Joy Christian has a new paper https://arxiv.org/abs/2204.10288 called:
Symmetric Derivation of the Singlet Correlations within a Quaternionic 3-sphere
In response to my note about Bell’s Blunder, Joy sent me the above paper which I read with interest to see what was new. I have long supported Joy’s approach to disprove Bell’s Theorem because I believed he was on the right track. I have communicated a few times with him and pointed out where I disagreed with his approach, but he only said there were no errors. I will use this opportunity to point out the problems I have in his approach.
Joy’s uses the same concept of random variables of +/-1 to calculate the spin correlation as Bell did. However, Joy’s thesis is Bell’s random variables reflect the geometry of the S2 sphere, that is our 3D space. Joy says, rather, they should reflect the geometry of the S3 hypersphere of quaternions, i.e. his equation (48). I agree. However, he then uses Bell’s definition of classical random variables to calculate the correlation by his Eq.(12): he uses only polarizations to calculate the correlation. If you look at Joy’s Equations (10), (11), (18), (26), (32), (33), they all give the random variables as +/-1, albeit with S3 origin and symmetry. I disagree with this approach as I point out in Bell’s Blunder. Polarization states only exist (i.e. +/-1) upon measurement. Joy assumes they are always polarized. He does not include any coherent terms.
By requiring spin to be always polarized, Joy gets to his Eq.(55) to (57) which account for the violation with the cosine term (-a.b) surviving and the two wedge products cancelling. The disagreement I have with this step follows. He introduces the Geometric Algebra expression for the bivector written as the dot and wedge product, his Equation (2) but then cancels the two wedge products. This is not correct in my view. Those wedges are coherences which he shows make NO contribution to the correlation. I disagree. They do contribute and lead to the violation of Bell’s Inequalities.
The second point is that the remaining -a.b term in Equation (57) cannot arise without non-local connectivity. That term can never come from a product state and can only arise from a persistent entangled state. In this way, Joy has introduced non-locality which accounts for the violation, the same as Bell did. Joy introduces non-local entanglement in his Equation (39). There is only one reason he introduce this, which is to cancel the wedges and keep the polarization. I call on Joy to justify that equation expresses a product state, rather than an entangled state.
The use of Bell’s random variables and his definition of correlation, Equation (12), is classical which cannot include coherences. I replace that definition of correlation by the quantum definition of the trace over the elements of reality A and B, just like von Neumann introduced (Von Neumann, John. Mathematical foundations of quantum mechanics. Princeton university press, 1955.) ,
<AB> = Tr[A^(adj)B rho]
Where rho is the state operator and adj is the adjoint operator. This is the quantum definition of correlation that includes coherence. Use of Eq.(12) cannot include coherence.
My final comment is a plea to Joy to try to simplify his expressions and shorten his papers. I think that he can reduce the equations a lot and express his ideas more concisely with less cumbersome equations. I think more people would be interested in Joy’s approach if his papers were more pedagogically and clearly written.
