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The following idiotic "critique" of my local model is by Richard D. Gill. He has posted it at this smear site:
https://pubpeer.com/publications/AEF49D ... BA21F824B4Richard D. Gill wrote:(The equation numbers refer to arXiv v4 of my paper:
https://arxiv.org/pdf/1405.2355v4.pdf)
According to (55) and (56), A(a, lambda) = lambda and B(b, lambda) = - lambda where lambda = +/-1. This should lead to E(a, b), computed in (60)-(68), equal to -1. But instead the author gets the result - a . b. How is it done?
Notice formula (58) where s_1 and s_2 are argued to be equal, leading in (59) to L(s_1, lambda)L(s_2, lambda) = -1. This result is then substituted inside a double limit as s_1 converges to a and s_2 converges to b in the transition from equation (62) to (63).
So s_1 and s_2 are equal yet converge to different limits a and b.
But that is not enough. A second trick is put into play a few lines later. According to (57) we should have L(a, lambda)L(b, lambda) = D(a)D(b) independent of lambda, which means that the step from (65) to (66) can't be correct.
To take that step he uses (50), but this contradicts (51) and (52). If L(a, lambda) = lambda I a and L(b, lambda) = lambda I b then L(a, lambda)L(b, lambda) = -ab independent of whether lambda = -1 or +1 (lambda and I both commute with a and b; lambda^2 = 1, I^2 = -1)."
Each of Gill's idiotic misrepresentations of my Clifford-algebraic calculation can be disposed off easily. They only expose his incompetence. Let me do that one by one:
"According to (55) and (56), A(a, lambda) = lambda and B(b, lambda) = - lambda where lambda = +/-1. This should lead to E(a, b), computed in (60)-(68), equal to -1. But instead the author gets the result - a . b. How is it done?”Eqs. (55) and (56) of my paper are trivially correct. But it is silly of Gill to write A(a, lambda) = lambda, because A = +/-1 is an outcome of a measurement result, whereas lambda is the orientation of the 3-sphere, which is the hidden variable in my model. And similarly it is silly of Gill to write B(b, lambda) = - lambda. Gill either deliberately writes this incorrectly to mislead the reader, or he is genuinely clueless about what is going on.
But it is true that A = +/-1 and B = -/+1 gives AB = -1 for a given lambda = +/-1. So naively, if one ignores the fact that lambda is an orientation of the 3-sphere and ignores the fact that initial spin-0 is conserved in any EPR-Bohm type experiment leading to my eq. (59), then one may mistakenly believe that E(a, b) = -1 always. But “E(a, b) = -1 always” can hold only if the conservation of spin angular momentum is violated. If one respects the conservation of spin angular momentum as one absolutely must, then inevitably E(a, b) = -a.b, as rigorously proven in eqs. (60) to (68) of my paper linked above. So there is absolutely no way to obtain anything other than the result E(a, b) = -a.b without either ignoring the geometry and topology of the 3-sphere or violating the conservation of angular momentum, or both.
“Notice formula (58) where s_1 and s_2 are argued to be equal, leading in (59) to L(s_1, lambda)L(s_2, lambda) = -1. This result is then substituted inside a double limit as s_1 converges to a and s_2 converges to b in the transition from equation (62) to (63).
So s_1 and s_2 are equal yet converge to different limits a and b.”There is nothing mysterious about this. To begin with, s_1 and s_2 are not just “argued to be equal”, they are necessarily equal because of the conservation of spin angular momentum. This is clear from eq. (58) and the fundamental duality relation between the vectors s_i and bivectors L_i. In the standard notation of geometric algebra the duality relation is given by a /\ b = lambda I.(a x b). Thus the product vector a x b is orthogonal to the plane a /\ b, with lambda fixing one of the two possible sides of the bivectorial plane. Eq. (59) thus follows trivially from the conservation of spin angular momentum. In fact, one can take eq. (59) as a statement of the conservation of spin angular momentum. See also the revised version of the paper (arXiv v5) I have linked below for a more precise explanation of these facts.
