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.