Q_{\textit{0.2}}
Q_{\textit{0.1}} \;with \; new\; notes^\textit{0}\;and\; equations\;to\;facilitate\;discussion.
Thanking ThomasT for his questions, billschnieder for his answers:
Toward \;Bell's \; (2004: 167) \;hope\; for\;a\;simple\;constructive\;model.
Q \in \{W, X, Y, Z\}.\;\;(1)^1
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
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
E(AB)_Q \equiv ((-1)^{2s} \cdot \int d\lambda\;\rho (\lambda )\;AB)_Q \;\;(4)^4
=((-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
= [(-1)^{2s}]_Q \cdot[ 2 \cdot P(B^+|Q,\,A^+) - 1].\;\;(6)^6
E(AB)_W = E(AB)_{'Malus'} = (cos[2 ({a}, {b})])/2.\;\;(7)^7
E(AB)_X = E(AB)_{'Stern-Gerlach'} = - ({a}\textbf{.}{b})/2.\;\;(8)^8
E(AB)_Y = E(AB)_{\textit{Aspect (2004)}} = cos[2 ({a}, {b})].\;\;(9)^9
E(AB)_Z = E(AB)_{\textit{EPRB/Bell (1964)}} = - {a}\textbf{.}{b}.\;\;(10)^{10}
((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}
((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}
QED: \;A \;simple \;constructive \;model \;delivers \;Bell's \;hope \;(2004: 167)!
Notes:
0. From
ThomasT's questions to
billschnieder's answers, this wholly classical analysis begins with the acceptance of Einstein-locality (EL). 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.]
Replacing RHS = by \neq, BT-inequalities may be seen in (9)-(10) above. But 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 !998: 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)-(4) show that Bell's important functional protocol [Bell 1964: (1), (2), (12)-(14)] may be satisfied: i.e., such functions exist. Moreover, (2)-(3) capture EL: which is all that is required for (4)-(10) to go through. 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 i-th particle-pair's HVs \lambda_i and \lambda_i': with the HVs here pair-wise drawn from infinite sets, no two pairs are the same; though W and X may be modified to improve this, somewhat.
\oplus = xor; exclusive-or.
a, b = arbitrary orientations: for W and X, in 2-space, orthogonal to the particles' line-of-flight; for Y and Z, in 3-space (from the spherical symmetry of the singlet state).
s = intrinsic spin, historically in units of h/2\pi. Units of h/4\pi would be better: 4\pi significant in terms of spherical symmetries in 3-space. (PS: A related thought for the critics: This classical analysis of four experiments, Q, yields the better value for unit spin angular momentum, h/4\pi. How come?)
\delta_{a} = Alice's device, its principal axis oriented a; etc.
\delta_{b}' = Bob's device, its principal axis oriented b; etc.
\delta_{a} \lambda\rightarrow \lambda_{a^+}\oplus \lambda_{a^-} = an Alice-device/particle interaction terminating when the particle's \lambda is transformed to \lambda_{a^+} xor \lambda_{a^-} (the device output correspondingly transformed to \pm1); 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).
\lambda_{a^+} xor \lambda_{a^-} = HV outcomes after device/particle interactions; etc. \lambda_{a^+} is parallel to a. For s = 1/2, \lambda_{a^-} is anti-parallel to a; for s = 1, \lambda_{a^-} is perpendicular to a; etc.
1. Re (1): The generality of Q and Malus' Method (
#6 below), enables this wholly classical analysis to go through. Q embraces:
W = 'Malus' (a classical experiment with photons) is Y with the source replaced by a classical one (the particles pair-wise correlated via identical linear-polarisations).
X = 'Stern-Gerlach' (a classical experiment with spin-half particles) is Z with the source replaced by a classical one (the particles pair-wise correlated via antiparallel spins).
Y = Aspect (2004).
Z = EPRB/Bell (1964).
2. Re (2): \equiv identifies relations drawn from Bell (1964). (2) & (3) correctly represent
Einstein-locality: a principle maintained throughout this classical analysis.
3. Re (3): Bell (1964) does not distinguish between \lambda and \lambda', and we introduce
s = intrinsic spin. (-1)^{2s} 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 (4): Integrating over \lambda, with \lambda' eliminated: hence the coefficient, per note at
#3.
