Sakurai, pages 228-230

[Alexandre de Castro]

Dear all,

I was reading Sakurai’s Modern Quantum Mechanics (specifically pages 228- 230), and wanted to share this argument with you to get your thoughts on it.

The starting point is the simplified version of Bell's inequality (based on Wigner’s model) that Sakurai presents for the spin singlet state. The classical relation imposed by local realism is expressed as:

$\sin^2(45^\circ) \le 2\sin^2(22.5^\circ)$

According to the formalism of quantum mechanics (Born's rule), we know the numerical result yields $0.5 > 0.2928$. This means the inequality is violated, proving non-locality. The direct connection between this scenario and the CHSH inequality is purely geometric: these angles ($22.5^\circ$ and $45^\circ$) are precisely the same ones that produce the maximum violation of the CHSH inequality ($S = 2\sqrt{2}$, exceeding the classical limit of 2). Currently, the literature treats both inequalities as equivalent in testing non-locality, merely changing the statistical approach: Wigner-Bell deals with counting probabilities, while CHSH deals with correlation expectation values.

Based on this, I would like to propose the following thought experiment:

What if we could construct an LHV model with uniformly distributed hidden variables that yields the opposite relation to Sakurai's ($\sin^2 45^\circ \ge 2\sin^2 22.5^\circ$)—meaning it respects the classical limit of Wigner-Bell—but still manages to violate the CHSH inequality?

From the standpoint of current mathematical formalism, this hypothesis faces a rigid barrier: Fine's Theorem. It establishes that if the underlying probabilities are classical and well-behaved, the resulting average (CHSH) is mathematically compelled to respect the limit of 2. However, if we accept the hypothetical premise that a uniform LHV model could bypass this algebraic constraint, the conceptual implications would be profound:

The CHSH inequality would be exposed as a flawed indicator. If a strictly local and realistic model  could exceed the limit of 2, a CHSH violation would no longer be definitive proof of quantum non-locality. Instead, it would be interpreted merely as a geometric artifact of statistical distributions—a false positive.

In short, if such an alternative mathematical model is viable, it would challenge the validity of the CHSH inequality as a boundary between the classical and quantum worlds.

I would highly appreciate your feedback.

Alexandre

[Mark Hadley]

Dear Alexandre,

You cannot violate CSHS with a local hidden variable model.

The main advantage of CSHS is that is very simple to derive. Excellent for teaching and ideal experiments. You are saying it places a stricter bound than other results, in which case it is a better inequality.

Fines theorem is not relevant, it is not an extra assumption. It's an extra result that also follows from the BI or CSHS assumptions.

There are only two(three) alternatives.  States can be described independently of the chosen measurements. In which case the results/ hidden variables/ propositions) etc are Boolean and we have classical physics. All bro abilities can be calculated from LHV by integrating over the possibilities.

Or states can only be described fully  if the measurement context is known. We then have a non Boolean structure ( the distributive law fails ) technically an orthomodular lattice of propositions. That is quantum theory. States are represented as spaces and subspaces if a Hilbert space over the field if complex numbers.

Three ( we have nonsense) 

Cheers

Mark

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[Richard Gill]

Dear Alexandre

Sakurai-Wigner is a version of Bell’s (1964) inequality which assumes that when the measurement directions are the same, the outcomes are equal and opposite.

Because of that, the case of measuring two particles in the singlet state can be converted to a story of measuring one particle. And this brings us back to an earlier result of Wigner.  

One is using the assumption that the measurement of one particle tells us what the outcome of the same measurement on the same particle is.

So you are talking about Bell (1964) not about the later Bell-CHSH inequality,

Bell’s three correlation inequality is in fact a special case of the CHSH four correlation inequality when one of the four correlations is assumed to equal +/-1

Fine’s theorem is a converse of the the Bell-CHSH result: local realism implies CHSH

Fine’s result is: all 8 one-sided CHSH inequalities hold, and no-signalling holds, implies that a LHV model fits exactly to the four correlations and marginal expectations.

So if a LHV model could be found violating CHSH then Bell’s theorem is wrong. 

But Bell’s theorem is true so a LHV model cannot be found violating CHSH.

I suggest you study the modern literature on Bell-CHSH-Fine.

Richard

PS Mark: I completely disagree with your characterisation of Fine’s theorem. Bell-CHSH was already there. LHV => CHSH. Fine proved  a converse. All CHSH inequalities hold AND no signalling holds => there exists a LHV model which fits to the data four 2x2 tables of a bivatiate probability distribution of binary variables.

