FOUNDATIONS OF QUANTUM MECHANICS

156 CHAPTER VII. BELL’S INEQUALITIES
The idea behind the requirement of outcome independence is that such a situation could only
occur because the HVT was incomplete; in a complete specification of the state of the pair of particles
which existed at the beginning of the trip also the color of the little balls should have been included,
even though the travelers did not know the color of their little ball. Then it automatically follows,
at given λ, that the little ball in New York is white and the observation in Tokyo provides no new
information.
2. Parameter independence, (VII. 62), means that the probability distribution of the outcomes
at A is independent of external changes at B, e.g. pointing the spin meter. The argumentation leading
to the assumption of parameter independence is generally associated with the possibility of signaling.
Suppose that, for example, adjustments ⃗ b and ⃗ b ′ existed such that
p ⃗a, ⃗ b
(a | λ) ≠ p ⃗a, ⃗ b ′ (a | λ), (VII. 70)
then, in principle, it is possible to instantaneously exchange signals between experimenters located
at A and B. Since the experimenter located at B can choose if he points his spin meter in the
direction ⃗ b or ⃗ b ′ , an experimenter located at A is able, if the source emits particle pairs in a pure
hidden - variables state λ, to register the relative frequency of outcomes of A and thereby retrieve
which adjustment has been chosen by the experimenter at B. Violation of parameter independence
therefore means that the HVT enables the instantaneous exchange of signals over arbitrarily large
distances.
3. Source independence, (VII. 63), means that the probability distribution over the hidden variable
describing the particle pair cannot depend on the measuring directions chosen by the experimenters.
The argumentation leading to the assumption of source independence is often described
in terms of the ‘free will’ of the experimenters. The experimenters are considered to be completely
‘free’ in their decision how to point their spin meters, and even to make their choice just at the last
moment, when the particles have long left the source. Therefore, the probability distribution ρ(λ),
which characterizes the source of the particle pairs, cannot depend on that.
Of course, here too it applies that violation of the requirement is logically conceivable. It is
possible that this freedom does not exist, and that at emitting the particles, the directions in which
the experimenters will measure have already been determined. It is also conceivable that by some
other cause a correlation exists between λ and the directions ⃗a and ⃗ b, influencing both. The first case,
in which all relevant factors of the EPR experiment are determined in advance and the experimenters
have no free will, is called super - determinism. Therefore, in a super - deterministic HVT the Bell
inequalities can be violated also.
VII. 5. 2
QUANTUM MECHANICS AS A STOCHASTIC HVT
Exclusively giving probability statements concerning outcomes of measurements, a stochastic
HVT conceptually differs less from quantum mechanics than other HVT’s. In fact we can, without
objection, take quantum mechanics itself as an example of a stochastic HVT by identifying λ with the
quantum mechanical state and Λ with the relevant Hilbert space. Since quantum mechanics does not
satisfy the Bell inequalities, it is interesting to examine which of the aforementioned requirements is
violated inevitably by quantum mechanics.