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

4.5 Exponential Decay and ‘Causality’ in Scattering 121
where S l (k) =e 2iδ l(k) is the corresponding diagonal element of the S-matrix.
For sufficiently large values of t, the factor e −iω(k)t oscillates rapidly with k.
This leads to destructive interference under the integral, except in regions of r
and t where the phase kr+ω(k)t (for the incoming wave), or kr−ω(k)t+2δ l (k)
(for the outgoing one), is almost independent of k over the width of the wave
packet f l (k) (whichmaybecenteredatk 0 , say). For the outgoing wave, for
example, this requirement means
d [
kr − ω(k)t +2δl (k) ]∣ ∣
∣k0 ≈ 0 =⇒ r ≈ dω(k 0)
t − 2 dδ l(k 0 )
. (4.48)
dk
dk 0 dk 0
A noticeable delay compared to propagation with the group velocity dω/dk
requires a large value of dδ l /dk, such as in the vicinity of a complex pole of δ l .
For sufficiently large times t, but not too large distances r from the scattering
center, and for initial momentum packets much wider than the size of the
imaginary part k 2 , only the pole contribution remains. For this one may write
S l (k) =e 2iδ l(k) ≈ k − k 1 − ik 2
k − k 1 +ik 2
, (4.49)
and hence k 0 = k 1 for the surviving wave packet that represents the decaying
state. In this spacetime region, the contribution of the pole to (4.49) is given
by its residue, whence
[
∫ ∞
ψ l (r, t) −→ −(−1) l k − k 1 − ik 2 e i
f l (k 1 )
t→∞
0 k − k 1 +ik 2
]
[
≈ (−1) l 2πk 2 f l (k 1 ) ei k 1r−ω(k 1)t
r
]
kr−ω(k)t
r
(
exp
[k 2
dk
r − dω(k 1)
t
dk 1
)]
(4.50)
(assuming k 2 ≪|k 1 |). In the last factor one recognizes the ‘imaginary part of
the energy’, γ = k 2 dω(k 1 )/dk 1 .
A positive delay (a ‘retardation’) of the scattered wave at the resonance
requires
(
)
dδ l
dk ≈−d k 2
k 2
− arctan =
dk
k − k 1 (k − k 1 ) 2 + k2
2 > 0 . (4.51)
The pole must therefore reside in the lower half-plane. This condition is often
referred to as causality in scattering, since the retardation specifies a direction
in time related to intuitive causality (Chap. 2). This position of the poles is
also used for deriving dispersion relations in T -orTCP-symmetric quantum
field theory. However, no time direction can be specified by the structure of
the S-matrix, since the latter is a consequence of the time-reversal-invariant
Hamiltonian. Exponential decay is a fact-like asymmetry that would be reversed,
using the same S-matrix, for scattering states with a time-reversed
boundary condition. So one would have to force the outgoing wave rather