FOUNDATIONS OF QUANTUM MECHANICS

160 CHAPTER VII. BELL’S INEQUALITIES
According to a HVT, at the moment a measurement is made the values of the spin quantities
are revealed. If we call these values w x (1) for the spin in x - direction of particle 1, etc, then, because
|GHZ⟩, (VII. 78), is a simultaneous eigenstate for the four quantities in C 8 , (VII. 77), it must
hold that
and
w x (1) w y (2) w y (3) = w y (1) w x (2) w y (3) = w y (1) w y (2) w x (3) = + 1, (VII. 79)
w x (1) w x (2) w x (3) = − 1. (VII. 80)
The product of these four factors is
(
wx (1) w y (2) w y (3) ) ( w y (1) w x (2) w y (3) ) ( w y (1) w y (2) w x (3) ) ( w x (1) w x (2) w x (3) )
= (+ 1) (+ 1) (+ 1) (− 1) = − 1. (VII. 81)
But if we consider the product as as a product of the 12 values of these spin quantities we find
w x (1) w y (2) w y (3) w y (1) w x (2) w y (3) w y (1) w y (2) w x (3) w x (1) w x (2) w x (3)
= w 2 x (1) w 2 y (1) w 2 x (2) w 2 y (2) w 2 x (3) w 2 y (3) = 1 6 = + 1, (VII. 82)
This leads to +1 = −1, which is, of course, an algebraical absurdity. And indeed, this is an algebraic
proof since it contains no probabilities or inequalities.
EXERCISE 35. What kind of HVT is excluded by the foregoing reasoning? Which postulates of
quantum mechanics are necessary to obtain the contradiction?
VII. 7
MISCELLANEA
Literature concerning the Bell inequalities has reached an extraordinarily large extent since the
seventies of the 20 th century, however, its growth has decreased in recent years. In conclusion of this
chapter we will briefly discuss some of the main topics.
VII. 7. 1
LOCALITY AND RELATIVITY
Although in these lecture notes we have restricted ourselves to non - relativistic quantum mechanics,
the speed of light did not play a role in our considerations, it is, of course, especially the
special theory of relativity which provides the inspiration to study the (im -) possibility of signaling.