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

4.5 Exponential Decay and ‘Causality’ in Scattering 119
4.5 Exponential Decay and ‘Causality’ in Scattering
There are only a few absolutely stable ‘particles’ (elementary quantum objects),
while all others are known as decaying on vastly different time scales.
In quantum theory, they may be described formally by means of complex energies.
ForaSchrödinger type time dependence e −iEt , a negative imaginary
part, E = E 0 − iγ with γ > 0, would lead to an exponentially decreasing
wave function. This does not just describe probabilities for different decay
times, since all parts of the wave function form one coherent superposition
(see below and Sect. 4.3.6). Even though microscopic, these objects have to be
regarded as open quantum systems. For example, an excited atom is coupled
to an initial vacuum (or a photon heat bath of zero temperature). Unbounded
space represents an ‘absorber’ of infinite capacity for the decay fragments.
The decaying system may also be described by means of an S-matrix for
the decay fragments, where unstable states show up as poles in the complex
energy or momentum plane. This S-matrix must represent the fundamental
(time-symmetric) dynamics. Exponential decay then seems to characterize
a fundamental direction in time (see Prigogine 1980, for example), similar
to Ritz’s retarded electrodynamics (Chap. 2). Since there are no energy eigenstates
with complex eigenvalues (sometimes called ‘Gamow vectors’) in Hilbert
space, this situation has even led to the proposal of ‘rigged Hilbert spaces’
(Böhm 1978). However, decaying systems may well be described in conventional
quantum mechanical terms, where the exponential time dependence
applies only approximately in a limited spacetime region.
Exponential decay of an arbitrary quantity A would be the consequence
of a constant loss rate, described by
dA
dt
= −λA , (4.44)
with λ>0. The absolute rate of change, dA/dt, is then completely determined
by A itself. This asymmetry under time reversal may be the consequence of
a special initial condition, similar to that characterizing irreversible master
equations. In particular, if A is a conserved quantity, any back-flow, must
be negligible. This condition represents a fact-like T-asymmetry that may be
explained by assuming a sufficiently large and initially empty reservoir (comparable
to the ‘irrelevant channel’ used in Sect. 3.2). If recurrence times are
sufficiently large, the exponential law (4.44) may remain an excellent approximation,
describing the decaying object for a very long time.
This disappearance of a ‘substance’ A from a given subsystem or region in
space is an entirely classical model. However, the time dependence (4.44) is
best known from radioactive decay in quantum theory, where A represents the
non-decay probability. It is then regarded as the standard example of quantum
indeterminism – usually understood as fundamental and law-like. This
interpretation of (4.44) would mean that decay events occur at unpredictable
though definite instants in time.