Bell's theorem mathematical content

[RockyMarciano]

Most discussions about Bell's theorem meaning get at some point entangled in semantic and philosophic debates that end up in confusion and disagreement. I wonder if it could be possible to avoid this by reducing the premise, the basic assumption to its bare-bones math content in algebraic/group theoretic/geometric terms trying to abstract it as much as possible from the physical or epistemologic/ontologic implications at a first stage.

I suggest referencing the original 1964 paper by Bell and its numbered equations, the basic premise is summarized in (1), and later in (14) with more pertinent details about the unit vector c.
IMO it establishes the spinorial representation of rotations(double universal cover SU(2)->SO(3) ) as something perfectly grounded on the mathematical structure and topology of the orientable manifold spaces admitting such spin structure, used in all classical physics' theories.
But maybe I'm missing something, or this is not the best way to characterize the main assumption from which the probabilities are constructed?

 

[stevendaryl]

Most discussions about Bell's theorem meaning get at some point entangled in semantic and philosophic debates that end up in confusion and disagreement. I wonder if it could be possible to avoid this by reducing the premise, the basic assumption to its bare-bones math content in algebraic/group theoretic/geometric terms trying to abstract it as much as possible from the physical or epistemologic/ontologic implications at a first stage.

I suggest referencing the original 1964 paper by Bell and its numbered equations, the basic premise is summarized in (1), and later in (14) with more pertinent details about the unit vector c.
IMO it establishes the spinorial representation of rotations(double universal cover SU(2)->SO(3) ) as something perfectly grounded on the mathematical structure and topology of the orientable manifold spaces admitting such spin structure, used in all classical physics' theories.
But maybe I'm missing something, or this is not the best way to characterize the main assumption from which the probabilities are constructed?

I'm a little confused by what you're saying. There are three different parts to Bell's proof that QM cannot be explained by a local hidden-variables theory:

1. Derive an inequality about the correlations of distant measurements.
2. Derive the quantum predictions for the correlation in the EPR experiment.
3. Show that 2 violates the inequality in 1.

Are you asking about #2: how to derive the quantum predictions for EPR? I would think so, because #1 has nothing to do with spinors. But if you're talking about #2, what do you mean "the main assumption from which the probabilities are constructed"? The derivation in #2 is just applying the rules of QM, for the special case of 2-component spinors. So I don't know what you would call "the main assumption".

 

[RockyMarciano]

I'm a little confused by what you're saying. There are three different parts to Bell's proof that QM cannot be explained by a local hidden-variables theory:

1. Derive an inequality about the correlations of distant measurements.
2. Derive the quantum predictions for the correlation in the EPR experiment.
3. Show that 2 violates the inequality in 1.

Are you asking about #2: how to derive the quantum predictions for EPR? I would think so, because #1 has nothing to do with spinors. But if you're talking about #2, what do you mean "the main assumption from which the probabilities are constructed"? The derivation in #2 is just applying the rules of QM, for the special case of 2-component spinors. So I don't know what you would call "the main assumption".

Actually my purpose was to center on the mathematical specification that leads to 1. That's why I mentioned equation (1) in Bell's paper. From that mathematical setting the probabilities and inequality are derived. so I figured it would be good to clarify this starting point avoiding preconceptions.

I get the impression the relation with (2 component) spinors comes from equation (1) in Bell's paper: ##A(a, \lambda)=\pm 1 B(b,\lambda)=\pm1## with outcomes A, B obtained from the iner product of spin component in the particular orientation and a unit vector) and the previous explanatory paragraphs, assuming the spin structure of the Pauli matrices representation for the measurements A and B, basically by implementing the results of spin measurements for arbitrary orientations, that is determining the isomorphism between SO(3) spatial rotations and SU(2) modulo plus or minus the identity. But if there is a more accurate description of the initial mathematical premise what would it be?

