Quantum Field Theory I

96 CHAPTER 3. INTERACTING QUANTUM FIELDS
After having warned the reader about the unsoundness of what follows, we
can proceed directly to the interpretation of the result for one-particle initial
state in terms of the decay rate: From the linear time dependence of the probability
density one can read out the probability density per unit time, this is
equal to the time derivative of the probability density and for the exponential
decay exp(−Γt) this derivative taken at t = 0 is nothing else but the decay rate
Γ (or dΓ if we are interested only in decays with specific final states).
The previous statement is such a dense pack of lies that it would be hardly
outmatched in an average election campaign. First of all, we did not get the
linear time dependence, since T is not the ”flowing time”, but rather a single
moment. Second, even if we could treat T as a variable, it definitely applies
only to large times and, as far as we can see now, has absolutely nothing to say
about infinitesimal times in the vicinity of t = 0. Third, if we took the linear
time dependence seriously, then why to speak about exponential decay. Indeed,
there is absolutely no indication of the exponential decay in our result.
Nevertheless, for the reasons unclear at this point (they are discussed in the
appendix ) the main conclusion of the above lamentable statement remains
valid: decay rate is given by the probability quasi-density divided by T. After
switching back to the infinite volume, which boils down to δ 4 V T (p f − p i ) →
δ 4 (p f −p i ), one obtains
dΓ = (2π) 4 δ 4 (P f −P i )
1 ∏
m
|M fi | 2
2E A
where E A is the energy of the decaying particle.
i=1
d 3 p i
(2π) 3 2E i
Remark: The exponential decay of unstable systems is a notoriously known
matter, perhaps too familiar to realize how non-trivial issue it becomes in the
quantum theory. Our primary understanding of the exponential decay is based
on the fact that exp(−Γt) is the solution of the simple differential equation
dN/dt = −ΓN(t), describing a population of individuals diminishing independently
of a) each other b) the previous history. In quantum mechanics, however,
the exponential decay should be an outcome of the completely different time evolution,
namely the one described by the Schrödinger equation.
Is it possible to have the exponential decay in the quantum mechanics, i.e. can
one get |〈ψ 0 |ψ(t)〉| 2 = e −Γt for |ψ(t)〉 being a solution of the Schrödinger equation
with the initial condition given by |ψ 0 〉 The answer is affirmative, e.g. one
can easily convince him/herself that for the initial state |ψ 0 〉 = ∑ c λ |ϕ λ 〉, where
H|ϕ λ 〉 = E λ |ϕ λ 〉, one obtains the exponential decay for ∑ |c λ | 2 δ(E −E λ ) def
=
p(E) = 1
2π
Γ
(E−E 0) 2 +Γ 2 /4
(the so-called Breit-Wigner distribution of energy in
the initial state).
The Breit-Wigner distribution could nicely explain the exponential decay in
quantum mechanics, were it not for the fact that this would require understanding
of why this specific distribution is so typical for quantum systems. And this
is very far from being obvious. Needless to say, the answer cannot be that the
Breit-Wigner is typical because the exponential decay is typical, since this would
immediately lead to a tautology. It would be nice to have a good understanding
for the exponential decay in the quantum mechanics, but (unfortunately) we are
not going to provide any.