The 64 thousand Euro challenge

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: The 64 thousand Euro challenge

Postby gill1109 » Sat Feb 20, 2021 12:53 am

Austin Fearnley wrote:I spent my time trying to get a simulation which worked to my own satisfaction. It needed some randomness in the measurement outcomes and retrocausality for the positron. It was never my intention to understand the full frills of Bell's Theorem as it is a no-go theorem. My focus was always on the loophole that is used by physics. I believe that I have done that and can now halt.

Could you post a reference to your simulation program (or email it privately)?
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Sat Feb 20, 2021 6:05 am

Richard wrote:
Austin Fearnley wrote:
I spent my time trying to get a simulation which worked to my own satisfaction. It needed some randomness in the measurement outcomes and retrocausality for the positron. It was never my intention to understand the full frills of Bell's Theorem as it is a no-go theorem. My focus was always on the loophole that is used by physics. I believe that I have done that and can now halt.

Could you post a reference to your simulation program (or email it privately)?


Hi Richard,
Treating these posts as if they were code, just when I think the next line is "END", you insert a line "NEXT i" or in other words "go back to the top of the thread".

Austin previously wrote:
I have already in the appendix of my June 2020 vixra paper https://vixra.org/abs/2101.0179 given my computer code for using my model to give Malus Law results in a particle-at-a-time simulation. In the main body of that paper I describe my classical model of the electron and the photon and show how that model can be used to give - cos theta for the Bell correlation as long as the antiparticles are travelling backwards in time.



Richard previously wrote:
Someone who is happy with backwards-in-time causation is easily pleased! I don't think that it *explains* anything; in fact, I would say that does not *explain* anything, because it obviously explains everything.
Anything that happens - then I can say, it was going to happen anyway! No need for physicists.


So what is the point of me writing a retrocausal simulation with all the Bell frills of randomisation etc which you will definitely not accept as valid because it is retrocausal?

Austin wrote previously :
I am trying to close down my physics work as I am feeling too old now and I have the feeling that what you are suggesting will involve me in a lot of computer programming work. I am prepared to do the work if it is meaningful. I don't know if it is meaningful unless I fathom why you are asking. I do not want to input a lot of effort if at the end you again simply say that you do not like the method.


Richard previously wrote:
I believe you haven't yet digested Bell's theorem.


Austin previously wrote:
It was never my intention to understand the full frills of Bell's Theorem as it is a no-go theorem. My focus was always on the loophole that is used by physics. I believe that I have done that and can now halt.
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Thu Feb 25, 2021 2:22 am

Austin Fearnley wrote:So what is the point of me writing a retrocausal simulation with all the Bell frills of randomisation etc which you will definitely not accept as valid because it is retrocausal?

The point is that I will be able to use it to show how unphysical retrocausality is.

Nature does not have to “use a loophole”. Nature does not have to operate according to the constraints of “local realism”. It operates according to quantum rules.
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Thu Feb 25, 2021 5:45 am

Postby gill1109 » Thu Feb 25, 2021 2:22 am
Austin Fearnley wrote:
So what is the point of me writing a retrocausal simulation with all the Bell frills of randomisation etc which you will definitely not accept as valid because it is retrocausal?

The point is that I will be able to use it to show how unphysical retrocausality is.

Nature does not have to “use a loophole”. Nature does not have to operate according to the constraints of “local realism”. It operates according to quantum rules.


Why do I need to write an unnecessary particle-at-a-time computer program. {Aside: see 'mutant app' 2020 at https://www.bbc.co.uk/news/education-53923279. So in 2021 English exams will instead be moderated without an app. Aka 'almost unmoderated'!}

As you already hold reverse time for positrons as unreal and/or unphysical and/or not local then what is in it for me? What possible extra good will it do for either of us to have such a computer program?

I possibly packed too much into my June 2020 paper at https://vixra.org/abs/2006.0160

There were three strands.
1. modelling classical electron and photon structures.
2. using those structures on a forwards-in-time Malus's Law experiment particle-at-a-time to obtain the Malus's Law formula for intensity of a polarised beam passing through a filter/or a S-G measurement.
3. mapping the forwards-in-time Malus experiment onto a retrocausal Bell experiment and converting the Malus intensities into the Bell correlation.

