FOUNDATIONS OF QUANTUM MECHANICS

122 CHAPTER V. HIDDEN VARIABLES
◃ Remark
Notice that the occurrence of non - maximal operators P |αi ⟩ is indeed essential, if P |αi ⟩ would be
maximal, C and D would commute, as we saw in section II. 4 on p. 30. M.J. Maczynski (1971) has
proved that if we exclusively consider maximal quantities, and therefore we would apply (V. 21) to
maximal quantities only, Kochen and Specker’s theorem is no longer valid, and in that case a HVT is
possible. ▹
An obvious expedient is to strictly constrain requirement (V. 21) to quantities which are measurable
within one context. In our example the projector P |α1 ⟩ is commeasurable with both C and D,
while mutually C and D are not commeasurable. Therefore, we have to distinguish between a value
assignment P |αi ⟩[λ] within the context of a measurement of C, and one within the context of a measurement
of D. We can think, for example, of a measurement of C and application of the function relation
P |α1 ⟩ = f(C), or of a measurement of D and application of the function relation P |α1 ⟩ = g(D).
More generally, suppose
A = f (C) = g(D) where [C, D] ≠ 0. (V. 41)
Then we distinguish the hidden variable quantities A C [λ] and A D [λ], where the index indicates the
context of measurement. If C and D do not commute there is, according to a contextual HVT, no
reason to assume that for all λ ∈ Λ it holds that
A C [λ] = A D [λ], (V. 42)
as is the case in every HVT we have considered so far.
Kochen and Specker do assume (V. 42), however, and find a contradiction with quantum mechanics.
The remedy is therefore to ‘split up’ all degenerate quantities by addition of the context in which
they are measured, as was firstly proposed by B.C. van Fraassen (1973). For the sake of convenience
we here assume that a measurement of a degenerated quantity always develops by means of the measurement
of a maximal quantity, which does not have to be split up. By definition we then have
A C [λ] = f ( C [λ] ) and A D [λ] = g ( D[λ] ) . (V. 43)
This yields a weaker form of (V. 21). Suppose A = f (C), B = g(C) and A = h(B) = h(g(C)),
then using (V. 43) we have
A C [λ] = h ( B C [λ] ) . (V. 44)
This consideration leads to a new postulate for a HVT, which, in case the HVT accommodates this
postulate, we call contextual.
CONTEXTUAL OBSERVABLES POSTULATE:
If A is a physical quantity which can be taken as a function of at least two other physical
quantities, for example A = f (C) and A = g (D), then, in the HVT, to A corresponds
a function A C : Λ → R iff quantity C is measured, and a function A D : Λ → R iff
quantity D is measured. If A, f(C) and g(D) are the corresponding quantum mechanical
operators, the following applies,
∀ λ ∈ Λ : A C [λ] = A D [λ] ⇐⇒ [C, D] = 0. (V. 45)