14 1. INTRODUCTION

Locality is easily understood in the naive functional integral picture.

Here, the theory is supposed to be described by a functional measure of the

form

eS(φ)/

dφ,

where dφ represents the non-existent Lebesgue measure on

C∞(M).

In this

picture, locality becomes the requirement that the action function S is a

local action functional.

Definition 5.0.1. A function

S :

C∞(M)

→ R[[ ]]

is a local action functional, if it can be written as a sum

S(φ) = Sk(φ)

where Sk(φ) is of the form

Sk(φ) =

M

(D1φ)(D2φ) · · · (Dkφ)dV olM

where Di are differential operators on M.

Thus, a local action functional S is of the form

S(φ) =

x∈M

L (φ)(x)

where the Lagrangian L (φ)(x) only depends on Taylor expansion of φ at x.

5.1. Of course, the naive functional integral picture doesn’t make sense.

If we want to give a definition of quantum field theory based on Wilson’s

ideas, we need a way to express the idea of locality in terms of the finite

energy effective actions

Seff

[Λ].

As Λ → ∞, the effective action

Seff

[Λ] is supposed to encode more and

more “fundamental” interactions. Thus, the first tentative definition is the

following.

Definition 5.1.1 (Tentative definition of asymptotic locality). A col-

lection of low-energy effective actions

Seff

[Λ] satisfying the renormalization

group equation is asymptotically local if there exists a large Λ asymptotic

expansion of the form

Seff

[Λ](φ) fi(Λ)Θi(φ)

where the Θi are local action functionals. (The Λ → ∞ limit of

Seff

[Λ] does

not exist, in general).

This asymptotic locality axiom turns out to be a good idea, but with a

fundamental problem. If we suppose that

Seff

[Λ] is close to being local for

some large Λ, then for all Λ Λ, the renormalization group equation implies

that

Seff

[Λ ] is entirely non-local. In other words, the renormalization group

flow is not compatible with the idea of locality.