FOUNDATIONS OF QUANTUM MECHANICS

V. 4. CONTEXTUAL HIDDEN VARIABLES 121
that via a detour we still impose a requirement for non - commeasurable quantities on the HVT. We
will consider this in detail now.
Suppose that operator A commutes with the maximal operators C 1 and C 2 , while [C 1 , C 2 ] ≠ 0.
Then we have
which implies
A = f (C 1 ) and A = g(C 2 ), (V. 34)
f (C 1 ) = g(C 2 ), (V. 35)
and we see that A is degenerate. Function rule (V. 21) leads to the same relation between the quantities
of the HVT,
yielding
A[λ] = f ( C 1 [λ] ) and A[λ] = g ( C 2 [λ] ) , (V. 36)
f ( C 1 [λ] ) = g ( C 2 [λ] ) . (V. 37)
Again, this is a relation between the value assignments to quantities which do not commute in quantum
mechanics, but the relation is not one - to - one, the functions f and g are not bijective.
It can be supposed that such a requirement is unreasonable is because such quantities are not
commeasurable. In other words, the structure of quantum mechanics, and particularly the proposition
that an operator can be a function of two non - commuting maximal operators, leads to relations
between quantities which cannot be measured in one single experiment.
The following is what occurs at the different decompositions of unity. Consider two bases, {|α j ⟩}
and {|β j ⟩}, in a Hilbert space H of dimension N > 2 and suppose that |α 1 ⟩ = |β 1 ⟩, while all other
basis vectors are different. Then we have
N∑
P |αj ⟩ = 11 =
j=1
N∑
P |βj ⟩ and P |α1 ⟩ = P |β1 ⟩. (V. 38)
j=1
Define, as follows, two maximal operators with all coefficients c j and d j distinct,
C :=
N∑
c j P |αj ⟩ and D :=
j=1
N∑
d j P |βj ⟩, (V. 39)
j=1
then it follows that
P |α1 ⟩ = f (C) = g(D). (V. 40)
This leads to a connection between the non - commuting operators C and D, and using (V. 21)
this leads to a connection between the corresponding representations C[λ] and D[λ] in the HVT. It is
this type of relations which the HVT cannot satisfy.