Bell's theorem refuted via elementary probability theory

[floccinauci]

This thread is depressing. A lot of bickering and talking past each other. A lot of unwarranted abuse heaped on Gordon for concepts that are not controversial.

In Bell's derivation. The is simply the expectation value for the paired-product of the outcomes an EPR experiment with Alice setting and Bob's setting , where the outcomes at Alice and Bob are represented mathematically as . An expectation value is a generalization of a weighted average calculated over any valid probability space of the experiment.

It is easy to envision three probability spaces:

1. The one used by Bell, which divides the outcomes based on outcomes hidden variables and corresponding probability functions , where resulting in the expectation value:



Note: I've fixed two notational problems in Bell's original paper: (1) The symbols and are labels, not variables since their values do not change during the experiment (2). I've used instead of to denote expectation, which is more standard.

2. An alternative probability space, defined over the actual outcome pairs obtained and the corresponding probabilities where . This yields the expectation of the paired product as:




Which is essentially Gordon's equation 5.

I fail to see the relevance of the discussion about the production of two functions to what is described in the 2-page paper linked above. Perhaps there could be some discussion about the relevance of invoking Malus law but the math up to at least equation 7 appears very valid.

[floccinauci]

3. The third probability space is based on the paired products and the corresponding probabilities . I'll leave the rest as an exercise for the reader.

All three representations of the expectation value of the paired-product are equivalent and analysis based on one should be consistent with all the others. As already mentioned, the more interesting part of the 2-page paper is the application of Malus law.

[Gordon Watson]

This thread is depressing. A lot of bickering and talking past each other. A lot of unwarranted abuse heaped on Gordon for concepts that are not controversial.

In Bell's derivation. The is simply the expectation value for the paired-product of the outcomes an EPR experiment with Alice setting and Bob's setting , where the outcomes at Alice and Bob are represented mathematically as . An expectation value is a generalization of a weighted average calculated over any valid probability space of the experiment.

It is easy to envision three probability spaces:

1. The one used by Bell, which divides the outcomes based on outcomes hidden variables and corresponding probability functions , where resulting in the expectation value:



Note: I've fixed two notational problems in Bell's original paper: (1) The symbols and are labels, not variables since their values do not change during the experiment (2). I've used instead of to denote expectation, which is more standard.

2. An alternative probability space, defined over the actual outcome pairs obtained and the corresponding probabilities where . This yields the expectation of the paired product as:




Which is essentially Gordon's equation 5.

I fail to see the relevance of the discussion about the production of two functions to what is described in the 2-page paper linked above. Perhaps there could be some discussion about the relevance of invoking Malus law but the math up to at least equation 7 appears very valid.

Thanks for the pleasant rationality, and the clue-full identifier.

A quick point of interest before I look in detail at improving my presentations in the light of your comments.

With otherwise-rational Bellians , in more technical discussions of Bell's 1964 eqn (14a):

which is my term for the unnumbered equation that follows his (14):

I like to hold the angles (a,b) and (a,c) constant .

That is (for me in this case): the unit-vectors a, b, and c are allowable variables in 3-space, bound only by the maintenance of (a,b) and (a,c) constant.

Under these conditions, Bellians need to recognise that the angle (b,c) is a variable. Thus the LHS of Bell's eqn (15) is a variable.

Example, E&OE: if I hold a and c constant such that (a,c)= π/3, and vary b with (a,b) constant at π/6: THEN (b,c) allowably varies from π/6 to π.

Point: many Bellians do not appreciate the spherical symmetry of the EPRB singlet-state. A fact which allows me to derive Bell's famous inequality -- BI, his 1964:(15) -- in both VALID and INVALID modes; though BI is, and remains, nonsense under EPRB and true local realism.

Thanks again; much appreciated; Gordon

[Gordon Watson]

3. The third probability space is based on the paired products and the corresponding probabilities . I'll leave the rest as an exercise for the reader.

All three representations of the expectation value of the paired-product are equivalent and analysis based on one should be consistent with all the others. As already mentioned, the more interesting part of the 2-page paper is the application of Malus law.

To be clear: Malus' Law (c1810, Paris) was heuristic in our search for the EPRB-related laws of nature.

Thus, from elementary probability theory, we know (in my notation):

(1). is, by a standard definition, the weighted average of the possible results,

and , weighted according to their probabilities. Note: with used for clarity as to where the results appear on our analysis.

(2). And:

(3). So, we have:

(3). But, from QM:

(4). So, by observation (influenced by Malus' Law), there being nowhere else to go: given (2) above.

and ; etc.***

All of which holds under the principle of true local realism (TLR).

