1. OVERVIEW 3

If we choose a section of each torsor T

(n+1)(E

,

Scl)

→ T

(n)(E

,

Scl)

we

find an isomorphism

T

(∞)(E

,

Scl)

∼

= series

Scl

+

S(1)

+

2S(2)

+ · · ·

where each

S(i)

is a local functional, that is, a functional which can be

written as the integral of a Lagrangian. Thus, this theorem allows one to

quantize the theory associated to any classical action functional

Scl.

How-

ever, there is an ambiguity to quantization: at each term in , we are free

to add an arbitrary local functional to our action.

1.3. The main results of this book are all stated in the context of this

theorem.

In Chapter 4, I give a definition of an action of the group R

0

on the

space of theories on

Rn.

This action is called the local renormalization group

flow, and is a fundamental part of the concept of renormalizability developed

by Wilson and others. The action of group R

0

on the space of theories on

Rn

simply arises from the action of this group on

Rn

by rescaling.

The coeﬃcients of the action of this local renormalization group flow on

any particular theory are the β functions of that theory. I include explicit

calculations of the β function of some simple theories, including the

φ4

theory on

R4.

This local renormalization group flow leads to a concept of renormaliz-

ability. Following Wilson and others, I say that a theory is perturbatively

renormalizable if it has “critical” scaling behaviour under the renormaliza-

tion group flow. This means that the theory is fixed under the renormal-

ization group flow except for logarithmic corrections. I then classify all

possible renormalizable scalar field theories, and find the expected answer.

For example, the only renormalizable scalar field theory in four dimensions,

invariant under isometries and under the transformation φ → −φ, is the

φ4

theory.

In Chapter 5, I show how to include gauge theories in my definition of

quantum field theory, using a natural synthesis of the Wilsonian effective

action picture and the Batalin-Vilkovisky formalism. Gauge symmetry, in

our set up, is expressed by the requirement that the effective action

Seff

[Λ]

at each energy Λ satisfies a certain scale Λ Batalin-Vilkovisky quantum

master equation. The renormalization group flow is compatible with the

Batalin-Vilkovisky quantum master equation: the flow from scale Λ to scale

Λ takes a solution of the scale Λ master equation to a solution to the scale

Λ equation.

I develop a cohomological approach to constructing theories which are

renormalizable and which satisfy the quantum master equation. Given any

classical gauge theory, satisfying the classical analog of renormalizability, I

prove a general theorem allowing one to construct a renormalizable quan-

tization, providing a certain cohomology group vanishes. The dimension