Bell's theorem refuted

Gordon Watson · May 14, 2012

Q^0.3

with extensive notes⁰to facilitate discussion.

Bell's theorem refuted: the simple constructive model that Bell wanted.

Gordon Watson

[math]Q \in \{W, X, Y, Z\}.\;\;(1)^1[/math]

[math]A({a},\lambda)_Q \equiv \pm 1 = ((\delta_{a} \lambda\rightarrow\lambda_{a^+}\oplus\lambda_{a^-}) \;cos[2s \cdot (a, \lambda_{a^+}\oplus\lambda_{a^-})])_Q.\;\;(2)^2[/math]

[math]B(b,\lambda')_Q = ((-1)^{2s} \cdot B(b, \lambda)_Q \equiv \pm 1 =((\delta_{b}' \lambda'\rightarrow \lambda'_{b^+}\oplus\lambda'_{b^-}) \;cos[2s\cdot (b, \lambda'_{b^+}\oplus\lambda'_{b^-})])_Q. \;\;\;(3)^3[/math]

[math]E(AB)_Q\equiv ((-1)^{2s} \cdot \int d\lambda\;\rho (\lambda)\;AB)_Q \;\;(4)^4[/math]

[math]=((-1)^{2s})_Q\cdot \int d\lambda \;\rho(\lambda) \;[P(A^+B^+|Q)-P(A^+B^-|Q)-P(A^-B^+|Q)+P(A^-B^-|Q)]\;\;(5)^5[/math]

[math]=[(-1)^{2s}]_Q \cdot[ 2 \cdot P(B^+|Q,\,A^+) - 1].\;\;(6)^6[/math]

[math]E(AB)_W= E(AB)_{'Malus'} = (cos[2 ({a}, {b})])/2.\;\;(7)^7[/math]

[math]E(AB)_X =E(AB)_{'Stern-Gerlach'} = - ({a}\textbf{.}{b})/2.\;\;(8)^8[/math]

[math]E(AB)_Y= E(AB)_{\textit{Aspect (2004)}} = cos[2 ({a}, {b})].\;\;(9)^9[/math]

[math]E(AB)_Z= E(AB)_{\textit{EPRB/Bell (1964)}} = - {a}\textbf{.}{b}.\;\;(10)^{10}[/math]

[math]((2s\cdot h/4\pi) \cdot (\delta_{a} \lambda\rightarrow \lambda_{a^+}\oplus\lambda_{a^-})\;cos[2s \cdot (a, \lambda_{a^+} \oplus\lambda_{a^-})])_Q = (\pm1)\cdot (s\cdot h/2\pi)_Q.\;\;(11a)^{11}[/math]

[math]((2s\cdot h/4\pi) \cdot (\delta_{b}' \lambda'\rightarrow\lambda'_{b^+}\oplus\lambda'_{b^-}) \;cos[2s \cdot (b,\lambda'_{b^+}\oplus\lambda'_{b^-})])_Q = (\pm1)\cdot (s\cdot h/2\pi)_Q.\;\;(11b)^{11}[/math]

QED: A simple constructive model delivers Bell's hope and refutes his theorem!

Notes:

0. This wholly classical analysis begins with the acceptance of Einstein-locality. It continues with Bell's hope: "... the explicit representation of quantum nonlocality [in 'the de Broglie-Bohm theory'] ... started a new wave of investigation in this area. Let us hope that these analyses also may one day be illuminated, perhaps harshly, by some simple constructive model. However that maybe, long may Louis de Broglie* continue to inspire those who suspect that what is proved by impossibility proofs is lack of imagination," (Bell 2004: 167). "To those for whom nonlocality is anathema, Bell's Theorem [bT] finally spells the death of the hidden variables program.³¹ But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible," (Mermin 1993: 814). [Emphasis, [.] and * added by GW.]

[math](9)-(10)[/math], with their RHS [math]=[/math], refute Bell's theorem: which requires RHS [math]\neq[/math]. So we side with Einstein, de Broglie, and the later Bell, against Bell's own 'impossibility' theorem. "For surely ... a guiding principle prevails? To wit: Physical reality makes sense and we can understand it. Or, to put it another way: Similar tests on similar things produce similar results, and similar tests on correlated things produce correlated results, without mystery. Let us see:" (Watson 1998: 814).

