CHAPTER 2

Theories, Lagrangians and counterterms

1. Introduction

In this chapter, we will make precise the definition of quantum field

theory we sketched in Chapter 1. Then, we will show the main theorem:

Theorem A. Let T

(n)(M)

denote the space of scalar field theories on

a manifold M, defined modulo

n+1.

Then T

(n+1)(M)

→ T

(n)(M)

is (in a canonical way) a principal bundle

for the space of local action functionals.

Further, T

(0)(M)

is canonically isomorphic to the space of local action

functionals which are at least cubic.

This theorem has a less natural formulation, depending on an additional

choice, that of a renormalization scheme. A renormalization scheme is an

object of a “motivic” nature, defined in Section 9.

Theorem B. The choice of a renormalization scheme leads to a section

of each principal bundle T

(n+1)(M)

→ T

(n)(M),

and thus to an isomor-

phism between the space of theories and the space of local action functionals

of the form

∑

iSi, where S0 is at least cubic.

1.1. Let me summarize the contents of this chapter.

The first few sections explain, in a leisurely fashion, the version of the

renormalization group flow we use throughout this book. Sections 2 and 4

introduce the heat kernel version of high-energy cut-off we will use through-

out the book. Section 3 contains a general discussion of Feynman graphs,

and explains how certain finite dimensional integrals can be written as sums

over graphs. Section 5 explains why infinities appear in the naive func-

tional integral formulation of quantum field theory. Section 6 shows how

the weights attached to Feynman graphs in functional integrals can be in-

terpreted geometrically, as integrals over spaces of maps from graphs to a

manifold.

In Section 7, we finally get to the precise definition of a quantum field

theory and the statement of the main theorem. Section 8 gives a variant of

this definition which doesn’t rely on the heat kernel, but instead works with

an arbitrary parametrix for the Laplacian. This variant definition is equiv-

alent to the one based on the heat kernel. Section 9 introduces the concept

of renormalization scheme, and shows how the choice of renormalization

scheme allows one to extract the singular part of the weights attached to

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http://dx.doi.org/10.1090/surv/170/02