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.
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