Quantum Field Theory I

3.1. NAIVE APPROACH 89
One may, of course, ignore this difference completely. And it is precisely this
ignorance, what constitutes the essence of our naive approach. Indeed, the core
of this approach is the work with the s-matrix. defined in terms of explicitly
known simple states, instead of dealing with the S-matrix, defined in terms of
the states experimentally accessible in the real world (with interactions). The
latter are usually not explicitly known, so the naivity simplifies life a lot.
What excuse do we have for such a simplification Well, if the interaction
may be viewed as only a small perturbation of the free theory — and we
have adopted this assumption already, namely in the perturbative treatment of
U (T,−T) operator — then one may hope that the difference between the two
sets ofstatesis negligible. Totakethis hopetooseriouslywouldbe indeed naive.
To ignore it completely would be a bit unwise. If nothing else, the s-matrix is
the zeroth order approximation to the S-matrix, since the unperturbed states
are the zeroth order approximation of the corresponding states in the full theory.
Moreover, the developments based upon the naive assumption tend to be
very useful, one can get pretty far using this assumption and almost everything
will survive the more rigorous treatment of the standard approach.
As to the calculation of the s-matrix elements, it follows the calculation
of the green functions very closely. One just uses the perturbative expansion
U (T,−T) = T exp{−i ∫ T
−T
dt H ′ I (t)} (see p.77) and the Wick’s theorem, which
is to be supplemented by 20
ϕ I (x)a + ⃗p,I (−T) = N{ϕ I (x)a + ⃗p,I (−T)}+ 1 √
2ω⃗p
e −ipx e −iω ⃗pT
a ⃗p,I (T)ϕ I (x) = N{a ⃗p,I (T)ϕ I (x)}+ 1 √
2ω⃗p
e ipx e −iω ⃗pT
or in the ”clipping notation”
+
ϕ I (x)a⃗p,I (−T) = √ 1 e −ipx e −iω ⃗pT
2ω⃗p
a ⃗p,I (T)ϕ I
(x) = 1 √
2ω⃗p
e ipx e −iω ⃗pT
Consequently, the s-matrix elements are obtained in almost the same way as are
the green functions, i.e. by means of the Feynman rules. The only difference is
the treatment of the external lines: instead of the factor d F (x−y), which was
present in the case of the green functions, the external legs provide the factors
e ∓ipx e −iω⃗pT for the s-matrix elements, where the upper and lower sign in the
exponent corresponds to the ingoing and outgoing particle respectively. (Note
that the √ 2ω ⃗p in the denominator is canceled by the √ 2ω ⃗p in the definition of
s fi .) In the Feynman rules the factor e −iω⃗pT is usually omitted, since it leads to
the pure phase factor exp{−iT ∑ n
i=1 ω ⃗p n
}, which is redundant for probability
densities, which we are interested in 21 .
20 Indeed, first of all one has ϕ I (x)a + ⃗p,I (t′ ) = N{ϕ I (x)a + ⃗p,I (t′ )}+[ϕ I (x),a + ⃗p,I (t′ )] and then
[ϕ I (x),a + ⃗p,I (t′ )] = ∫ d 3 p ′
(2π) 3 e i⃗p′ .⃗x
√
2ω⃗p ′ [a ⃗p ′ ,I(t),a + ⃗p,I (t′ )] = ∫ d 3 p ′
√
2ω⃗p ′ ei(⃗p′ .⃗x−ω ⃗p ′t+ω ⃗p t ′ ) δ(⃗p − ⃗p ′ ) and
the same gymnastics (with an additional substitution p ′ → −p ′ in the integral) is performed
for [a ⃗p,I (t ′ ),ϕ I (x)].
21 Omission of the phase factor is truly welcome, otherwise we should bother about the ill-