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

where the sum is over an orthonormal basis of eigenfunctions 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.
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