A classical challenge to Bell's Theorem?

[Gordon Watson]

... $e$ also corresponds to $+1$. ...

= $e$, an EPR-epr, corresponds to $+1$: GW translation.​

Bill: Since $+1 = A^+$, I believe the above quote puts you clearly in Bell's camp re EPR eprs. To pinpoint this (and identify a consequent difficulty with Bell's theorem), we follow words from Bell (2004):

"To explain this dénouement without mathematics I cannot do better than follow d'Espagnat (1979; 1979a)," Bell (2004: 147). Where we find:

"These conclusions require a subtle but important extension of the meaning assigned to a notation such as $A^+$. Whereas previously $A^+$ was merely one possible outcome of a measurement made on a particle, it is converted by this [Bell-style] argument into an attribute of the particle itself [prior to measurement]. To be explicit, if some unmeasured proton has the property that a measurement along the axis $A^+$ would give the definite result $A^+$, then that proton is said to have the property $A^+$. In other words, the physicist has been led to the conclusion that both protons in each pair have definite spin components at all times," d'Espagnat (1979: 134), with GW [.] and emphasis added.

"The key point is the definition of "property $A^+$,"" d'Espagnat (1980: 9).​

So here's the difficulty (a product of naive-realism, it seems to me): ... "to have property $A^+$ ... at all times" is to imply that the pristine proton was polarized $A^+$prior-to-its-test. HOWEVER: as Bell (2004: 82, his emphasis) acknowledges:

"... each particle, considered separately IS unpolarized ...".​

So we have a contradiction; one supporting my view that the "corresponds" in the EPR definition gives rise to problems. SO, to be clear: I do not support Bell's "interpretation" of what EPR meant; if that is what Bell is doing. But it is your interpretation, according to my careful reading:

So we differ now re ERP's eprs; but we can, surely, agree soon?

PS: If that IS what EPR meant, then I'm disagreeing with them too: See my previous post.

 

[harrylin]

[..] if some unmeasured proton has the property that a measurement along the axis $A^+$ would give the definite result $A^+$, then that proton is said to have the property $A^+$. In other words, the physicist has been led to the conclusion that both protons in each pair have definite spin components at all times," d'Espagnat (1979: 134), with GW [.] and emphasis added.
[..]
So here's the difficulty (a product of naive-realism, it seems to me): ... "to have property $A^+$ ... at all times" is to imply that the pristine proton was polarized $A^+$prior-to-its-test. [..]

Hi I have no idea what the two of you mean with "EPR-epr" (maybe you can write it out?).
Anyway, I find it a weird approach to QM if one assumes that a resulting feature of a measurement must necessarily have existed before the measurement - and that's certainly not what EPR meant. They certainly knew that a measurement is an interaction which affects that what is measured.

 

[billschnieder]

Hi I have no idea what the two of you mean with "EPR-epr" (maybe you can write it out?).

Einstein-Podolsky-Rosen's element of physical reality. Sorry

 

[billschnieder]

= $e$, an EPR-epr, corresponds to $+1$: GW translation.​

Bill: Since $+1 = A^+$, I believe the above quote puts you clearly in Bell's camp re EPR eprs. To pinpoint this (and identify a consequent difficulty with Bell's theorem), we follow words from Bell (2004):

"To explain this dénouement without mathematics I cannot do better than follow d'Espagnat (1979; 1979a)," Bell (2004: 147). Where we find:

"These conclusions require a subtle but important extension of the meaning assigned to a notation such as $A^+$. Whereas previously $A^+$ was merely one possible outcome of a measurement made on a particle, it is converted by this [Bell-style] argument into an attribute of the particle itself [prior to measurement]. To be explicit, if some unmeasured proton has the property that a measurement along the axis $A^+$ would give the definite result $A^+$, then that proton is said to have the property $A^+$. In other words, the physicist has been led to the conclusion that both protons in each pair have definite spin components at all times," d'Espagnat (1979: 134), with GW [.] and emphasis added.

"The key point is the definition of "property $A^+$,"" d'Espagnat (1980: 9).​

So here's the difficulty (a product of naive-realism, it seems to me): ... "to have property $A^+$ ... at all times" is to imply that the pristine proton was polarized $A^+$prior-to-its-test. HOWEVER: as Bell (2004: 82, his emphasis) acknowledges:

"... each particle, considered separately IS unpolarized ...".​

So we have a contradiction; one supporting my view that the "corresponds" in the EPR definition gives rise to problems. SO, to be clear: I do not support Bell's "interpretation" of what EPR meant; if that is what Bell is doing. But it is your interpretation, according to my careful reading:

So we differ now re ERP's eprs; but we can, surely, agree soon?

PS: If that IS what EPR meant, then I'm disagreeing with them too: See my previous post.

I do not agree with this. I have repeatedly highlighted that "corresponds" is not the same as "equivalent" or "the same as". If only you would take up the simple example I proposed earlier, all of this will become clearer but I do not know how else to explain myself but certainly you have misunderstood me.

 

[Gordon Watson]

I do not agree with this. I have repeatedly highlighted that "corresponds" is not the same as "equivalent" or "the same as". If only you would take up the simple example I proposed earlier, all of this will become clearer but I do not know how else to explain myself but certainly you have misunderstood me.

Bill, please: would you mind expanding on what exactly it is that you disagree with? I take it you agree that I have correctly cited Bell's position? Then, after that, all your "expansion" requires, it seems to me, is to state what $e$ is IN your terms.


