9
Quantization of Gauge Fields
We will now turn to the problem of the quantization of gauge theories. We
will begin with the simplest gauge theory, the free electromagnetic field.
This is an
abelian
gauge theory. After that we will discuss at length the
quantization of non-abelian gauge fields. Unlike abelian theories, such as the
free electromagnetic field, even in the absence of matter fields non-abelian
gauge theories are not free fields and have highly non-trivial dynamics.
9.1 Canonical Quantization of the Free Electromagnetic Field
Maxwell’s theory was the first field theory to be quantized. However, the
quantization procedure involves a number of subtleties not shared by the
other problems that we have considered so far. The issue is the fact that this
theory has a local gauge invariance. Unlike systems which only have global
symmetries, not all the classical configurations of vector potentials represent
physically distinct states. It could be argued that one should abandon the
picture based on the vector potential and go back to a picture based on
electric and magnetic fields instead. However, there is no local Lagrangian
that can describe the time evolution of the system now. Furthermore is not
clear which fields,
E
or
B
(or some other field) plays the role of coordinates
and which can play the role of momenta. For that reason, one sticks to
the Lagrangian formulation with the vector potential
A
μ
as its independent
coordinate-like variable.
The Lagrangian for Maxwell’s theory
L
=
−
1
4
F
μ
ν
F
μ
ν
(9.1)

9.1 Canonical Quantization of the Free Electromagnetic Field
261
where
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
, can be written in the form
L
=
1
2
(
E
2
−
B
2
)
(9.2)
where
E
j
=
−
∂
0
A
j
−
∂
j
A
0
B
j
=
−
ϵ
jk
ℓ
∂
k
A
ℓ
(9.3)
The electric field
E
j
and the space components of the vector potential
A
j
form a canonical pair since, by definition, the momentum
Π
j
conjugate to
A
j
is
Π
j
(
x
) =
∂
L
δ∂
0
A
j
(
x
)
=
∂
0
A
j
+
∂
j
A
0
=
−
E
j
(9.4)
Notice that since
L
does not contain any terms which include
∂
0
A
0
, the
momentum
Π
0
, conjugate to
A
0
, vanishes
Π
0
=
δ
L
δ∂
0
A
0
= 0
(9.5)
A consequence of this result is that
A
0
is essentially arbitrary and it plays
the role of a Lagrange multiplier. Indeed it is always possible to find a gauge
transformation
φ
A
′
0
=
A
0
+
∂
0
φ
A
′
j
=
A
j
−
∂
j
φ
(9.6)
such that
A
′
0
= 0. The solution is
∂
0
φ
=
−
A
0
(9.7)
which is consistent provided that
A
0
vanishes both in the remote part and
in the remote future,
x
0
→
±
∞
.
The canonical formalism can be applied to Maxwell’s electrodynamics if
we notice that the fields
A
j
(
x
) and
Π
j
′
(
x
′
) obey the equal-time Poisson
Brackets
{
A
j
(
x
)
,
Π
j
′
(
x
′
)
}
P B
=
δ
jj
′
δ
3
(
x
−
x
′
)
(9.8)
or, in terms of the electric field
E
,
{
A
j
(
x
)
, E
j
′
(
x
′
)
}
P B
=
−
δ
jj
′
δ
3
(
x
−
x
′
)
(9.9)
The classical Hamiltonian density is defined in the usual manner
H
=
Π
j
∂
0
A
j
−
L
(9.10)

262
Quantization of Gauge Fields
We find
H
(
x
) =
1
2
(
E
2
+
B
2
)
−
A
0
(
x
)
▽
·
E
(
x
)
(9.11)
Except for the last term, this is the usual answer. It is easy to see that the