TLR is the union of true locality (or relativistic causality: no influence propagates superluminally, after Einstein) and true realism (or non-naive realism: some existents change interactively, after Bohr).***

*** Edit: as shown elsewhere, our probabilities hold under local relations between latent (pristine; ie, devoid of direct interaction) properties and revealed (via interaction) properties.

If (akin to EPR), without interaction, we can predict with certainty the result of a physical experiment, then there exist latent properties that mediate this result.

HTH; Gordon
.

[Gordon Watson]

If (akin to EPR), without interaction, we can predict with certainty the result of a physical experiment, then there exist latent properties that mediate this result.
.

EDIT, with apologies:

If, without any way disturbing a system [say, in Bob's locale], we can predict with certainty the result of that system's test via , which may be a disturbance: then local existents and mediate this result. That is, : where and are existents in Bob's locale.

Thus: our extension of Malus' Law does not depart from the nature of his original law: local existents determine local results.

TLR; Gordon
.

[floccinauci]

and ; etc.***

All of which holds under the principle of true local realism (TLR).

Please elaborate. The above is too terse to be convincing to anyone.

[Gordon Watson]

.
EDITED, 24 Nov 2020 (and some typos fixed), in response to floccinauci's request for elaboration. See foot.

... As already mentioned, the more interesting part of the 2-page paper is the application of Malus law.

To be clear: Malus' Law (c1810, Paris) was heuristic in our search for the EPRB-related laws of nature.

Thus, from elementary probability theory, we know (in my notation):

(1). is, by a standard definition, the weighted average of the possible results,

and , weighted according to their probabilities. Note: for clarity, we use to show where the results appear in our analysis.

(2). And:

(3). So, we have:

EDIT (3a). Since there are only two results, and , there can only be two probability-weighted components.

(4). But, from QM:

via the following analysis.

EDIT (4a). Elaborating: Here, via , we have the expected result of the EPRB experiment.

And thus, akin to Malus' technique, we observe the consequences forced upon us:



Forced, since: (i) There are only two results, and , there can only be two probability-weighted components.

(ii) Those two probabilities sum to unity: as in (2). (iii) The weighted results sum to : as in (3).

(5). So, by observation (influenced by Malus' Law), there being nowhere else to go: given (2)-(4a) above:

(6). and ; etc.***

(7). All of which holds under the principle of true local realism (TLR). TLR is the union of true locality (or relativistic causality: no influence propagates superluminally, after Einstein) and true realism (or non-naive realism: some existents change interactively, after Bohr).***

***
(8). As shown elsewhere, our probabilities hold under local relations between local existents. For:

(9). If, without any way disturbing a system [say, in Bob's locale], we can predict with certainty the result of that system's test via , which may be a disturbance: then local existents and mediate this result. That is, : where and are existents in Bob's locale.

(10). Thus: our extension of Malus' Law does not depart from the nature of his original law: local existents determine local results.

HTH; Gordon
.

(6). and ; etc.***

(7). All of which holds under the principle of true local realism (TLR).

Please elaborate. The above is too terse to be convincing to anyone.

Many thanks. I've fixed some typos and elaborated above. HTH; Gordon
.

[floccinauci]

Hardly an elaboration:

e·lab·o·rate
verb
/əˈlabəˌrāt/ develop or present (a theory, policy, or system) in detail.

Please explain in detail why equation (6) is consistent with local realism. You haven't done this.

[gill1109]

Hardly an elaboration:

e·lab·o·rate
verb
/əˈlabəˌrāt/ develop or present (a theory, policy, or system) in detail.

Please explain in detail why equation (6) is consistent with local realism. You haven't done this.

“floccinauci”, past experience suggests that Gordon is never going to do what you ask of him. Or rather: he will do it, but the detail won’t help. On the contrary, it will give you a headache.

Gordon starts by assuming that the QM predictions are correct. I think that he considers that this satisfies *his* principle of “true local realism” because Alice and Bob each do measurements locally and it is realism to accept the QM predictions. Gordon knows enough elementary probability theory in order to write the expectation of AB as the probability the product is +1 minus the probability it is -1. He knows that those probabilities add to +1. He knows the trigonometric formula for expressing a cosine in terms of cosine and sine of half the angle.

That’s all there is here. Gordon says Bell was wrong but really he has no interest in what Bell was trying to do, since for him, under Gordon’s own TRUE local realism, everything is crystal clear.

