CHAPTER 1

Introduction

1. Overview

Quantum field theory has been wildly successful as a framework for the

study of high-energy particle physics. In addition, the ideas and techniques

of quantum field theory have had a profound influence on the development

of mathematics.

There is no broad consensus in the mathematics community, however,

as to what quantum field theory actually is.

This book develops another point of view on perturbative quantum field

theory, based on a novel axiomatic formulation.

Most axiomatic formulations of quantum field theory in the literature

start from the Hamiltonian formulation of field theory. Thus, the Segal

(Seg99) axioms for field theory propose that one assigns a Hilbert space of

states to a closed Riemannian manifold of dimension d − 1, and a unitary

operator between Hilbert spaces to a d-dimensional manifold with boundary.

In the case when the d- dimensional manifold is of the form M × [0,t], we

should view the corresponding operator as time evolution.

The Haag-Kastler (Haa92) axioms also start from the Hamiltonian for-

mulation, but in a slightly different way. They take as the primary object

not the Hilbert space, but rather a C algebra, which will act on a vacuum

Hilbert space.

I believe that the Lagrangian formulation of quantum field theory, using

Feynman’s sum over histories, is more fundamental. The axiomatic frame-

work developed in this book is based on the Lagrangian formalism, and on

the ideas of low-energy effective field theory developed by Kadanoff (Kad66),

Wilson (Wil71), Polchinski (Pol84) and others.

1.1. The idea of the definition of quantum field theory I use is very

simple. Let us assume that we are limited, by the power of our detectors,

to studying physical phenomena that occur below a certain energy, say Λ.

The part of physics that is visible to a detector of resolution Λ we will call

the low-energy effective field theory. This low-energy effective field theory

is succinctly encoded by the energy Λ version of the Lagrangian, which is

called the low-energy effective action

Seff

[Λ].

The notorious infinities of quantum field theory only occur if we con-

sider phenomena of arbitrarily high energy. Thus, if we restrict attention to

1

http://dx.doi.org/10.1090/surv/170/01