 

[rubi]

Bell's theorem (in the CHSH version for simplicity) at full rigor goes as follows:

Theorem (Bell). Let $(\Lambda,\Sigma,\mu)$ be a probability space and $A_\alpha, A_{\alpha'}, B_\beta, B_{\beta'} : \Lambda\rightarrow\{-1,1\}$ be random variables on $(\Lambda,\Sigma,\mu)$. Then $$\left|\left<A_{\alpha} B_{\beta}\right>+\left<A_{\alpha} B_{\beta'}\right>+\left<A_{\alpha'} B_{\beta}\right>-\left<A_{\alpha'} B_{\beta'}\right>\right| \leq 2 \text{.}$$
Proof. For every $\lambda$, either $\left|B_{\beta}(\lambda)+B_{\beta'}(\lambda)\right|=2$ and $\left|B_{\beta}(\lambda)-B_{\beta'}(\lambda)\right|=0$, or exchange $2$ and $0$. Hence, we have:
$\left|\left<A_{\alpha} B_{\beta}\right>+\left<A_{\alpha} B_{\beta'}\right>+\left<A_{\alpha'} B_{\beta}\right>-\left<A_{\alpha'} B_{\beta'}\right>\right|$
$=\left|\int_\Lambda A_{\alpha}(\lambda) B_{\beta}(\lambda)\,\mathrm d\mu(\lambda)+\int_\Lambda A_{\alpha}(\lambda) B_{\beta'}(\lambda)\,\mathrm d\mu(\lambda)+\int_\Lambda A_{\alpha'}(\lambda) B_{\beta}(\lambda)\,\mathrm d\mu(\lambda)-\int_\Lambda A_{\alpha'}(\lambda) B_{\beta'}(\lambda)\,\mathrm d\mu(\lambda)\right|$
$=\left|\int_\Lambda \left(A_{\alpha}(\lambda) B_{\beta}(\lambda)+A_{\alpha}(\lambda) B_{\beta'}(\lambda)+A_{\alpha'}(\lambda) B_{\beta}(\lambda)-A_{\alpha'}(\lambda) B_{\beta'}(\lambda)\right)\,\mathrm d\mu(\lambda)\right|$
$\leq \int_\Lambda \left|A_{\alpha}(\lambda) B_{\beta}(\lambda)+A_{\alpha}(\lambda) B_{\beta'}(\lambda)+A_{\alpha'}(\lambda) B_{\beta}(\lambda)-A_{\alpha'}(\lambda) B_{\beta'}(\lambda)\right|\,\mathrm d\mu(\lambda)$
$\leq \int_\Lambda \left|A_{\alpha}(\lambda)\right| \left|B_{\beta}(\lambda)+B_{\beta'}(\lambda)\right|\,\mathrm d\mu(\lambda)+\int_\Lambda\left|A_{\alpha'}(\lambda)\right| \left|B_{\beta}(\lambda)-B_{\beta'}(\lambda)\right|\,\mathrm d\mu(\lambda)$
$= \int_\Lambda 2\,\mathrm d\mu(\lambda) + \int_\Lambda 0\,\mathrm d\mu(\lambda)= 2$

That's all. No fancy assumptions about spinors or group theory are needed. QM violates the assumptions of the theorem by not requiring the observables $\hat A_\alpha$, $\hat A_{\alpha'}$, $\hat B_\beta$ and $\hat B_{\beta'}$ to be representable as random variables $A_\alpha, A_{\alpha'}, B_\beta, B_{\beta'} : \Lambda\rightarrow\{-1,1\}$ on a probability space. Non-local theories like BM violate the assumptions by introducing dependence on non-local parameters ($A_{\alpha\beta}, A_{\alpha'\beta}, \ldots, B_{\alpha\beta}, B_{\alpha'\beta},\ldots : \Lambda\rightarrow\{-1,1\}$).

 

[RockyMarciano]

Let $(\Lambda,\Sigma,\mu)$ be a probability space and $A_\alpha, A_{\alpha'}, B_\beta, B_{\beta'} : \Lambda\rightarrow\{-1,1\}$ be random variables on $(\Lambda,\Sigma,\mu)$.

As I said for the purposes of this thread I am interested in describing mathematically, in the simplest manner without loss of generality, exactly what leads to the construction of the space of probabilities in the quote above that is used to construct the Bell inequality just by applying classical probabilities so that the expectation value when measuring A and B can be written $E=\int d\lambda p(\lambda) A(\vec a,\lambda) B(\vec b,\lambda)$, and to QM predictions by applying the Born rule.

No fancy assumptions about spinors or group theory are needed.

The above quoted probability space and random variables assignment can be described for single measurements without loss of generality by the double cover mapping from the unit 3-sphere to the Bloch sphere, that identifies mixed states(interior of the Bloch sphere with pure states(at the sphere's surface) in quantum mechanical terms and is also compatible with the EPR experiment.

