3Symmetries of the path integral
III Advanced Quantum Field Theory
3.2 The Ward–Takahashi identity
We focus on an important example of this. The QED action is given by
S[A, ψ] =
Z
d
d
x
1
4
F
µν
F
µν
+ i
¯
ψ
/
Dψ + m
¯
ψψ
.
This is invariant under the global transformations
ψ(x) 7→ e
iα
ψ(x), A
µ
(x) 7→ A
µ
(x)
for constant
α ∈ R/
2
πZ
. The path integral measure is also invariant under this
transformation provided we integrate over equal numbers of
ψ
and
¯
ψ
modes in
the regularization.
In this case, the classical current is given by
j
µ
(x) =
¯
ψ(x)γ
µ
ψ(x).
We will assume that D
ψ
D
¯
ψ
is invariant under a position-dependent
α
. This is a
reasonable assumption to make, if we want our measure to be gauge invariant.
In this case, the classical current is also the quantum current.
Noting that the infinitesimal change in
ψ
is just (proportional to)
ψ
itself,
the Ward identity applied to hψ(x
1
)
¯
ψ(x
2
)i gives
∂
µ
hj
µ
(x)ψ(x
1
)
¯
ψ(x
2
)i = −δ
4
(x − x
1
)hψ(x
1
)
¯
ψ(x
2
)i + δ
4
(x − x
2
)hψ(x
1
)
¯
ψ(x
2
)i.
We now try to understand what these individual terms mean. We first understand
the correlators hψ(x
1
)
¯
ψ(x
2
)i.
Recall that when we did perturbation theory, the propagator was defined as
the Fourier transform of the free theory correlator
hψ
(
x
1
)
¯
ψ
(
x
2
)
i
. This is given
by
D(k
1
, k
2
) =
Z
d
4
x
1
d
4
x
2
e
ik
1
·x
1
e
−ik
2
·x
2
hψ(x
1
)
¯
ψ(x
2
)i
=
Z
d
4
y d
4
x
2
e
i(k
1
−k
2
)·x
2
e
ik
1
·y
hψ(y)
¯
ψ(0)i
= δ
4
(k
1
− k
2
)
Z
d
4
y e
ik·y
hψ(y)
¯
ψ(0)i.
Thus, we can interpret the interacting correlator
hψ
(
x
1
)
¯
ψ
(
x
2
)
i
as the propa-
gator with “quantum corrections” due to the interacting field.
Definition (Exact propagator). The exact (electron) propagator is defined by
S(k) =
Z
d
4
y e
ik·y
hψ(y)
¯
ψ(0)i,
evaluated in the full, interacting theory.
Usually, we don’t want to evaluate this directly. Just as we can compute the
sum over all diagrams by computing the sum over all connected diagrams, then
take exp, in this case, one useful notion is a one-particle irreducible graph.
Definition
(One-particle irreducible graph)
.
A one-particle irreducible graph for
hψ
¯
ψi
is a connected Feynman diagram (in momentum space) with two external
vertices
¯
ψ
and
ψ
such that the graph cannot be disconnected by the removal of
one internal line.
This definition is rather abstract, but we can look at some examples to see
what this actually means.
Example. The following are one-particle irreducible graphs:
γ
¯
ψ ψ
γ
¯
ψ ψ
while the following is not:
¯
ψ ψ
We will write
1PI
= Σ(
/
k)
for the sum of all contributions due to one-particle irreducible graphs. This is
known as the electron self-energy. Note that we do not include the contributions
of the propagators connecting us to the external legs. It is not difficult to see
that any Feynman diagram with external vertices
¯
ψ, ψ is just a bunch of 1PI’s
joined together. Thus, we can expand
S(k) ∼
¯
ψ ψ
¯
ψ
ψ
+
¯
ψ ψ
¯
ψ
ψ
1
i
/
k + m
+ quantum corrections.
with the quantum corrections given by
¯
ψ ψ
¯
ψ
ψ
=
¯
ψ ψ
1PI
¯
ψ
ψ
+
¯
ψ ψ
1PI 1PI
¯
ψ
ψ
+ ···
This sum is easy to perform. The diagram with
n
many 1PI’s has contributions
from
n
many 1PI’s and
n
+ 1 many propagators. Also, momentum contribution
forces them to all have the same momentum. So we simply have a geometric
series
S(k) ∼
1
i
/
k + m
+
1
i
/
k + m
Σ(
/
k)
1
i
/
k + m
+
1
i
/
k + m
Σ(
/
k)
1
i
/
k + m
Σ(
/
k)
1
i
/
k + m
+ ···
=
1
i
/
k + m − Σ(
/
k)
.
We can interpret this result as saying integrating out the virtual photons
gives us a shift in the kinetic term by Σ(
/
k).
We now move on to study the other term. It was the expectation
hj
µ
(x)ψ(x
1
)
¯
ψ(x
2
)i.
