FOUNDATIONS OF QUANTUM MECHANICS

150 CHAPTER VII. BELL’S INEQUALITIES
EXERCISE 32. What type of HVT is excluded by Wigner’s reasoning?
◃ Remark
Wigner (1970) makes the observation that the HVT would have been possible if the terms in (VII. 44)
had been sin 1 2 θ instead of sin2 1 2θ. Apparently, our world depends on such ‘minimal’ mathematical
differences. ▹
VII. 4
THE DERIVATION OF EBERHARD AND STAPP
In the previous derivations of the Bell inequalities hidden variables were assumed, which represent
properties of the pair of particles and determine the outcomes of measurements of all physical
quantities. As a consequence, in this HVT a joint probability is defined for the values of non -
commuting quantities also, as we saw in Wigner’s derivation. This follows from the fact that at
given λ both A(⃗a, λ) and A(⃗a ′ , λ) are fixed, for example
p ( A(⃗a) = 1 ∧ A(⃗a ′ ) = 1 ) ∫
= ρ(λ) dλ, (VII. 49)
∆
where ∆ ⊂ Λ is the area in which both A(⃗a, λ) = 1 and A(⃗a ′ , λ) = 1. Since quantum mechanics
does not acknowledge such ‘simultaneous probabilities’ for non - commuting quantities, the quantities
not being simultaneously measurable, it could be suspected that this property of the HVT is the main
reason for the deviation from quantum mechanics, instead of locality or determinism.
In the next derivation of the Bell inequality, given by P. Eberhard and H. Stapp (1977), the existence
of hidden variables is not assumed. They claim that the Bell inequality follows from an assumption
of locality only. However, what will be shown to be necessary in this derivation, is the
assumption that we can speak reasonably about the outcomes of measurements which have not actually
been carried out.
THE EBERHARD - STAPP THEOREM:
Quantum mechanics is a non - local theory.
Proof
Consider again the EPRB experiment. Let ⃗a and ⃗a ′ be two readings of the spin meter at A, and ⃗ b
and ⃗ b ′ likewise at B. We can carry out four experiments:
I : ⃗a, ⃗ b II : ⃗a, ⃗ b ′ III : ⃗a ′ , ⃗ b IV : ⃗a ′ , ⃗ b ′ . (VII. 50)
Define, for the n th pair of particles, a n (I) as the outcome of a spin measurement in the direction ⃗a
of the particle traveling to A while the meter at A points in the direction ⃗a, while at the other
particle, which travels to B, spin in the direction ⃗ b is measured; this gives a n (I) = ±1 for
experiment I and likewise for a n (II), a n ′ (III), a n ′ (IV), b n (I), b n ′ (II), b n (III) and b n ′ (IV).
These values represent outcomes of measurements of actual or possible measurements, not actual
properties of the particles which also exist if they are not measured.