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
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