If we could agree with this description, or any other that anyone suggested, we'd have a good way to avoid the usual problematic word descriptions of the conditions of the theorem in terms of philosophically charged and semantically ambiguous terms like "local","realism", "definiteness" etc...

 

[stevendaryl]

The above quoted probability space and random variables assignment can be described for single measurements without loss of generality by the double cover mapping from the unit 3-sphere to the Bloch sphere, that identifies mixed states(interior of the Bloch sphere with pure states(at the sphere's surface) in quantum mechanical terms and is also compatible with the EPR experiment.

If we could agree with this description, or any other that anyone suggested, we'd have a good way to avoid the usual problematic word descriptions of the conditions of the theorem in terms of philosophically charged and semantically ambiguous terms like "local","realism", "definiteness" etc...

I don't understand what you're talking about. What Bell is assuming is that

1. There is some variable $\lambda$ (ranging over some set of values $\Lambda$) with a corresponding probability distribution $p(\lambda)$.
   
2. There are two functions $A(\lambda, \vec{a})$ and $B(\lambda, \vec{b})$ that for any value $\lambda$ in $\Lambda$, and for any unit vectors $\vec{a}$ and $\vec{b}$:

• $A(\lambda, \vec{a}) = \pm 1$
• $B(\lambda, \vec{b}) = \pm 1$

What does that have to do with the Bloch sphere?

There is a technical assumption about measurability involved, which was pointed out by Pitowsky, which is that for all $\vec{a}, \vec{b}, X, Y$

$\{ \lambda | A(\lambda, \vec{a}) = X \wedge B(\lambda, \vec{b}) = Y \}$

is a measurable set.

 

[rubi]

As I said for the purposes of this thread I am interested in describing mathematically, in the simplest manner without loss of generality, exactly what leads to the construction of the space of probabilities in the quote above that is used to construct the Bell inequality just by applying classical probabilities so that the expectation value when measuring A and B can be written $E=\int d\lambda p(\lambda) A(\vec a,\lambda) B(\vec b,\lambda)$, and to QM predictions by applying the Born rule.

There is no construction of a space of probabilities or anything like that. The assumptions just state that we are in the setting of classical probability theory and the expression for the correlations is just a definition. This suffices to prove the inequality.

The above quoted probability space and random variables assignment can be described for single measurements without loss of generality by the double cover mapping from the unit 3-sphere to the Bloch sphere, that identifies mixed states(interior of the Bloch sphere with pure states(at the sphere's surface) in quantum mechanical terms and is also compatible with the EPR experiment.

Sorry, but it doesn't appear like know what these words mean. At least you are using them in a way that makes no sense.

If we could agree with this description, or any other that anyone suggested, we'd have a good way to avoid the usual problematic word descriptions of the conditions of the theorem in terms of philosophically charged and semantically ambiguous terms like "local","realism", "definiteness" etc...

The theorem I stated above contains absolutely no reference to any philosophical terms. It's all precise mathematics.

There is a technical assumption about measurability involved, which was pointed out by Pitowsky, which is that for all $\vec{a}, \vec{b}, X, Y$

$\{ \lambda | A(\lambda, \vec{a}) = X \wedge B(\lambda, \vec{b}) = Y \}$

is a measurable set.

This technical condition is contained in my post through the requirement that $A_\alpha,\ldots$ should be random variables. Random variables are required to be measurable functions, which means that the preimages of measurable sets are measurable. The sets you wrote down are exactly those preimages.

 

[RockyMarciano]

I don't understand what you're talking about. What Bell is assuming is that

1. There is some variable $\lambda$ (ranging over some set of values $\Lambda$) with a corresponding probability distribution $p(\lambda)$.
   
2. There are two functions $A(\lambda, \vec{a})$ and $B(\lambda, \vec{b})$ that for any value $\lambda$ in $\Lambda$, and for any unit vectors $\vec{a}$ and $\vec{b}$:

• $A(\lambda, \vec{a}) = \pm 1$
• $B(\lambda, \vec{b}) = \pm 1$

What does that have to do with the Bloch sphere?

There is no construction of a space of probabilities or anything like that. The assumptions just state that we are in the setting of classical probability theory and the expression for the correlations is just a definition.

