Dear Cristi,
I am grateful that you are making the effort to understand my theory, and will try hard to assist you. It is only people like you who can actually end up accepting my theory.
Before describing the local deflection formula, let's discuss the inputs. Alice will input a, representing the orientation of her Stern-Gerlach magnetic field axis, and Bob will input b, the orientation of his device. The spins input to each are assumed to be random, and denoted by λ for Alice and the anti-correlated spin -λ for Bob.
ALL of these parameters, a, b, and λ are generated as random unit vectors in a Bloch sphere.
As you now understand, the local angle θ = (λ,a) between the local spin and Alice's (any) local SG device initially precesses, then aligns with the field, and the energy of precession is exchanged with (converted to) a vertical component of velocity, hence kinetic energy. The vertical deflection is shown to be proportional to θ, with a contribution x = X(1-cos θ) as given in my equation (4).
Now consider that the maximum force of the gradient on the dipole occurs when the dipole is aligned with the field, and this can be shown to be X where X is the first term in parentheses in equation (4). This occurs when θ = 0. So the maximum deflection will occur if the particle enters the device aligned with the field and will be X. If the particle is initially not aligned, then the deflection will be less than maximum by the amount x (eqn 4). Thus the maximum minus the θ-dependent contribution is
X - X(1-cos θ) = X cos θ.
This is the θ-dependent deflection Bob and Alice will calculate according to my energy-exchange theory. [ Noting that Alice's angle θ = (λ,a) is different from Bob's angle θ = (-λ,b)].
This formula yields a number, essentially, X cos θ, which will be sent to the statistical unit which accepts Alice's number A(λ,a) [and similarly Bob's number B(-λ,b)] and stores them as a pair for later statistical processing. As I emphasized in my previous answer, although the number was "derived from" a and λ it does not contain a or λ and therefore Alice's a and Bob's b are never present in one place, as they are in the quantum mechanical calculation of the expectation value. Thus the model is truly local.
So Alice's A output and Bob's B output, neither of which are +1 or -1 as Bell requires, are multiplied to obtain the number AB, and we ask what is the 'average' or 'expectation value' of the ABs?
The definition of this expectation value is
< AB > = Sum [ p(AB) * (AB) ] over all i
where the sum is over all ABs. [I calculate this expectation value based on 10,000 random spins for every pair of settings a and b.]
Now the AB values are easy - they are computed by simply taking Alice's deflection reading and Bob's and multiplying the two together to obtain AB. But what is the probability distribution of these ABs? As the 10,000 spins per pair (a,b) are generated randomly, the AB values cannot be calculated by a closed form analytic formula, but they are very easily grouped into bins in the manner of multichannel analyzer measurements, and from the distribution of the numbers in the bins, a probability distribution is easily generated. This is the p(AB) for a given AB over all ABs, 10,000 per (a,b) in the case shown in figure 7. This is done for 300 different values of the angle (a,b) which is labeled θ in the figure.
To summarize: if all spins entered the SG device aligned with the local field, the gradient-based force would be maximum, and the deflections would all go to the same point on the screen, normalized to +1 or -1. This is what Bell assumes. From this simple model Bell is unable to derive quantum correlations.
But the spins do not enter aligned. They enter with a random orientation, so the deflection is not maximum, X, but is proportional to the angle between the spin and the axis, and is X cos θ. This, not +1 or -1, is the deflection A (or B) sent by Alice (or Bob) to the statistical unit. For a given pair of settings (a,b) 10,000 spins are calculated, and 10,000 numbers AB are generated, and plugged into the sum, weighted by their probability p(AB). This entirely numerical procedure produces the correlation shown, -a.b, which Bell claims to be impossible.
It is a very simple classical model, which produces the quantum correlation, based on the 'real' physics of the inhomogeneous field (i.e., nonzero gradient) instead of Bell's 'unreal' constant field model which does fail to produce the quantum correlation.
Let me thank you once again for giving my theory this much attention. I am honored.
Edwin Eugene Klingman
