0Introduction

0.2 Building a quantum field theory
We now try to give a very brief outline of how one does quantum field theory.
Roughly, we follow the steps below:
(i)
We pick a space to represent our “universe”. This will always be a manifold,
but we usually impose additional structure on it.
In particle physics, we often pick the manifold to be a 4-dimensional
pseudo-Riemannian manifold of signature +
−−
. Usually, we in
fact pick (M, g) = (R
4
, η) where η is the usual Minkowski metric.
In condensed matter physics, we often choose (
M, g
) = (
R
3
, δ
), where
δ is the flat Euclidean metric.
In string theory, we have fields living on Riemann surface Σ (e.g.
sphere, torus). Instead of specifying a metric, we only specify the
conformal equivalence class [
g
] of the metric, i.e. the metric up to a
scalar factor.
In QFT for knots, we pick
M
to be some oriented 3-manifold, e.g.
S
3
,
but with no metric.
In this course, we will usually take (
M, g
) = (
R
d
, δ
) for some
d
, where
δ is again the flat, Euclidean metric.
We might think this is a very sensible and easy choice, because we are
all used to working in (
R
d
, δ
). However, mathematically, this space is
non-compact, and it will lead to a lot of annoying things happening.
(ii)
We pick some fields. The simplest choice is just a function
φ
:
M R
or
C
. This is a scalar field. Slightly more generally, we can also have
φ : M N for some other manifold N. We call N the target space.
For example, quantum mechanics is a quantum field theory. Here we
choose
M
to be some interval
M
=
I
= [0
,
1], which we think of as time,
and the field is a map φ : I R
3
. We think of this as a path in R
3
.
φ
In string theory, we often have fields
φ
: Σ
N
, where
N
is a Calabi–Yau
manifold.
In pion physics,
π
(
x
) describes a map
φ
: (
R
4
, η
)
G/H
, where
G
and
H
are Lie groups.
Of course, we can go beyond scalar fields. We can also have fields with
non-zero spin such as fermions or gauge fields, e.g. a connection on a
principal
G
-bundle, as we will figure out later. These are fields that carry
a non-trivial representation of the Lorentz group. There is a whole load of
things we can choose for our field.
Whatever field we choose, we let
C
be the space of all field configurations,
i.e. a point
φ C
represents a picture of our field across all of
M
. Thus
C
is some form of function space and will typically be infinite dimensional.
This infinite-dimensionality is what makes QFT hard, and also what makes
it interesting.
(iii)
We choose an action. An action is just a function
S
:
C R
. You tell
me what the field looks like, and I will give you a number, the value of
the action. We often choose our action to be local, in the sense that we
assume there exists some function L(φ, φ, ···) such that
S[φ] =
Z
M
d
4
x
g L(φ(x), φ(x), ···).
The physics motivation behind this choice is obvious we don’t want
what is happening over here to depend on what is happening at the far
side of Pluto. However, this assumption is actually rather suspicious, and
we will revisit this later.
For example, we have certainly met actions that look like
S[φ] =
Z
d
4
x
1
2
(φ)
2
+
m
2
2
φ
2
+
λ
4!
φ
4
,
and for gauge fields we might have seen
S[A] =
1
4
Z
d
4
x F
µν
F
µν
.
If we have a coupled fermion field, we might have
S[A, ψ] =
1
4
Z
d
4
x F
µν
F
µν
+
¯
ψ(
/
D + m)ψ.
But recall when we first encountered Lagrangians in classical dynamics,
we worked with lots of different Lagrangians. We can do whatever thing
we like, make the particle roll down the hill and jump into space etc, and
we get to deal with a whole family of different Lagrangians. But when we
come to quantum field theory, the choices seem to be rather restrictive.
Why can’t we choose something like
S[A] =
Z
F
2
+ F
4
+ cosh(F
2
) + ···?
It turns out we can, and in fact we must. We will have to work with
something much more complicated.
But then what were we doing in the QFT course? Did we just waste time
coming up with tools that just work with these very specific examples? It
turns out not. We will see that there are very good reasons to study these
specific actions.
(iv)
What do we compute? In this course, the main object we’ll study is the
partition function
Z =
Z
C
Dφ e
S[φ]/~
,
which is some sort of integral over the space of all fields. Note that the
minus sign in the exponential is for the Euclidean signature. If we are not
Euclidean, we get some i’s instead.
We will see that the factor of
e
S[φ]/~
means that as
~
0, the dominant
contribution to the partition function comes from stationary points of
S
[
φ
]
over
C
, and this starts to bring us back to the classical theory of fields.
The effect of
e
S[φ]/~
is to try to suppress “wild” contributions to
Z
, e.g.
where φ is varying very rapidly or φ takes very large values.
Heuristically, just as in statistical physics, we have a competition between
the two factors D
φ
and
e
S[φ]/~
. The action part tries to suppress crazy
things, but how well this happens depends on how much crazy things are
happening, which is measured by the measure D
φ
. We can think of this
Dφ as “entropy”.
However, the problem is, the measure D
φ
on
C
doesn’t actually exist!
Understanding what we mean by this path integral, and what this measure
actually is, and how we can actually compute this thing, and how this has
got to do with the canonical quantization operators we had previously, is
the main focus of the first part of this course.