I would think that implicit in those assumptions is the possibility that whatever mathematical space of whatever theory those assumptions apply to, it allows for the EPR experiment elements and outcomes to be possible, right? For instance it is implicit in the assumptions the orientability of any theory's manifold that we might want to consider as compatible with the inequality derived from those assumptions.
And since the assumptions were constructed to be able to meet the requirements of the EPR experiment that involved the concept of spin,and measurements in arbitrary orientations in three dimensions of the said spin components, I don't think it is so outrageous to infer that the mathematical space of the theories that are compatible with the assumptions in the theorem must admit a spin structure. Also admitting classical probabilities is implicit as is in any mathematical construct that has the usual logic as foundation.

And that is what goes into the above quoted assumptions and definitions implicitly, since otherwise they wouldn't serve the purpose of minimal requirements to generalize the EPR experiment and test those minimal mathematical assumptions to describe the gedanken experiment and their probabilistic consequences against different theories predictions and real experiments.

This seems very basic but if we can't develop an understanding of the point of departure of the OP it gets really hard to move forward.

 

[HARSHARAJ]

I have recently done a research work on hidden variables in QM, this is a reference, quite simply written and explained.
Lorenzo Maccone, A simple proof of Bell's inequality, American Journal of Physics 81, 854 (2013).

 

[stevendaryl]

I would think that implicit in those assumptions is the possibility that whatever mathematical space of whatever theory those assumptions apply to, it allows for the EPR experiment elements and outcomes to be possible, right? For instance it is implicit in the assumptions the orientability of any theory's manifold that we might want to consider as compatible with the inequality derived from those assumptions.

I'm having trouble understanding what you're talking about. Bell's theorem doesn't have anything to do with orientability of manifolds. It doesn't mention manifolds at all. It does assume that there is a notion of events being causally separated. That is, it assumes that there is no causal influence of Alice's measurement on Bob's result, or vice-versa. Yes, it's certainly possible that this assumption is false for actual EPR experiments, because of some weirdness such as FTL communication, or wormholes, or back-in-time causation.

 

[HARSHARAJ]

I would like to mention something. Bell's Whole paper, in a nutshell is like: "Let's calculated expectation value of product of two components of spin(sigma1.unit vec a and sigma2.unit vec b)with a local hidden variable model. Now let's allow the magnets to rotate freely so that by any means measurement done at one won't have any causal effect on the other(experimental locality condition), we also take a pair of entangled particles, a pair of spin half particles in the singlet spin state. Now 1. if EPR were to be correct then these local hidden variables had determined the properties of the particles prior to the experiment, since they were focusing on the causal relationship, and hence the expectation value must be if not equal but an approximation of the original result, but 2. if they are wrong then it will be the opposite, and as Bell showed that 2nd one is the right choice". Of course there is a non local action going on and Bell's paper was chiefly to frame a model as per EPR thought and to test it. That it, a simple 5 page paper.

 

[Mentz114]

I have recently done a research work on hidden variables in QM, this is a reference, quite simply written and explained.
Lorenzo Maccone, A simple proof of Bell's inequality, American Journal of Physics 81, 854 (2013).

Also here https://arxiv.org/abs/1212.5214

 

[HARSHARAJ]

Also here https://arxiv.org/abs/1212.5214

Yes, it's available at the library too, that one was the exact citation by the way, but nevertheless, it was one of the primary paper i read to get into the subject, and it's quite simply written though the vein diagram explaining was a bit superfluous, I came up with a little math for the same, but point is if one goes by I think subsection is "Bell's Theorem" and define two terms and goes through the appendix the whole inequality will be cleared out.

 

[RockyMarciano]

I'm having trouble understanding what you're talking about. Bell's theorem doesn't have anything to do with orientability of manifolds. It doesn't mention manifolds at all. It does assume that there is a notion of events being causally separated. That is, it assumes that there is no causal influence of Alice's measurement on Bob's result, or vice-versa. Yes, it's certainly possible that this assumption is false for actual EPR experiments, because of some weirdness such as FTL communication, or wormholes, or back-in-time causation.

The theorem simply stated: " No 'local realist' physical theory can ever reproduce all QM's predictions". And my goal in this thread is just to pinpoint exactly what is the minimal mathematical structure of the physical theory referred to as "local realist" in the theorem, which is what the theorem assumes as premise. I really don't know what is hard to understand here.
Sure, nowhere in the theorem are manifolds explicitly mentioned, here is where the concept of implicit enters. You don't think the physical theories alluded in the theorem need the concept of manifold or space? That would be odd, both classical mechanics and relativity to mention some use such mathematical concepts.
How do you suggest to have components of spin measured in different orientations if that space is not orientable? What kind of EPR physical theory would that be?

