Quantum Field Theory I

3.1. NAIVE APPROACH 93
3.1.3 Cross-sections and decay rates
Because of the relativistic normalization of states 〈⃗p|⃗p ′ 〉 = 2E ⃗p (2π) 3 δ 3 (⃗p−⃗p ′ )
the S-matrix (s-matrix) 24 elements do not give directly the probability amplitudes.
To take care of this, one has to use the properly normalized vectors
(2π) −3/2 (2E) −1/2 |⃗p〉, which leads to the probability amplitude equal to S fi
multiplied by ∏ n
j=1 (2π)−3/2 (2E j ) −1/2 . Taking the module squared one obtains
probability density = |S fi | 2 n
∏
j=1
1
(2π) 3 2E j
This expression presents an unexpected problem. The point is that S fi turns
out to contain δ-functions, and so |S fi | 2 involves the ill-defined square of the δ-
function. Theδ-functionsoriginatefromthenormalizationofstatesandtheyare
potentially present in any calculation which involves states normalized to the δ-
function. Inmanycasesoneisluckyenoughnottoencounteranysuchδ-function
in the result, but sometimes one is faced with the problem of dealing with
δ-functions in probability amplitudes. The problem, when present, is usually
treatedeitherbyswitchingtothe finitevolumenormalization,orbyexploitation
of the specific set of basis vectors (”wave-packets” rather than ”plane waves”).
There are two typical δ-functions occurring in S fi . The first one is just the
normalization δ-function, symbolically written as δ(f −i), which represents the
complete result in the case of the free Hamiltonian (for initial and final states
being eigenstates of the free Hamiltonian). In the perturbative calculations,
this δ(f −i) remains always there as the lowest order result. It is therefore a
common habit to split the S-matrix as (the factor i is purely formal)
S fi = δ(f −i)+iT fi
and to treat the corresponding process in terms of T fi , which contains the complete
information on transition probability for any |f〉 ≠ |i〉. The probability for
|f〉 = |i〉 can be obtained from the normalization condition (for the probability).
Doing so, one effectively avoids the square of δ(f −i) in calculations.
But even the T-matrix is not ”δ-free”, it contains the momentum conservation
δ-function. Indeed, both the (full) Hamiltonian and the 3-momentum
operator commute with the time-evolution operator 0 = [P µ ,U (T,−T)]. When
sandwiched between some 4-momentum eigenstates 25 〈f| and |i〉, this implies
(p µ f −pµ i )〈f|U (T,−T)|i〉 = 0, and consequently (pµ f −pµ i )S fi = 0. The same, of
course, must hold for T fi . Now any (generalized) function of f and i, vanishing
for p µ f ≠ pµ i , is either a non-singular function (finite value for p f = p i ), or a
distribution proportional to δ 4 (p f −p i ), or even a more singular function (proportional
to some derivative of δ 4 (p f −p i )). We will assume proportionality to
the δ-function
T fi = (2π) 4 δ 4 (p f −p i )M fi
where (2π) 4 is a commonly used factor. Such an assumption turns out to lead to
the finite result. We will ignorethe othertwo possibilities, since if the δ-function
provides a final result, they would lead to either zero or infinite results.
24 All definitions of this paragraph are formulated for the S-matrix, but they apply equally
well for the s-matrix, e.g. one have s fi = δ(f −i)+it fi and t fi = (2π) 4 δ 4 ( p f −p i
) mfi .
25 The notational mismatch, introduced at this very moment, is to be discussed in a while.