6Non-abelian gauge theory
III Advanced Quantum Field Theory
6.2 Yang–Mills theory
At the classical level, Yang–Mills is an example of non-abelian gauge theory
defined by the action
S[∇] =
1
2g
2
Y M
Z
M
(F
µν
, F
µν
)
√
gd
d
x,
where (
·, ·
) denotes the Killing form on the Lie algebra
g
of the gauge group,
and
g
2
Y M
is the coupling constant. For flat space, we have
√
g
= 1, and we will
drop that term.
For example, if G = SU(n), we usually choose a basis such that
(t
a
, t
b
) =
1
2
δ
ab
,
and on a local U ⊆ M, we have
S[∇] =
1
4g
2
Y M
Z
F
a
µν
F
b,µν
δ
ab
d
d
x,
with
F
a
µν
= ∂
µ
A
a
ν
− ∂
ν
A
a
µ
+ f
a
bc
A
b
µ
A
c
ν
.
Thus, Yang–Mills theory is the natural generalization of Maxwell theory to the
non-Abelian case.
Note that the action is treated as a function of the connection, and not the
curvature, just like in Maxwell’s theory. Schematically, we have
F
2
∼ (dA + A
2
)
2
∼ (dA)
2
+ A
2
dA + A
4
.
So as mentioned, there are non-trivial interactions even in the absence of charged
matter. This self-interaction is proportional to the structure constants
f
a
bc
. So
in the abelian case, these do not appear.
At the level of the classical field equations, if we vary our connection by
∇ 7→ ∇ + δa, where δa is a matrix-valued 1-form, then
δF
∇
= F
∇+δa
− F
∇
= ∇
[µ
δa
ν]
dx
µ
dx
ν
.
In other words,
δF
µν
= ∂
[µ
δa
ν]
+ [A
µ
, δa
ν
].
The Yang–Mills equation we obtain from extremizing with respect to these
variations is
0 = δS[∇] =
1
g
2
Y M
Z
(δF
µν
, F
µν
) d
d
x =
1
g
2
Y M
(∇
µ
δa
ν
, F
µν
) d
d
x = 0.
So we get the Yang–Mills equation
∇
µ
F
µν
= ∂
µ
F
µν
+ [A
µ
, F
µν
] = 0.
This is just like Gauss’ equation. Recall we also had the Bianchi identity
∇ ∧ F = 0,
which gives
∇
µ
F
νλ
+ ∇
ν
F
λµ
+ ∇
λ
F
µν
= 0,
similar to Maxwell’s equations.
But unlike Maxwell’s equations, these are non-linear PDE’s for
A
. We no
longer have the principle of superposition. This is much more similar to general
relativity. In general relativity, we had some non-linear PDE’s we had to solve
for the metric or the connection.
We all know some solutions to Einstein’s field equations, say black holes and
the Schwarzschild metric. We also know many solutions to Maxwell’s equations.
But most people can’t write down a non-trivial solution to Yang–Mills equations.
This is not because Yang–Mills is harder to solve. If you ask anyone who
does numerics, Yang–Mills is much more pleasant to work with. The real reason
is that electromagnetism and general relativity were around for quite a long time,
and we had a lot of time to understand the solutions. Moreover, these solutions
have very direct relations to things we can observe at everyday energy scales.
However, this is not true for Yang–Mills. It doesn’t really describe everyday
phenomena, and thus less people care.
Note that the action contains no mass terms for A, i.e. there is no A
2
term.
So
A
should describe a massless particle, and this gives rise to long-range force,
just like Coulomb or gravitational forces. When Yang–Mills first introduced this
theory, Pauli objected to this, because we are introducing some new long-range
force, but we don’t see any!
To sort-of explain this, the coupling constant
g
2
Y M
plays no role in the
classical (pure) theory. Of course, it will play a role if we couple it to matter.
However, in the quantum theory,
g
2
Y M
appears together with
~
as
g
2
Y M
~
. So
the classical theory is only a reasonable approximation to the physics only if
g
2
Y M
∼
0. Skipping ahead of the story, we will see that
g
2
Y M
is marginally
relevant. So at low energies, the classical theory is not a good approximation for
the actual quantum theory.
So let’s look at quantum Yang–Mills theory.