I'm avoiding your simple example (for now) because it is EPR that we need to be satisfied with. SO EVEN IF I analysed your example, the underlined question would still remain (for me, at least). So, please, a direct answer in your terms re $e$ and we're finished, I'm believing. Thanks, GW.

PS: I am also believing that Bell's interpretation of an EPR-epr is more common than many realize. And I'm taking it that one of your "disagrees" is that you do NOT agree with him?

PPS: There may be subtle differences, and NOT SO subtle differences, between "corresponds", "equivalent" and "the same as" --- for IT seems to me that it very much depends on the CONTEXT! That's why the EPR context is my primary focus here (for now). Me not wishing to be stubborn, just insistent, on that one point.

 

[Gordon Watson]

... I find it a weird approach to QM if one assumes that a resulting feature of a measurement must necessarily have existed before the measurement - and that's certainly not what EPR meant. They certainly knew that a measurement is an interaction which affects that what is measured.

Hi Harald, I agree: "Disturbance" was well-known and accepted from the early days of QM. YET we find it occasionally ignored in modern times. So interesting questions arise: Did EPR ignore it?* Did Bell ignore it (per citations given above)? Or was he presenting what he thought were EPR's views!? And where does Herbert sit? Cheers, G

*PS: Note that EPR start their definition of an epr with: "If, without in any way disturbing a system, we can predict ..." But did they make their predictions correctly on the basis of what had been revealed by a disturbed system?

As you can see in my dialogue/struggles with Bill, I do not see this EPR-epr business to be as straight-forward as many suggest: And that is why I favour my $e$ and $q$ analysis (above).

 

[Gordon Watson]

$Q_{\textit{0.2}}$

$Q_{\textit{0.1}} \;with \; new\; notes^\textit{0}\;and\; equations\;to\;facilitate\;discussion.$

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

 

[Gordon Watson]

... There is no requirement whatsoever that any theory must be causal to "match our experience". All that is required is that there be agreement between theory and observation. (Emphasis added, GW)

DrC, not wishing to side-track the "Foundations" thread, I'd welcome your elaboration of the above emphasised sentence in the context of this thread:

1. How do you reconcile your support for Bell's "theorem" with the emphasised sentence?

2. Given your early oppositional remarks to the theme of this thread, how do you now (see sentence) reconcile that opposition with the agreement between theory and observation that is here: https://www.physicsforums.com/showpost.php?p=3905795&postcount=287.

3. And, please, relevant to both Q1 and Q2: ... especially having regard to Bell's own view of his "theorem" (Bell 2004: 167; Mermin 1993: 814: see extracts in Post #287 immediately above)?

Thanks, GW

 

[Gordon Watson]

Another quick update: the extension to spinor electrodynamics (which is more realistic than scalar electrodynamics) has been described in a short article in Journal of Physics: Conference Series ( http://dx.doi.org/10.1088/1742-6596/361/1/012037 - free access):

"2. After introduction of a complex 4-potential (producing the same electromagnetic field as the standard real 4-potential), the spinor field can be algebraically eliminated from spinor electrodynamics; the resulting equations describe independent evolution of the electromagnetic field.

3. The resulting theories for the electromagnetic field can be embedded into quantum field
theories."

The details can be found in the references of the article.

Hi Andrey, and congratulations on the publication of another advance in your work. However, with respect to the passage copied below AND your concern about breaching Bell inequalities, I suggest that you need to carefully distinguish this dichotomy, imho:

The (1) "violation of a Bell inequality" is NOT the same as (2) "falsifying local realism".

I am certain that valid experiments (and good theory) will continue to deliver (1): a violation of Bell inequalities. I am confident that no experiments will ever falsify (2): local realism (properly defined).

To these ends, and to this latter end in particular, I'd welcome your comments on the breaching of Bell inequalities AND the explicit local realism (and any other matter) in https://www.physicsforums.com/showpost.php?p=3905795&postcount=287

PS: As previously discussed, I believe that the BOLD-ed sentence below greatly weakens your work. Me believing it to be a FALSE hope (as opposed to Bell's positive one, as discussed and delivered in the above link).

With best regards,

Gordon
....

From http://iopscience.iop.org/1742-6596/361/1/012037/pdf/1742-6596_361_1_012037.pdf -- "Of course, the Bell inequalities cannot be violated in such a theory. But there are some reasons to believe these inequalities cannot be violated either in experiments or in quantum theory. Indeed, there seems to be a consensus among experts that “a conclusive experiment falsifying in an absolutely uncontroversial way local realism is still missing” [4]. On the other hand, to prove theoretically that the inequalities can be violated in quantum theory, one needs to use the projection postulate (loosely speaking, the postulate states that if some value of an observable is measured, the resulting state is an eigenstate of the relevant operator with the relevant eigenvalue). However, such postulate, strictly speaking, is in contradiction with the standard unitary evolution of the larger quantum system that includes the measured system and the measurement device, as such postulate introduces irreversibility and turns a superposition of states into their mixture. Therefore, mutually contradictory assumptions are required to prove the Bell theorem, so it is on shaky grounds both theoretically and experimentally and can be circumvented if, for instance, the projection postulate is rejected. [Emphasis added by GW: other issues arising not addressed here.]​

 

[rlduncan]

Hi Gordon,

Re: Post # 287

I am glad to see that someone has incorporated Malus in a classical way and compared to the qm predictions. If possible, could you summarize and/or elaborate in more detail?

 

[Doc Al]

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