7Gauge theories
III Symmetries, Fields and Particles
7.1 Electromagnetism and U(1) gauge symmetry
In electromagnetism, we had two fields
E
and
B
. There are four Maxwell’s
equations governing how they behave. Two of them specify the evolution of the
field and how they interact with matter, while the other two just tells us we can
write the field in terms of a scalar and a vector potential Φ
, A
. Explicitly, they
are related by
E = −∇Φ +
∂A
∂t
B = ∇ × A.
The choice of potentials is not unique. We know that
E
and
B
are invariant
under the transformations
Φ 7→ Φ +
∂α
∂t
A 7→ A + ∇α
for any gauge function α = α(x, t).
Now we believe in relativity, so we should write things as 4-vectors. It so
happens that there are four degrees of freedom in the potentials, so we can
produce a 4-vector
a
µ
=
Φ
A
i
.
We can then succinctly write the gauge transformations as
a
µ
7→ a
µ
+ ∂
µ
α.
We can recover our fields via something known as the electromagnetic field tensor,
given by
f
µν
= ∂
µ
a
ν
− ∂
ν
a
µ
.
It is now easy to see that
f
is invariant under gauge transformations. We can
expand the definitions out to find that we actually have
f
µν
=
0 E
x
E
y
E
z
−E
x
0 −B
z
B
y
−E
y
B
z
0 −B
x
−E
z
−B
y
B
x
0
So we do recover the electric and magnetic fields this way.
The free field theory of electromagnetism then has a Lagrangian of
L
EM
= −
1
4g
2
f
µν
f
µν
=
1
2g
2
(E
2
− B
2
).
For reasons we will see later on, it is convenient to rescale the a
µ
by
A
µ
= −ia
µ
∈ iR = L(U(1))
and write
F
µν
= −if
µν
= ∂
µ
A
ν
− ∂
ν
A
µ
.
Now the magic happens when we want to couple this to matter.
Suppose we have a complex scalar field φ : R
3,1
→ C with Lagrangian
L
φ
= ∂
µ
φ
∗
∂
µ
φ − W (φ
∗
φ).
The interesting thing is that this has a global U(1) symmetry by
φ 7→ gφ
φ
∗
7→ g
−1
φ
∗
for g = e
iδ
∈ U(1), i.e. the Lagrangian is invariant under this action.
In general, it is more convenient to talk about infinitesimal transformations
Consider an element
g = exp(X) ≈ 1 + X,
where we think of X as “small”. In our case, we have X ∈ u(1)
∼
=
iR, and
φ 7→ φ + δ
X
φ,
φ
∗
7→ φ
∗
+ δ
X
φ
∗
,
where
δ
X
φ = Xφ
δ
X
φ
∗
= −Xφ
∗
.
Now this is a global symmetry, i.e. this is a symmetry if we do the same
transformation at all points in the space. Since we have
δ
X
L
φ
= 0,
we get a conserved charge. What if we want a local symmetry? We want to have
a different transformation at every point in space, i.e. we now have a function
g : R
3,1
→ U(1),
and we consider the transformations given by
φ(x) 7→ g(x)φ(x)
φ
∗
(x) 7→ g
−1
(x)φ
∗
(x).
This is in general no longer a symmetry. Under an infinitesimal variation
X : R
3,1
→ u(1), we have
δ
X
φ = Xφ.
So the derivative transforms as
δ
X
(∂
µ
φ) = ∂
µ
(δ
X
φ) = (∂
µ
X)φ + X∂
µ
φ.
This is bad. What we really want is for this to transform like φ, so that
∂
µ
φ 7→ g(x)∂
µ
φ.
Then the term ∂
µ
φ
∗
∂
µ
φ will be preserved.
It turns out the solution is to couple
φ
with
A
µ
. Recall that both of these
things had gauge transformations. We now demand that under any gauge
transformation, both should transform the same way. So from now on, a gauge
transformation X : R
3,1
→ u(1) transforms both φ and A
µ
by
φ 7→ φ + Xφ
A
µ
7→ A
µ
− ∂
µ
X.
However, this does not fix the problem, since everything we know so far that
involves the potential
A
is invariant under gauge transformation. We now do
the funny thing. We introduce something known as the covariant derivative:
D
µ
= ∂
µ
+ A
µ
,
Similar to the case of general relativity (if you are doing that course), the
covariant derivative is the “right” notion of derivative we should use whenever
we want to differentiate fields that are coupled with
A
. We shall now check that
this derivative transforms in the same way as φ. Indeed, we have
δ
X
(D
µ
φ) = δ
X
(∂
µ
φ + A
µ
φ)
= ∂
µ
(δ
X
φ) + A
µ
δ
X
φ − ∂
µ
Xφ
= X∂
µ
φ + XA
µ
φ
= XD
µ
φ.
This implies that the kinetic term
(D
µ
φ)
∗
D
µ
φ
is gauge invariant. So we can put together a gauge-invariant Lagrangian
L = −
1
4g
2
F
µν
F
µν
+ (D
µ
φ)
∗
(D
µ
φ) − W (φ
∗
φ).
So what the electromagnetic potential
A
gives us is a covariant derivative D
which then allows us to “gauge” these complex fields to give them a larger
symmetry group.