6Non-abelian gauge theory
III Advanced Quantum Field Theory
6.3 Quantum Yang–Mills theory
Our first thought to construct a path integral for Yang–Mills may be to compute
the partition function
Z
naive
?
=
Z
A
DA e
−S
Y M
[A]
,
where
A
is the “space of all connections” on some fixed principal
G
-bundle
P → M
. If we were more sophisticated, we might try to sum over all possible
principal G-bundles.
But this is not really correct. We claim that no matter how hard we try to
make sense of path integrals, this integral must diverge.
We first think about what this path integral can be. For any two connections
∇, ∇
0
, it is straightforward to check that
∇
t
= t∇ + (1 − t)∇
0
is a connection of
P
. Consequently, we can find a path between any two
connections on
P
. Furthermore, we saw before that the difference
∇
0
− ∇ ∈
Ω
1
M
(
g
). This says that
A
is an (infinite-dimensional) affine space modelled on
Ω
1
M
(
g
)
∼
=
T
∇
A
. This is like a vector space, but there is no preferred origin 0.
There is even a flat metric on A, given by
ds
2
A
=
Z
M
(δA
µ
, δA
µ
) d
d
x,
i.e. given any two tangent vectors
a
1
, a
2
∈
Ω
1
M
(
g
)
∼
=
T
∇
A
, we have an inner
product
ha
1
, a
2
i
∇
=
Z
M
(a
1µ
, a
µ
2
) d
d
x.
Importantly, this is independent of the choice of ∇.
This all is trying to say that this
A
is nice and simple. Heuristically, we
imagine the path integral measure is the natural “
L
2
” measure on
A
as an affine
space. Of course, this measure doesn’t exist, because it is infinite dimensional.
But the idea is that this is just the same as working with a scalar field.
Despite this niceness, S
Y M
[∇] is degenerate along gauge orbits. The action
is invariant under gauge transformations (i.e. automorphisms of the principal
G
-bundle), and so we are counting each connection infinitely many times. In
fact, the group
G
of all gauge transformations is an infinite dimensional space
that is (locally)
Maps
(
M, G
), and even for compact
M
and
G
, the volume of
this space diverges.
Instead, we should take the integral over all connections modulo gauge
transformations:
Z
Y M
=
Z
A/G
dµ; e
−S
Y M
[∇]/~
,
where
A/G
is the space of all connections modulo gauge transformation, and d
µ
is some sort of measure. Note that this means there is no such thing as “gauge
symmetry” in nature. We have quotiented out by the gauge transformations
in the path integral. Rather, gauge transformations are a redundancy in our
description.
But we now have a more complicated problem. We have no idea what the
space
A/G
looks like. Thus, even formally, we don’t understand what d
µ
on this
space should be.
In electromagnetism, we handled this problem by “picking a gauge”. We are
going to do exactly the same here, but the non-linearities of our non-abelian
theory means this is more involved. We need to summon ghosts.