Kiefer C. Quantum gravity

308 INTERPRETATION
10.1.1 Decoherence in quantum mechanics
If quantum theory is universally valid, every system should be described in quantum
terms, and it would be inconsistent to draw an aprioriborder line between
a quantum system and a classical apparatus. John von Neumann was the first
who analysed in 1932 the measurement process within quantum mechanics; see
von Neumann (1932). He considers the coupling of a system (S) to an apparatus
(A), see Fig. 10.1.
S
✲
A
Fig. 10.1. Original form of the von Neumann measurement model.
If the states of the measured system that are discriminated by the apparatus
are denoted by |n〉 (e.g. spin up and spin down), an appropriate interaction
Hamiltonian has the form
H int = ∑ n
|n〉〈n|⊗Ân . (10.1)
The operators Ân act on the states of the apparatus and are rather arbitrary, but
must, of course, depend on the ‘quantum number’ n. Equation (10.1) describes
an ‘ideal’ interaction during which the apparatus becomes correlated with the
system state, without changing the latter. There is thus no disturbance of the
system by the apparatus—on the contrary, the apparatus is disturbed by the
system (in order to yield a measurement result).
If the measured system is initially in the state |n〉 and the device in some
initial state |Φ 0 〉, the evolution according to the Schrödinger equation with the
Hamiltonian (10.1) reads
( )
t
|n〉|Φ 0 〉 −→ exp (−iH int t) |n〉|Φ 0 〉 = |n〉 exp −iÂnt |Φ 0 〉
≡|n〉|Φ n (t)〉 . (10.2)
The resulting apparatus states |Φ n (t)〉 are often called ‘pointer states’. A process
analogous to (10.2) can also be formulated in classical physics. The essential
new quantum features now come into play when one considers a superposition
of different eigenstates (of the measured ‘observable’) as the initial state. The
linearity of time evolution immediately leads to
( ) ∑
t
c n |n〉 |Φ 0 〉 −→ ∑ c n |n〉|Φ n (t)〉 . (10.3)
n
n
But this state is a superposition of macroscopic measurement results (of which
Schrödinger’s cat is just one drastic example)! To avoid such a bizarre state,