7. A DEFINITION OF A QUANTUM FIELD THEORY 53

Then, the correlation function for the effective interaction I[L] is defined

by

EI[L](f1, . . . , fn) =

γ∈Γn

1

Aut(γ)

n−g(γ)−1Cγ(I[L])(f1,...,fn).

Unlike the heuristic graphical expansion we gave for the correlation functions

of the

φ3

theory, this expression is well-defined.

We should interpret this expansion as saying that we can compute the

correlation functions from the effective interaction I[L] by allowing particles

to propagate in the usual way, and to interact by I[L]; except that in between

any two interactions, particles must travel for a proper time of at least L.

This accounts for the fact that edges which join to internal vertices are

labelled by P (L, ∞).

If we have a collection {I[L] | L ∈ (0, ∞)} of effective interactions, then

the renormalization group equation is equivalent to the statement that all

the correlation functions constructed from I[L] using the prescription given

above are independent of L.

7. A definition of a quantum field theory

7.1. Now we have some preliminary definitions and an understanding

of why the terms in the graphical expansion of a functional integral diverge.

This book will describe a method for renormalizing these functional integrals

to yield a finite answer.

This section will give a formal definition of a quantum field theory, based

on Wilson’s philosophy of the effective action; and a precise statement of the

main theorem, which says roughly that there’s a bijection between theories

and Lagrangians.

Definition 7.1.1. A local action functional I ∈

O(C∞(M

)) is a func-

tional which arises as an integral of some Lagrangian. More precisely, if we

Taylor expand I as I =

∑

k

Ik where

Ik(λa) =

λkIk(a)

(so that Ik is homogeneous of degree k of the variable a ∈

C∞(M)),

then Ik

must be of the form

Ik(a) =

s

j=1

M

D1,j(a) · · · Dk,j(a)

where Di,j are arbitrary differential operators on M.

Let

Oloc(C∞(M

)) ⊂

O(C∞(M

)) be the subspace of local action function-

als.

As before, let

Oloc(C∞(M +

))[[ ]] ⊂

Oloc(C∞(M

))[[ ]]

be the subspace of those local action functionals which are at least cubic

modulo .