4 1. INTRODUCTION
of the space of possible renormalizable quantizations is given by a different
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
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
We will assume that we
have an action functional of the form
where L (φ) is a Lagrangian. A typical Lagrangian of interest would be
L (φ) = −
where D is the Lorentzian analog of the Laplacian operator.
A field φ ∈
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 ∈
each weighted by
An observable – a measurement one can make – is a function
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.