12 1. INTRODUCTION
Definition 3.6.1. A parametrix for the Laplacian D on a manifold is
a symmetric distribution P on M
such that (D ⊗1)P − δM is a smooth
function on M
(where δM refers to the δ-distribution on the diagonal of
This condition implies that the operator ΦP :
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) =
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, ∞) =
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
[P ] for each
parametrix. If P, P are two different parametrices, then
[P ] and
must be related by a certain renormalization group equation, which expresses
[P ] in terms of a sum over graphs whose vertices are labelled by
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
[Λ](φ) = −
φ D φ +
for φ ∈
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.