Now the index 1 and 2 refers to the observation stations 1 and 2 of Alice and Bob. The measurement functions A(a, lambda) and B(b, lambda), in the standard local-realistic framework of Bell-1964, are defined by my eqs. (55) and (56). They correspond to two separate detection processes of Alice and Bob, by two different detectors, at two remote stations of Alice and Bob. There is therefore nothing mysterious about s_1 converging to the limit a, which is a detection direction chosen by Alice, and s_2 converging to limit b, which is a detection direction chosen by Bob. So two separate physical detection processes are taking place, at two separate remote observation stations. In summary, initially the two constituent spins, s_1 and s_2 are equal to each other, s_1 = s_2, because of the conservation of spin angular momentum of the initial spin-0 of the neutral pion (see Fig. 2 in the paper). These spins then go their separate ways towards the detectors of Alice and Bob. The detection processes of these spins are then defined as limiting processes in eqs. (55) and (56). In fact, my eqs. (55) and (56) are exactly the same as what Bell wrote down in eq. (9) of his 1964 paper, but my normalized spin components along the directions a and b are expressed as limits. But that is just a mathematically different expression for defining the same number (a normalized spin component). You can see this in more detail in eqs. (57) to (64) of the revised paper I have posted below, by comparing those with Bell’s eq. (9). I think Gill is capable of understanding this, but he disingenuously pretends to show that there is something fishy going on.
“But that is not enough. A second trick is put into play a few lines later. According to (57) we should have L(a, lambda)L(b, lambda) = D(a)D(b) independent of lambda, which means that the step from (65) to (66) can't be correct.”There are no “tricks” in my paper. Gill uses such rhetorical devices deliberately to mislead the reader. For instance, lambda is not just a number. It is the orientation of the 3-sphere. It cannot disappear and reappear as Gill manipulates it to be. Nor does Gill respect the fact that L’s are bivectors, which are thus non-commutative, Clifford-algebraic objects. So, for instance, eq. (57) by no means say that
L(a, lambda)L(b, lambda) = D(a)D(b)
for both orientations lambda = +1 and lambda = -1. Let me stress again: lambda is the overall orientation (or parity) of the 3-sphere, which changes the handedness of the bivectors L(a) and L(b) with respect to the detectors. Gill is either genuinely ignorant to not understand this, or deliberately pretends to not understand it. In any case, the correct equations for lambda = +1 and lambda = -1 cases are the following:
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
but
L(a, lambda = -1)L(b, lambda = -1) = D(b) D(a),
because the change of the overall orientation of the 3-sphere changes the handedness of the spins L(a) and L(b) with respect to the detectors D(a) and D(b). Recall also that orientation of a manifold is a relative concept. See, for example, the explicit definition of orientation given in
my IJTP paper.
So, contrary to Gill’s silly claim, the product D(a) D(b) is not independent of lambda, but its dependence on the orientation lambda is more subtle than Gill realizes. Orientation changes the order of the product. Consequently, there is nothing wrong with eq. (65) and (66). See also eqs. (70), (71), and (72) in the linked paper.
“To take that step he uses (50), but this contradicts (51) and (52). If L(a, lambda) = lambda I a and L(b, lambda) = lambda I b then L(a, lambda)L(b, lambda) = -ab independent of whether lambda = -1 or +1 (lambda and I both commute with a and b; lambda^2 = 1, I^2 = -1)."Eq. (50) does not contradict eqs. (51) and (52). Gill’s argument here is quite silly. It shows that he has no understanding of even the most basic concepts in geometric (or Clifford) algebra, let alone those with respect to relative changes in the orientation of this algebra. Now L(a, lambda) L(b, lambda) = -ab is correct. But how on earth can –ab be independent of the orientation lambda if the product on LHS of the equation is with respect to a given lambda? In fact what he has missed again is the correct understanding of how the duality works between the vectors a and b and the bivectors L(a) and L(b) for different choices of lambda. It is quite funny how Gill manipulates lambda as if it was just a number, like +/-1. It is not just a number. It represents the orientation of the 3-sphere, taken as a physical space. To see his mistake, let us see what –ab actually is in terms of lambda. –ab is the geometric or Clifford product between the vectors a and b. We can expand that product as
-ab = -a.b - a /\ b.
But as we saw above a /\ b = lambda I.(a x b). So the geometric product –ab is by no means independent of lambda. It very much depends on the choice of lambda.
It is extremely unfortunate that physics has been paying such a high price for decades for the ignorance and incompetence of academic terrorist like Richard D. Gill.
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Note that I have now updated the article on the
arXiv and added the following paragraph to it which includes eight new equations:
https://arxiv.org/abs/1405.2355.
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