54 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Thus, local action functionals are the same as Lagrangians modulo those

Lagrangians which are a total derivative.

Definition 7.1.2. A perturbative quantum field theory, with space of

fields

C∞(M)

and kinetic action −

1

2

φ, (D

+m2)φ

, is given by a set of

effective interactions I[L] ∈

O+(C∞(M

))[[ ]] for all L ∈ (0, ∞], such that

(1) The renormalization group equation

I[L] = W (P (ε, L),I[ε])

is satisfied, for all ε, L ∈ (0, ∞].

(2) For each i, k, there is a small L asymptotic expansion

Ii,k[L]

r∈Z≥0

gr(L)Φr

where gr(L) ∈

C∞((0,

∞)L) and Φr ∈

Oloc(C∞(M

)).

Let T

(∞)

denote the set of perturbative quantum field theories, and let T

(n)

denote the set of theories defined modulo

n+1.

Thus, T

(∞)

= lim

←−

T

(n).

Let me explain more precisely what I mean by saying there is a small L

asymptotic expansion

Ii,k[L]

j∈Z≥0

gr(L)Φr.

Without loss of generality, we can require that the local action functionals

Φr appearing here are homogeneous of degree k in the field a.

Then, the statement that there is such an asymptotic expansion means

that there is a non-decreasing sequence dR ∈ Z, tending to infinity, such

that for all R, and for all fields a ∈

C∞(M),

lim

L→0

L−dR

Ii,k[L](a) −

R

r=0

gr(L)Φr(a) = 0.

In other words, we are asking that the asymptotic expansion exists in the

weak topology on

Hom(C∞(M)⊗k,

R)Sk .

The main theorem of this chapter (in the case of scalar field theories) is

the following.

Theorem A. Let T

(n)

denote the set of perturbative quantum field the-

ories defined modulo

n+1.

Then T

(n+1)

is, in a canonical way, a principal

bundle over T

(n)

for the abelian group

Oloc(C∞(M

)) of local action func-

tionals on M.

Further, T (0) is canonically isomorphic to the space Oloc(C∞(M

+

)) of

local action functionals which are at least cubic.

There is a variant of this theorem, which states that there is a bijection

between theories and Lagrangians once we choose a renormalization scheme,

which is a way to extract the singular part of certain functions of one vari-

able. The concept of renormalization scheme will be discussed in Section 9;