FOUNDATIONS OF QUANTUM MECHANICS

VII. 7. MISCELLANEA 161
Therefore, it is interesting to consider the EPRB experiment schematically in a Minkowski diagram,
figure VII. 9.
ct
A
B
λ
Figure VII. 9: Minkowski diagram of the EPRB experiment, where λ is in the past light cones of both
A and B
A natural requirement of locality for a relativistic stochastic HVT is that the probability of an
outcome A depends exclusively on the variables which specify the state in the past light cone of
the measuring event at A, and likewise for B. Bell has called it local causality. We have seen that
quantum mechanics is not a local causal theory. Indeed, the probability of an outcome at A cannot be
influenced by the choice of the direction of measurement ⃗ b at B, but with the outcome at B, which
can be registered there, a prediction can be done by an observer at B concerning the particle at A
which an observer at A can not do, even if he has complete knowledge of the state in the past light
cone of A.
x
VII. 7. 2
LOCALITY VERSUS CONDITIONAL INDEPENDENCE
A problem that is brought up in some publications, e.g. Fine (1982), De Muynck (1986, 1996),
is, to what extent locality is necessary to derive the Bell inequalities. The authors argue that in
‘requirements of locality’ only a special form of statistic independence is expressed. The distance
between the measuring apparatuses is in absolutely no way manifest in the requirement. Although
‘locality’ is a term which seems to presuppose a space - time, such space - times are conspicuous by
their absence in relevant locality assumptions, they all are probability statements without reference to
space or time.
Indeed, strictly speaking one cannot say that these assumptions express a requirement of locality.
It could be possible to expect an analogous independence for a hypothetical pair of particles, for
example a photon and a gluon, which absolutely cannot interact with each other, but are located very
close to each other. The essence is that in a local theory the large distance between the particles can
be taken to be a sufficient, but not necessary condition for the absence of interactions.
The requirement of outcome independence in the HVT is not a representation of the requirement
of locality, it has only been motivated by it. The conclusion that is sometimes drawn from this, that
apparently locality itself is irrelevant for the Bell inequality, is, however, incorrect. Factual violation
of the Bell inequality means that every stochastic HVT satisfying the factorizability as formulated in
section VII. 5 is excluded, and therefore, also the local versions are excluded.