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
denote the space of scalar field theories on
a manifold M, defined modulo
Then T
is (in a canonical way) a principal bundle
for the space of local action functionals.
Further, T
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
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
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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