Computational Physics

15.3 Interacting fields and renormalisation 473

Thed-dimensional delta-function reflects the energy–momentum conservation of
the scattering process, which is related to the space-time translation invariance of
the Green’s function.
For the free field theory it is found that

G(k,−k)=

1

k^2 +m^2

(15.31)

which leads to the real-space form:

G(x−x′)=

e−|x−x

′|m

|x−x′|η

; large|x−x′|, (15.32)

withη=(d− 1 )/2. We see that the Green’s function has a finite correlation
lengthξ= 1 /m. Higher-order correlation functions for the free field theory can
be calculated usingWick’s theorem: correlation functions with an odd number of
φ-fields vanish, but if they contain an even number of fields, they can be written as
a symmetric sum over products of pair-correlation functions, for example

G(x 1 ,x 2 ,x 3 ,x 4 )=〈φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 )〉

=〈φ(x 1 )φ(x 2 )〉〈φ(x 3 )φ(x 4 )〉+〈φ(x 1 )φ(x 3 )〉〈φ(x 2 )φ(x 4 )〉

+〈φ(x 1 )φ(x 4 )〉〈φ(x 2 )φ(x 3 )〉. (15.33)

In fact, it is well known that for stochastic variables with a Gaussian distribution,
all higher moments can be formulated similarly in terms of the second moment.
Wick’s theorem is a generalisation of this result.

15.3 Interacting fields and renormalisation

The free field theory can be solved analytically: all the Green’s functions can be
given in closed form. This is no longer the case when we are dealing with interacting
fields. If we add, for example, a termgφ^4 to the free field Lagrangian, the only way
to proceed analytically is by performing a perturbative analysis in the coupling
constantg. It turns out that this gives rise to rather difficult problems. The terms
in the perturbation series involve integrals over some momentum coordinates, and
these integrals diverge! Obviously our predictions for physical quantities must be
finite numbers, so we seem to be in serious trouble. Since this occurs in most
quantum field theories as soon as we introduce interactions, it is a fundamental
problem which needs to be faced.
To get a handle on the divergences, one starts by controlling them in some suitable
fashion. One way to do this is by cutting off the momentum integrations at some
large but finite value. This renders all the integrals occurring in the perturbation
series finite, but physical quantities depend on the (arbitrary) cut-off which is still