5. Re (5): P denotes Probability. A^+ denotes A = +1, etc. The expansion is from classical probability theory:
causal-independence and logical-dependence carefully distinguished. The probability-coefficients +1, -1, -1, +1 (respectively), represent the relevant A\cdot B product: each built from the relevant Einstein-local (causally-independent) values for A and B.
The reduction (5)-(6) follows, (A1)-(A4), each step from classical probability theory; \int d\lambda \;\rho(\lambda) = 1. From (5):
P(A^+B^+|Q) -P(A^+B^-|Q)-P(A^-B^+|Q)+P(A^-B^-|Q)\;\;(A1)
=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)
=[P(B^+|Q,A^+)-P(B^-|Q,A^+)-P(B^+|Q,A^-)+P(B^-|Q,A^-)]/2\;\;(A3)
= 2 \cdot P(B^+|Q,\,A^+) - 1.\;\;(A4)
NB: In (A2), with random variables: P(A^+|Q)=P(A^-|Q)=P(B^+|Q)= P(B^-|Q) = 1/2.\;\;(A5)
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 P(B^+|Q,\,A^+).
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 (6) to (7)-(10).
NB: P(B^+|Q,\,A^+) = P(\delta_{b}' \lambda_i'\rightarrow \lambda'_{b^+}|Q,\,\delta_{a} \lambda_i\rightarrow \lambda_{a^+}): a prediction of the normalised frequency with which Bob's result is +1, given that Alice's result is +1; see also (11).
7. Re (7): Within Malus' capabilities, W would show (from observation):
P(B^+|W,\,A^+) = [cos^2 ({a}, {b}) + 1/2]/2= ([cos^2 [s \cdot ({a}, {b})] + 1/2]/2)_W \;\;(A6) 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, X would show (from observation):
P(B^+|X,\,A^+) = ([cos^2 [s \cdot ({a}, {b})] + 1/2]/2)_X = [cos^2 [({a}, {b})/2] + 1/2]/2\;\;(A7): whence (8), from (6). Alternatively, they could derive the same result (without experiment) by including their discovery, s = 1/2, in Malus' Law.
9. Re (9): Conducted by Aspect (2004), Y would show (from observation):
P(B^+|Y,\,A^+) = cos^2 [s\cdot({a}, {b})]_Y = cos^2 ({a}, {b})\;\;(A8): whence (9), from (6). To see this, Aspect (2004: (3)) has (in our notation):
P(A^+B^+|Y) = [cos^2 ({a}, {b})]/2 = P(A^+|Y)P(B^+|Y, A^+) = P(B^+|Y, A^+)/2 (A9), from (A5); whence
P(B^+|Y, A^+) = cos^2 ({a}, {b}).\;\;(A8)
10. Re (10): Analysed by Bell (1964), Z would show (from observation):
P(B^+|Z,\,A^+) = cos^2 [s\cdot({a}, {b})]_Z = cos^2 [({a}, {b})/2]\;\;(A10): whence (10), from (6). Unlike Aspect (2004), Bell (1964) does not derive subsidiary probabilities. Instead, Bell (1964: (3)) has (in our notation):
E(AB)_Z = -({a}. {b}) = -[ 2 \cdot P(B^+|Z,\,A^+) - 1] (A11), from (6), with s = 1/2; whence
P(B^+|Z, A^+) = cos^2 [({a}, {b})/2].\;\;(A10)
11. Re (11): With s\cdot h a driver, the dynamic-process
((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)
terminates when the trig-argument is 0 or ∏; the move to such an argument determined by this fact: one of \lambda_{a^+} xor \lambda_{a^-} is a certain terminus, the other impossible: a "push-me/pull-you" dynamic on the \lambda_i 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: with Planck's constant h confined to the outer extremities on both sides of (11). Thus all the maths is classical: LHS-(s\cdot h) drives the particle/device interaction; RHS-(s\cdot h) is a potential driver for a next interaction.
References:
Aspect (2004):
http://arxiv.org/abs/quant-ph/0402001
Bell (1964):
http://www.scribd.com/doc/51171189/Bell-1964-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, I understand that one reviewer was a former student of de Broglie.
With questions, typos, improvements, critical comments, etc., most welcome,
GW
(as opposed to Bell's positive one, as discussed and delivered in the above link).