FOUNDATIONS OF QUANTUM MECHANICS

V. 2. NON - CONTEXTUAL HIDDEN VARIABLES 111
In the HVT, a pure physical state corresponds to a single ‘point’ λ ∈ Λ. We assume that the
system is always in one of these states λ ∈ Λ, even though we do not know in which one. A general,
mixed state is a probability distribution over Λ. For any given λ every physical quantity A has an
exact value, denoted by A[λ], which is revealed upon measurement of A, and therefore a physical
quantity A can be represented as a real function on the space A : Λ → R.
Furthermore, every quantity represented by quantum mechanics has to have a counterpart in the
HVT. If such a quantity, corresponds to the function A : Λ → R the values A [λ] can take are
the eigenvalues of the self - adjoint operator A : H → H which, according to quantum mechanics,
corresponds to quantity A.
It is also required that every quantum mechanical state can be represented in the HVT; for every
state operator W there must be a corresponding probability distribution ρ W over Λ. It is, however,
not necessary that pure quantum states correspond to pure hidden variable states, the idea being that
the HVT allows for a more detailed, complete description of the system. Neither is it necessary that
every probability distribution on Λ corresponds to a state operator, the HVT could easily be a theory
richer than quantum mechanics.
The requirement that the HVT has to reproduce the empirical statements of quantum mechanics
is now expressed in the requirement that the expectation values of quantity A belonging to a physical
system in a physical state, corresponding in the HVT to ρ W , and in quantum mechanics to W , coincide,
∫
⟨A⟩ ρW := A[λ] ρ W (λ) dλ = Tr A W, (V. 1)
Λ
where ρ W : Λ → [0, ∞) is a probability density,
∫
ρ W (λ) dλ = 1. (V. 2)
Λ
For a pure state |ψ⟩, (V. 1) reduces to
∫
A[λ] ρ ψ (λ) dλ = ⟨ψ | A | ψ⟩. (V. 3)
Λ
In the discrete case the integrals are replaced by summations.
Summary
An non - contextual HVT is any theory meeting the following requirements.
(i) Every physical state of a physical system corresponds to a probability distribution ρ over Λ.
This is the state postulate.
(ii) Every physical quantity A corresponds to a function A : Λ → R, λ ↦→ A[λ]. This is the
observables postulate.