FOUNDATIONS OF QUANTUM MECHANICS

dµ A(⃗a, λ, µ) dν B( ⃗ b, λ, ν) ρ(λ, µ, ν)
VII. 3. WIGNER’S DERIVATION 147
The expectation value in this HVT is
∫ ∫
∫
E(⃗a, ⃗ b) = dλ
Λ Λ a Λ b
∫
= ⟨A(⃗a, λ)⟩ ⟨B( ⃗ b, λ)⟩ ρ(λ) dλ, (VII. 26)
Λ
which is an ‘averaged’ version of (VII. 4). With (VII. 25) we see that
∫
|E(⃗a, ⃗ b) − E(⃗a, ⃗ b ′ )| = |⟨A(⃗a, λ)⟩ ( ⟨B( ⃗ b, λ)⟩ − ⟨B( ⃗ b ′ , λ)⟩ ) | ρ(λ) dλ
Λ
∫
|⟨B( ⃗ b, λ)⟩ − ⟨B( ⃗ b ′ , λ)⟩| ρ(λ) dλ. (VII. 27)
Λ
Likewise we have
|E(⃗a ′ , ⃗ b) + E(⃗a ′ , ⃗ b ′ )|
∫
Λ
|⟨B( ⃗ b, λ)⟩ + ⟨B( ⃗ b ′ , λ)⟩| ρ(λ) dλ, (VII. 28)
and therefore
|E(⃗a, ⃗ b) − E(⃗a, ⃗ b ′ )| + |E(⃗a ′ , ⃗ b) + E(⃗a ′ , ⃗ b ′ )| 2, (VII. 29)
since |x + y| + |x − y| 2 if |x| 1 and |y| 1. We see that (VII. 29) is, indeed, the Bell
inequality (VII. 13).
For ⃗a ′ = ⃗ b ′ and the assumption of perfect anti - correlation E ( ⃗ b ′ , ⃗ b ′ ) = −1, from inequality
(VII. 13) follows the original Bell inequality (VII. 10). But, as we showed, (VII. 13) remains
valid under the weaker conditions (VII. 25). □
◃ Remark
It is not necessary to assume mutual independence for µ and λ or for ν and λ as in (VII. 22), the
result (VII. 25) also follows when we make the weaker assumption that the conditional probability
distributions of the apparatuses factorize the conjoint probability distribution ρ,
ρ(λ, µ, ν) = ρ(λ) ρ 1 (µ | λ) ρ 2 (ν | λ). ▹ (VII. 30)
VII. 3
WIGNER’S DERIVATION
E.P. Wigner (1970) was the first to give an elegant derivation of a Bell inequality in terms
of probabilities. We again consider the EPRB experiment from section VII. 1. Using three directions,
⃗n 1 , ⃗n 2 , ⃗n 3 ∈ R 3 , define
σ i := ⃗n i · ⃗σ and τ i := ⃗n i · ⃗τ with i ∈ {1, 2, 3}. (VII. 31)