The Physical Basis of The Direction of Time (The Frontiers ...

120 4 The Quantum Mechanical Arrow of Time
The decay law (4.44) defines an elementary master equation (3.48) with
a Green’s function Ĝret simply given by the decay rate λ (see Sect. 4.1.2).
Its foundation on time-symmetric fundamental dynamics (such as a universal
Schrödinger equation) requires quite analogous assumptions, for example the
negligibility of any back-flow into ‘doorway states’ that are directly coupled to
A (see Fig. 3.4). Therefore, a conserved quantity has to disappear fast enough
from such doorway states into ‘deeper’ (dynamically more distant) states,
which must form a large reservoir.
A simple model is provided by the T -symmetric finite reaction chain
dA n
dt
= −(λ n + λ n−1 )A n + λ n A n+1 + λ n−1 A n−1 , (4.45)
with n =0,...,N, λ −1 = λ N = 0, and the (improbable) initial condition
A n≠0 ≈ 0. n = 1 represents here the doorway channel of Sect. 3.2. For λ 0 ≪
λ n≠0 , one obtains
dA 0
≈−λ 0 A 0 , (4.46)
dt
as long as A 1 ≪ A 0 . This requires only λt ≪ N, rather than λt ≪ 1, since all
A n≠0 will relax into partial equilibrium A n≠0 ≈ A 1 on a short time scale (or
just propagate away for N →∞).
Exponential decay can similarly be described by a deterministic wave equation
on a continuum, where the small transition rate λ 0 is replaced by a potential
barrier. It is irrelevant that the Schrödinger equation does here not
describe the conserved quantity (‘probability’) itself. An overall time dependence
according to a complex energy eigenvalue, ψ(t) ∝ exp[−i(E 0 − iγ)t],
would not be compatible with unitarity, but it may well represent an approximation
that is valid in a bounded though growing spacetime region (Khalfin
1958, Petzold 1959, Peres 1980a) – similar to the reaction chain (4.45). Distant
regions in space form a large reservoir.
In scattering theory, unstable states correspond to poles of the S-matrix
S nn ′(k), analytically continued into the complex plane, at points k = k 1 −
ik 2 in the lower right half-plane (k 1 > 0andk 2 > 0), where k is the wave
number, k 2 = k1 2 − k2 2 − 2ik 1 k 2 =2mE. In the restricted spacetime region,
where exponential behavior is observed after the incoming waves producing
the decaying system have ceased, the wave function is dominated by the Breit–
Wigner part (i.e., the pole contribution). This requires a (positive) time delay
during the scattering process, which must be described by the relevant partial
wave ψ l (r, t)Y lm (θ, φ). Its radial factor ψ l (r, t) may be expanded in terms of
energy eigenstates, ψ (k)
l
(r, t) :=φ k,l (r)e −iω(k)t , in the form
ψ l (r, t) =
−→
r→∞
∫ ∞
0
∫ ∞
0
f l (k)ψ (k)
l
(r, t)dk
f l (k) e−ikr − (−1) l S l (k)e ikr
e −iω(k)t dk, (4.47)
r