3Interacting fields
III Quantum Field Theory
3.6 Correlation functions and vacuum bubbles
The
S
-matrix elements are good and physical, because they correspond to
probabilities of certain events happening. However, we are not always interested
in these quantities. For example, we might want to ask the conductivity of some
quantum-field-theoretic object. To do these computations, quantities known as
correlation functions are useful. However, we are not going to use these objects
in this course, so we are just going to state, rather than justify many of the
results, and you are free to skip this chapter.
Before that, we need to study the vacuum. Previously, we have been working
with the vacuum of the free theory |0i. This satisfies the boring relation
H
0
|0i = 0.
However, when we introduce an interaction term, this is no longer the vacuum.
Instead we have an interacting vacuum |Ωi, satisfying
H |Ωi = 0.
As before, we normalize the vacuum so that
hΩ|Ωi = 1.
Concretely, this interaction vacuum can be obtained by starting with a free
vacuum and then letting it evolve for infinite time.
Lemma. The free vacuum and interacting vacuum are related via
|Ωi =
1
hΩ|0i
U
I
(t, −∞) |0i =
1
hΩ|0i
U
S
(t, −∞) |0i.
Similarly, we have
hΩ| =
1
hΩ|0i
h0|U(∞, t).
Proof.
Note that we have the last equality because
U
I
(
t, −∞
) and
U
S
(
t, −∞
)
differs by factors of e
−iHt
0
which acts as the identity on |0i.
Consider an arbitrary state |Ψi. We want to show that
hΨ|U(t, −∞) |0i = hΨ|ΩihΩ|0i.
The trick is to consider a complete set of eigenstates
{
Ω
, |xi}
for
H
with energies
E
x
. Then we have
U(t, t
0
) |xi = e
iE
x
(t
0
−t)
|xi.
Also, we know that
1 = |ΩihΩ| +
Z
dx |xihx|.
So we have
hΨ|U(t, −∞) |0i = hΨ|U(t, −∞)
|ΩihΩ| +
Z
dx |xihx|
|0i
= hΨ|ΩihΩ|0i + lim
t
0
→−∞
Z
dx e
iE
x
(t
0
−t)
hΨ|xihx|0i.
We now claim that the second term vanishes in the limit. As in most of the
things in the course, we do not have a rigorous justification for this, but it is not
too far-stretched to say so since for any well-behaved (genuine) function
f
(
x
),
the Riemann-Lebesgue lemma tells us that for any fixed a, b ∈ R, we have
lim
µ→∞
Z
b
a
f(x)e
iµx
dx = 0.
If this were to hold, then
hΨ|U(t, −∞) |0i = hΨ|ΩihΩ|0i.
So the result follows. The other direction follows similarly.
Now given that we have this vacuum, we can define the correlation function.
Definition
(Correlation/Green’s function)
.
The correlation or Green’s function
is defined as
G
(n)
(x
1
, ··· , x
n
) = hΩ|T φ
H
(x
1
) ···φ
H
(x
n
) |Ωi,
where φ
H
denotes the operators in the Heisenberg picture.
How can we compute these things? We claim the following:
Proposition.
G
(n)
(x
1
, ··· , x
n
) =
h0|T φ
I
(x
1
) ···φ
I
(x
n
)S |0i
h0|S |0i
.
Proof.
We can wlog consider the specific example
t
1
> t
2
> ··· > t
n
. We start
by working on the denominator. We have
h0|S |0i = h0|U(∞, t)U(t, −∞) |0i = h0|ΩihΩ|ΩihΩ|0i.
For the numerator, we note that S can be written as
S = U
I
(∞, t
1
)U
I
(t
1
, t
2
) ···U
I
(t
n
, −∞).
So after time-ordering, we know the numerator of the right hand side is
h0|U
I
(∞, t
1
)φ
I
(x
1
)U
I
(t
1
, t
2
)φ
I
(x
2
) ···φ
I
(n)U
I
(t
n
, −∞) |0i
h0|U
I
(∞, t
1
)φ
I
(x
1
)U
I
(t
1
, t
2
)φ
I
(x
2
) ···φ
I
(n)U
I
(t
n
, −∞) |0i
= h0|U
I
(∞, t
1
)φ
H
(x
1
) ···φ
H
(x
n
)U
I
(t
n
, −∞) |0i
= h0|ΩihΩ|T φ
H
(x
1
) ···φ
H
(x
n
) |ΩihΩ|0i.
So the result follows.
Now what does this quantity tell us? It turns out these have some really nice
physical interpretation. Let’s look at the terms
h0|T φ
I
(
x
1
)
···φ
I
(
x
n
)
S |0i
and
h0|S |0i individually and see what they tell us.
For simplicity, we will work with the
φ
4
theory, so that we only have a single
φ
field, and we will, without risk of confusion, draw all diagrams with solid lines.
Looking at
h0|S |0i
, we are looking at all transitions from
|0i
to
|0i
. The
Feynman diagrams corresponding to these would look like
1 vertex 2 vertex
These are known as vacuum bubbles. Then
h0|S |0i
is the sum of the amplitudes
of all these vacuum bubbles.
While this sounds complicated, a miracle occurs. It happens that the different
combinatoric factors piece together nicely so that we have
h0|S |0i = exp(all distinct (connected) vacuum bubbles).
Similarly, magic tells us that
h0|T φ
I
(x
1
) ···φ
I
(x
n
)S |0i =
X
connected diagrams
with n loose ends
h0|S |0i.
So what
G
(n)
(
x
1
, ··· , x
n
) really tells us is the sum of connected diagrams modulo
these silly vacuum bubbles.
Example.
The diagrams that correspond to
G
(4)
(
x
1
, ··· , x
4
) include things
that look like
x
1
x
2
x
3
x
4
x
1
x
4
x
3
x
2
x
1
x
2
x
3
x
4
Note that we define “connected” to mean every line is connected to some of the
end points in some way, rather than everything being connected to everything.
We can come up with analogous Feynman rules to figure out the contribution
of all of these terms.
There is also another way we can think about these Green’s function. Consider
the theory with a source J(x) added, so that
H = H
0
+ H
int
− J(x)φ(x).
This
J
is a fixed background function, called a source in analogy with electro-
magnetism.
Consider the interaction picture, but now we choose (
H
0
+
H
int
) to be the
“free” part, and −Jφ as the “interaction” piece.
Now the “vacuum” we use is not
|0i
but
|Ωi
, since this is the “free vacuum”
for our theory. We define
W [J] = hΩ|U
I
(−∞, ∞) |Ωi.
This is then a functional in J. We can compute
W [J] = hΩ|U
I
(−∞, ∞) |Ωi
= hΩ|T exp
Z
d
4
x − J(x)φ
H
(x)
|Ωi
= 1 +
∞
X
n=1
(−1)
n
n!
Z
d
4
x
1
···d
4
x
n
J(x
1
) ···J(x
n
)G
(n)
(x
1
, ··· , x
n
).
Thus by general variational calculus, we know
G
(n)
(x
1
, ··· , x
n
) = (−1)
n
δ
n
W [J]
δJ(x
1
) ···δJ(x
n
)
J=0
.
Thus we call W [J] a generating function for the function G
(n)
(x
1
, ··· , x
n
).