I simulated the Malus experiment as particle-at-a-time to understand how the particles structures worked. It was not necessary for me to perform the Bell analysis particle-at-a-time as it was so easy to see that it worked for beams of particles. It really is very simple to see that the Bell correlation is obtainable for beams and for particles.

But as you do not see retrocausality as real etc (insert any number of woolly philosphical words here) there is no point in my writing a new program.

Further, is is very clear that the retrocausal method can cope with any randomising of detector settings even when randomising for individual pairs during times of flight. So again I do not see the point of random settings. But if they are done, just use coin flips rather than particle measurements. At least for a simulation as the particles are not 'real' in a simulation ... there is no point in that level of complexity or pretended reality in a simulation.

Finally, can you remind me why particle-at-a-time simulations are so important, please. I now have it firmly in my head that my simulation, using Malus beam intensities, works for the retrocausal Bell experiment using beams too. For Malus one naturally uses beams to prove the statistical Malus's Law. It should be equally valid to use retrocausal beams to prove the statistical Bell correlation result.

BTW I did no completely rule out writing a program ... though feel it is a complete waste of my time as I would spend lots of time whereas you have already dismissed it.
Austin previously wrote:
I believe that my description is complete. It is not a computer program but describes a program that could be written, and a Bell's experiment program with randomised S-G settings before each measurement. If that is not believable enough then I will write one in June 2021 after I have had another break.


Maybe I will defer writing the program to December 2021.
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Fri Feb 26, 2021 6:28 am

Austin Fearnley wrote:Can you remind me why particle-at-a-time simulations are so important, please. I now have it firmly in my head that my simulation, using Malus beam intensities, works for the retrocausal Bell experiment using beams too. For Malus one naturally uses beams to prove the statistical Malus's Law. It should be equally valid to use retrocausal beams to prove the statistical Bell correlation result.

Austin, you really need to read "Bertlmann's socks". There are no particles in a loophole-free Bell experiment. There are time slots - at two well-separated locations. Within those time slots, binary inputs "a", "b" are inserted into an apparatus and binary outputs "x", "y" are recorded. The distance between the labs and the length of the time-slots is such that there is no way Alice's input "a" could be available at Bob's lab before his output "y" has been recorded. According to quantum theory it is possible to observe certain correlations or more precisely conditional probability distributions p(x, y|a, b). Quantum theory tells us what those probability distributions might be, and quantum engineering succeeds in realising them. In particular, quantum theory - and quantum engineering - allow the realisation of certain probability distributions which cannot be realised by a non-conspiratorial local realistic theory. Some people find this hard to believe. If you find it easy to believe, then good for you.

The title of this thread is "The 64 thousand Euro challenge". The challenge was to imitate quantum predictions (and experimental findings) in a computer simulation experiment on two separate computers. Alice's and Bob's computers are in separate rooms and not connected to one another in any way. Each receives a stream of settings and outputs a stream of outcomes. The challenge is to come up with computer simulation programs which reliably, verifiably, reproducibly, reproduce the quantum correlations. Some people think it can be done and spend a lot of time writing programs and writing papers about models. I know it can't be done.

I guess you agree that it can't be done but couldn't care less anyway.

I'm not saying that there is any benefit for you, to write programs to illustrate how retrocausality would fit into this picture. Apparently, you are happy with the idea that clicks of photodetectors have influence backwards in time on the results of coin tosses There is only a benefit for those who do understand Bell's theorem and are interested in whether your picture of the physics is non-local, non-realistic, or conspiratorial.
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Fri Feb 26, 2021 10:34 am

Richard wrote:
Austin, you really need to read "Bertlmann's socks". There are no particles in a loophole-free Bell experiment. There are time slots - at two well-separated locations. Within those time slots, binary inputs "a", "b" are inserted into an apparatus and binary outputs "x", "y" are recorded. The distance between the labs and the length of the time-slots is such that there is no way Alice's input "a" could be available at Bob's lab before his output "y" has been recorded.