Taking maths to be the best logic, with probability theory the best maths in the face of uncertainty, we eliminate unnecessary uncertainty at the outset: capture Einstein-locality: the foundation from which (4)-(10) proceed. That is: (4)-(6) proceed from classical probability theory; (7)-(10) follow from Malus' Method (see #6 below). (11) then provides the physics that underlies the logic here: every relevant element of the physical reality having a counterpart in the theory.

' = a prime, identifies an item in, or headed for, Bob's locale. Their removal from "hidden-variables" (HVs) follows from the initial correlation (via recognised mechanisms) of the [math]i[/math]-th particle-pair's HVs [math]\lambda_i[/math] and [math]\lambda_i'[/math] -- with the HVs here pair-wise drawn from infinite sets, no two pairs are the same; though [math]W[/math] and [math]X[/math] may be modified to improve this, somewhat.

[math]\oplus[/math] = xor; exclusive-or.

[math]a, b[/math] = arbitrary orientations: for [math]W[/math] and [math]X[/math], in 2-space, orthogonal to the particles' line-of-flight; for [math]Y[/math] and [math]Z[/math], in 3-space (from the spherical symmetry of the singlet state).

[math]s[/math] = intrinsic spin, historically in units of [math]h/2\pi[/math]. (NB: This classical analysis of four experiments, [math]Q[/math], yields the better value for unit spin angular momentum, [math]h/4\pi[/math]: significant in terms of spherical symmetries in 3-space.)

[math]\delta_{a}[/math] = Alice's device, its principal axis oriented [math]a[/math]; etc.

[math]\delta_{b}'[/math] = Bob's device, its principal axis oriented [math]b[/math]; etc.

[math]\delta_{a}\lambda\rightarrow \lambda_{a^+}\oplus \lambda_{a^-}[/math] = an Alice-device/particle interaction terminating when the particle's [math]\lambda[/math] is transformed to [math] \lambda_{a^+}[/math] xor [math] \lambda_{a^-}[/math] (the device output correspondingly transformed to [math]\pm1[/math]); etc. This may be seen as "a development towards greater physical precision … to have the [so-called] 'jump' in the equations and not just the talk," Bell (2004: 118), "so that it would come about as dynamical process in dynamically defined conditions." This latter hope being delivered expressly, and smoothly, in [math](11)[/math].

[math]\lambda_{a^+}[/math] xor [math] \lambda_{a^-}[/math] = HV outcomes after device/particle interactions; etc. [math] \lambda_{a^+}[/math] is parallel to [math]a[/math]. For [math]s = 1/2[/math], [math] \lambda_{a^-}[/math] is anti-parallel to [math]a[/math]; for [math]s = 1[/math], [math] \lambda_{a^-}[/math] is perpendicular to [math]a[/math]; etc.

1. Re [math](1)[/math]: The generality of [math]Q[/math] and Malus' Method (#6 below), enables this wholly classical analysis to go through. [math]Q[/math] embraces:

[math]W[/math] = 'Malus' (a classical experiment with photons) is [math]Y[/math] with the source replaced by a classical one (the particles pair-wise correlated via identical linear-polarisations).

[math]X[/math] = 'Stern-Gerlach' (a classical experiment with spin-half particles) is [math]Z[/math] with the source replaced by a classical one (the particles pair-wise correlated via antiparallel spins).

[math]Y[/math] = Aspect (2004).

[math]Z[/math] = EPRB/Bell (1964).

2. Re [math](2)[/math]: [math]\equiv[/math] identifies relations drawn from Bell (1964). (2) & (3) correctly represent Einstein-locality: a principle maintained throughout this classical analysis.

3. Re [math](3)[/math]: Bell (1964) does not distinguish between [math]\lambda[/math] and [math]\lambda'[/math], and we introduce s = intrinsic spin. [math](-1)^{2s}[/math] thus arises from Q embracing spin-1/2 and spin-1 particles: in some ways a complication, it brings out the unity of the classical approach used here.

4. Re [math](4)[/math]: Integrating over [math]\lambda[/math], with [math]\lambda'[/math] eliminated: hence the coefficient, per note at #3.

