FOUNDATIONS OF QUANTUM MECHANICS

112 CHAPTER V. HIDDEN VARIABLES
(iii) The range of A : Λ → R coincides with the spectrum of the self - adjoint operator A which,
according to quantum mechanics, corresponds to quantity A.
The expectation value of A when the physical system is in the state ρ W which, according
to quantum mechanics, corresponds to the state operator W , equals the quantum mechanical
expression for the expectation value
⟨A⟩ ρW :=
∫
Λ
A[λ] ρ W (λ) dλ = Tr AW.
We will call this last requirement (iii) the reproduction criterion.
Since all probabilities in quantum mechanics can be written as Tr PW , with P ∈ P(H), it follows
that all probability distributions in quantum mechanics coincide with the corresponding probability
distributions in the HVT.
We can now ask whether it is possible to construct a HVT satisfying the above requirements. The
answer is that it is indeed possible, even in a quite trivial way, by choosing Λ large enough. We
illustrate this by means of a simple example.
Suppose there are only three quantities A, B, C, with possible values {a 1 }, {b 1 , b 2 }, {c 1 , c 2 } and
represented by functions A, B, C : Λ → R. The possible value combinations are
(a 1 , b 1 , c 1 ), (a 1 , b 1 , c 2 ), (a 1 , b 2 , c 1 ), (a 1 , b 2 , c 2 ). (V. 4)
We now construct a space Λ by identifying every value combination with a point of Λ. If we denote
these points by λ 1 , λ 2 , λ 3 and λ 4 , then
A[λ 1 ] = a 1 , B[λ 3 ] = b 2 , C [λ 4 ] = c 2 , etc. (V. 5)
When there are more quantities, we extend Λ correspondingly.
We have to introduce a probability measure
µ : F (Λ) → [0, 1] with
∑
µ(λ j ) = 1 (V. 6)
j
such that (V. 1) is satisfied. In our case Λ is discrete and consists of four points only, as a result of
which the integral (V. 1) becomes a sum. For example, to quantity B it must apply that
Tr B W =
4∑
B[λ j ] µ W (λ j )
j=1
This is satisfied by
= b 1
(
µW (λ 1 ) + µ W (λ 2 ) ) + b 2
(
µW (λ 3 ) + µ W (λ 4 ) ) . (V. 7)
µ W (a i , b j , c k ) = Tr P ai W Tr P bj W Tr P ck W, (V. 8)