[Alexandre de Castro]

Mark, Richard, 

QM predictions for the singlet state flow so naturally from the Wigner-Bell inequality. Yet, at the exact same time, this very same state violates the CHSH inequality.

[Bryan Sanctuary]

 Actually, Alexandre, you can violate the CHSH if you treat spin as really spinning, by that I mean it is a rotor.  Then, rather than Boolean pairs from the beginning, (as Bell does), you have a quaternion.  That carries a common phase between A and B, so if you keep the quaternion up to the detector (that then gives Boolean clicks), you can indeed violate BI.  

The violation is a long-range phase correlation, (like a laser, or superfluid) that is not seen in the individual  Boolean outcomes, but only after statistical analysis of the data, post-experiment.

Also Fine's theorem is not only relevant, it is pivotal.  In 1982, when Fine published, it changed the focus of Bell's work. 

Finally, no Hilbert spaces are needed to fully describe the results.  The reason is very simple:  Dirac decided on the matter-antimatter solution which is Clifford algebra Cl(1,3).  There is a perfectly acceptable second solution that Dirac did not know, and I published in 2024. That is, the algebra is Cl(2,2).  I am simply using the second, and that is why I disagree with almost everything Mark says.

Best wishes

Bryan

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[Alexandre de Castro]

Ok Brain,
it would be like considering a coin flip. The spinning coin represents a quantum state (continuum), and when it lands, a classical state (Boolean).

[Bryan Sanctuary]

Hi Alexandre

Lots of Laughter.  My name is not BRAIN, but Bryan.  

Your penny analogy is really very relevant:  spin a coin, and yes, it goes into a boolean head or tail.  BUT something very interesting happens before it reaches the final equilibrium state: it goes into a resonance between the angular momentum and gravity, so it presecces on the table top in that well known resonance state before collapsing.  That happens in classical mechanics a lot and is a major area in mechanical engineering (catastrophic failure due to internal resonances)  

https://youtu.be/XggxeuFDaDU?si=v2ZK2lS8FTq7Bim-

Imagine if there was no friction and the system is conservative.  It drops down from spinning, and if conditions are right, it gets locked in that resonance state.  It can transition to the ground state, but that comes from internal energy.

The bivector spin is the same: it has a natural resonance from the balance of external torque and the internal bivector (just like an engine).

So the penny state gives a view of the quantum state of a spinning bivector.

Bryan

[Richard Gill]

Dear Alexandre

QM predicts that the statistics of certain ideal measurements on a system prepared in the singlet state will violate Wigner-Bell-Sakurai, and CHSH. There is no contradiction. 

QM violates local realism.

Experiment + CHSH gives strong evidence (but not definitive proof) that physical reality does not obey local realism. QM does however fit very well. Obviously, physics is never finished. So far, QM has stood up to every test.

Richard

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On 13 Jul 2026, at 22:32, Alexandre de Castro <alx...@gmail.com> wrote:



[Alexandre de Castro]

Sorry, Bryan, that was a typo. 

If your bivector is similar to a spinning coin, I believe you're on the right track. I also did some calculations to show that a "spinning coin" violates the CHSH inequality. The problem is that extensive and relatively complex calculations are always open to refutation. 

[Alexandre de Castro]

There really is a problem, Richard. 

It's possible to show this in a very clear way.

[Richard Gill]

There is a problem with local realism. It can be shown in a very clear way, if you are familiar with probability theory and with the statistical theory of causality. See for instance my short paper

https://arxiv.org/abs/2211.05569

By the way, congratulations on the wonderful community you have created with your Google group! I’m so glad you are still involved in the discussions as well as being owner/manager.

Sent from my iPad

On 14 Jul 2026, at 07:09, Alexandre de Castro <alx...@gmail.com> wrote:



[Bryan Sanctuary]

Alexandre

Richard's remark is the Party Line.  There are two points:

1. No one can explain nonlocaity.  It is accepted as revelation in physics

2. Reality exists in R^3, and so that is where spin needs to be formulated--like a coin

If you use Cl(2,2) and not Cl(1,3) then 1 and 2 above melt in your hands.

Bryan

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[Richard Gill]

Dear Bryan

No-one can explain non-locality. No one can explain quantum mechanics.

Your theory is no better. You posit formulas for mean value and expectation value - you mimic the usual Born rules.  These formulas appear in your theory as axioms. As assumptions. Your model does not explain why a particular spin ends up triggering  “+” or a “-“ on a detector.

That’s why I’m know I will be amused when we see how you solve the challenge that arises when you try to illustrate your theory (which, as I said, is incomplete) on a single classical computer.