5. Re [math](5)[/math]: [math]P[/math] denotes Probability. [math]A^+[/math] denotes [math]A =+1[/math], etc. The expansion is from classical probability theory: causal-independence and logical-dependence carefully distinguished. The probability-coefficients [math]+1, -1, -1, +1[/math] (respectively), represent the relevant [math]A\cdot B[/math] product: each built from the relevant Einstein-local (causally-independent) values for [math]A[/math] and [math]B[/math]. The reduction [math](5)-(6)[/math] follows, each step from classical probability theory; [math]\int d\lambda \;\rho(\lambda) = 1[/math]. From [math](5)[/math]:

[math]P(A^+B^+|Q)-P(A^+B^-|Q)-P(A^-B^+|Q)+P(A^-B^-|Q)\;\;(A1)[/math]

[math]=P(A^+|Q)P(B^+|Q,A^+)-P(A^+|Q)P(B^-|Q,A^+)-P(A^-|Q)P(B^+|Q,A^-)+P(A^-|Q)P(B^-|Q,A^-)\;\;(A2)[/math]

[math]=[P(B^+|Q,A^+)-P(B^-|Q,A^+)-P(B^+|Q,A^-)+P(B^-|Q,A^-)]/2\;\;(A3)[/math]

NB: In [math](A2)[/math], with random variables: [math]P(A^+|Q) = P(A^-|Q)=P(B^+|Q)=P(B^-|Q) = 1/2.\;\;(A5)[/math]

6. Re [math](6)[/math]: [math](6)[/math], or variants, allows the application of Malus' Method, as follows: Following Malus' example (ca 1810), we would study the results of experiments and write equations to capture the underlying generalities: here [math]P(B^+|Q,\,A^+)[/math]. However, since no Q is experimentally available to us, we here derive (from theory), the expected observable probabilities: representing observations that could and would be made from real experiments, after Malus. Footnotes #7-10 below show the observations that lead from [math](6)[/math] to [math](7)-(10)[/math].

NB: [math]P(B^+|Q,\,A^+)= P(\delta_{b}' \lambda_i'\rightarrow \lambda'_{b^+}|Q,\,\delta_{a}\lambda_i\rightarrow \lambda_{a^+}) =[/math] a prediction of the normalised frequency with which Bob's result is [math]+1[/math], given that Alice's result is [math]+1[/math]; see also [math](11)[/math].

7. Re [math](7)[/math]: Within Malus' capabilities, W would show (from observation): [math]P(B^+|W,\,A^+)= [cos^2 ({a}, {b}) + 1/2]/2= ([cos^2 [s \cdot ({a}, {b})] + 1/2]/2)_W\;\;(A6)[/math] in modern terms: whence [math](7)[/math], from [math](6)[/math]. Alternatively, he could derive the same result (without experiment) from his famous Law.

8. Re [math](8)[/math]: Within Stern & Gerlach's capabilities, X would show (from observation):

[math]P(B^+|X,\,A^+)= ([cos^2

[s \cdot ({a}, {b})]

+ 1/2]/2)_X = [cos^2 [({a}, {b})/2] +1/2]/2\;\;(A7)[/math]: whence [math](8)[/math], from [math](6)[/math]. Alternatively, they could derive the same result (without experiment) by including their discovery, [math]s =1/2[/math], in Malus' Law.

9. Re [math](9)[/math]: From Aspect (2004), Y would show (from observation):

[math]P(B^+|Y,\,A^+)= cos^2 [s\cdot({a}, {b})]_Y = cos^2 ({a}, {b})\;\;(A8)[/math]: whence [math](9)[/math],from [math](6)[/math]. To see this, Aspect (2004: (3)) has (in our notation):

[math]P(A^+B^+|Y)= [cos^2 ({a}, {b})]/2 = P(A^+|Y)P(B^+|Y, A^+) = P(B^+|Y, A^+)/2[/math] [math](A9)[/math], from [math](A5)[/math]; whence [math]P(B^+|Y,A^+) = cos^2 ({a}, {b}).\;\;(A8)[/math]

