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].