 

[rubi]

The theorem simply stated: " No 'local realist' physical theory can ever reproduce all QM's predictions". And my goal in this thread is just to pinpoint exactly what is the minimal mathematical structure of the physical theory referred to as "local realist" in the theorem, which is what the theorem assumes as premise.

I explained it in my post #4. The minimal mathematical structure required is precisely:

Let $(\Lambda,\Sigma,\mu)$ be a probability space and $A_\alpha, A_{\alpha'}, B_\beta, B_{\beta'} : \Lambda\rightarrow\{-1,1\}$ be random variables on $(\Lambda,\Sigma,\mu)$.

As you can see, it says nothing about manifolds or spin structures or orientations or other fancy mathematics (all of which you have been using in non-sensible way). If you think any additional assumption is needed, then you should be able to point to a line of the proof in my post #4, which you think doesn't follow from the assumptions I quoted.

 

[stevendaryl]

The theorem simply stated: " No 'local realist' physical theory can ever reproduce all QM's predictions". And my goal in this thread is just to pinpoint exactly what is the minimal mathematical structure of the physical theory referred to as "local realist" in the theorem, which is what the theorem assumes as premise.

Well, it has nothing to do with orientability of manifolds.

 

[RockyMarciano]

I explained it in my post #4. The minimal mathematical structure required is precisely:

No, I guess you are not paying attention to what I'm writing. The probability space is indeed enough to derive the inequality. But I'm not talking about that or the theorem's proof.
I'm talking about the physical theories referred to in the theorem, the probabillty space and definition of the variables describe concisely the experiment but it is not enough to describe the putative physical theories in which such experiment can take place themselves, rather they limit the possible structures of those physical theories as those in which experiments describable with that probability space and variables can take place, that's what this thread is about.

As you can see, it says nothing about manifolds or spin structures or orientations or other fancy mathematics (all of which you have been using in non-sensible way). If you think any additional assumption is needed, then you should be able to point to a line of the proof in my post #4, which you think doesn't follow from the assumptions I quoted.

I'm sorry that you consider objects like manifolds that have been around in physics for over a century as fancy. Not much I can do about it but advise you to read.

Well, it has nothing to do with orientability of manifolds.

Hmm... if you insist that physical theories modeled by non-orientable manifolds are compatible with EPR experiments...good luck to you.

 

[rubi]

I'm talking about the physical theories referred to in the theorem

The theorem does not refer to any theories. It's a mathematical theorem about certain functions on a probability space.

the probabillty space and definition of the variables describe concisely the experiment but it is not enough to describe the putative physical theories in which such experiment can take place, that's what this thread is about.

The inequality holds in any theory, which has four functions $A_{\alpha},A_{\alpha'},B_{\beta}, B_{\beta'}$ with values $\pm 1$. Yes, it is really that simple. The values $\pm 1$ don't even need to refer to spin measurements. They can might as well be answers to yes/no questions or whatever you can think of that can assume values $\pm 1$ or be encoded as values $\pm 1$.

I'm sorry that you consider objects like manifolds that have been around in physics for over a century as fancy. Not much I can do about it but advise you to read.

I suppose you should find them fancy, since I can tell from your posts that you have exactly zero clue what these words mean. Otherwise, you would not use them it such a completely meaningless way. None of your sentences makes any mathematical sense, not even with lots of goodwill. So really you should be doing much more reading (browsing Wikipedia and randomly picking up words won't suffice).

Hmm... if you insist that physical theories modeled by non-orientable manifolds are compatible with EPR experiments...good luck to you.

Yes, it is perfectly possible to perform an EPR experiment on a non-orientable spacetime.

 

[stevendaryl]

Hmm... if you insist that physical theories modeled by non-orientable manifolds are compatible with EPR experiments...good luck to you.

I insist on avoiding using the phrase "non-orientable manifold" when talking about Bell's theorem.

 

[stevendaryl]

Yes, it is perfectly possible to perform an EPR experiment on a non-orientable spacetime.

I'm wondering if he has been reading Joy Christian.

 

[RockyMarciano]

The theorem does not refer to any theories. It's a mathematical theorem about certain functions on a probability space.

Boy, you don't even have this right, so it's worthless to keep discussing ,it is not a mathematical theorem, it is a physics no-go theorem with the usual rigour required of those in mathematical physics.

 

[rubi]

I'm wondering if he has been reading Joy Christian.

Now that you mention it, this might be the case.