10. Re [math](10)[/math]: From Bell (1964), Z would show (from observation):

[math]P(B^+|Z,\,A^+)= cos^2 [s\cdot({a}, {b})]_Z = cos^2 [({a}, {b})/2]\;\;(A10)[/math]: whence [math](10)[/math], from [math](6)[/math]. Unlike Aspect (2004), Bell (1964) does not derive subsidiary probabilities. Instead, Bell (1964: (3)) has (in our notation): [math]E(AB)_Z= -({a}. {b}) = -[ 2 \cdot P(B^+|Z,\,A^+) - 1][/math] [math](A11)[/math], from [math](6)[/math], with [math]s = 1/2[/math]; whence [math]P(B^+|Z,A^+) = cos^2 [({a}, {b})/2].\;\;(A10)[/math]

11. Re [math](11)[/math]: With [math]s\cdot h[/math] a driver, the dynamic-process

[math]((2s\cdot h/4\pi)\cdot(\delta _{a}\lambda \rightarrow \lambda_{a^+}\oplus\lambda_{a^-})\;cos[2s \cdot (a, \lambda_{a^+} \oplus\lambda_{a^-})])_Q\;(A12)[/math]

terminates when the trignometric-argument is 0 or ∏; the move to such an argument determined by these facts: one of [math]\lambda_{a^+}[/math] xor [math]\lambda_{a^-}[/math] is an

impossible

terminus, the other

certain

: a "push-me/pull-you" dynamic on the [math]\lambda_i[/math] under test; a smooth determined classical-style transition as opposed to a 'quantum jump'; etc. [math](11)[/math] thus provides the physics that underlies the logic here: every relevant element of the physical reality has a counterpart in the theory. Further, with Planck's constant [math]h[/math] confined to the outer extremities on both sides of [math](11)[/math],

all the maths is classical

. And while LHS-[math](s\cdot h)[/math] drives the particle/device interaction, the emergent RHS-[math](s\cdot h)[/math] is a potential driver for a subsequent interaction.

References:

Aspect (2004):

http://arxiv.org/abs/quant-ph/0402001
Bell (1964):

http://www.scribd.co...-Bell-s-Theorem
Bell (2004): Speakable and Unspeakable in Quantum Mechanics; 2nd edition. CUP, Cambridge.
Mermin (1993):

Rev. Mod. Phys.

65

, 3, 803-815. Footnote #31: "Many people contend that Bell's Theorem demonstrates nonlocality independent of a hidden-variables program, but there is no general agreement about this."
Watson (1998):

Phys. Essays,

11

, 3, 413-421. See also ERRATUM:

Phys. Essays,

12

, 1, 191. A peer-reviewed* draft of ideas here, its exposition clouded by the formalism and type-setting errors. *However, completing the circle, one reviewer was a former student and close associate of de Broglie.

With questions, typos, improvements, critical comments, etc., most welcome,

Gordon

Edited May 14, 2012 by Gordon Watson

hypervalent_iodine · May 14, 2012

!

Moderator Note

Gordon Watson,

I've split the above post from the Bell's theorem thread as we do not allow for personal hypotheses to be posted as answers to mainstream science questions.

In future, please keep the speculative material here, in the speculations forum.

Gordon Watson · May 14, 2012

!

Moderator Note

Gordon Watson, I've split the above post from the Bell's theorem thread as we do not allow for personal hypotheses to be posted as answers to mainstream science questions. ...

Speculative? Reproducing all the results of QM, my submission, a peer-reviewed/published theory: fully in line with Bell's own views! Sir; expect a vigorous defence!

Thanks, Gordon

Edited May 14, 2012 by Gordon Watson

hypervalent_iodine · May 14, 2012

I'm not a physicist and make no claim to be at all proficient in it. When you start a thread with,

"Bell's theorem refuted: the simple constructive model that Bell wanted.

Gordon Watson"

I make the assumption that these are your own speculations and thus, belong in the Speculations forum. If it turns out it belongs in the Science forums, I'll happily move it back.

Gordon Watson · May 14, 2012

I'm not a physicist and make no claim to be at all proficient in it. When you start a thread with,

"Bell's theorem refuted: the simple constructive model that Bell wanted.