Austin previously wrote:
I completely accept that I cannot get -0.707 for a forwards in time Bell correlation for the detector settings of 0 deg for Alice and 45 deg for Bob. I have never bothered with simulating a CHSH design set up as I believe it is a waste of time in a simulation. I have normally used a single setting for each detector of Alice and Bob.

Similarly, I am not interested in Bell-like experiments that do not involve particles. Nowadays, I am only interested in the physics, not the maths/stats for its own sake.

Also, your wording ["there is no way"]in the final sentence is probably only interpretable in a forwards-in-time experiment because it is not true for positrons which are moving backwards in time. As I previously wrote:
Austin previously wrote:
The backwards-in-time causality means that (using the positron's time frame) Alice's positron has unknown polarisation before she measures it, as it is coming in from somewhere in the universe. After Alice's measurement the positron has a polarisation vector along or against Alice's random setting. When the positron reaches the Oven it still has the polarisation angle from Alice's setting. After the Oven, the positron is replaced by an entangled electron still with the polarisation angle from Alice's setting, but travelling forwards-in-time to Bob's device. So retrocausality is essential, as is Malus's Law. Malus's Law on its own is not enough.

Richard wrote:
Apparently, you are happy with the idea that clicks of photodetectors have influence backwards in time on the results of coin tosses There is only a benefit for those who do understand Bell's theorem and are interested in whether your picture of the physics is non-local, non-realistic, or conspiratorial.

I find it hard to know why you mention influence on coin tosses. Are you referring to detector setting values or Alice/Bob's measurement outcomes? As far as I can see there is no influence on detector settings but there is an influence on on measurement outcomes. Not a complete determination as there is a random element to the measurement outcomes. The influence is in accordance with the polarisation angles while the random element is because of hidden variables which are relatively less important. The -cos theta result is achieved despite the hidden variables rather than because of them. But the polarisation angles of the particles are what determine the Bell correlation. Only with retrocausality, of course.
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Fri Feb 26, 2021 12:07 pm

Austin Fearnley wrote:I find it hard to know why you mention influence on coin tosses. Are you referring to detector setting values or Alice/Bob's measurement outcomes? As far as I can see there is no influence on detector settings but there is an influence on measurement outcomes. Not a complete determination as there is a random element to the measurement outcomes. The influence is in accordance with the polarisation angles while the random element is because of hidden variables which are relatively less important. The -cos theta result is achieved despite the hidden variables rather than because of them. But the polarisation angles of the particles are what determine the Bell correlation. Only with retrocausality, of course.

Hard to know why I mention influence on coin tosses? It's easy to know, but you are wilfully blind to Bell's theorem. You don't see because you don't want to.

If your model reproduces the singlet correlations and if it is local and realist then it is conspiratorial. That means that the settings are not chosen independently of the physics which goes on in the process leading to the detectors clicking in one channel or the other. Alice's detector already knows the outcome of Bob's coin toss since both were fixed at the time of the big bang. Sure.
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Fri Feb 26, 2021 1:08 pm

Hi Richard
Richard wrote:
Hard to know why I mention influence on coin tosses? It's easy to know, but you are wilfully blind to Bell's theorem. You don't see because you don't want to.

If your model reproduces the singlet correlations and if it is local and realist then it is conspiratorial. That means that the settings are not chosen independently of the physics which goes on in the process leading to the detectors clicking in one channel or the other. Alice's detector already knows the outcome of Bob's coin toss since both were fixed at the time of the big bang. Sure.


I believe that I am open to learning and I certainly do not want to hold on to false ideas. However, I genuinely cannot see a retrocausal mechanism for what you are suggesting. It seems impossible to me. I have reproduced the Bell correlation using backwards-in-time positrons. No need for the Big Bang. No need to affect the detector settings.

Say we conduct a one-pair-at-a-time Bell experiment with a long time in between pairs. Say Alice measures a positron. How can that one positron decide the measurement setting of either detector? How does it do it? IMO I have thrown out spooky entanglement. Is there more spookiness at work here that needs to be cleared out?