Boy, you don't even have this right, so it's worthless to keep discussing ,it is not a mathematical theorem, it is a physics no-go theorem with the usual rigour required of those in mathematical physics.

This is a pointless semantic distinction. It's clearly a mathematical theorem, even though it has applications in physics. Mathematicians knew about it 100 years before Bell rediscovered it and understood its implications for physics.

 

[RockyMarciano]

I'm wondering if he has been reading Joy Christian.

Is that a name or are you bringing religion into this?

 

[Misha]

...it is not a mathematical theorem, it is a physics no-go theorem with the usual rigour required of those in mathematical physics.

What do you mean by a 'no-go theorem'? Bell's inequality is not necessarily broken by quantum systems.

 

[RockyMarciano]

This is a pointless semantic distinction. It's clearly a mathematical theorem, even though it has applications in physics. Mathematicians knew about it 100 years before Bell rediscovered it and understood its implications for physics.

LOL, now this actually sounds creepy/crackpotty.

 

[berkeman]

Thread closed temporarily for Moderation...

EDIT -- Thread re-opened. Please try to stay on-topic. Thanks.

 

[rubi]

LOL, now this actually sounds creepy/crackpotty.

I don't know what's funny about it and why it sounds creepy/crackpotty. It happens all the time in science that theorems are forgotten and rediscovered later. See this paper. And given your meaningless concatenation of mathematical words in your previous posts, you should be very careful with calling others crackpots.

But anyway, it's completely irrelevant whether you want to call it a physical theorem or a mathematical theorem. It's a theorem and it can be applied to many situations. All you need is four random variables with values $\pm 1$. It is trivial to come up with completely pathological theories in which Bell's inequality is satisfied. Here is one: $\Lambda=\{0\}$, $\Sigma=\mathcal P(\Lambda)$, $\mu=\delta_0$, $A_\alpha(0)=1$, $A_{\alpha'}(0)=-1$, $B_\beta(0)=1$ and $B_{\beta'}(0)=1$. As you can see, I did not need to say anything about manifolds or spin structures or any other fancy math to come up with this theory.

 

[Misha]

It's a theorem and it can be applied to many situations.

As I understand, OP sees Bell's inequality as a statement about dynamics and, therefore, questions the procedure of identifying a quantum system and respective Bell's inequality to be violated. Say, if Bell's inequality deals with the space of states $\{-1, 1 \}$, it seems to call for some sort of spin dynamics but then it raises other questions and so on. I think, I can see where this comes from, as I had similar troubles with this whole Bell's business. Actually, I still have but on a slightly different level.

Probably, the picture will be clearer if the main statements are formulated slightly different.

Bell's inequality. For any four functions $A_i, B_i$ with values in the interval $[-c,c]$, that is $A_i : V_i \to [-c, c]$ and so on, the following inequality holds
$$
|F| \leq 2c^2,
$$
where $F = A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2$.

Violation of Bell's inequality for operators. There exist such four operators $\widehat{A}_i, \widehat{B}_i$ with $||\widehat{A}_i|| \leq c$ and $||\widehat{B}_i|| \leq c$ that
$$
||\widehat{F}|| > 2c^2,
$$
where $\widehat{F} = \widehat{A}_1 \widehat{B}_1 + \widehat{A}_1 \widehat{B}_2 + \widehat{A}_2 \widehat{B}_1 - \widehat{A}_2 \widehat{B}_2$.

 

[zonde]

Maybe it's trivial, but think it is good idea to put conditions of Bell theorem into mathematical form before doing the same thing with assumptions. So I propose these conditions:

1. We can produce series of paired events $A_\alpha$ and $B_\beta$ where $\alpha$ and $\beta$ are freely changeable parameters. $A_\alpha$ and $B_\beta$ each can be split into two subevents: $A_\alpha=\pm 1$ and $B_\beta=\pm 1$.
2. For every i-th pair of events $A_{\alpha i}$ and $B_{\beta i}$ in series, $P(A_{\alpha i}=+1, B_{\beta i}=+1)=0$ and $P(A_{\alpha i}=-1, B_{\beta i}=-1)=0$ whenever $\alpha = \beta$.

@RockyMarciano, do these seem fine to you?

 

[RockyMarciano]

(For the benefit of civilized debate I'll comply with the admins demand to stay on topic therefore ignoring petty provocations)

It is trivial to come up with completely pathological theories in which Bell's inequality is satisfied.