If you really have nothing to say, elaborating what you said only makes it less clear that you actually had nothing to say.

I think everything useful has been said by someone called “Mikko” on the Quora discussion on the viXra abstract page, https://vixra.org/abs/2011.0073

If you want to see the same discussion again, look at the Quora discussion on Gordon’s previous try.
https://vixra.org/abs/2010.0068

Or Gordon’s previous try before that.
https://vixra.org/abs/1909.0216

[Gordon Watson]

...

Please explain in detail why equation (6) is consistent with local realism. You haven't done this.

1. The three asterisks *** refer to elsewhere; which is here:

viewtopic.php?f=6&t=451&start=80#p11973

2. Consistent with TLR, I there provide Bell's "impossible" functions .

3. To understand my functions, see the schematics in (7)-(9) and the inner workings of Alice's detector in paragraph 2.3.

4. The relevant integrals, under TLR, are (19)-(26). (I prefer to work with ; as in recent discussion).

5. Thus, via TLR, (24) is equivalent to the expression you are seeking.

6. In the context of paragraph 3.1, the relevant equivalence relation -- like -- is clear from the context.

7. So alternative short-form expressions for my functions are:



.

8. To be compared with Bell's inadequate (1964) sgn-functions.

Gordon
.

[gill1109]

...

Please explain in detail why equation (6) is consistent with local realism. You haven't done this.

1. The three asterisks *** refer to elsewhere; which is here:

viewtopic.php?f=6&t=451&start=80#p11973

2. Consistent with TLR, I there provide Bell's "impossible" functions .

3. To understand my functions, see the schematics in (7)-(9) and the inner workings of Alice's detector in paragraph 2.3.

4. The relevant integrals, under TLR, are (19)-(26). (I prefer to work with ; as in recent discussion).

5. Thus, via TLR, (24) is equivalent to the expression you are seeking.

6. In the context of paragraph 3.1, the relevant equivalence relation -- like -- is clear from the context.

7. So alternative short-form expressions for my functions are:



.

8. To be compared with Bell's inadequate (1964) sgn-functions.

Gordon
.

Your formulas under 7. are still meaningless. The equivalence relation referred to as “tilde” is not clear to anybody from the context.

The meaning of a, b with superscript +/- is not clear from the context.

If one evaluates “lambda tilde a” where tilde is an equivalence relation on some set, and lambda, a are elements of that set, then the outcome is T or F (true or false). You can’t take an inner product of T or F with a vector.

Your scribblings are comprehensible to nobody but yourself. You are living under a private delusion. I’m sorry for you, nobody else active on this forum lives in *your* world. But hopefully your world is populated with beings who do understand you, so everything is OK.

[floccinauci]

Your formulas under 7. are still meaningless. The equivalence relation referred to as “tilde” is not clear to anybody from the context.

The meaning of a, b with superscript +/- is not clear from the context.

If one evaluates “lambda tilde a” where tilde is an equivalence relation on some set, and lambda, a are elements of that set, then the outcome is T or F (true or false). You can’t take an inner product of T or F with a vector.

Your scribblings are comprehensible to nobody but yourself.

I have to agree with this. The purpose of writing is to make concepts that are already clear to the author to become understandable to the audience. I wouldn't conclude necessarily that the ideas behind the scribblings are rubbish. Rather the presentation is lacking in clarity so far.

What the heck is



With the understanding that a function has three aspects:
- inputs: domain
- transformation
- outputs: range

Please explain using words (no symbols please) in detail for the A(.) case above how the inputs are transformed to produce the outputs. If you could do that, then someone else may be able to help you formulate a better notation for concisely representing it. Your choice of notation is horrendous.

[Gordon Watson]

Your formulas under 7. are still meaningless. The equivalence relation referred to as “tilde” is not clear to anybody from the context.

The meaning of a, b with superscript +/- is not clear from the context.

If one evaluates “lambda tilde a” where tilde is an equivalence relation on some set, and lambda, a are elements of that set, then the outcome is T or F (true or false). You can’t take an inner product of T or F with a vector.

Your scribblings are comprehensible to nobody but yourself.

I have to agree with this. The purpose of writing is to make concepts that are already clear to the author to become understandable to the audience. I wouldn't conclude necessarily that the ideas behind the scribblings are rubbish. Rather the presentation is lacking in clarity so far.

What the heck is

(X), say.

With the understanding that a function has three aspects:
- inputs: domain
- transformation
- outputs: range

Please explain using words (no symbols please) in detail for the A(.) case above how the inputs are transformed to produce the outputs. If you could do that, then someone else may be able to help you formulate a better notation for concisely representing it. Your choice of notation is horrendous.

