4Wilsonian renormalization
III Advanced Quantum Field Theory
4.5 Taking the continuum limit
So far, we assumed we started with an effective theory at a high energy scale Λ
0
,
and studied the behaviour of the couplings as we flow down to low energies. This
is pretty much what we do in, say, condensed matter physics. We have some
detailed description of the system, and then we want to know what happens
when we zoom out. Since we have a fixed lattice of atoms, there is a natural
energy scale Λ
0
to cut off at, based on the spacing and phonon modes of the
lattice.
However, in high energy physics, we want to do the opposite. We instead
want to use what we know about the low energy version of the system, and then
project and figure out what the high energy theory is. In other words, we are
trying to take the continuum limit Λ
0
→ ∞.
What do we actually mean by that? Suppose our theory is defined at a
critical point g
∗
i
and some cutoff Λ
0
. Then by definition, in our path integral
Z(Λ
0
, g
∗
i
) =
Z
C
∞
(M)
≤Λ
0
Dϕ e
−S
Λ
0
[ϕ,g
∗
i
]
,
No matter what values of Λ
0
we pick (while keeping the
g
∗
i
fixed), we are going
to get the same path integral, and obtain the same answers, as that is what
“critical point” means. In particular, we are free to take the limit Λ
0
→ ∞
, and
then we are now integrating over “all paths”.
What if we don’t start at a critical point? Suppose we start somewhere on
the critical surface,
{g
i
}
. We keep the same constants, but raise the value of Λ
0
.
What does the effective theory at a scale Λ look like? As we increase Λ
0
, the
amount of “energy scale” we have to flow down to get to Λ increases. So as we
raise Λ
0
, the coupling constants at scale Λ flow towards the critical point. As
we take this continuum limit Λ
0
→
0, we end up at a critical point, namely a
conformal field theory. This is perhaps a Gaussian, which is not very interesting,
but at least we got something.
However, suppose our theory has some relevant operators turned on. Then
as we take the limit Λ
0
→ ∞
, the coupling constants of our theory diverges!
This sounds bad.
It might seem a bit weird that we fix the constants and raise the values of Λ
0
.
However, sometimes, this is a reasonable thing to do. For example, if we think
in terms of the “probing distances” of the theory, as we previously discussed,
then this is equivalent to taking the same theory but “zooming out” and probing
it at larger and larger distances. It turns out, when we do perturbation theory,
the “naive” thing to do is to do exactly this. Of course, we now know that the
right thing to do is that we should change our couplings as we raise Λ
0
, so as to
give the same physical predictions at any fixed scale Λ
<
Λ
0
. In other words, we
are trying to backtrace the renormalization group flow to see where we came
from! This is what we are going to study in the next chapter.