We note that using the definition of D, our classical action can be written as
S[A, ψ] =
Z
d
d
x
1
4
F
µν
F
µν
+
¯
ψ
/
∂ψ + j
µ
A
µ
+ m
¯
ψψ.
In position space, this gives interaction vertices of the form
x
1
x
x
2
Again, we want to consider quantum corrections to this interaction vertex. It
turns out the interesting correlation function is exactly hj
µ
(x)ψ(x
1
)
¯
ψ(x
2
)i.
This might seem a bit odd. Why do we not just look at the vertex itself, and
just consider
hj
µ
i
? Looking at
ψj
µ
¯
ψ
instead corresponds to including including
the propagators coming from the external legs. The point is that there can be
photons that stretch across the vertex, looking like
x
1
x
x
2
So when doing computations, we must involve the two external electron prop-
agators as well. (We do not include the photon propagator. We explore what
happens when we do that in the example sheet)
We again take the Fourier transform of the correlator, and define
Definition
(Exact electromagnetic vertex)
.
The exact electromagnetic vertex
Γ
µ
(k
1
, k
2
) is defined by
δ
4
(p + k
1
− k
2
)S(k
1
)Γ
µ
(k
1
, k
2
)S(k
2
)
=
Z
d
4
x d
4
x
1
d
4
x
2
hj
µ
(x)ψ(x
1
)
¯
ψ(x
2
)ie
ip·x
e
ik
1
·x
1
e
−ik
2
·x
2
.
Note that we divided out the
S
(
k
1
) and
S
(
k
2
) in the definition of Γ
µ
(
k
1
, k
2
),
because ultimately, we are really just interested in the vertex itself.
Can we figure out what this Γ is? Up to first order, we have
hψ(x
1
)j
µ
(x)
¯
ψ(x
2
)i ∼ hψ(x
1
)
¯
ψ(x)iγ
µ
hψ(x)
¯
ψ(x
2
)i + quantum corrections.
So in momentum space, after dividing out by the exact propagators, we obtain
Γ
µ
(k
1
, k
2
) = γ
µ
+ quantum corrections.
This first order term corresponds to diagrams that do not include photons going
across the two propagators, and just corresponds to the classical
γ
µ
inside the
definition of j
µ
. The quantum corrections are the interesting parts.
In the case of the exact electron propagator, we had this clever idea of
one-particle irreducible graphs that allowed us to simplify the propagator compu-
tations. Do we have a similar clever idea here? Unfortunately, we don’t. But we
don’t have to! The Ward identity relates the exact vertex to the exact electron
propagator.
Taking the Fourier transform of the Ward identity, and dropping some
δ-functions, we obtain
(k
1
− k
2
)
µ
S(k
1
)Γ
µ
(k
1
, k
2
)S(k
2
) = iS(k
1
) − iS(k
2
).
Recall that
S
(
k
i
) are matrices in spinor space, and we wrote them as
1
···
. So it is
easy to invert them, and we find
(k
1
− k
2
)
µ
Γ
µ
(k
1
, k
2
) = iS
−1
(k
2
) − iS
−1
(k
1
)
= −i(i
/
k
1
+ m − Σ(
/
k
1
) − i
/
k
2
− m + Σ(
/
k
2
))
= (k
1
− k
2
)
µ
γ
µ
+ i(Σ(
/
k
1
) − Σ(
/
k
2
)).
This gives us an explicit expression for the quantum corrections of the exact
vertex Γ
µ
in terms of the quantum corrections of the exact propagator S(k).
Note that very little of this calculation relied on what field we actually worked
with. We could have included more fields in the theory, and everything would
still go through. We might obtain a different value of Σ(
/
k
), but this relation
between the quantum corrections of Γ
µ
and the quantum corrections of
S
still
holds.
What is the “philosophical” meaning of this? Recall that the contributions
to the propagator comes from the
¯
ψ
/
∂ψ
term, while the contributions to the
vertex comes from the
¯
ψ
/
Aψ term. The fact that their quantum corrections are
correlated in such a simple way suggests that our quantum theory treats the
¯
ψ
/
Dψ
term as a whole, and so they receive the “same” quantum corrections. In
other words, the quantum theory respects gauge transformations. When we first
studied QED, we didn’t understand renormalization very well, and the Ward
identity provided a sanity check that we didn’t mess up the gauge invariance of
our theory when regularizing.
How could we have messed up? In the derivations, there was one crucial
assumption we made, namely that D
ψ
D
¯
ψ
is invariant under position-dependent
transformations
ψ
(
x
)
7→ e
iα(x)
ψ
(
x
). This was needed for
j
µ
to be the classical
current. This is true if we regularized by sampling our field at different points
in space, as long as we included the same number of ψ and
¯
ψ terms.
However, historically, this is not what we used. Instead, we imposed cutoffs
in the Fourier modes, asking
k
2
≤
Λ
0
. This is not compatible with arbitrary
changes
ψ
(
x
)
7→ e
iα(x)
ψ
(
x
), as we can introduce some really high frequency
changes in ψ by picking a wild α.