10 1. INTRODUCTION
length-scale L. This object encodes all physical phenomena occurring at
lengths greater than L. (The effective interaction can also be considered in
the energy-scale picture also: the relationship between the effective action
Seff
[Λ] and the effective interaction
Ieff
[Λ] is simply
Seff
[Λ](φ) = −
1
2
M
φ D φ +
Ieff
[Λ](φ)
for fields φ ∈
C∞(M)[0,Λ).
The reason for introducing the effective interac-
tion is that the world-line version of the renormalization group flow is better
expressed in these terms.
In the world-line picture of physics at lengths greater than L, we can
only consider paths which evolve for a proper time greater than L, and
then interact via
Ieff
[L]. All processes which involve particles moving for a
proper time of less than L between interactions are assumed to be subsumed
into
Ieff
[L].
The renormalization group equation for these effective interactions can
be described by saying that quantities we compute using this prescription
are independent of L. That is,
Definition 3.5.1. A collection of effective interactions
Ieff
[L] satisfies
the renormalization group equation if, when we compute correlation func-
tions using
Ieff
[L] as our interaction, and allow particles to travel for a
proper time of at least L between any two interactions, the result is indepen-
dent of L.
If one works out what this means, one sees that the scale L effective in-
teraction
Ieff
[L] can be constructed in terms of
Ieff
[ε] by allowing particles
to travel along paths with proper-time between ε and L, and then interact
using
Ieff
[ε].
More formally,
Ieff
[L] can be expressed as a sum over Feynman graphs,
where the edges are labelled by the propagator
P (ε, L) =
L
ε
e−τm2
Kτ
and where the vertices are labelled by
Ieff
[ε].
This effective interaction
Ieff
[L] is an -dependent functional on the
space
C∞(M)
of fields. We can expand
Ieff
[L] as a formal power series
Ieff
[L] =
i,k≥0
iIi,k
eff
[L]
where
Ii,k
eff
[L] :
C∞(M)
→ R
is homogeneous of order k. Thus, we can think of Ii,k[L] as being a symmetric
linear map
Ii,k
eff
[L] :
C∞(M)⊗k
→ R.
