* * *
Well, Lockyer seems to have finally recognized his error. I hope he has the courage to admit his error in public (I expect no such decency from Gill and Moldoveanu).
Let me spell out the key point explicitly here so there remains no doubt in anyone's mind. The mathematical demonstration is exceedingly simple and easy to follow.
What we want to show is that the first equation above (as written by me in
these two
papers) forms a right-handed system; and the second equation (again, as written by me in
these two
papers) forms a left-handed system. Gill, Moldoveanu, and Lockyer falsely claim that I have made a sign error on the RHS of the second equation.
To check the handedness of these equations, let us set the angle between
and
to be 90 degrees and define
for this case. Now multiply (using the geometric product) the first equation, on both sides, by
, from the left. Since we have set
and
to be orthogonal to each other, and since all bivectors such as
square to
, the first equation (in this special case of orthogonal
and
reduces to
Now if we follow the same procedure for the second equation -- this time using the bivector
, it also reduces to
So far so good. Both equations, as long as they remain "unaware" of each other, can be taken to represent a right-handed system, because their RHS equals to
.
But now suppose we wish to compare the two systems, as done in the successive trials of EPR-B type experiments ( is the spin "up", or "down" in a given trial? ). Then we must find a functional mapping between the two equations. But that is a trivial task in the present set up, since we see that the respective bivectors are related as
for
any directional vector
. If we now substitute the above mapping in the second of the two equations we have derived, we obtain at once, for this
second system,
Thus we see at once that the second equation represents a left-handed system
with respect to the first system, because now the RHS of this equation equals to
.
In conclusion, a sign mistake has indeed been made for the past several years, but it is made by Gill, Moldoveanu, and Lockyer. They should go back to their schools.