10 1. INTRODUCTION length-scale L. This object encodes all physical phenomena occurring at lengths greater than L. (The effective interaction can also be considered in the energy-scale picture also: the relationship between the effective action Seff[Λ] and the effective interaction Ieff[Λ] is simply Seff[Λ](φ) = − 1 2 M φ D φ + Ieff[Λ](φ) for fields φ ∈ C∞(M)[0,Λ). The reason for introducing the effective interac- tion is that the world-line version of the renormalization group flow is better expressed in these terms. In the world-line picture of physics at lengths greater than L, we can only consider paths which evolve for a proper time greater than L, and then interact via Ieff[L]. All processes which involve particles moving for a proper time of less than L between interactions are assumed to be subsumed into Ieff[L]. The renormalization group equation for these effective interactions can be described by saying that quantities we compute using this prescription are independent of L. That is, Definition 3.5.1. A collection of effective interactions Ieff[L] satisfies the renormalization group equation if, when we compute correlation func- tions using Ieff[L] as our interaction, and allow particles to travel for a proper time of at least L between any two interactions, the result is indepen- dent of L. If one works out what this means, one sees that the scale L effective in- teraction Ieff[L] can be constructed in terms of Ieff[ε] by allowing particles to travel along paths with proper-time between ε and L, and then interact using Ieff[ε]. More formally, Ieff[L] can be expressed as a sum over Feynman graphs, where the edges are labelled by the propagator P (ε, L) = L ε e−τm2K τ and where the vertices are labelled by Ieff[ε]. This effective interaction Ieff[L] is an -dependent functional on the space C∞(M) of fields. We can expand Ieff[L] as a formal power series Ieff[L] = i,k≥0 i I eff i,k [L] where Ieff[L] i,k : C∞(M) → R is homogeneous of order k. Thus, we can think of Ii,k[L] as being a symmetric linear map Ieff[L] i,k : C∞(M)⊗k → R.

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