Gordon Watson

Many thanks for the links; and for catching my boo-boo! Ouch! Wrong one: there's 2! My apologies! Sincerely, "Private Panic, esquire."

Q0.3b Q0.3a with new notes0 and formatting to facilitate discussion. Bell's theorem refuted via 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. 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 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.] 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: similar tests on similar things produce similar results and similar tests on correlated things produce correlated results, without mystery. Let us see:" (after 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: (2)-(3) capture Einstein-locality: the foundation from which (4)-(10) proceed. That is: (4)-(6) follow from classical probability theory; (7)-(10) flow from Malus' Method (see #6 below). (11) then provides the physics that underlies the logic here: every relevant element of the physical reality has a counterpart in the theory. Symbols: [math]\prime[/math] = a prime: it identifies an item headed for, or in Bob's locale. A prime's removal [math][e.g., \lambda_i' = -\lambda_i][/math] follows from the initial correlation (via recognised mechanisms) of the [math]i[/math]-th particle-pair's [math]\lambda_i[/math] and [math]\lambda_i'[/math]. With the [math]\lambda[/math]-s 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: Our 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 (11). [math]\lambda[/math] = a vector in 3-space = generic representation of the orientation of the total spin of each particle. In (11), [math]\lambda[/math] is coupled to Planck's constant to represent a physically-significant dynamical-variable; etc: so a vector is all that is required. NB: No "hidden variables" appear here: the natural physical variables of the experiment are quite sufficient; though their values may be unknown to us, hence the use of probability theory. [math]\lambda_{a^+}[/math] xor [math] \lambda_{a^-}[/math] = outcomes after device/particle interactions; etc. [math]\lambda_{a^+}[/math] is parallel to [math]a[/math]. [math]\lambda_{a^-}[/math] is anti-parallel to [math]a[/math] for [math]s = 1/2[/math]; and perpendicular to [math]a[/math] for [math]s= 1[/math]; etc. 1. Re (1): The generality of [math]Q[/math] and Malus' Method (see #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 (2): [math]\equiv[/math] identifies relations drawn from Bell (1964). (2)-(3) correctly represent Einstein-locality: a principle maintained and endorsed throughout this classical analysis. 3. Re (3): Bell (1964) does not distinguish between [math]\lambda[/math] & [math]\lambda'[/math] and we introduce s = intrinsic spin. [math](-1)^{2s}[/math] thus arises from [math]Q[/math] 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 (4): Integrating over [math]\lambda[/math], with [math]\lambda'[/math] eliminated: hence the coefficient, per note at #3. 5. Re (5): [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 (5)-(6) follows, each step from classical probability theory; [math]\int d\lambda \;\rho(\lambda) = 1[/math]. Thus, from (5): [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 (A2), with random variables: [math]P(A^+|Q) = P(A^-|Q)=P(B^+|Q)=P(B^-|Q) = 1/2.\;\;(A5)[/math] 6. Re (6): (6), 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 [math]Q[/math] 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 (6) to (7)-(10). Malus' Method proceeds directly from (6). (6a) is the generalised expectation for experiments [math]Q[/math]. [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 (11). 7. Re (7): Within Malus' capabilities, [math]W[/math] 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 (7), from (6). Alternatively, he could derive the same result (without experiment) from his famous Law. 8. Re (8): Within Stern & Gerlach's capabilities, [math]X[/math] 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 (8), from (6). Alternatively, they could derive the same result (without experiment) by including their discovery, [math]s =1/2[/math], in Malus' Law. 9. Re (9): From Aspect (2004), [math]Y[/math] would show (from observation): [math]P(B^+|Y,\,A^+)= cos^2 [s\cdot({a}, {b})]_Y = cos^2 ({a}, {b})\;\;(A8)[/math]: whence (9), from (6). 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 (A5); whence [math]P(B^+|Y,A^+) = cos^2 ({a}, {b}).\;\;(A8)[/math] 10. Re (10): From Bell (1964), [math]Z[/math] would show (from observation): [math]P(B^+|Z,\,A^+)= cos^2 [s\cdot({a}, {b})]_Z = cos^2 [({a}, {b})/2]\;\;(A10)[/math]: whence (10), from (6). 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 (6), with [math]s = 1/2[/math]; whence [math]P(B^+|Z,A^+) = cos^2 [({a}, {b})/2].\;\;(A10)[/math] 11. Re (11): 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. (11) 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 (11), all the intermediate 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. Conclusions: This wholly classical analysis delivers significant results, with no new physical experiments required. Proceeding from an acceptance of Einstein-locality (like Bell), Bell's theorem is refuted via undergraduate maths and logic. Bell's "perhaps harshly" is thus vindicated; perhaps more than expected: Bell (2004: 167). The theory here is in line with Bell's hope for a simple constructive model: see (11); i.e., in line with Bell's (often ignored) disquiet with his theorem and associated analyses: Bell (2004: 167). The theory thus delivers Bell's hope to see quantum "jumps" in the equations and not just in the talk: Bell (2004: 118). Our results apply equally to experiments with more than two particles; e.g., CRB, GHSZ, GHZ: delivering every QM outcome exactly. 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. To be clear: The theory's experimental predictions coincide exactly with those of standard quantum mechanics: with no departures. This study also shows that QM's "projection postulate" and "unitary evolution" are not contradictory (as commonly believed): this conclusion is supported by our access to a very different physically-significant interpretation; see (11). We conclude, via the rigour of under-graduate mathematics and logic: Einstein-locality prevails over non-locality. Non-locality is the misleading name given to the misunderstood mechanics associated with breaches of Bell's theorem. Bell's theorem is the incorrect claim that equations (2)-(4) cannot deliver (9)-(10). Bell was correct in hoping for a simple constructive model to illuminate analyses such as his. That model is here. 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 Watson

