FOUNDATIONS OF QUANTUM MECHANICS

VII
BELL’S INEQUALITIES
There is hardly a paper - nor was there any during the past two and a half decades -
which deals with the foundations of quantum mechanics and does not refer to the work
of John Stewart Bell.
Bell’s theorem is the most profound discovery of science.
— Max Jammer
— Henry Stapp
[. . . ] Bell is generally credited with having brought down a purely philosophical issue
from the lofty realms of abstract speculation to the tangible reach of empirical investigation
and of having thereby established what has been called ‘experimental metaphysics’.
— Max Jammer
The ‘Bell inequalities’ is a generic term for inequalities in terms of measurable physical quantities
which are satisfied by hidden variables theories, but are violated by quantum mechanics. We will
derive several Bell inequalities, belonging to different types of hidden variables theories. This
also includes indeterministic, stochastic HVT’s, which fell outside the scope of chapter V.
VII. 1
LOCAL DETERMINISTIC HIDDEN VARIABLES
VII. 1. 1
DERIVATION OF THE FIRST BELL INEQUALITY
Returning to the hidden variables theories, HVT’s, we focus our attention at a specific experiment.
In the article ‘On the Einstein Podolsky Rosen paradox’ (1964), J.S. Bell examines the EPR experiment,
discussed in section I. 2, in a version which was given by Bohm and Aharonov (Bohm 1957),
also called the EPRB experiment. Bohm and Aharonov proposed an experiment in which two spin
1/2 particles are prepared in the singlet state and, next, move apart in opposite directions. After they
are separated, the spin of each of the particles is measured in an arbitrary direction, where the spin of
particle 1 is measured in direction ⃗a and the remote particle 2 in direction ⃗ b, as in figure III. 3, p. 73.
In this experiment, one can follow the same argument as EPR. Using the notation of section III. 6,
if measurement of ⃗σ 1 · ⃗a yields the value +1 then, for the singlet state, measurement of ⃗σ 2 · ⃗a must
yield the value −1 and vice versa.
Since the result of a measurement of a spin component of the one particle can be predicted with
certainty by measuring the same component of the other particle, whereas the particles are far away