4 1. INTRODUCTION

of the space of possible renormalizable quantizations is given by a different

cohomology group.

In Chapter 6, I apply this general theorem to prove renormalizability

of pure Yang-Mills theory. To apply the general theorem to this example,

one needs to calculate the cohomology groups controlling obstructions and

deformations. This turns out to be a lengthy (if straightforward) exercise in

Gel’fand-Fuchs Lie algebra cohomology.

Thus, in the approach to quantum field theory presented here, to prove

renormalizability of a particular theory, one simply has to calculate the

appropriate cohomology groups. No manipulation of Feynman graphs is

required.

2. Functional integrals in quantum field theory

Let us now turn to giving a detailed overview of the results of this book.

First I will review, at a basic level, some ideas from the functional inte-

gral point of view on quantum field theory.

2.1. Let M be a manifold with a metric of Lorentzian signature. We

will think of M as space-time. Let us consider a quantum field theory of a

single scalar field φ : M → R.

The space of fields of the theory is

C∞(M).

We will assume that we

have an action functional of the form

S(φ) =

x∈M

L (φ)(x)

where L (φ) is a Lagrangian. A typical Lagrangian of interest would be

L (φ) = −

1

2

φ(D

+m2)φ

+

1

4!

φ4

where D is the Lorentzian analog of the Laplacian operator.

A field φ ∈

C∞(M,

R) can describes one possible history of the universe

in this simple model.

Feynman’s sum-over-histories approach to quantum field theory says

that the universe is in a quantum superposition of all states φ ∈

C∞(M,

R),

each weighted by

eiS(φ)/

.

An observable – a measurement one can make – is a function

O :

C∞(M,

R) → C.

If x ∈ M, we have an observable Ox defined by evaluating a field at x:

Ox(φ) = φ(x).

More generally, we can consider observables that are polynomial functions

of the values of φ and its derivatives at some point x ∈ M. Observables of

this form can be thought of as the possible observations that an observer at

the point x in the space-time manifold M can make.