No breaks in equations were required or requested. The line spacing via <RETURN> key (and every other method tried) was always outside the [math] tags. Only today did it work!? I'll look into your "align" suggestion. Thanks.

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 Q0.3above. It was posted (in haste as the editing time elapsed), after many hours of formatting failures. There are no major errors there, afaik, but Q0.4 will be much improved in style and clarity. Q0.3a Q0.3 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 Q0.3 draft. To avoid past problems re formatting, I want to be sure the maths presentation is stable. 0Notes to follow, more anon, Gordon

Q0.4 Q0.3 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})].\;\;(7)^7[/math] [math]E(AB)_X =E(AB)_{'Stern-Gerlach'} = - \tfrac{1}{2} {a}\textbf{.}{b}.\;\;(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! 1. My cdot [math](\cdot)[/math] puzzle may be seen in the different cdot spacings in (6a) compared to (11a) and (11b). Why do the spacings differ when the codes appear to be identical? 2. How do I now line-space these equations neatly? A single "hit return button" doesn't do it. Thanks, Gordon

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

Hi to all, especially Cap'n Refsmmat for the warm welcome. By way of introduction, I'm not much at talking about myself. But it seems to me that SFN offers the possibility of excellent soft-ware and even better friends to help discuss and publish scientific articles. As to publishing: a wide range of readers can help to find typos, errors, improvements, etc. If they survive, such articles are then freely available on-line to all: and may be readily up-dated and discussed in associated forums. As an engineer, Einstein is my pin-up boy and Bell's theorem my current focus. Few realise that Bell, dissatisfied* with his theorem, hoped that a simple constructive model would resolve the issue satisfactorily. Such a model exists, as will become clearer when I master the soft-ware here. Being a slow learner don't help. The draft-article, classified as Speculative, is pitched at the level of under-grad maths and logic. Ciao, for now, Gordon Watson * "... 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 may be, 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 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). 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."

Thanks. And using google to find the FAQs: (link to the wrong SFN forum removed by moderator)

Are these links correct? Seem to link to Forums page; not specifically to FAQ or BB code?

Thanks. Sorry. No. The LaTeX was fine: the problem occurred when formatting the spacing between the LaTeX-formatted equations; trying for the "normal" uniform spacing -- knowing about your point -- which spacing is still not there. But I left it because edit time was running out. I'm gonna look at the source coding (better if an expert did THAT but) BUT I'm confident there was a bug to do with the "return" key action: which is what I normally use for spacings? In short: Just look at the grouping there and realise I wanted it like (1)-(6) right through. AND THAT at one stage they were all like that EXCEPT one: It was then, "adjusting" that ONE, that the drama unfolded; including in lower paragraph spacings. Help. Thanks.

Thanks for this! Then I next I had big problems when formatting equations in http://www.sciencefo...ed/#entry677673 A small change to an equation at the top produced consequences in paragraphs at the bottom of the page. Also, attempting to have uniform equation spacings changed the text spacing in paragraphs further down. I'm doing something wrong here. No problems from old forum; must be some rules I'm missing? More help would be appreciated. With thanks again, Gordon YES, it was as simple as that! Great! Many thanks; and wondering if you can help with my page formatting problems, please? See my earlier thankful reply to Cap. Thanks again, Gordon

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.

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

What is the EDIT time-limit please? The "timed-out from editing" limit after posting. Is there a latex sand-box (or external link) that matches the SFN coding requirement? I'm familiar with itex. And http://www.codecogs....x/eqneditor.php But not all codings that work at "other physics forums" are happy here: see example below. Help! Thanks, Gordon ((2s\cdoth/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\cdoth/2\pi)_Q.\;\;(11a)^{11} ((2s\cdoth/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\cdoth/2\pi)_Q.\;\;(11b)^{11} [math]((2s\cdoth/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\cdoth/2\pi)_Q.\;\;(11b)^{11}[/math]

Q0.3 with extensive notes0 to 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

I'll be happy to point you to some good references AND present an alternative view, a neglected one: one in line with Bell's own views; one that shoots him down! So I'll be back. In meantime here's something to ponder: My wholly classical analysis of Bell's theorem 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 may be, 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 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). [All emphasis and [.] added by GW.] 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 (1999): 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 colleague of de Broglie. More soon, Gordon