FrediFizzx wrote:In the paper that this thread is about (stay on topic, Richard), I don't understand eq. 19 to 22. Maybe you can explain more about them?
Thanks for the question, Fredi. I should probably have prefaced CLR with this health warning: Despite the fact that CLR employs undergrad maths and logic, disentangling quantum entanglements may cause dizziness. See a doctor if the pain persists.
Let me add (with apologies) that I'll need a doctor myself after pecking out this interim reply on a small-screen computer!
As a prelim to explanation: From Section 2, we want the CLR/EPRB dynamics to do the talking: not words. In Appendix A.7, there's a brief comment on CLR dynamics: with particular reference to the IMPORTANT "If … Then …" component of its maths. Overarching the dynamics and the maths is this fact: each step in each equation is physically significant: so when you can't see a way through, ask a question re the physics applying at that specific point.
Further, compared to QM: CLR is based on functions (which are more general than operators) in R^3 (more real than Hilbert space). ... ... . So:
(19) is CLR's response to Bell (1964) as reflected in (1) and LHS (2). For (19) is built from CLR's version of (1) -- ie, (16)-(18) -- and then LHS (2).
(20) is self-explanatory -- as a function-set -- from (19), by observation. Thus (20) is a simple representation of EPRB as we seek to understand its correlated outputs. For each function in (20) -- like each function in CLR -- has dynamical consequences; ie, in (20), we see Alice's and Bob's SGDs at work via {SGD(a), SGD(b')| EPRB}; for (20) is just the set {Q, R, Q', R'| EPRB}.
Now, when we bring (16) and (17) together to determine <AB>, there is just one independent variable in the resulting combination: for λ + λ' = 0 in each EPRB spin-conserving decay. Which is very convenient because one independent variable allows us to apply the maths/implication of If … Then …. to record/employ a fact about the "dependent variable".
For what is revealed by a test on the independent variable has immediate
factual consequences for what is
then revealed about the twinned "dependent" variable. (Which is not some super-luminal/non-local effect BUT late/delayed FACTUAL news about a pristine property that was in existence immediately after the relevant spin-conserving decay; a property of one twin, NOW revealed by a test on the other twin. A property hidden until revealed by testing; such is the nature of many beables.)
So, in (21) we see the consequence of having one independent variable in our function-set: we can eliminate the Q that acts on the eliminated "dependent" variable.* So, the selected independent variable λ' (your choice; but the paper has pedagogically selected λ' ) in its response function R' is acted on by its Q', to reveal standard SGD(b') outcomes ±1; to THUS reveal the IF component of our maths: In this composite case, λ' = ±b'. The THEN then follows in the remaining response function: (-λ'.a) = (-(±b').a),
respectively. So the reducing full-function-set consists of {(-(±b').a)(±1)} = {-b'.a} = -b'.a: QED. (Your understanding perhaps not helped here by my shortcut with ±b' combined. SO, as a good exercise, I suggest that you do them separately.)
Thus (22) follows: with this expanded note to the introductory phrase (for aid/clarity): "or, equivalently, completing (19) WITH THE SUPERFLUOUS* Q-FUNCTION DELETED, as at the start of (21): ...
Somewhat surprisingly, I believe that such analysis is clearer with increasing numbers of particles: so you may find that wresting with (26)-(37) is a better introduction-to/explanation-of (19)-(22). Particularly if you choose a different beable; contrary to the example in the paper. The guiding light being that here is only one correct answer; as per the paper.
NOW: What is the physical significance of CLR's reduction of (19) to (21) or (22)?
Under CLR's If/Then maths: IF pristine λ' is revealed by Bob's test to be equivalent to +b' -- THEN it is a fact (a pre-existing one) that pristine λ (= -λ') is equivalent to -b'. A result that can be experimentally confirmed by an Alice test employing SGD(-b) -- with no prime here because this is an Alice setting! But independent of Alice doing such a test: λs equivalent to -b' (= -b) will give a spread of results under an Alice test with her SGD set at anything other that ±b.
This spread is demonstrated in (23)-(24) where the familiar trig functions are seen. In other words; the set of λ
~ -b makes law-based transitions to ±a under Alice's test with SGD(a). This spread is understandable because the SGD squeeze-function maps an infinity of λs to just two DECs; so our factual knowledge is limited to those two facts -- plus the distributive LAW.
Concluding, for now, Fredi: I'm not sure how helpful the above will be. But -- if you have any interest in Bell's EPR work -- I encourage you to understand every move in every CLR equation. And I'm happy to help; at every step: together finding improvements and clarifications along the way.
PS: Moreover, by way of encouragement; apart from the fact that you'll be on Einstein's side: I doubt we'll find errors! For every CLR result accords with the experimental evidence: CLR at the same time revealing (from first principles) a chain of errors in Bell's work -- for which there is neither experimental confirmation nor logical justification.
* PPS: For the mathematically inclined, the CLR maths proceeds equally satisfactorily if the superfluous Q is NOT removed and the distributive is used. In that case, in EPRB, a selected Q-function is allowed to operate on the other Q-function by becoming a Q-functional. The selection of such a functional, it will be seen, is equivalent to the elimination of the redundant Q.
With best regards; and thanks again; Gordon
Posted: Tue Apr 29, 2014 12:10 am