FOUNDATIONS OF QUANTUM MECHANICS

120 CHAPTER V. HIDDEN VARIABLES
with θ the angle between |ψ⟩ and |χ⟩, see figure V. 4.
ψ
1
θ
χ
cos 2 θ
0
Figure V. 4: µ(P i ) = cos 2 θ
In the appendix of these lecture notes, p. 183, ff., we will prove that, if we assign to each point
of the upper half of a unit sphere a non - negative real number such that 1 is assigned to the ’north
pole’, 0 is assigned to the ’equator’ and the sum of the values of each orthogonal triad in this half
sphere is 1, there is only one possible value assignment and that is the quantum mechanical one,
i.e., in accordance with cos 2 θ.
◃ Remarks
First, illustrations of Kochen and Specker’s theorem are easy to find for Hilbert spaces of dimension
larger than 3, for example 8, in which case a handful of quantities suffices, see Mermin (1993). We
will come back to that in section VII. 6. Second, when restricted to rational angles between spin
vectors, no contradiction with quantum mechanics can be obtained, as D.A. Meyer (1999) proved. ▹
V. 3. 1 SUMMARY
According to Kochen and Specker’s theorem, a HVT satisfying the state postulate and the observables
postulate, p. 111 (i) and (ii), together with the function rule (V. 21), is contradictory to the state
postulate and the observables postulate of quantum mechanics if dim H > 2, although for Hilbert
spaces with dim H 2 it is possible. This conclusion shows how stringent the vector space structure
of quantum mechanics is, and in particular, the fact that there are many different decompositions of
unity forms a heavy barrier for a HVT.
V. 4 CONTEXTUAL HIDDEN VARIABLES
Essential for Kochen and Specker’s proof is the fact that a 1 - dimensional projector can be part
of several decompositions of unity. This is possible as long as the projectors are not maximal, i.e.,
if dim H > 2. The existence of degenerated projectors, apart from unity, is essential for the proof of
Kochen and Specker, and for this reason it does not hold in a 2 - dimensional H where all projectors,
except 11, are maximal. By means of degenerated projectors also non - commuting operators become
connected to each other. By the requirement (V. 21) this is transferred to the quantities of the HVT, so