44 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

This is the propagator with an infrared cut-off L and an ultraviolet cut-off

ε. Here ε and L are length scales rather than energy scales; length behaves

as the inverse to energy. Thus, ε is the high-energy cut-off and L is the

low-energy cut-off.

The propagator P (ε, L) damps down the high energy modes in the prop-

agator P . Indeed,

P (ε, L) =

i

e−ελi

−

e−Lλi

λi

2

+ m2

ei ⊗ ei

so that the coeﬃcient of ei ⊗ ei decays as λi

−2e−ελi

for λi large.

Because P (ε, L) is a smooth function on M × M, as long as ε 0,

the expression W (P (ε, L),I) is well-defined for all I ∈

O+(C∞(M

))[[ ]].

(Recall the superscript + means that I must be at least cubic modulo .)

Definition 4.2.1. The map

O+(C∞(M

))[[ ]] →

O+(C∞(M

))[[ ]]

I → W (P (ε, L),I)

is defined to be the renormalization group flow from length scale ε to length

scale L.

From now on, we will be using this length-scale version of the renormal-

ization group flow.

Figure 2 illustrates the first few terms of the renormalization group flow

from scale ε to scale L.

As the effective interaction I[L] varies smoothly with L, there is an infin-

itesimal form of the renormalization group equation, which is a differential

equation in I[L]. This is illustrated in figure 3.

The expression for the propagator in terms of the heat kernel has a

very natural geometric/physical interpretation, which will be explained in

Section 6.

5. Singularities in Feynman graphs

5.1. In this section, we will consider explicitly some of the simple Feyn-

man graphs appearing in W (P (ε, L),I) where I(φ) =

1

3! M

φ3,

and try to

take the limit as ε → 0. We will see that, for graphs which are not trees,

the limit in general won’t exist. The

1

3!

present in the interaction term sim-

plifies the combinatorics of the Feynman diagram expansion. The Feynman

diagrams we will consider are all trivalent, and as explained in Section 3,

each vertex is labelled by the linear map

C∞(M)⊗3

→R

φ1 ⊗ φ2 ⊗ φ3 →

∂

∂φ1

∂

∂φ2

∂

∂φ3

I

=

M

φ1(x)φ2(x)φ3(x)