Master Dissertation

Chapter 3
The Scattering Matrix
As the notion of the scattering matrix, or the S-matrix for short, plays the
central role in scattering theory I will give a brief introduction to the
development of it in the case of quantum mechanics. Later our main aim
will be to construct the S-matrix of quantum electrodynamics, QED, by
perturbation theory. First a definition:
Definition 3.1. Let H be a Hilbert space. Then a two-parameter family
U(s, t), (s, t) ∈ R 2 of bounded operators on H is called a unitary
propagator, if the following is satisfied:
(1) U(s, t) is unitary for all s, t ∈ R,
(2) U(t, t) = 1 for all t ∈ R,
(3) U(r, s)U(s, t) = U(r, t) for all r, s, t ∈ R,
(4) The map (s, t) ↦→ U(s, t)ψ is continuous for all ψ ∈ H.
Consider a quantum mechanical system described by the Hamiltonian
H = H0 + V (t), where H0 is the free Hamiltonian and V is the
time-dependent interaction. Then the time evolution is given by a unitary
propagator in the sense that
ψ(t) = U(t, s)ψ(s) 1 . (3.1)
Definition 3.2. Let U(s, t) be a unitary propagator then we define the
wave operators as follows
Win = s- lim
t→−∞ U(t, 0)∗ e −iH0t = s- lim
t→−∞ U(0, t)e−iH0t .
Wout = s- lim
t→∞ U(t, 0) ∗ e −iH0t = s- lim
t→∞ U(0, t)e −iH0t .
Provided the strong limits exist.
1 We remember that if H is time independent then then time evolution is given by the
unitary transform ψ(t) = e −iHt ψ(t0).
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