1Classical field theory
III Quantum Field Theory
1.3 Symmetries and Noether’s theorem for field theories
As in the case of classical dynamics, we get a Noether’s theorem that tells us
symmetries of the Lagrangian give us conserved quantities. However, we have
to be careful here. If we want to treat space and time equally, saying that a
quantity “does not change in time” is bad. Instead, what we have is a conserved
current, which is a 4-vector. Then given any choice of spacetime frame, we can
integrate this conserved current over all space at each time (with respect to the
frame), and this quantity will be time-invariant.
Theorem
(Noether’s theorem)
.
Every continuous symmetry of
L
gives rise to
a conserved current j
µ
(x) such that the equation of motion implies that
∂
µ
j
µ
= 0.
More explicitly, this gives
∂
0
j
0
+ ∇ · j = 0.
A conserved current gives rise to a conserved charge
Q =
Z
R
3
j
0
d
3
x,
since
dQ
dt
=
Z
R
3
dj
0
dt
d
3
x
= −
Z
R
3
∇ · j d
3
x
= 0,
assuming that j
i
→ 0 as |x| → ∞.
Proof.
Consider making an arbitrary transformation of the field
φ
a
7→ φ
a
+
δφ
a
.
We then have
δL =
∂L
∂φ
a
δφ
a
+
∂L
∂(∂
µ
φ
a
)
δ(∂
µ
φ
a
)
=
∂L
∂φ
a
− ∂
µ
∂L
∂(∂
µ
φ
a
)
δφ
a
+ ∂
µ
∂L
∂(∂
µ
φ
a
)
δφ
a
.
When the equations of motion are satisfied, we know the first term always
vanishes. So we are left with
δL = ∂
µ
∂L
∂(∂
µ
φ
a
)
δφ
a
.
If the specific transformation
δφ
a
=
X
a
we are considering is a symmetry, then
δL
= 0 (this is the definition of a symmetry). In this case, we can define a
conserved current by
j
µ
=
∂L
∂(∂
µ
φ
a
)
X
a
,
and by the equations above, this is actually conserved.
We can have a slight generalization where we relax the condition for a
symmetry and still get a conserved current. We say that
X
a
is a symmetry if
δL
=
∂
µ
F
µ
(
φ
) for some
F
µ
(
φ
), i.e. a total derivative. Replaying the calculations,
we get
j
µ
=
∂L
∂(∂
µ
φ
a
)
X
a
− F
µ
.
Example
(Space-time invariance)
.
Recall that in classical dynamics, spatial
invariance implies the conservation of momentum, and invariance wrt to time
translation implies the conservation of energy. We’ll see something similar in
field theory. Consider x
µ
7→ x
µ
− ε
µ
. Then we obtain
φ
a
(x) 7→ φ
a
(x) + ε
ν
∂
ν
φ
a
(x).
A Lagrangian that has no explicit x
µ
dependence transforms as
L(x) 7→ L(x) + ε
ν
∂
ν
L(x),
giving rise to 4 currents — one for each ν = 0, 1, 2, 3. We have
(j
µ
)
ν
=
∂L
∂(∂
µ
φ
a
)
∂
ν
φ
a
− δ
µ
ν
L,
This particular current is called
T
µ
ν
, the energy-momentum tensor. This satisfies
∂
µ
T
µ
ν
= 0,
We obtain conserved quantities, namely the energy
E =
Z
d
3
x T
00
,
and the total momentum
P
i
=
Z
d
3
x T
0i
.
Example. Consider the Klein–Gordon field, with
L =
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
φ
2
.
We then obtain
T
µν
= ∂
µ
φ∂
ν
φ − η
µν
L.
So we have
E =
Z
d
3
x
1
2
˙
φ
2
+
1
2
(∇φ)
2
+
1
2
m
2
φ
2
.
The momentum is given by
P
i
=
Z
d
3
x
˙
φ∂
i
φ.
In this example,
T
µν
comes out symmetric in
µ
and
ν
. In general, it would
not be, but we can always massage it into a symmetric form by adding
σ
µν
= T
µν
+ ∂
ρ
Γ
ρµν
with Γ
ρµν
a tensor antisymmetric in ρ and µ. Then we have
∂
µ
∂
ρ
Γ
ρµν
= 0.
So this σ
µν
is also invariant.
A symmetric energy-momentum tensor of this form is actually useful, and is
found on the RHS of Einstein’s field equation.
Example (Internal symmetries). Consider a complex scalar field
ψ(x) =
1
√
2
(φ
1
(x) + iφ
2
(x)),
where φ
1
and φ
2
are real scalar fields. We put
L = ∂
µ
ψ
∗
∂
µ
ψ −V (ψ
∗
ψ),
where
V (ψ
∗
ψ) = m
2
ψ
∗
ψ +
λ
2
(ψ
∗
ψ)
2
+ ···
is some potential term.
To find the equations of motion, if we do all the complex analysis required,
we will figure that we will obtain the same equations as the real case if we treat
ψ and ψ
∗
as independent variables. In this case, we obtain
∂
µ
∂
µ
ψ + m
2
ψ + λ(ψ
∗
ψ)ψ + ··· = 0
and its complex conjugate. The
L
has a symmetry given by
ψ 7→ e
iα
ψ
. Infinites-
imally, we have δψ = iαψ, and δψ
∗
= −iαψ
∗
.
This gives a current
j
µ
= i(∂
µ
ψ
∗
)ψ −i(∂
µ
ψ)ψ
∗
.
We will later see that associated charges of this type have an interpretation of
electric charge (or particle number, e.g. baryon number or lepton number).
Note that this symmetry is an abelian symmetry, since it is a symmetry under
the action of the abelian group U(1). There is a generalization to a non-abelian
case.
Example
(Non-abelian internal symmetries)
.
Suppose we have a theory with
many fields, with the Lagrangian given by
L =
1
2
N
X
a=1
∂
µ
φ
a
∂
µ
φ
a
−
1
2
N
X
a=1
m
2
φ
2
a
− g
N
X
a=1
φ
2
a
!
2
.
This theory is invariant under the bigger symmetry group
G
=
SO
(
N
). If we
view the fields as components of complex fields, then we have a symmetry under
U(
N/
2) or even
SU
(
N/
2). For example, the symmetry group
SU
(3) gives the
8-fold way.
Example.
There is a nice trick to determine the conserved current when our
infinitesimal transformation is given by
δφ
=
αφ
for some real constant
α
.
Consider the case where we have an arbitrary perturbation
α
=
α
(
x
). In this
case,
δL
is no longer invariant, but we know that whatever formula we manage
to come up with, it has to vanish when
α
is constant. Assuming that we only
have first-order derivatives, the change in Lagrangian must be of the form
δL = (∂
µ
α(x))h
µ
(φ).
for some h
µ
. We claim that h
µ
is the conserved current. Indeed, we have
δS =
Z
d
4
x δL = −
Z
d
4
x α(x)∂
µ
h
µ
,
using integration by parts. We know that if the equations of motion are satisfied,
then this vanishes for any
α
(
x
) as long as it vanishes at infinity (or the boundary).
So we must have
∂
µ
h
µ
= 0.