my R scripts. I guess my R's pi it is about as accurate as Minkwe's Python's pi. R on my computer represents real numbers in 64 bits numbers: IEC 60559 floating-point (double precision) arithmetic. The smallest real number epsilon such that 1 + epsilon is larger than 1 is 2.220446e-16.

What about:

"But there could be even simpler reasons for the discrepancies. One possible reason is that my analytical model includes the states in the analysis when |cos(a - e)| = 1/2 sin^2(t), whereas Michel's simulation excludes the states from the analysis when |cos(a - e)| = 1/2 sin^2(t). So there is a minute statistical discrepancy here between my analytical model and Michel's simulation. I am not qualified to judge how this discrepancy may be playing out in various simulations."

We could change "less than or equal" to "less than" in the code and see if it made any difference. I believe we wouldn't see a difference. The chance that two independent random numbers between 0 and 1 are equal (in R or Python on a present day machine - I have a recent MacBook Pro, max memory size and processor speed) is about 10 to the minus 16.

[Joy Christian]

We could change "less than or equal" to "less than" in the code and see if it made any difference. I believe we wouldn't see a difference. The chance that two independent random numbers between 0 and 1 are equal (in R or Python on a present day machine - I have a recent MacBook Pro, max memory size and processor speed) is about 10 to the minus 16.

OK, then, let us try a more sophisticated test. Here is a Mathematica version of Michel's code: http://libertesphilosophica.info/Minkwe_Sim_J_Reed.pdf.

As you may know, Mathematica is an "interpreted" language. It would be good to find out whether the discrepancies you are seeing (at a higher level of accuracy) persists in a language like Mathematica. We would then have to take the discripancies much more seriously.

[gill1109]

Python and R are also interpreted languages. Moreover, both their definitions and their implementations are open and free. I don't like to use Mathematica. Steve Wolfram is a gangster.

The discrepancy between my S^1 and S^2 models is easy to understand. Let e be a uniform random point on the unit sphere S^2, to be thought of as a subset of R^3. Let a be a point on the intersection between the equatorial plane and S^2, i.e., a point on S^1, the unit circle in the equatorial plane of R^3. Let f be the orthogonal projection of e onto the equatorial plane (so *into* the unit disk). The inner product of f with a is different from that of e with a. So the acceptance criterium is different. We compare the absolute value of the inner product with 1/2 sin^2 theta_0 where theta_0 is drawn uniformly at random between 0 and pi/2. This is the only difference between my two simulations.

[Joy Christian]

Python and R are also interpreted languages. Moreover, both their definitions and their implementations are open and free. I don't like to use Mathematica. Steve Wolfram is a gangster.

The discrepancy between my S^1 and S^2 models is easy to understand. Let e be a uniform random point on the unit sphere S^2, to be thought of as a subset of R^3. Let a be a point on the intersection between the equatorial plane and S^2, i.e., a point on S^1, the unit circle in the equatorial plane of R^3. Let f be the orthogonal projection of e onto the equatorial plane (so *into* the unit disk). The inner product of f with a is different from that of e with a. So the acceptance criterium is different. We compare the absolute value of the inner product with 1/2 sin^2 theta_0 where theta_0 is drawn uniformly at random between 0 and pi/2. This is the only difference between my two simulations.

It is irrelevant for science who or what Stephen Wolfram is.

I am also not too concerned about the S^1 simulation and its relation to the S^2 simulation. The real issue for me is the discrepancy you are seeing between your S^2 simulation in R and the prediction of the cosine function by my analytical model (as well as by the recipe called quantum mechanics).

So the real question for me is: Is the minute discrepancy between the S^2 simulation in R and the cosine correlation a real discrepancy? I very much doubt it.

I wonder what difference does it make if \theta_o is taken in the full range of [0, 2\pi] instead in the range of [0, \pi/2]. I do not believe it will make any difference (because of the codomain [0, 1] of the sine function), but just in case there is something in R that is seeing the difference.