5. LOCALITY 15
This problem, however, is an artifact of the particular form of the renor-
malization group equation we are using. The notion of “energy” is very
non-local: high-energy eigenvalues of the Laplacian are spread out all over
the manifold. Things work much better if we use the version of the renor-
malization group flow based on length rather than energy.
The length-based version of the renormalization group flow was sketched
earlier. It will be described in detail in Chapter 2, and used throughout the
rest of the book.
This length scale version of the renormalization group equation is essen-
tially equivalent to the version based on energy, in the following sense:
Any solution to the length-scale RGE can be translated
into a solution to the energy-scale RGE and
conversely3.
Under this transformation, large length will correspond (roughly) to low
energy, and vice-versa.
The great advantage of working with length scales, however, is that one
can make sense of locality. Unlike the energy-scale renormalization group
flow, the length-scale renormalization group flow diffuses from local to non-
local. We have seen earlier that it is more convenient to describe the length-
scale version of an effective field theory by an effective interaction
Ieff [L]
rather than by an effective action. If the length-scale L effective interaction
Ieff
[L] is close to being local, then
Ieff
[L + ε] is slightly less local, and so
on.
As L 0, we approach more “fundamental” interactions. Thus, the
locality axiom should say that
Ieff
[L] becomes more and more local as
L 0. Thus, one can correct the tentative definition asymptotic locality to
the following:
Definition 5.1.2 (Asymptotic locality). A collection of low-energy ef-
fective actions
Ieff
[L] satisfying the length-scale version of the renormaliza-
tion group equation is asymptotically local if there exists a small L asymp-
totic expansion of the form
Seff
[L](φ) fi(L)Θi(φ)
where the Θi are local action functionals. (The actual L 0 limit will not
exist, in general).
Because solutions to the length scale and energy scale RGEs are in bi-
jection, this definition applies to solutions to the energy scale RGE as well.
We can now update our definition of quantum field theory:
Definition 5.1.3. A (continuum) quantum field theory is:
(1) An effective action
Seff
[Λ] :
C∞(M)[0,Λ]
R[[ ]]
3The
converse requires some growth conditions on the energy-scale effective actions
Seff
[Λ].
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