I have not taken on board the full Bell's Theorem paraphernalia as I am only interested in the minimum requirement for the physics. That may be because my background in physics is many Susskind courses entitled: 'the theoretical minimum' and I took that to heart. He said his courses were for people who do not have much time [left :) .... his joke ] . Also the Bell saga has been running for 50 years or more and so I wanted to sidestep the full details as they did not seem to be clearing up the issue.

Similarly, I have not come to my retrocausal view by reading the philosphical papers on retrocausality. I cannot abide wish-washy terms and I bypass them as inessentials if possible. (If there is another retrocausal paper where the Bell correlation is actually calculated, I would be pleased to have a reference to it.) Maybe that is why I was unclear about positrons in a Bell experiment spookily setting the detector polarisation vectors. I came to retrocausality through drawing Feynman diagrams for particle interactions. I also drew many Feynman diagrams where the preons were interacting, and that is a lot of preons in a single diagram of a particle interaction. With preons going forwards in time and antipreons going backwards in time.

Best wishes, and assuring you that I am reporting what I believe to be true.

PS Alice's detector detects an electron arriving with a polarisation vector (that is, the electron's polarisation vector) aligned with Bob's detector angle because the measurement by Bob's detector changed the positron's polarisation vector (that is, the positron's polarisatin vector) to align with or against Bob's detector's polarisation angle. That is the correct way around. The detectors change the particles' polarisation vectors. That is not spooky. The particles do not change the detector setting angles. Now if that were to happen it would be very spooky.
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Fri Feb 26, 2021 9:11 pm

Bell’s theorem says: QM is incompatible with locality+realism+no_conspiracy. OK, if you don’t manipulate the detector setting, then you must violate realism or locality (or both). I would like to understand which.

The problem must be locality. By backwards-in-time action you change things far away, before it should have been possible. Thank you! Mystery resolved.

Retrocausality induces non-locality.
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Sat Feb 27, 2021 2:53 am

Richard wrote:
Bell’s theorem says: QM is incompatible with locality+realism+no_conspiracy. OK, if you don’t manipulate the detector setting, then you must violate realism or locality (or both). I would like to understand which.

The problem must be locality. By backwards-in-time action you change things far away, before it should have been possible. Thank you! Mystery resolved.

Retrocausality induces non-locality.

OK. I believe that my retro-positron method does not affect any detector settings. So that leaves the wishy-washy terms locality and realism.

You somewhere described a local process as contained within a game of life program where all movements of a micro or unit block/entity are not to exceed one unit square in one unit time interval. Normally time is only forwards in this game. I can imagine that you can have aggregations of unit blocks to make macro bodies. My retro-positron method would need to amend this game so that antimatter micro blocks could not move more than one square in one unit of time but where time is moving backwards [always backwards, not back and forth]. Also micro blocks gravitationally repel one another so macro aggregates of antimatter blocks are disallowed. So this game would have locally-moving matter microparticles and locally-moving micro antimatter particles. And also locally-moving macro matter but NO locally-moving macro antimatter. I guess one could say that this is a new rule for locality which keeps the main rules of locality while incorporating backwards in time motions, too. I look at this as locality.

Note that if Alice measures a positron then Bob measures the partner electron, and does not measure the same positron. In my model, the electron is not the same partner positron switching time directions. So no particle is in two different places at the same time. The whole 'entanglement' idea of a measurement on one particle instantly determining the exact measurement outcome on the entangled partner is also thrown out by my method.

So I maintain that my method may appear to be local. In physics, though, on the smallest scale of distance, any two nearby points are separated by quantum foam which interferes with accessibility if not locality. Also, as the distance scale gets smaller the energy associated with that distance gets larger, leading to requirements of renormalisation to divide out the infinities of energy. So I do not claim full, i.e. 'quantum foam', locality.

That leaves the most wishy-washy term 'realism'. Well, having backwards-in-time motion of positrons does appear to defy the current idea of realism. So I guess that is what I would pick as the area of incompatibility. And I have previously accepted Bell's Theorem as true if limited to forwards-in-time particle motions.
Austin previously wrote:
I completely accept that I cannot get -0.707 for a forwards in time Bell correlation for the detector settings of 0 deg for Alice and 45 deg for Bob. I have never bothered with simulating a CHSH design set up as I believe it is a waste of time in a simulation. I have normally used a single setting for each detector of Alice and Bob.

