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.
