2Chiral and gauge symmetries
III The Standard Model
2.2 Gauge symmetry
Another important aspect of the Standard Model is the notion of a gauge
symmetry. Classically, the Dirac equation has the gauge symmetry
ψ(x) 7→ e
iα
ψ(x)
for any constant
α
, i.e. this transformation leaves all observable physics un-
changed. However, if we allow
α
to vary with
x
, then unsurprisingly, the kinetic
term in the Dirac Lagrangian is no longer invariant. In particular, it transforms
as
¯
ψi
/
∂ψ 7→
¯
ψi
/
∂ψ −
¯
ψγ
µ
ψ∂
µ
α(x).
To fix this problem, we introduce a gauge covariant derivative D
µ
that transforms
as
D
µ
ψ(x) 7→ e
iα(x)
D
µ
ψ(x).
Then we find that
¯
ψi
/
Dψ transforms as
¯
ψi
/
Dψ 7→
¯
ψi
/
Dψ.
So if we replace every
∂
µ
with D
µ
in the Lagrangian, then we obtain a gauge
invariant theory.
To do this, we introduce a gauge field A
µ
(x), and then define
D
µ
ψ(x) = (∂
µ
+ igA
µ
)ψ(x).
We then assert that under a gauge transformation
α
(
x
), the gauge field
A
µ
transforms as
A
µ
7→ A
µ
−
1
g
∂
µ
α(x).
It is then a routine exercise to check that D
µ
transforms as claimed.
If we want to think of
A
µ
(
x
) as some physical field, then it should have a
kinetic term. The canonical choice is
L
G
= −
1
4
F
µν
F
µν
,
where
F
µν
= ∂
µ
A
ν
− ∂
ν
A
µ
=
1
ig
[D
µ
, D
ν
].
We call this a U(1) gauge theory, because
e
iα
is an element of U(1). Officially,
A
µ
is an element of the Lie algebra
u
(1), but it is isomorphic to
R
, so we did
not bother to make this distinction.
What we have here is a rather simple overview of how gauge theory works.
In reality, we find the weak field couples only with left-handed fields, and we
have to modify the construction accordingly.
Moreover, once we step out of the world of electromagnetism, we have to
work with a more complicated gauge group. In particular, the gauge group will
be non-abelian. Thus, the Lie algebra
g
has a non-trivial bracket, and it turns
out the right general formulation should include some brackets in
F
µν
and the
transformation rule for A
µ
. We will leave these for a later time.