FOUNDATIONS OF QUANTUM MECHANICS

V. 4. CONTEXTUAL HIDDEN VARIABLES 123
Although splitting up quantities is a natural consequence of the idea of commeasurability, it means
giving up a one - to - one relation between the quantities of quantum mechanics and those of the HVT in
a very drastic manner; since the operator P |α1 ⟩ is part of infinitely many decompositions of unity, there
are infinitely many contexts in which P |α1 ⟩ can be measured.
The idea that the context of the measurement must be taken into the consideration can already be
found in Bell (1966). In this article, which was actually written earlier than his famous article with the
Bell inequality, Bell makes some observations concerning the requirements which could be imposed
to a contextual HVT. They have to have a spatial meaning and enable us to interpolate a space - time
picture, preferably causally, between the preparation and the measurement of states.
He then considers Bohm’ s theory of the quantum potential, see chapter VI, and shows that this
theory is not local. He wonders if every HVT of quantum mechanics must have this non - local character
(Bell 1966, p. 452),
However, it must be stressed that, to the present writer’s knowledge, there is no proof that
any hidden variable account of quantum mechanics must have this extraordinary character.
It would therefore be interesting, perhaps, to pursue some further “impossibility
proofs,” replacing the arbitrary axioms objected to above by some condition of locality,
or of separability of distant systems.
Meanwhile, still before the delayed publication of his article, Bell (1964) himself had found such a
proof.
Now we will show how the idea of locality can be brought to expression in a contextual HVT with
‘split’ quantities. Consider a composite system with Hilbert space H = H I ⊗ H II and an operator of
the form A ⊗ 11 where A is maximal in H I . Then the operator A ⊗ 11 is not maximal in H, and
A ⊗ 11 = f (X), (V. 46)
where X is some maximal operator on H. Especially consider an X of the form
X = X I ⊗ X II . (V. 47)
Suppose there is no interaction, or not anymore, between the systems I and II. Then we can raise the
question if X II must be taken to belong to the context of A ⊗ 11.
Consider a second maximal operator
Y = X I ⊗ Y II (V. 48)
which only differs from X in the last factor. We then have
A ⊗ 11 = f (X) = g(Y ). (V. 49)
A requirement of locality is now that
(A ⊗ 11) XI ⊗ X II
[λ] = (A ⊗ 11) XI ⊗ Y II
[λ], (V. 50)
in other words, a change in that what is measured of system II, does not result in a splitting of
quantities of system I. A contextual HVT satisfying (V. 50) is called local.