FOUNDATIONS OF QUANTUM MECHANICS

114 CHAPTER V. HIDDEN VARIABLES
To illustrate this we assume that in our example C = A+B so that c 1 = a 1 +b 1 and c 2 = a 1 +b 2 .
Now the possible value combinations in the HVT are
(a 1 , b 1 , a 1 + b 1 ), (a 1 , b 1 , a 1 + b 2 ), (a 1 , b 2 , a 1 + b 1 ), (a 1 , b 2 , a 1 + b 2 ), (V. 16)
and we see that (A + B) [λ] is not equal to A [λ] + B [λ] for all λ. Nevertheless, the HVT succeeded
in reproducing, by construction, all quantum mechanical expectation values, in other words,
the HVT reproduces the relation
⟨ψ | A + B | ψ⟩ = ⟨ψ | A | ψ⟩ + ⟨ψ | B | ψ⟩, (V. 17)
without requiring
(A + B)[λ] = A[λ] + B[λ]. (V. 18)
If we would require (V. 18), Λ would only consist of the points (a 1 , b 1 , a 1 + b 1 ) and (a 1 , b 2 , a 1 + b 2 )
which is, of course, a strong restriction.
In the very first proof of the impossibility of a HVT, that is, of the insolubility of the completeness
problem, given by Von Neumann (1932), the requirement (V. 18) was indeed imposed on the HVT.
Von Neumann required (V. 18) for every hidden variable state, in particular also for pure hidden
variable states, which means that (V. 18) must apply to all λ ∈ Λ. We don’t need to discuss Von
Neumann’s elaborate proof of this claim in detail, since J.S. Bell (1966) has shown this impossibility
by means of a very simple example.
Since the values of A[λ] etc. have to be the eigenvalues of the corresponding operators, it can be
seen immediately that this requirement cannot be satisfied in general. Consider for example the Pauli
matrices
σ x =
( ) 0 1
, σ
1 0 y =
( ) 0 − i
i 0
and σ x + σ y =
( )
0 1 − i
. (V. 19)
1 + i 0
The eigenvalues σ x and σ y are ±1, but the eigenvalues of σ x + σ y are ± √ 2, and therefore, (V. 18)
cannot be satisfied.
Bell argued that the requirement (V. 18) is physically unreasonable. For instance, measuring
σ x ,σ y and σ x +σ y requires three different measurement apparatuses, for example three Stern - Gerlach
magnets in three different orientations. There is absolutely no reason to assume that an algebraical
link would exist between the individual outcomes of these measurements. The fact that in quantum
mechanics the relation (V. 17) exists for pure states, even in case A and B do not commute, must be
considered as a particular property of quantum mechanics.
Since the requirement (V. 18) is unreasonably strong, one can wonder whether there are other,
reasonable, requirements which can be imposed to a HVT in order to find acceptable solutions of the
completeness problem. This brings us to the next section.