Here is the context:

https://vixra.org/abs/2011.0073

Title: Bell's Theorem Refuted via True Local Realism

Abstract Bell's theorem has been described as the most profound discovery of science; one of the few essential discoveries of 20th Century physics; indecipherable to non-mathematicians. However, taking elementary analysis to be an adequate logic here, we refute Bell's theorem, correct his inequality and identify his error. Further, we do this under the principle of true local realism, the union of true locality (or relativistic causality: no influence propagates superluminally) and true realism (or non-naive realism: some existents change interactively). We thus lay the foundation for a more complete physical theory: one in line with Einstein's ideas and Bell's hopes. Let's see.

Gordon
.

Here is the backgound:

Richard Gill and local could not (or would not) evaluate the integral in (19). So, seeking to make things clearer, I offered (X), for comparison with Bell's 1964:(4).

That is, my

(X)

versus Bell's

. (Y)

Me thinking that if you understood (Y) then you'd understand (X): me taking that it that they had read my draft.

THUS: Since (X) does not appear in https://vixra.org/abs/2011.0073, I expected it to be fringe issue for floccinauci: me taking that they too had read my draft.

SO NOW: (X) has a LHS, a middle, and a RHS. The LHS and RHS are from Bell (1964): so I take it that neither they nor their contents need further explanation.

Then, since is a unit-vector: in paragraph 2.2, I define as a particle's post-interaction spin-axis.

So the middle of (X) reads as follows: if lambda is equivalent to under the equivalence relation defined in paragraph 3.1 -- which, given the equivalence, is safely suppressed in the short-form notation -- then:

; etc.

It should then be clear that (X) satisfies all the conditions of a function. QED.

So, side issues aside:

1. The question then remains: Does the integral in (19) proceed as claimed under TLR?

I maintain that it does.

2. Since my choice of notation is said to be horrendous, I'd appreciate pointers to several examples.

Gordon
.

[floccinauci]

What the heck is



With the understanding that a function has three aspects:
- inputs: domain
- transformation
- outputs: range

Please explain using words (no symbols please) in detail for the A(.) case above how the inputs are transformed to produce the outputs. If you could do that, then someone else may be able to help you formulate a better notation for concisely representing it. Your choice of notation is horrendous.

I'm trying to help you but you are not making it easy.

[FrediFizzx]

What the heck is



With the understanding that a function has three aspects:
- inputs: domain
- transformation
- outputs: range

Please explain using words (no symbols please) in detail for the A(.) case above how the inputs are transformed to produce the outputs. If you could do that, then someone else may be able to help you formulate a better notation for concisely representing it. Your choice of notation is horrendous.

I'm trying to help you but you are not making it easy.

That one is easy. He just means lambda is +/-a. Basically a description of the polarizer action.
.

[gill1109]

What the heck is



With the understanding that a function has three aspects:
- inputs: domain
- transformation
- outputs: range

Please explain using words (no symbols please) in detail for the A(.) case above how the inputs are transformed to produce the outputs. If you could do that, then someone else may be able to help you formulate a better notation for concisely representing it. Your choice of notation is horrendous.

I'm trying to help you but you are not making it easy.

That one is easy. He just means lambda is +/-a. Basically a description of the polarizer action.
.

Fred, you are a genius. And we just learned from Gordon himself that the equivalence relation is “a is equivalent to lambda” if and only if A(a, lambda) = 1. So it’s a relation which depends on which function A we are talking about.

a superscript + or - means: a, or -a. He could just as well have said “x”. He means “any direction, but later I will take it to be either Alice’s setting or its negative”. Gordon needs to learn to use the universal quantifiers “for all”, and “there exists”; and have names for sets of things, e.g. the set of all directions.

We still did not learn how to take the dot product of ‘true’ or ‘false’ with a vector. But that was not Gordon’s intention. Fred has a good suggestion, I think, of what Gordon may well have meant: take the dot product of either a or -a with a itself, depending on whether A(a, lambda) = +1 or -1. He means replace lambda with a or -a. Then take the dot product with a.

Since the vector a has length one, this means that A(a, lambda) = A(a, lambda) = +/-1. It doesn’t tell us which answer we’ll get, because Gordon has not told us what he has in mind for the function “A”.

He writes some known stuff in very private notation, but does not do anything interesting with it at all, at least, not yet.

But we are making progress, at last, thanks to the combined efforts of many friends!

