FOUNDATIONS OF QUANTUM MECHANICS

VIII. 4. THE MEASUREMENT PROBLEM IN THE NARROW SENSE 177
as |c j | ̸= |c k | for j ≠ k. Therefore it is possible for every state |ψ⟩ for which this holds to exactly
indicate the potential corresponding properties. A generalization to mixed states can be achieved by
taking the spectral decomposition of W of the composite system as the preferred decomposition, the
Schmidt decomposition (VIII. 21) is then found for the special case of pure states.
The idea that the meaning of the state vector can be exclusively formulated in terms of measurements
is rejected, the state vector describes factual properties. The description by the wave function
is, however, incomplete, |ψ⟩ determines the possibilities and the probabilities of the possibilities, but
the real physical situation is not determined. Quantum mechanics is fundamentally indeterministic
because sometimes one possibility, at other times another one occurs.
Moreover, in this interpretation the ‘only if’ part of the property postulate is rejected, if a system
is in an eigenstate it has indeed the corresponding eigenvalue, but not ‘only if’; a system which is
in a superposition of eigenstates, (VIII. 21), nevertheless has one of the properties. In the first case
a composite physical system necessarily has the property, in the second case contingently. In logic
the italicized words are called ‘modalities’, hence the name modal interpretation. The projection
postulate is now superfluous.
If, however, the singlet state, being a state of a composite system also, is considered in the modal
interpretation, this interpretation tells us less than quantum mechanics with the property postulate
does.
◃ Remarks
In this interpretation, the metastability or possibly permanent nature of the quantities of system 2 plays
no role in attributing properties. Another point in this interpretation is that, besides the Schrödinger
dynamics for the state, there seems to be a need for a dynamics describing how properties change in
time. Several attempts have been made to that end. ▹
EXERCISE 38. What does quantum mechanics with the property postulate say about the EPRB
experiment, p. 139, that the modal interpretation does not say, and why? Does it help to couple a
measuring apparatus to the composite system of the two spin particles?
VIII. 4. 8
DECOHERENCE
Finally we will discuss the option which is possibly supported by the majority of physicists, see
H.J. Groenewold (1946), K. Gottfried (1989), N.G. van Kampen (1988), W.H. Zurek (1981 and 1982).
Bell (1990) named this option the For All Practical Purposes solution, briefly FAPP. The idea is to
show that the difference between the pure state (VIII. 15) and the mixed state (VIII. 16) is hardly
perceptible in practice.
A measuring apparatus is a macroscopic system which is in continuous interaction with its surroundings.
A more realistic representation of the measurement process will therefore be of the