I believe there is no need for me to write that program after all ... Calculations of the Bell correlation for my retro-positron method are, I believe, adequately shown in my June 2020 paper: https://vixra.org/abs/2006.0160
'Adequate' because my particle structures downplay the importance of hidden variables.
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Sat Feb 27, 2021 10:08 pm

The words ‘local’ and ‘realism’ are not wishy-washy. You want to blur their meaning! It’s not necessary. Mathematically, I like your ideas.
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Sun Feb 28, 2021 3:46 am

Richard wrote:
The words ‘local’ and ‘realism’ are not wishy-washy. You want to blur their meaning! It’s not necessary. Mathematically, I like your ideas.


Hi Richard,

OK. I will tone down. But locality+realism+no_conspiracy is used as an arbitration method when it is too subjective for my liking.

There are a few more points.
1. It is simple to fill in a 2x2 measurement results table for a Bell simulation using a retro-positron method with beams. It starts with one beam of positrons moving backwards in time. That splits into two beams, one goes to Alice and one to Bob. Those researchers make measurements (either +1 or -1) on individual positrons. That makes four polarised beams to keep track of. Those beams are still polarised on getting back to the source and are replaced by four beams of forwards-in-time beams of electrons heading to the other/second researcher. The four electron beams are polarised identically to the positron polarisations. The second researcher make measurements which splits the beams again, from four beams to eight. An individual beam has measurements which all go into one cell of the 2x2 results table. Measurements of positrons are based on unpolarised incoming particles and so the + and - numbers are evenly split. That make four beams of electrons coming from the source with 25% of particles in each beam, on average. These four beams are already polarised on arrival and so their numbers passing and failing the second filter are not 50/50. The Malus's Law formula (adapted for electrons by halving the angle) gives either cos^2 (theta/2) or 1 - cos^2 (theta/2) as the intensity of the beam passing/failing the second detector filter. So the 2x2 cells are filled by these numbers. If the 2x2 cell is then normalised to have total 1, the 2x2 cell will have correlation of - cos theta. Where theta is the difference angles of the two detector settings.

This is so clear that I do not see the worth of making another particle-at-a-time program. I already made a particle-at-a-time simulation for the Malus experiment. My Bell experiment simulation is merely Malus with a time twist.

2. A point about simulations versus real experments. I can see people might be concerned about mystical connections from the Big Bang in a real experiment, but surely this does not apply in a simulation. There are no actual positrons in the simulation travelling backwards in time (subject to correction by computer experts ...). Everything in the simulation is travelling forwards in time. So cannot we rule out suspicious real-world effects during a simulation? And if it can work inside the limited and constrained 'box' of a simulation, then why bother to look for mysticism in the real experiment. (That IMO applies mainly to entanglement, but also to locality.)

3. I see Bell's Theorem as very important in making one look to see how nature works differently from what one expected classically. But I think I now know the answer physically as well as mathematically.

4. In mid 2020 I revised my structure for the electron and photon to make it conform to Malus's Law results. I can now apply my new structure to ask myself 'what is a hidden variable'. As an analogy a hidden variable is like a phase angle. It is like a vector which is continuously changing direction though conforming to a rule. A precession rule or something similar. So the actual direction of the hidden variable depends on time. A measurement at any instant depends on the direction of the hidden variable at that instant. I really have ignored theoretical Bell formulae in my work, but I guess that λ should be replaced by λ(t) to take into account the dynamic nature of the hidden variable. Wiki says that "The idea persisted, however, that the electron in fact has a definite position and spin". I agree that my electron does NOT have a definite static hidden variable direction. 'Spin' is a tricky and distracting word in this context because all electrons have either spin +0.5 or spin -0.5, and these values do not change during time of flight. So 'spin' +0.5 or -0.5 does not help a lot in considering hidden variables. But 'phase' does change during flight and, according to Feynman, variations in phase, at measurements, can lead to interference patterns. If time is taken out of the formulae then what is left is an average value of the hidden variable, i.e. the polarisation angle of the particle. This does not give enough information to determine an individual measurement but does give enough statistical information to determine the average measurement for a group of particles with that polarisation vector. (So the polarisation vector is good enough for Malus's Law and good enough for a retro-positron version of Bell.)

