1QFT in zero dimensions
III Advanced Quantum Field Theory
1.2 Interacting theories
Physically interesting theories contain interactions, i.e.
S
(
φ
) is non-quadratic.
Typically, if we are trying to compute
Z
d
n
φ P (φ) exp
−
S(φ)
~
,
these things involve transcendental functions and, except in very special circum-
stances, is just too hard. We usually cannot perform these integrals analytically.
Naturally, in perturbation theory, we would thus want to approximate
Z =
Z
d
n
φ exp
−
S(φ)
~
by some series. Unfortunately, the integral very likely diverges if
~ <
0, as
usually
S
(
φ
)
→ ∞
as
φ → ±∞
. So it can’t have a Taylor series expansion
around
~
= 0, as such expansions have to be valid in a disk in the complex plane.
The best we can hope for is an asymptotic series.
Recall that a series
∞
X
n=0
f
n
(~)
is an asymptotic series for
Z
(
~
) if for any fixed
N ∈ N
, if we write
Z
N
(
~
) for
the first N terms on the RHS, then
lim
~→0
+
|Z(~) − Z
N
(~)|
~
N
→ 0.
Thus as
~ →
0
+
, we get an arbitrarily good approximation to what we really
wanted from any finite number of terms. But the series will in general diverge if
try to fix
~ ∈ R
>0
, and include increasingly many terms. Most of the expansions
we do in quantum field theories are of this nature.
We will assume standard results about asymptotic series. Suppose
S
(
φ
) is
smooth with a global minimum at φ = φ
0
∈ R
n
, where the Hessian
∂
2
S
∂φ
a
∂φ
b
φ
0
is positive definite. Then by Laplace’s method/Watson’s lemma, we have an
asymptotic series of the form
Z(~) ∼ (2π~)
n/2
exp
−
S(φ
0
)
~
p
det ∂
a
∂
b
S(φ
0
)
1 + A~ + B~
2
+ ···
.
We will not prove this, but the proof is available in any standard asymptotic
methods textbook. The leading term involves the action evaluated on the
classical solution
φ
0
, and is known as the semiclassical term. The remaining
terms are called the quantum correction.
In Quantum Field Theory last term, the tree diagrams we worked with were
just about calculating the leading term. So we weren’t actually doing quantum
field theory.
Example. Let’s consider a single scalar field φ with action
S(φ) =
m
2
2
φ
2
+
λ
4!
φ
4
,
where
m
2
, λ >
0. The action has a unique global minimum at
φ
0
= 0. The
action evaluated at
φ
0
= 0 vanishes, and
∂
2
S
=
m
2
. So the leading term in the
asymptotic expansion of Z(~, mλ) is
(2π~)
1/2
m
.
Further, we can find the whole series expansion by
Z(~, m, λ) =
Z
R
dφ exp
−1
~
m
2
2
φ
2
+
λ
4!
φ
4
=
√
2~
m
Z
d
˜
φ exp
−
˜
φ
2
exp
−
4λ~
4!m
4
˜
φ
4
∼
√
2~
m
Z
d
˜
φ e
−
˜
φ
2
N
X
n=0
1
n!
−4λ~
4!m
4
n
˜
φ
4n
=
√
2~
m
N
X
n=0
−4λ~
4!m
4
n
1
n!
Z
d
˜
φ e
−
˜
φ
2
˜
φ
4n
=
√
2~
m
N
X
n=0
−4λ~
4!m
4
n
1
n!
Γ
2n +
1
2
.
The last line uses the definition of the Gamma function.
We can plug in the value of the Γ function to get
Z(~, m, λ) ∼
√
2π~
m
N
X
n=0
−~λ
m
4
n
1
(4!)
n
n!
(4n)!
4
n
(2n)!
=
√
2π~
m
1 −
~λ
8m
4
+
35
384
~
2
λ
2
m
8
+ ···
.
We will get to understand these coefficients much more in terms of Feynman
diagrams later.
Note that apart from the factor
√
2π~
m
, the series depends on (
~, λ
) only
through the product
~λ
. So we can equally view it as an asymptotic series in
the coupling constant
λ
. This view has the benefit that we can allow ourselves
to set ~ = 1, which is what we are usually going to do.
As we emphasized previously, we get an asymptotic series, not a Taylor series.
Why is this so?
When we first started our series expansion, we wrote
exp
(
−S
(
φ
)
/~
) as a
series in
λ
. This is absolutely fine, as
exp
is a very well-behaved function when
it comes to power series. The problem comes when we want to interchange the
integral with the sum. We know this is allowed only if the sum is absolutely
convergent, but it is not, as the integral does not converge for negative
~
. Thus,
the best we can do is that for any
N
, we truncate the sum at
N
, and then do
the exchange. This is legal since finite sums always commute with integrals.
We can in fact see the divergence of the asymptotic series for finite (
~λ
)
>
0
from the series itself, using Stirling’s approximation. Recall we have
n! ∼ e
n log n
.
So via a straightforward manipulation, we have
1
(4!)
n
n!
(4n)!
4
n
(2n)!
∼ e
n log n
.
So the coefficients go faster than exponentially, and thus the series has vanishing
radius of convergence.
We can actually plot out the partial sums. For example, we can pick
~
=
m
= 1,
λ
= 0
.
1, and let
Z
n
be the
n
th partial sum of the series. We can start
plotting this for n up to 40 (the red line is the true value):
0 5 10 15 20 25 30 35 40
2.46
2.48
2.5
2.52
2.54
n
Z
n
After the initial few terms, it seems like the sum has converged, and indeed
to the true value. Up to around
n
= 30, there is no visible change, and indeed,
the differences tend to be of the order
∼
10
−7
. However, after around 33, the
sum starts to change drastically.
We can try to continue plotting. Since the values become ridiculously large,
we have to plot this in a logarithmic scale, and thus we plot
|Z
n
|
instead. Do
note that for sufficiently large
n
, the actual sign flips every time we increment
n! This thing actually diverges really badly.
0 10 20 30 40 50 60
10
0
10
3
10
6
10
9
10
12
n
|Z
n
|
We can also see other weird phenomena here. Suppose we decided that we
have
m
2
<
0 instead (which is a rather weird thing to do). Then it is clear that
Z
(
m
2
, λ
) still exists, as the
φ
4
eventually dominates, and so what happens at
the
φ
2
term doesn’t really matter. However, the asymptotic series is invalid in
this region.
Indeed, if we look at the fourth line of our derivation, each term in the
asymptotic series will be obtained by integrating
e
˜
φ
2
˜
φ
4n
, instead of integrating
e
−
˜
φ
2
˜
φ
4n
, and each of these integrals would diverge.
Fundamentally, this is since when
m
2
<
0, the global minimum of
S
(
φ
) are
now at
φ
0
= ±
r
6m
2
λ
.
φ
S(φ)
φ
0
Our old minimum
φ
= 0 is now a (local) maximum! This is the wrong point to
expand about! Fields with
m
2
<
0 are called tachyons, and these are always
signs of some form of instability.
Actions whose minima occur at non-zero field values are often associated
with spontaneous symmetry breaking. Our original theory has a
Z/
2 symmetry,
namely under the transformation
x ↔ −x
. But once we pick a minimum
φ
0
to expand about, we have broken the symmetry. This is an interesting kind of
symmetry breaking because the asymmetry doesn’t really exist in our theory.
It’s just our arbitrary choice of minimum that breaks it.