6Quantum electrodynamics
III Quantum Field Theory
6.3 Coupling to matter in classical field theory
We now move on to couple our EM field with matter. We first do it in the clear
and sensible universe of classical field theory. We will tackle two cases — in the
first case, we do coupling with fermions. In the second, we couple with a mere
complex scalar field. It should be clear one how one can generalize these further
to other fields.
Suppose the resulting coupled Lagrangian looked like
L = −
1
4
F
µν
F
µν
− A
µ
j
µ
,
plus some kinetic and self-interaction terms for the field we are coupling with.
Then the equations of motion give us
∂
µ
F
µν
= j
ν
.
Since F
µν
is anti-symmetric, we know we must have
0 = ∂
µ
∂
ν
F
µν
= ∂
ν
j
ν
.
So we know
j
µ
is a conserved current. So to couple the EM field to matter, we
need to find some conserved current.
Coupling with fermions
Suppose we had a spinor field ψ with Lagrangian
L =
¯
ψ(i
/
∂ − m)ψ.
This has an internal symmetry
ψ 7→ e
−iαψ
¯
ψ 7→ e
iα
¯
ψ,
and this gives rise to a conserved current
j
µ
=
¯
ψγ
µ
ψ.
So let’s try
L = −
1
4
F
µν
F
µν
+
¯
ψ(i
/
∂ − m)ψ −e
¯
ψγ
µ
A
µ
ψ,
where e is some coupling factor.
Have we lost gauge invariance now that we have an extra term? If we just
substituted
A
µ
for
A
µ
+
∂
µ
λ
, then obviously the Lagrangian changes. However,
something deep happens. When we couple the two terms, we don’t just add a
factor into the Lagrangian. We have secretly introduced a new gauge symmetry
to the fermion field, and now when we take gauge transformations, we have to
transform both fields at the same time.
To see this better, we rewrite the Lagrangian as
L = −
1
4
F
µν
F
µν
+
¯
ψ(i
/
D − m)ψ,
where D is the covariant derivative given by
D
µ
ψ = ∂
µ
ψ + ieA
µ
ψ.
We now claim that L is invariant under the simultaneous transformations
A
µ
7→ A
µ
+ ∂
µ
λ(x)
ψ 7→ e
−ieλ(x)
ψ.
To check that this is indeed invariant, we only have to check that
¯
ψ
/
Dψ
term.
We look at how D
µ
ψ transforms. We have
D
µ
ψ = ∂
µ
ψ + ieA
µ
ψ
7→ ∂
µ
(e
−ieλ(x)
ψ) + ie(A
µ
+ ∂
µ
λ(x))e
−ieλ(x)
ψ
= e
−ieλ(x)
D
µ
ψ.
So we have
¯
ψ
/
Dψ 7→
¯
ψ
/
Dψ,
i.e. this is gauge invariant.
So as before, we can use the gauge freedom to remove unphysical states after
quantization. The coupling constant
e
has the interpretation of electric charge,
as we can see from the equation of motion
∂
µ
F
µν
= ej
ν
.
In electromagnetism, j
0
is the charge density, but after quantization, we have
Q = e
Z
d
3
x
¯
ψγ
0
ψ
=
Z
d
3
p
(2π)
3
(b
s†
p
b
s
p
− c
s†
p
c
s
p
)
= e × (number of electrons − number of anti-electrons),
with an implicit sum over the spin
s
. So this is the total charge of the electrons,
where anti-electrons have the opposite charge!
For QED, we usually write e in terms of the fine structure constant
α =
e
2
4π
≈
1
137
for an electron.
Coupling with complex scalars
Now let’s try to couple with a scalar fields. For a real scalar field, there is no
suitable conserved current to couple
A
µ
to. For a complex scalar field
ϕ
, we can
use the current coming from ϕ → e
iα
ϕ, namely
j
µ
= (∂
µ
ϕ)
∗
ϕ − ϕ
∗
∂
µ
ϕ.
We try the obvious thing with an interaction term
L
int
= −i[(∂
µ
ϕ)
∗
ϕ − ϕ
∗
∂
µ
ϕ]A
µ
This doesn’t work. The problem is that we have introduced a new term
j
µ
A
µ
to
the Lagrangian. If we compute the conserved current due to
ϕ 7→ e
iα
ϕ
under
the new Lagrangian, it turns out we will obtain a different
j
µ
, which means in
this theory, we are not really coupling to the conserved current, and we find
ourselves going in circles.
To solve this problem, we can try random things and eventually come up
with the extra term we need to add to the system to make it consistent, and
then be perpetually puzzled about why that worked. Alternatively, we can also
learn from what we did in the previous example. What we did, at the end, was
to invent a new covariant derivative:
D
µ
ϕ = ∂
µ
ϕ + ieA
µ
ϕ,
and then replace all occurrences of
∂
µ
with D
µ
. Under a simultaneous gauge
transformation
A
µ
7→ A
µ
+ ∂
µ
λ(x)
ϕ 7→ e
−ieλ(x)
ϕ,
the covariant derivative transforms as
D
µ
ϕ 7→ e
iλ(x)
D
µ
ϕ,
as before. So we can construct a gauge invariant Lagrangian by
L = −
1
4
F
µν
F
µν
+ (D
µ
ϕ)
†
(D
µ
ϕ) − m
2
ϕ
∗
ϕ.
The current is the same thing as we’ve had before, except we have
j
µ
= i(D
µ
ϕ)
†
ϕ − ϕ
†
D
µ
ϕ.
In general, for any field
φ
taking values in a complex vector space, we have a
U(1) gauge symmetry
φ 7→ e
iλ(x)
φ.
Then we can couple with the EM field by replacing
∂
µ
φ 7→ D
µ
φ = ∂
µ
φ + ieλ(x)A
µ
φ.
This process of just taking the old Lagrangian and then replacing partial deriva-
tives with covariant derivatives is known as minimal coupling. More details on
how this works can be found in the Symmetries, Fields and Particles course.