FOUNDATIONS OF QUANTUM MECHANICS

182 CHAPTER VIII. THE MEASUREMENT PROBLEM
THE WIGNER - ARAKI - YANASE THEOREM:
The evolution U(τ), which brings about the measurement transition (VIII. 4) when measuring
physical quantity A, is possible iff A commutes with all additive conserved quantities
of the composite system of object system and measuring apparatus. In other words,
conserved physical quantities which are not additive, additive physical quantities which
are not conserved, and physical quantities which are neither conserved nor additive, cannot
be measured exactly.
Proof
Here we will only prove the ‘if’ - part of the theorem. Let B be an additive conserved quantity of
the composite system SM, i.e., B is, by definition, of the form
B = B 1 ⊗ 11 + 11 ⊗ B 2 , (VIII. 34)
which is conserved. This means that B commutes with the Hamiltonian H of the composite
system,
[B, H] = 0. (VIII. 35)
Then B also commutes with every function of H and therefore with U (τ) = e − i H τ ,
[B, U (τ)] = 0 =⇒ B = U † (τ) B U (τ). (VIII. 36)
Consider the matrix element
B jk := ⟨a j | ⊗ ⟨r 0 | B |a k ⟩ ⊗ |r 0 ⟩. (VIII. 37)
On the one hand, because of the additivity of B, (VIII. 34), we have
B jk = ⟨a j | ⊗ ⟨r 0 | (B 1 ⊗ 11 + 11 ⊗ B 2 ) |a k ⟩ ⊗ |r 0 ⟩
= ⟨a j | B 1 | a k ⟩ + δ jk ⟨r 0 | B 2 | r 0 ⟩, (VIII. 38)
while on the other hand, using (VIII. 36), we see that
B jk = ⟨a j | ⊗ ⟨r 0 | U † (τ) B U (τ) |a j ⟩ ⊗ |r 0 ⟩ = ⟨a j | ⊗ ⟨r j | B | a k ⟩ ⊗ |r k ⟩
= δ jk ⟨a j | B 1 | a k ⟩ + δ jk ⟨r j | B 2 | r k ⟩. (VIII. 39)
Comparison of these two results shows that
⟨a j | B 1 | a k ⟩ = 0 for j ≠ k, (VIII. 40)
which means that in the basis {|a j ⟩} of H S , B 1 is in diagonal form and therefore A commutes
with B 1 . □
This theorem shows that the scheme (VIII. 4) can, strictly speaking, apply only to measurement of
quantities which commute with all additive conserved quantities. However, the measurement scheme
remains approximately valid if the value of the conserved quantity is large, which will easily be the
case for macroscopic apparatuses. We therefore see that, whereas the U (t) in (VIII. 2) exists, a more
concrete interpretation can come across problems. The shortcomings of the conventional formalism
of quantum mechanics with regard to giving a faithful description of the measurement process, has
lead to interesting extensions of the formalism, see, for instance, Busch, Lahti and Mittelstaedt (1991).