[floccinauci]

That one is easy. He just means lambda is +/-a. Basically a description of the polarizer action.
.

Progress. Then:

where


But we need to see the details of the function

[FrediFizzx]

That one is easy. He just means lambda is +/-a. Basically a description of the polarizer action.
.

Progress. Then:

where


But we need to see the details of the function

Well, I would have to say that it looks like lambda would have to be the particle spin vector and what we have here is the polarizer action. So, pick a function for that.
.

[gill1109]

That one is easy. He just means lambda is +/-a. Basically a description of the polarizer action.
.

Progress. Then:

where


But we need to see the details of the function

Well, I would have to say that it looks like lambda would have to be the particle spin vector and what we have here is the polarizer action. So, pick a function for that.
.

Fred and “floccinauchi”: Gordon’s mysterious equation tells us no more and no less than that A(a, lambda) = A(a, lambda). Not very illuminating. The rest of the paper does not contain a specification of any function A. Instead it contains an argument that Bell’s derivation of Bell’s original (three correlation) inequality is wrong. Gordon’s argument is easily seen to be wrong. It has been discussed repeatedly in the comments on three viXra papers by Gordon with very similar titles and similar content. Someone called “mikko” explains carefully what is wrong. For instance, https://vixra.org/abs/2011.0073 , most recently:

Bell's theorem says that it is not possible to construct a hidden variable theory that predicts the same as quantum mechanics for every experiment. The article does not express the theorem in this way but uses instead a mathematical expression that captures the core idea for a particular kind of experiment: it is not possible to construct functions ρ, A, and B so that

∫ dλ ρ(λ) A(a,λ) B(b,λ) = a · b for every pair of directions (or unit vectors) a and b.

Indeed, no such functions are constructed in the article. Instead, already exposed fallacies from earlier articles are reused

In the "Analysis" section a large number of mathematical and other symbols are introduced. Apparently the intent is to make the mathematics and pseudo-mathematics so hard to read that the holes in the logic are not seen. However, it is easy to see that no construction is presented for ρ, A, or B. The main content of the section is to present an experiment where Bell's theorem can be demostrated. The presentation is correct but better presentations of the same experiment can be found elsewhere, starting with Bell's original article (1964) or even earlier in D. BOHM and Y. AHARONOV, Phys. Rev. 108, 1070 (1957).

The analysis of the proposed experiment described in the "Analysis" section is attempted in the "Refutation" section, which pretends to refute Bell's theorem. In equations 16 and 17 the quantum mechanical probabilities that any hidden variable theory should reproduce are presented. In equation 24 these probabilities are used as if they followed from previous equations. In reality the well known fallacy known as "begging the question" is applied. This presentation is essentially the same as in the article https://vixra.org/abs/2010.... so the comments on that page apply here, too.

The "Conclusions" section is mainly other than conculsions from earlier sections and the rest is based on fallacies.

The "Appendix" claims to refute Bell's inequality

| ∫ dλ ρ(λ) A(a,λ) A(b,λ) - ∫ dλ ρ(λ) A(a,λ) A(c,λ) | ≤ 1 + ∫ dλ ρ(λ) A(b,λ) A(c,λ),

which is a theorem of elementary probability theory, one of "conditions of possible experience" as Boole called them, applied to the experiment discussed in both Bell's and this article. The fallacy is simply to assume for the integrals values that are not obtained from any λ and A and pretend that they could refute the inequality theorem. The same fallacy is also used in https://vixra.org/abs/1812.... and exposed in discussion on that page.

[Gordon Watson]

What the heck is



With the understanding that a function has three aspects:
- inputs: domain
- transformation
- outputs: range

Please explain using words (no symbols please) in detail for the A(.) case above how the inputs are transformed to produce the outputs. If you could do that, then someone else may be able to help you formulate a better notation for concisely representing it. Your choice of notation is horrendous.

I'm trying to help you but you are not making it easy.

Please be more specific and elaborate on your view that the notation is horrendous.

I am always happy to make improvements and clarify things; and such here might clarify matters more efficiently.

Note that the notation in (19) foreshadows this from you:

1. The one used by Bell, which divides the outcomes based on outcomes hidden variables and corresponding probability functions , where resulting in the expectation value:



Note: I've fixed two notational problems in Bell's original paper: (1) The symbols and are labels, not variables since their values do not change during the experiment (2). I've used instead of to denote expectation, which is more standard.

But you depart from this below. So, as I'll comment there, we need to find which is the more helpful notation.

Gordon
.