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

[Penergy[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.