5. You referred me to Gerard t'Hooft.

Slide 5 of his notes at:
http://www.emfcsc.infn.it/issp2017/docs ... tHooft.pdf

shows a very interesting analogy with Black Holes. It is mostly beyond my understanding of course. IMO his region III is analogous to our macro space time and his 'local time' is the time of the thermodynamic arrow of time. His region I could be analogous to micro particle time. His region II could correspond to the reversed time direction of antiparticles; he names these time arrows 'distant time'. Just an interesting analogy and possible link between BHs and spacetimes of particles.

Note that if a positron is an electron travelling backwards in time, then the reverse time must be a property of a particle's spacetime and not a property of the particle itself. That is what led me to the importance of particle spacetime and S^3.

Note also that slide #35 has a timelike Moibus strip which is what I am thinking of for my version of S^3. If space is closed with a point at infinity, then time may similarly be closed at infinity.

I had (I have stopped now!) been wondering how to preserve times' arrows if wending one's way round a Moibus timelike strip. Gerard seems (???) to be saying something similar in slide #36
"Demanding that the external observer chooses the point where the Hamilton
density switches sign as being on the horizon, gives us a good practical
definition for the entire Hamiltonian."

Best wishes
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Mon Mar 01, 2021 3:32 am

I use Bell’s theorem not to find out how Nature works, but how to find out where are the mistakes in works of those who argue that Bell’s theorem is false. They typically come up either with a mathematical model or a computer simulation program, or both, which they claim provides a counter-example to the theorem.

I don’t do this out of some religious crusade. On the contrary, I want to understand Bell’s work better. This has led to strengthened forms of Bell’s theorem, better understanding of loopholes and how to avoid them, etc.
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Re: The 64 thousand Euro challenge

Postby Austin Fearnley » Mon Mar 01, 2021 6:53 am

Hi Richard

I think we agree and can stop here? I have not disagreed with Bell's Theorem for many years, and have worked with my slimmed down (practical simulation) version of it where it is impossible to break the Bell Inequalities in a (forwards in time) Bell simulation. My early problem was not believing that real experiments were achieving - cos theta. Must be a loophole somewhere etc. Only when I eventually accepted that the real experiments might actually be achieving -cos theta could I begin to look for other options. Retro-positrons may seem weird but IMO they have made sense of many other features features and made my model conform more to QM: for example there is only one |up> electron; the hidden variable (phase) vector cannot be used to identify a particular electron; and more.

I took up particle physics on retirement to get a break from maths/stats. I did not realise then how important stats is in QM. The phase (=hidden variable) is sort of raw data. The polarisation vector is a summary statistic (the average) of the phase. Anyway, I now want a break from particle physics, and I am not tempted to look at the theory of Bell's Theorem myself. But I wish you well in your task.
Austin Fearnley
 

Re: The 64 thousand Euro challenge

Postby gill1109 » Wed Mar 03, 2021 2:30 am

Austin Fearnley wrote:Hi Richard

I think we agree and can stop here? I have not disagreed with Bell's Theorem for many years, and have worked with my slimmed down (practical simulation) version of it where it is impossible to break the Bell Inequalities in a (forwards in time) Bell simulation. My early problem was not believing that real experiments were achieving - cos theta. Must be a loophole somewhere etc. Only when I eventually accepted that the real experiments might actually be achieving -cos theta could I begin to look for other options. Retro-positrons may seem weird but IMO they have made sense of many other features features and made my model conform more to QM: for example there is only one |up> electron; the hidden variable (phase) vector cannot be used to identify a particular electron; and more.

I took up particle physics on retirement to get a break from maths/stats. I did not realise then how important stats is in QM. The phase (=hidden variable) is sort of raw data. The polarisation vector is a summary statistic (the average) of the phase. Anyway, I now want a break from particle physics, and I am not tempted to look at the theory of Bell's Theorem myself. But I wish you well in your task.

I agree!
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