2Free field theory
III Quantum Field Theory
2.6 Propagators
We now introduce the very important idea of a propagator. Suppose we are
living in the vacuum. We want to know the probability of a particle appearing
at
x
and then subsequently being destroyed at
y
. To quantify this, we introduce
the propagator.
Definition
(Propagator)
.
The propagator of a real scalar field
φ
is defined to
be
D(x − y) = h0|φ(x)φ(y) |0i.
When we study interactions later, we will see that the propagator indeed
tells us the probability of a particle “propagating” from x to y.
It is not difficult to compute the value of the propagator. We have
h0|φ(x)φ(y) |0i =
Z
d
3
p
(2π)
3
d
3
p
0
(2π)
3
1
p
4E
p
E
0
p
h0|a
p
a
†
p
0
|0ie
−ip·x+ip
0
·y
.
with all other terms in
φ
(
x
)
φ
(
y
) vanishing because they annihilate the vacuum.
We now use the fact that
h0|a
p
a
†
p
0
|0i = h0|[a
p
, a
†
p
0
] |0i = i(2π)
3
δ
3
(p − p
0
),
since a
†
p
0
a
p
|0i = 0. So we have
Proposition.
D(x − y) =
Z
d
3
p
(2π)
3
1
2E
p
e
−ip·(x−y)
.
The propagator relates to the ∆(
x −y
) we previously defined by the following
simple relationship:
Proposition. We have
∆(x − y) = D(x − y) − D(y − x).
Proof.
∆(x − y) = [φ(x), φ(y)] = h0|[φ(x), φ(y)] |0i = D(x − y) − D(y − x),
where the second equality follows as [
φ
(
x
)
, φ
(
y
)] is just an ordinary function.
For a space-like separation (x − y)
2
< 0, one can show that it decays as
D(x − y) ∼ e
−m(|x−y|)
.
This is non-zero! So the field “leaks” out of the light cone a bit. However,
since there is no Lorentz-invariant way to order the events, if a particle can
travel in a spacelike direction
x → y
, it can just as easily travel in the other
direction. So we know would expect
D
(
x −y
) =
D
(
y −x
), and in a measurement,
both amplitudes cancel. This indeed agrees with our previous computation that
∆(x − y) = 0.
Feynman propagator
As we will later see, it turns out that what actually is useful is not the above
propagator, but the Feynman propagator:
Definition (Feynman propagator). The Feynman propagator is
∆
F
(x − y) = h0|T φ(x)φ(y) |0i =
(
h0|φ(x)φ(y) |0i x
0
> y
0
h0|φ(y)φ(x) |0i y
0
> x
0
Note that it seems like this is not a Lorentz-invariant quantity, because we
are directly comparing
x
0
and
y
0
. However, this is actually not a problem, since
when
x
and
y
are space-like separated, then
φ
(
x
) and
φ
(
y
) commute, and the
two quantities agree.
Proposition. We have
∆
F
(x − y) =
Z
d
4
p
(2π)
4
i
p
2
− m
2
e
−ip·(x−y)
.
This expression is a priori ill-defined since for each
p
, the integrand over
p
0
has
a pole whenever (
p
0
)
2
=
p
2
+
m
2
. So we need a prescription for avoiding this.
We replace this with a complex contour integral with contour given by
−E
p
E
p
Proof.
To compare with our previous computations of
D
(
x −y
), we evaluate the
p
0
integral for each p. Writing
1
p
2
− m
2
=
1
(p
0
)
2
− E
2
p
=
1
(p
0
− E
p
)(p
0
+ E
p
)
,
we see that the residue of the pole at p
0
= ±E
p
is ±
1
2E
p
.
When
x
0
> y
0
, we close the contour in the lower plane
p
0
→ −i∞
, so
e
−p
0
(x
0
−t
0
)
→ e
−∞
= 0. Then
R
d
p
0
picks up the residue at
p
0
=
E
p
. So the
Feynman propagator is
∆
F
(x − y) =
Z
d
3
p
(2π)
4
(−2πi)
2E
p
ie
−iE
p
(x
0
−y
0
)+ip·(x−y)
=
Z
d
3
p
(2π
3
)
1
2E
p
e
−ip·(x−y)
= D(x − y)
When x
0
< y
0
, we close the contour in the upper-half plane. Then we have
∆
F
(x − y) =
Z
d
3
p
(2π)
4
2πi
−2E
p
ie
iE
p
(x
0
−y
0
)+ip·(x−y)
=
Z
d
3
p
(2π)
3
1
2E
p
e
−iE
p
(y
0
−x
0
)−ip·(y−x)
We again use the trick of flipping the sign of p to obtain
=
Z
d
3
p
(2π)
3
1
2E
p
e
−ip·(y−x)
= D(y −x).
Usually, instead of specifying the contour, we write
∆
F
(x − y) =
Z
d
4
p
(2π)
4
ie
−ip·(x−y)
p
2
− m
2
+ iε
,
where
ε
is taken to be small, or infinitesimal. Then the poles are now located at
−E
p
E
p
So integrating along the real axis would give us the contour we had above. This
is known about the “iε-prescription”.
The propagator is in fact the Green’s function of the Klein–Gordon operator:
(∂
2
t
− ∇
2
+ m
2
)∆
F
(x − y) =
Z
d
4
p
(2π)
4
i
p
2
− m
2
(−p
2
+ m
2
)e
−ip·(x−y)
= −i
Z
d
4
p
(2π)
4
e
−ip·(x−y)
= −iδ
4
(x − y).
Other propagators
For some purposes, it’s useful to pick other contours, e.g. the retarded Green’s
function defined as follows:
−E
p
E
p
This can be given in terms of operators by
Definition
(Retarded Green’s function)
.
The retarded Green’s function is given
by
∆
R
(x − y) =
(
[φ(x), φ(y)] x
0
> y
0
0 y
0
> x
0
This is useful if we have some initial field configuration, and we want to see
how it evolves in the presence of some source. This solves the “inhomogeneous
Klein–Gordon equation”, i.e. an equation of the form
∂
µ
∂
µ
φ(x) + m
2
φ(x) = J(x),
where J(x) is some background function.
One also defines the advanced Green’s function which vanishes when
y
0
< x
0
instead. This is useful if we know the end-point of a field configuration and want
to know where it came from.
However, in general, the Feynman propagator is the most useful in quantum
field theory, and we will only have to use this.