You could in theory (according to QM) do it on a pair of separated quantum computers with a lot of pre-established quantum entanglement saved in their quantum memories.

Richard

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On 14 Jul 2026, at 10:54, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Bryan Sanctuary]

Richard,

"No-one can explain non-locality. No one can explain quantum mechanics."  Very negative.  However, I can and do explain them both.  I do not posit anything, I simply start with a classical bivector and do classical mechanics.  That is very natural and for that I explain away nonlocality and understand that spin has a classical origin.

You have not come close to challenging that. 

Bryan

[Richard Gill]

Bryan, you clearly have not carefully read what I emailed to you.

Bye for now

[Bart Jongejan]

Dear all,

I read "spinning coin" in the current discussion as an analogy to spin, I think. That reminds me of the two articles that I uploaded about 25 years ago, wherein I propose a highly speculative geometric model for a spinning particle that might be able to somehow explain CHSH. Have a look and let me know what you think.

https://arxiv.org/pdf/quant-ph/0007010 (even talks about a spinning coin!)

https://arxiv.org/pdf/quant-ph/0105020

I still think these papers are worth the ink, but they don't go all the way to a full theory that replaces or supplements QM. I hope clever people I can get some inspiration from them. I believe GR is more fundamental than QM, so these papers might be of interest for especially those who have the same opinion.

My contribution Sec. 5.3 in https://arxiv.org/pdf/2605.13154 is less controversial, I hope. The matter in the earlier papers can be read as a sequel to Sec 5.3 in the latter paper. It is like Star Wars: first the end, then back to the beginning.

Bart Jongejan

 

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[Richard Gill]

Dear Bart

The first of those papers of yours has the following title and abstract:

Space-Time Structure as Hidden Variable

Bart Jongejan

EPR correlations exist and can be observed independently of any a priori given frame of reference. We can even construct a frame of reference that is based on these correlations. This observation-based frame of reference is equivalent to the customary a priori given frame of reference of the laboratory when describing real EPR experiments. J.S. Bell has argued that local hidden parameter theories that reproduce the predictions of Quantum Mechanics cannot exist, but the counterfactual reasoning leading to Bell's conclusion is physically meaningless if the frame of reference that is based on EPR-correlations is accepted as the backdrop for EPR-type experiments. The refutal have [sic. Should be: refutation of] Bell's proof opens up for the construction of a viable hidden parameter theory. A model of a spin h/2 particle in terms of a non-flat metric of space-time is shown to be able to reproduce the predictions of quantum mechanics in the Bohm-Aharonov version of the EPR experiment, without introducing non-locality.

Yes, Bell assumes a classical 3+1 dimensional space-time. You can disagree with the assumption. Disagreeing with the assumption does not make his proof invalid. Your construction confirms his findings.

I like it!

Richard

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On 14 Jul 2026, at 17:15, Bart Jongejan <bart.j...@gmail.com> wrote:



[Richard Gill]

Dear Bart

Your second paper has abstract as follows:

Does Bell's theorem apply if perceived pseudo-Euclidean space is emergent?

Bart Jongejan

Einstein, Podolsky and Rosen (EPR) showed that it is possible to predict with certainty the value of a property without disturbing the object in question. In contrast, Quantum Mechanics (QM) holds that if different measurement setups cannot coexist, then predictions about those can neither. Using an EPR-inspired experiment with distantly separated measurements on pairs of entangled spinning particles, Bell proved that no local hidden variable (HV) theory can describe reality in more detail than QM. However, it is possible to conceive a viable HV theory based on the assumption that the perceived structure of spacetime is emergent from a hidden curved spacetime. According to this theory, locality can be maintained for each of the measurements while what is perceived as non-locality can be ascribed to the emergence of spacetime correlations between the instruments of the two parties. The theory predicts correlations that agree with QM, provided that the hidden spacetime has three spatial dimensions. If it had fewer than three dimensions, the CHSH inequality would not be violated and if more, Tsirelson's bound would be violated. According to this HV theory, the laboratory frame of reference is a corollary of correlations of the type that are the subject of Bell's thought experiment.

This is a fascinating idea. I do not think your theory is viable, because of some subtle mathematical issues. Which I tried to explain in our recent three author joint paper preprint. I need to rewrite the section where I criticise your very intriguing discovery.

But I recommend everyone to read it and give us their own thoughts.

Richard

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On 14 Jul 2026, at 17:15, Bart Jongejan <bart.j...@gmail.com> wrote: