***
For the past six years Gill, Lockyer and Moldoveanu have repeatedly and loudly claimed that I have made a sign error at the very heart of my local-realistic 3-sphere model for the EPR-Bohm correlations. For example, in the abstract of his supposed "critique" of my model Richard D. Gill declared that my "...research program has been set up around an elaborately hidden but trivial mistake." His silly claim is of course demonstrably false and has been made up for the political and sociological reasons to protect his vested interests in the industry of Bell's theorem, which has led Gill to bombard the President of my college and other departmental heads at Oxford University with a series of malicious letters about me, have my published papers retracted, and spread all kinds of rumors about my work and me personally.
But thanks to the enormous mediation efforts for the past five months by our own Jay R. Yablon at Retraction Watch, their sign error has now been brought out very clearly. My purpose here is to explain as clearly as possible the sign error that Gill, Lockyer and Moldoveanu have been making for the past six years and pinning it on me, either for malicious reasons or for the reasons of breathtaking incompetence. In fact I have exposed their error already in the appendix of my Refutation of Gill.
To understand their error, recall that within the 8D Clifford algebra of the orthogonal directions in the physical 3D space, one may consider the 4D subalgebra of the right-handed (or counterclockwise) bivectors { alpha_i = + e_j e_k } and left-handed (or clockwise) bivectors { beta_i = - e_j e_k }, with the index i standing for the orthogonal vector-like coordinates x, y, or z permuted cyclically. Now it is easy to find from textbooks that alpha_i and beta_i satisfy the following pair of equations:
alpha_i alpha_j = - delta_ij - epsilon_ijk alpha_k
and
beta_i beta_j = - delta_ij + epsilon_ijk beta_k,
where delta_ij is the Kronecker delta and epsilon_ijk is the Levi Civita symbol. The important sign to note here is the + sign on the RHS of the equation for beta_i.
Note that the above are equations for bivectors. Now in Clifford algebra, by convention, the duality relation between vectors and bivectors is always expressed with the standard trivector, I = e_x e_y e_z ; just as in vector algebra, by convention, cross product is always understood with the right-hand rule. As in eq. (A.11) of my Refutation of Gill I have linked above, the duality relation between the left-handed vector basis { - e_i } and left-handed bivector basis { - e_j e_k } is expressed as
beta_i = - e_j e_k = I . ( - e_i ).
This can now be compared unambiguously with the following duality relation between right-handed vector basis { +e_i } and right-handed bivector basis { +e_j e_k }:
alpha_i = +e_j e_k = I . ( +e_i ).
Next, using the above duality relation the very first equation for alpha_i written above can be rewritten as
( I . e_i ) ( I . e_j ) = - delta_ij - epsilon_ijk ( I . e_k ).
This equation is not in dispute. It is a standard equation that everyone agrees with. But using the duality relation for beta_i we can also rewrite the beta_i equation as
[ I . ( - e_i ) ] [ I . ( - e_j ) ] = - delta_ij + epsilon_ijk [ I . ( - e_k ) ] ,
which, due to cancellations of the minus signs, again reduces to the same familiar identity
( I . e_i ) ( I . e_j ) = - delta_ij - epsilon_ijk ( I . e_k ).
This is the correct identity inside the mirror. But since this is identical to the identity derived for alpha_i, Gill, Lockyer and Moldoveanu claim that my program fails.
Here my use of the word "mirror" is of course metaphorical, standing for the left-handed set of vector coordinates. With that in mind, here is the problem with their claim. Experiments in physics are not done inside the mirror. The experiments are done outside the mirror, with fixed coordinates set up using only the right-handed vector basis { +e_i }. Therefore we must translate the above geometrical identity from inside the mirror to outside the mirror. In other words, we must translate the last identity, currently written in terms of left-handed vector basis { - e_i }, to an identity written entirely in terms of right-handed vector basis { +e_i }. Gill, Lockyer and Moldoveanu, however, have been neglecting this important step required for consistent physics. There are at least two ways to accomplish this task. One of them is used in my Refutation of Gill linked above, which is +I --> -I. The other way is self-evident: - e_i --> +e_i , for all i. But whichever method we use, we end up with
( I . e_i ) ( I . e_j ) = - delta_ij + epsilon_ijk ( I . e_k ).
This is now the correct equation for the left-handed bivectors beta_i outside the mirror, and it is no longer identical to the equation for the right-handed bivectors alpha_i. Both identity-equations are now outside the mirror. Thus we can now compare them unambiguously, take average of the two, or whatever we like, because they are both expressed in terms of the right-handed coordinates conventionally used by the experimenters in their laboratories. Hence we can derive E(a, b) = -a.b, say by averaging over a large number of alternative products (I . e_i) (I . e_j), because now there is a sign difference between the two hidden variable possibilities:
( I . e_i ) ( I . e_j ) = - delta_ij - epsilon_ijk ( I . e_k )
OR
( I . e_i ) ( I . e_j ) = - delta_ij + epsilon_ijk ( I . e_k ).
This is not the first time their sign error has been pointed out to Gill, Lockyer and Moldoveanu. But perhaps the above is the most clear-cut exposition of their error.
Joy Christian
***