Gordon Watson"

I make the assumption that these are your own speculations and thus, belong in the Speculations forum. If it turns out it belongs in the Science forums, I'll happily move it back.

Fair enough. The title was chosen to emphasise a neglected fact: Bell was dissatisfied with his theorem (with citations given in the text). He hoped for a simple constructive model; I have one.

**swansont** · May 14, 2012

Speculative? Reproducing all the results of QM, my submission, a peer-reviewed/published theory

Physics Essays? Yes, speculative.

Gordon Watson · May 16, 2012

Physics Essays? Yes, speculative.

Hi swansont, thanks for having a look at my draft. I welcome any and all comments about it; as some say: better dead than not-read. Please also understand that I welcome its acceptance into the Speculations Forum; and I accept that it is up to me to clearly defend my case there; meaningfully contributing to SFN in other forums, etc, and having some good clean fun at the same time. So, in this latter regard, two thoughts in reply: .

1. (Based on a true story): When the police (think you) found me in a crack-joint (think Physics Essays) and suspected that I was a crack-head (think crack-pot): Was I a crack-head (think crack-pot) or a little-known world-leader in one small speciality for the treatment of addiction (think Bell's theorem)?

2. Some science forums accept journals from the Thomson-Reuter Master-Journal-List. Physics Essays was there, the last time I looked. (But: NO problem, I know what you mean!)

PS: What about we set a time limit for my spell in Speculations Forum (think Purgatory): if no significant error is found within two weeks of me fixing the formatting (say; others have been trying for much longer), could I come out then?

With best regards, and thanks again, Gordon

Edited May 16, 2012 by Gordon Watson

**swansont** · May 16, 2012

PS: What about we set a time limit for my spell in Speculations Forum (think Purgatory): if no significant error is found within two weeks of me fixing the formatting (say; others have been trying for much longer), could I come out then?

I have my doubts as to whether this would become accepted physics in two weeks.

As I'm sure you're aware, in science the burden of proof is on the presenter. It's more than readers not finding any errors. You must propose tests that can falsify the hypothesis and pass those tests, and differentiate it from the concept it's attempting to replace. It's going to take some significant results to replace the accepted notion that Bell's theorem is valid.

Gordon Watson · May 17, 2012

I have my doubts as to whether this would become accepted physics in two weeks.

As I'm sure you're aware, in science the burden of proof is on the presenter. It's more than readers not finding any errors. You must propose tests that can falsify the hypothesis and pass those tests, and differentiate it from the concept it's attempting to replace. It's going to take some significant results to replace the accepted notion that Bell's theorem is valid.

Accepted, of course! BUT I thought Purgatory at SFN might be an improvement on that elsewhere: "a place of temporary punishment, cleansing those destined for heaven but not quite ready for it." Me thinking that the improvement might be an intermediate stage, like, in the true scientific spirit: "Let's crack this one together; it's got most of us beat!" (More in hope than expectation -- a bit like John Bell --it seems?)

Getting serious, here are "some significant results" -- emphasising that NO new physical experiments are required -- the analysis being WHOLLY CLASSICAL throughout:

1. Here's the first hypothesis, easily tested: Proceeding from an acceptance of Einstein-locality (just like Bell), Bell's theorem is refuted via undergraduate maths and logic. (Calling all undergrad mathematicians, classical physicists, quantum physicists, logicians!)

2. Second hypothesis: The theory is in line with Bell's hope for a simple constructive model; i.e., in line with Bell's own (often over-looked) dissatisfaction with his famous theorem AND the associated analyses.

3. Third: The theory delivers Bell's hope to see quantum "jumps" in the equations and not just in the talk, Bell (2004: 118).

4. The result applies equally to experiments with more than two particles, e.g., GHSZ, GHZ, CRB (delivering EVERY QM outcome exactly). To be clear: The theory's experimental predictions exactly coincide with those of standard quantum mechanics, and no departures are predicted.

5. Getting ahead of ourselves AND NOT part of the preliminaries above: The result has consequences for those who believe that QM's "projection postulate" and "unitary evolution" are contradictory.

SO, in brief, the theory is not an attack on QM. It is an attack on Bell's theorem, which (as Peres and others agree) is NO part of quantum theory: as a little study here will show.