Once again you are missing my point and intention in this thread. And you are confusing something trivial like the proof of the derivation of the Bell inequality from certain probabilistic conditions with Bell's theorem. Finding pathological situations that satisfy the inequality is totally orthogonal to what I'm saying.

Once again this thread is not about that, it discusses something previous to laying out the abstract probabilistic conditions that a theory fulfilling the EPR experiment must have. What the theorem says is not that all theories satisfying the inequality must be of certain type, but rather that those that want to replicate all QM predictions (confirmed empirically) must violate them, the theorem is only relevant for those physical theories that violate the inequalities because experiment says that they are the only ones viable(and what better way to identify them than characterizing mathematically the minimal common denominator of those that are not viable to discard them immediately?) , so your trivial example has nothing to do with the theorem nor with what I'm discussing in this thread.
The mathematical requirements I'm after belong to the physical theories' spaces that fulfill EPR assumptions and are obviously a subset of all those conceivable math constructs that could satisfy the probabilistic inequalities.

 

[stevendaryl]

Once again this thread is not about that, it discusses something previous to laying out the abstract probabilistic conditions that a theory fulfilling the EPR experiment must have. What the theorem says is not that all theories satisfying the inequality must be of certain type,

No, that's backwards. It proves that all theories of a certain type satisfy the inequality.

but rather that those that want to replicate all QM predictions (confirmed empirically) must violate them

He proved that all theories of a certain type satisfy a particular inequality. EPR experiments do not satisfy that inequality. Therefore, EPR experiments cannot be explained by a theory of that type.

There is nothing in Bell's argument that relies on any property of quantum mechanics. The fact that EPR violates his inequality is an empirical matter.

 

[RockyMarciano]

Maybe it's trivial, but think it is good idea to put conditions of Bell theorem into mathematical form before doing the same thing with assumptions. So I propose these conditions:

1. We can produce series of paired events $A_\alpha$ and $B_\beta$ where $\alpha$ and $\beta$ are freely changeable parameters. $A_\alpha$ and $B_\beta$ each can be split into two subevents: $A_\alpha=\pm 1$ and $B_\beta=\pm 1$.
2. For every i-th pair of events $A_{\alpha i}$ and $B_{\beta i}$ in series, $P(A_{\alpha i}=+1, B_{\beta i}=+1)=0$ and $P(A_{\alpha i}=-1, B_{\beta i}=-1)=0$ whenever $\alpha = \beta$.

@RockyMarciano, do these seem fine to you?

Again, the goal of this thread is not directly concerned with the probabilities that lead to the Bell inequalities other than for the fact that all the physical theories with the minimal common mathematical description I'm trying to reach consensus about will certainly satisfy the inequalities when using classicall probabilities on them. The advantage is that by identifying the principal features of this physical theories spaces and mathematical objects in them that directly lead to satisfying the inequality(because that is a feature of EPR math models) it would be so much easier to discard from the beguinning a lot of mathematical models for physics that are not compatible with experiment.

 

[stevendaryl]

Again, the goal of this thread is not directly concerned with the probabilities that lead to the Bell inequalities

Your first post asked what is "the best way to characterize the main assumption from which the probabilities are constructed?"

 

[RockyMarciano]

Wait, first let's clarify what I mean by EPR experiment and EPR mode, I refer to Einstein's so called "local realist" model, and to the gedanken experiment operational requirements and assumptions not to the actual Aspect type experiments outcomes showing the violations.

No, that's backwards. It proves that all theories of a certain type satisfy the inequality.

And therefore that those that don't satisfy them like the QM predictions or those appparently compatible with nature don't. This is what I'm saying. It is not backwards.

He proved that all theories of a certain type satisfy a particular inequality. EPR experiments do not satisfy that inequality. Therefore, EPR experiments cannot be explained by a theory of that type.

Sure, look tho the possible source of confusion here above.

There is nothing in Bell's argument that relies on any property of quantum mechanics. The fact that EPR violates his inequality is an empirical matter.

Of course. Again there might be some trivial misunderstanding here, see the first commnet of this post.

 

[zonde]

Wait, first let's clarify what I mean by EPR experiment and EPR mode, I refer to Einstein's so called "local realist" model, and to the gedanken experiment operational requirements and assumptions not to the actual Aspect type experiments outcomes showing the violations.

Yes, please clarify these operational requirements of gedanken experiment in mathematical form as it seems you are not satisfied with my attempt in post #29.