2 1. INTRODUCTION

phenomena occurring at energies less than Λ, we can compute any quantity

we would like in terms of the effective action

Seff

[Λ].

If Λ Λ, then the energy Λ effective field theory can be deduced

from knowledge of the energy Λ effective field theory. This leads to an

equation expressing the scale Λ effective action

Seff

[Λ ] in terms of the

scale Λ effective action

Seff

[Λ]. This equation is called the renormalization

group equation.

If we do have a continuum quantum field theory (whatever that is!)

we should, in particular, have a low-energy effective field theory for every

energy. This leads to our definition : a continuum quantum field theory is

a sequence of low-energy effective actions Seff [Λ], for all Λ ∞, which are

related by the renormalization group flow. In addition, we require that the

Seff

[Λ] satisfy a locality axiom, which says that the effective actions

Seff

[Λ]

become more and more local as Λ → ∞.

This definition aims to be as parsimonious as possible. The only as-

sumptions I am making about the nature of quantum field theory are the

following:

(1) The action principle: physics at every energy scale is described by a

Lagrangian, according to Feynman’s sum-over-histories philosophy.

(2) Locality: in the limit as energy scales go to infinity, interactions

between fields occur at points.

1.2. In this book, I develop complete foundations for perturbative quan-

tum field theory in Riemannian signature, on any manifold, using this defi-

nition.

The first significant theorem I prove is an existence result: there are as

many quantum field theories, using this definition, as there are Lagrangians.

Let me state this theorem more precisely. Throughout the book, I will

treat as a formal parameter; all quantities will be formal power series in

. Setting to zero amounts to passing to the classical limit.

Let us fix a classical action functional

Scl

on some space of fields E ,

which is assumed to be the space of global sections of a vector bundle on a

manifold M

1.

Let T

(n)(E

,

Scl)

be the space of quantizations of the classical

theory that are defined modulo

n+1.

Then,

Theorem 1.2.1.

T

(n+1)(E

,

Scl)

→ T

(n)(E

,

Scl)

is a torsor for the abelian group of Lagrangians under addition (modulo those

Lagrangians which are a total derivative).

Thus, any quantization defined to order n in can be lifted to a quan-

tization defined to order n + 1 in , but there is no canonical lift; any two

lifts differ by the addition of a Lagrangian.

1The

classical action needs to satisfy some non-degeneracy conditions