16 1. INTRODUCTION

for all Λ ∈ (0, ∞). More precisely,

Seff

[Λ] should be a formal

power series both in the field φ ∈

C∞(M)[0,Λ]

and in the variable

.

(2) Modulo , each

Seff

[Λ] must be of the form

Seff

[Λ](φ) = −

1

2

M

φ D φ + cubic and higher terms.

where D is the positive-definite Laplacian. (If we want to consider

a massive scalar field theory, we can replace D by D

+m2).

(3) If Λ Λ,

Seff

[Λ ] is determined from

Seff

[Λ] by the renormaliza-

tion group equation (which makes sense in the formal power series

setting).

(4) The effective actions

Seff

[Λ], when translated into a solution to

the length-scale version of the RGE, satisfy the asymptotic locality

axiom.

Since solutions to the energy and length-scale versions of the RGE are

equivalent, one can base this definition entirely on the length-scale version

of the RGE. We will do this in the body of the book.

Earlier we sketched a very general form of the RGE, which uses an

arbitrary parametrix to define a “scale” of the theory. In Chapter 2, Section

8, we will give a definition of a quantum field theory based on arbitrary

parametrices, and we will show that this definition is equivalent to the one

described above.

6. The main theorem

Now we are ready to state the first main result of this book.

Theorem A. Let T

(n)

denote the set of theories defined modulo

n+1.

Then, T

(n+1)

is a principal bundle over T

(n)

for the abelian group of local

action functionals S :

C∞(M)

→ R.

Recall that a functional S is a local action functional if it is of the form

S(φ) =

M

L (φ)

where L is a Lagrangian. The abelian group of local action functionals is

the same as that of Lagrangians up to the addition of a Lagrangian which

is a total derivative.

Choosing a section of each principal bundle T

(n+1)

→ T

(n)

yields an

isomorphism between the space of theories and the space of series in whose

coeﬃcients are local action functionals.

A variant theorem allows one to get a bijection between theories and

local action functionals, once one has made an additional universal (but

unnatural) choice, that of a renormalization scheme. A renormalization