PS: Apologies for the poor formatting of Q_0.3above. It was posted (in haste as the editing time elapsed), after many hours of formatting failures. There are no major errors there, afaik, but Q_0.4 will be much improved in style and clarity.

Q_0.3a

Q_0.3 with new notes⁰ and formatting to facilitate discussion.

Bell's theorem refuted: the simple constructive model that Bell wanted.

[math]Q \in \{W, X, Y,Z\}.\;\;(1)^1[/math]

[math]A({a},\lambda)_Q\equiv \pm 1 = ((\delta_{a}\lambda\rightarrow\lambda_{a^+}\oplus\lambda_{a^-}) \;cos[2s \cdot (a,\lambda_{a^+}\oplus\lambda_{a^-})])_Q.\;\;(2)^2[/math]

[math]B(b,\lambda')_Q =((-1)^{2s} \cdot B(b, \lambda)_Q \equiv \pm 1 = ((\delta_{b}'\lambda'\rightarrow\lambda'_{b^+}\oplus\lambda'_{b^-}) \;cos[2s \cdot(b,\lambda'_{b^+}\oplus\lambda'_{b^-})])_Q. \;\;(3)^3[/math]

[math]E(AB)_Q\equiv((-1)^{2s}\cdot\int d\lambda\;\rho (\lambda)\;AB)_Q \;\;(4)^4[/math]

[math]=((-1)^{2s})_Q\cdot\int d\lambda \;\rho(\lambda)\;[P(A^+B^+|Q)-P(A^+B^-|Q)-P(A^-B^+|Q)+P(A^-B^-|Q)]\;\;(5)^5[/math]

[math]=[(-1)^{2s}]_Q\cdot[ 2\cdot P(B^+|Q,\,A^+) - 1]\;\;(6)^6[/math]

[math]=[(-1)^{2s}]_Q\cdot(cos[2s\cdot(a, b)]_Q - \tfrac{1}{2} cos[2s\cdot(a, b)]_{W,X}).\;\;(6a)^6[/math]

[math]E(AB)_W= E(AB)_{'Malus'} = \tfrac{1}{2} cos[2 ({a},{b})] =[/math] Correct classical result. [math]\;\;(7)^7[/math]

[math]E(AB)_X =E(AB)_{'Stern-Gerlach'} = - \tfrac{1}{2} {a}\textbf{.}{b} =[/math] Correct classical result. [math]\;\;(8)^8[/math]

[math]E(AB)_Y=E(AB)_{\textit{Aspect (2004)}} = cos[2 ({a}, {b})] =[/math] Bell's theorem refuted. [math]\;\;(9)^9[/math]

[math]E(AB)_Z=E(AB)_{\textit{EPRB/Bell (1964)}} = - {a}\textbf{.}{b}=[/math] Bell's theorem refuted. [math]\;\;(10)^{10}[/math]

[math]((2s\cdot h/4\pi)\cdot(\delta_{a} \lambda\rightarrow \lambda_{a^+}\oplus\lambda_{a^-})\;cos[2s\cdot(a, \lambda_{a^+} \oplus\lambda_{a^-})])_Q = (\pm1)\cdot(s\cdot h/2\pi)_Q.\;\;(11a)^{11}[/math]

[math]((2s\cdot h/4\pi)\cdot(\delta_{b}' \lambda'\rightarrow\lambda'_{b^+}\oplus\lambda'_{b^-})\;cos[2s\cdot(b,\lambda'_{b^+}\oplus\lambda'_{b^-})])_Q = (\pm1)\cdot(s\cdot h/2\pi)_Q.\;\;(11b)^{11}[/math]

QED: A simple constructive model refutes Bell's theorem and realises his hope.

PS: This draft is a re-formatted improvement on the Q_0.3 draft. To avoid past problems re formatting, I want to be sure the maths presentation is stable.

⁰Notes to follow, more anon, Gordon

Edited May 16, 2012 by Gordon Watson

Gordon Watson · May 17, 2012

Q_0.3b

Q_0.3a with new notes⁰ and formatting to facilitate discussion.

Bell's theorem refuted via the simple constructive model that Bell wanted.