6 1. INTRODUCTION

The only difference between this picture and the quantum field theory for-

mulation is that we have replaced i by T .

If we consider the limiting case, when the temperature T in our statis-

tical system is zero, then the system is “frozen” in some extremum of the

action functional S(φ). In the dictionary between quantum field theory and

statistical mechanics, the zero temperature limit corresponds to classical

field theory. In classical field theory, the system is frozen at a solution to

the classical equations of motion.

Throughout this book, I will work perturbatively. In the vocabulary

of statistical field theory, this means that we will take the temperature

parameter T to be infinitesimally small, and treat everything as a formal

power series in T . Since T is very small, the system will be given by a small

excitation of an extremum of the action functional.

In the language of quantum field theory, working perturbatively means

we treat as a formal parameter. This means we are considering small

quantum fluctuations of a given solution to the classical equations of mo-

tions.

Throughout the book, I will work in Riemannian signature, but will

otherwise use the vocabulary of quantum field theory. Our sign conventions

are such that can be identified with the negative of the temperature.

3. Wilsonian low energy theories

Wilson (Wil71; Wil72), Kadanoff (Kad66), Polchinski (Pol84) and others

have studied the part of a quantum field theory which is seen by detectors

which can only measure phenomena of energy below some fixed Λ. This

part of the theory is called the low-energy effective theory.

There are many ways to define “low energy”. I will start by giving a

definition which is conceptually very simple, but diﬃcult to work with. In

this definition, the low energy fields are those functions on our manifold M

which are sums of low-energy eigenvectors of the Laplacian.

In the body of the book, I will use a definition of effective field theory

based on length rather than energy. The great advantage of this defini-

tion is that it relates better to the concept of locality. I will explain the

renormalization group flow from the length-scale point of view shortly.

In this introduction, I will only discuss scalar field theories on compact

Riemannian manifolds. This is purely for expository purposes. In the body

of the book I will work with a general class of theories on a possibly non-

compact manifold, although always in Riemannian signature.

3.1. Let M be a compact Riemannian manifold. For any subset I ⊂

[0, ∞), let

C∞(M)I

⊂

C∞(M)

denote the space of functions which are sums

of eigenfunctions of the Laplacian with eigenvalue in I. Thus,

C∞(M)≤Λ

denotes the space of functions that are sums of eigenfunctions with eigen-

value ≤ Λ. We can think of

C∞(M)≤Λ

as the space of fields with energy at

most Λ.