
#217
Apr3012, 11:01 AM

P: 3,178





#218
Apr3012, 11:07 AM

P: 678





#219
Apr3012, 11:29 AM

P: 269





#220
Apr3012, 11:55 AM

P: 269

Now if you allow A and B to take different values depending on the values of a and b respectively, things will become slightly more interesting. 



#221
Apr3012, 12:05 PM

P: 678





#222
Apr3012, 01:03 PM

P: 678





#223
Apr3012, 02:08 PM

P: 678

Therefore adding the "a" and "b" labels? We have:
[itex]E(A_aB_b)_V = 2 \cdot P(B^+_bV,\,A^+_a)  1 [/itex] [itex]E(A_aB_b)_V = 2 \cdot P(B^+_bV,\,A^_a) + 1 [/itex] and 



#224
Apr3012, 02:28 PM

P: 678

[itex]A(a, \lambda) = \int d\lambda\; \delta (\lambda  a) \cos[2s\cdot(a,\lambda)] = \pm 1[/itex] where (a,λ) is the angle between the vectors a and λ. What does XOR add to the whole thing? Do you mean [itex] a \in [a^+, a^][/itex]? Am I understanding you correctly? 



#225
Apr3012, 04:39 PM

P: 375

1. You should always feel free to pick me up on my formatting. I thought it would be clear that [2s(a, λ)] was the argument of a trig function; (a, λ) is exactly as you say. 2. So your CENTRAL expansion is correct. 3. BUT your RHS ±1 is incorrect. You only have only delivered +1. That's where XOR comes in: to deliver the PLUS XOR the MINUS one. 4. a is a, the orientation of the principal axis of Alice's device. 5. The set {a^{+}, a^{}} represents the orientations that λ may be transformed to via the particle/device interaction: δ_{a}λ → a^{+}°a^{}. ALT: δ_{a}λ → {a^{+}, a^{}} for W, X, Y, Z. With apologies for the hurried shorthand. ....... EDIT: Bill, et al., to avoid the XOR: [itex]A({\textbf{a}}, \lambda) = \int d\lambda\; (\delta_{\textbf{a}} \lambda \rightarrow \left \{{\textbf{a}}^+ , {\textbf{a}}^ \right \}) \cos[2s\cdot({\textbf{a}}, \lambda)] = \pm 1.[/itex] Though the insistence on a (aBOLD) throughout does nothing to match the prettiness in Bill's offering!? Apart from that deficiency, an advantage of this format is this (imho): 1. A function is a process that transforms an element of a set into exactly one element of another set. 2. The Alicedevice/hiddenvariable interaction ([itex]\delta_{\textbf{a}} \lambda[/itex]) is a process that transforms ([itex]\rightarrow[/itex]) an element of a set ([itex]\lambda \in \Lambda[/itex]) into exactly one element of a dichotomic set [itex](\left \{{\textbf{a}}^+ , {\textbf{a}}^ \right \})[/itex]. ....... PS: It was a mistake for me to introduce this maths distraction as a point of interest for DK, etc. The maths here goes through on the basis that Bell's A and B are sound representations of Einsteinlocality AND we accept that such functions exist. For me, DK's important contribution is an adequate approximation to Bell's A and B: it was on that basis that we are where we are now. We can come back to these deltas later, if need be. 



#226
Apr3012, 07:36 PM

P: 269

But it is not that simple: if you change the value of a and leave b intact then you are allowed to assign different values to A(a,λi), but you must keep the same values for B(b,λi) and P(λiV), because these do not depend on a. Same for B and b. Even better way of doing that would be to define a={a_{j}}, b={b_{k}} [itex]A^{\lambda_i}_{a_j}=A(a_j,\lambda_i)=\pm 1[/itex] and similarly for B. Then your result for E(AB) will be local realistic. Unfortunately you will not be able to reproduce QM predictions with it. 



#227
Apr3012, 08:00 PM

P: 678

[itex]A(a,\lambda_1) = +1 \;,\; A(a,\lambda_2) = +1 \;,\; A(a,\lambda_3) = 1 \;,\; A(a,\lambda_4) = 1 \;,\; [/itex] The outcome of the function depends on BOTH "a" and "λ" ! The way you should read this is: λ1 is the hidden variable which together with the setting "a" results in a +1 outcome. Change the *value* of "a" and of course you will need a different *value* for λ1 to obtain the result +1. This is not different for Bell's case. The only outcomes possible are ±1. For any given *value* of "a", λ1 represents the corresponding *value* of λ that together give you a +1 outcomes. Don't you see that? [itex]E(A_aB_b)_V = 2 \cdot P(B^+_bV,\,A^+_a)  1[/itex] Different *values* for "a" and "b" give you different *values* for [itex]P(B^+_bV,\,A^+_a)[/itex] and therefore a different value for [itex]E(A_aB_b)_V[/itex] I do not see a legitimate criticism here. What you are saying does not make any sense. 



