0Introduction
III Advanced Quantum Field Theory
0.1 What is quantum field theory
What is Quantum Field Theory? The first answer we might give is — it’s just a
quantum version of a field theory (duh!). We know the world is described by
fields, e.g. electromagnetic fields, and the world is quantum. So naturally we
want to find a quantum version of field theory. Indeed, this is what drove the
study of quantum field theory historically.
But there are other things we can use quantum field theory for, and they
are not so obviously fields. In a lot of condensed matter applications, we can
use quantum field theory techniques to study, say, vibrations of a crystal, and
its quanta are known as “phonons”. More generally, we can use quantum field
theory to study things like phase transitions (e.g. boiling water). This is not so
obvious a place you think quantum field theory will happen, but it is. Some
people even use quantum field theory techniques to study the spread of diseases
in a population!
We can also use quantum field theory to study problems in mathematics!
QFT is used to study knot invariants, which are ways to assign labels to different
types of knots to figure out if they are the same. Donaldson won a fields medal
for showing that there are inequivalent differentiable structures on
R
4
, and this
used techniques coming from quantum Yang–Mills theory as well.
These are not things we would traditionally think of as quantum field theory.
In this course, we are not going to be calculating, say, Higgs corrections to
γγ-interactions. Instead, we try to understand better this machinery known as
quantum field theory.