12 1. INTRODUCTION Definition 3.6.1. A parametrix for the Laplacian D on a manifold is a symmetric distribution P on M 2 such that (D ⊗1)P − δM is a smooth function on M 2 (where δM refers to the δ-distribution on the diagonal of M. This condition implies that the operator ΦP : C∞(M) → C∞(M) asso- ciated to the kernel P is an inverse for D, up to smoothing operators: both P ◦ D − Id and D ◦P − Id are smoothing operators. If Kt is the heat kernel for the Laplacian D, then Plength(0,L) = L 0 Ktdt is a parametrix. This family of parametrices arises when one considers the world-line picture of quantum field theory. Similarly, we can define an energy-scale parametrix Penergy[Λ, ∞) = λ≥Λ 1 λ eλ ⊗ eλ where the sum is over an orthonormal basis of eigenfunctions eλ for the Laplacian D, with eigenvalue λ. Thus, we see that in either the world-line picture or the momentum scale picture one has a family of parametrices ( Plength(0,L) and Penergy[Λ, ∞), respectively) which converge (in the L → 0 and Λ → ∞ limits, respectively) to the zero distribution. The renormalization group equation in either case is written in terms of the one-parameter family of parametrices. This suggests a more general version of the renormalization group flow, where an arbitrary parametrix P is viewed as defining a “scale” of the the- ory. In this picture, one should have an effective action Ieff[P ] for each parametrix. If P, P are two different parametrices, then Ieff[P ] and Ieff[P ] must be related by a certain renormalization group equation, which expresses Ieff[P ] in terms of a sum over graphs whose vertices are labelled by Ieff[P ] and whose edges are labelled by P − P . If we restrict such a family of effective interactions to parametrices of the form Plength(0,L) one finds a solution to the world-line version of the renormalization group equation. If we only consider parametrices of the form Penergy[Λ, ∞), and then define Seff[Λ](φ) = − 1 2 M φ D φ + Ieff[P energy [Λ, ∞)](φ), for φ ∈ C∞(M)[0,Λ), one finds a solution to the energy scale version of the renormalization group flow. A general definition of a quantum field theory along these lines is ex- plained in detail in Chapter 2, Section 8. This definition is equivalent to one based only on the world-line version of the renormalization group flow, which is the definition used for most of the book.

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