#228
Apr3012, 08:19 PM

P: 269

Sorry, I can't even begin to discuss your 'equation', it makes me cringe. Did you finally figure out why ∫f(λ)dλ is not a function of λ? I suggest you do that before embarking on a quest to resolve nature's deepest mysteries. 



#229
Apr3012, 08:39 PM

P: 678





#230
Apr3012, 09:03 PM

P: 269

I'm outta here. Good luck with your quest. 



#231
Apr3012, 11:20 PM

P: 375

Just be sure to recall that your "havingago" got us started on this path. That's something! (Maybe one for the grandkids? ) PS: I thought you and Bill were doing a good job. So come back soon ... now that you're a little wiser (as am I), thanks to Bill. PS: I've improved that cringemaking expression; see http://www.physicsforums.com/showpos...&postcount=225 GW 



#232
Apr3012, 11:39 PM

P: 1,583





#233
May112, 03:34 AM

P: 375

If you refer to the "Malus Method" as the main issue, keep in mind that its use is still limited (here) to classical analysis with a focus on ontology (i.e., the nature of λ, the HVs; the nature of particle/device interactions  δ_{a}λ, δ_{b}λ'  from a classical point of view). In that way it differs from some "QM Methods". And in that way it DOES provide "a mechanism": for the method itself was prompted by the search for the "underlying" mechanics; and it would not be up for discussion if nothing of interest had been found: the interesting point in the OP being that of finding functions satisfying Bell's A and B. In brief, the mechanics goes thus: The HVcarrying particles, their HVs pairwise correlated by recognised mechanisms, separate and fly to Alice and Bob. Interaction with the respective devices leads to a local transformation of each HV, most clearly seen in W where photons (initially pairwise linearlypolarised identically) are transformed into pairs with different linearpolarisations. (Representing a fact accepted early in the foundations of QM: a "measurement" perturbs the measured object.) ... ... ... Since the classical analysis is straightforward, and Einsteinlocal (but see below), I suggest you study it and then see how it applies to your interest in Herbert's Paradox and its mechanics. If you ensure that every step in your classical analysis satisfies Einsteinlocality, the accompanying part of the analysis MUST relate to determining the distribution of the Einsteinlocal outcomes. That brings in probability theory ("maths is the best logic") to derive the frequencies that will be found experimentally. And, classically, you need to clearly distinguish between causal independence and logical dependence. So you should check to see how EL is dealt with (once and for all, at the start of the analysis), and then ensure that the remaining classical maths is focussed on determining the frequencies of the various outcomes that will be found experimentally: with no unintended disruption or fiddlingwith EL; nor cheating. As to what Bell proved, it is my opinion that he proved that EPR elements of physical reality are untenable. (A conclusion I support.) So, imho, it is possible to see EL maintained in Bell's work, and popular ideas about reality condemned. Do you wonder then: Where does the classical analysis here depart from Bell's analysis? You will see that nowhere here, classically, do we address a third device, at orientation c, in the same context as discussing an experiment with Alice (deviceorientation a) and Bob (b).** That move by Bell, it seems to me, confirms his focus on EPR elements of reality. For, otherwise, he needs must recognise that a measurement locally perturbs the measured system ... and until that perturbation, EPR elements of reality (generally) do not exist (IMHO). Or, to put it another way: the move to c follows from an acceptance of EPR's epr; though there may be other views of reality that also permit it ... remembering that Bell's theorem is not a property of quantum theory (Peres 1995, 162), so it is not unreasonable to examine the extent to which it is NOT a property of classical theory. PS: Discussion of this line would be best in a new thread, it seems to me. (The focus here should be on finding errors in the classical approach.) ** That is: The classical analysis ranges over (a, b), (b, c), (a, c); reflecting all possible real experiments, but no impossible ones. Also: The HVs are classically sourced from infinite sets so that (here, in this case) no two pairs of particles are the same (P = 0). With thanks again, GW 



#234
May112, 08:51 AM

P: 678




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