FOUNDATIONS OF QUANTUM MECHANICS

146 CHAPTER VII. BELL’S INEQUALITIES
suitably chosen spin directions. This enables an experimental test of these statements, and therefore
of the correctness of the philosophical bases of both theories. A. Shimony (1989) spoke, concerning
the experimental testing of the Bell inequalities, of ‘experimental metaphysics’.
However, the question of experimental testing puts the derivation of the Bell inequalities in another
perspective. We no longer want to compare a HVT with quantum mechanics, but with experimental
results. In this respect (VII. 6), implying perfect anti - correlation when ⃗a = ⃗ b, is overly
idealized. In a real experiment the particle detectors are not perfectly efficient, in the sense that not
all particles are registered. Imagine a detector which, even if A(⃗a, λ) = 1, sometimes gives 0, i.e. not
measured, or even −1, i.e. wrongly measured. Moreover, in a contextual HVT the outcomes could also
be dependent of the measuring context, i.e. of (possibly hidden) variables of the detectors. But also in
this generalized situation it is possible to derive the inequality (VII. 13) from a locality assumption.
We will show this by proving the next theorem.
BELL’S SECOND THEOREM:
A local deterministic contextual HVT is empirically inconsistent with quantum mechanics.
Proof
Assume that the quantities A and B are functions of three arguments,
A = A(⃗a, λ, µ), B = B( ⃗ b, λ, ν) where A, B ∈ {− 1, 1}. (VII. 21)
Here the local deterministic character of the HVT is expressed; the outcome of the measurement
at the measuring apparatus measuring ⃗a · ⃗σ is determined by λ ∈ Λ, describing the source, by the
local hidden variables of that measuring device, expressed symbolically by µ ∈ Λ a , and by the
position ⃗a of the meter pointer. Therefore, the requirement of locality is that A does not depend
on ⃗ b and ν, and B does not depend on ⃗a and µ. We also assume that the hidden variables of the
apparatuses are independent of each other and of λ,
Defining
ρ(λ, µ, ν) = ρ(λ) ρ 1 (µ) ρ 2 (ν). (VII. 22)
and
⟨A(⃗a, λ)⟩ :=
⟨B( ⃗ b, λ)⟩ :=
∫
A(⃗a, λ, µ) ρ 1 (µ) dµ (VII. 23)
Λ a
∫
B( ⃗ b, λ, ν) ρ 2 (ν) dν, (VII. 24)
Λ b
we have, instead of assumption (VII. 2), the much weaker requirements
|⟨A(⃗a, λ)⟩| 1 and |⟨B( ⃗ b, λ)⟩| 1, (VII. 25)
and we will show now that from this it is again possible to derive the Bell inequality (VII. 13).