4. SHARP AND SMOOTH CUT-OFFS 43

By analogy with the case of finite-dimensional integrals, we have the

formal identity

W (P, I) (a) = log

φ∈C∞(M)

exp −

1

2

φ, (D

+m2)φ

+

1

I(φ + a)

Both sides of this equation are ill-defined. The propagator P is not a smooth

function on

C∞(M×M),

but has singularities along the diagonal; this means

that W (P, I) is not well defined. And, of course, the integral on the right

hand side is infinite dimensional.

In a similar way, we have the following (actual) identity, for any func-

tional I ∈ O+(C∞(M ))[[ ]]:

(†) W

(

P[Λ

,Λ)

, I

)

(a)

= log

φ∈C∞(M)[Λ

,Λ)

exp −

1

2

φ, (D

+m2)φ

+

1

I(φ + a) .

Both sides of this identity are well-defined; the propagator P[Λ

,Λ)

is a smooth

function on M × M, so that W

(

P[Λ ,Λ),I

)

is well-defined. The right hand

side is a finite dimensional integral.

The equation (†) says that the map

O+(C∞(M

))[[ ]] →

O+(C∞(M

))[[ ]]

I → W

(

P[Λ ,Λ),I

)

is the renormalization group flow from energy Λ to energy Λ .

4.2. In this book we will use a cut-off based on the heat kernel, rather

than the cut-off based on eigenvalues of the Laplacian described above.

For l ∈ R 0, let

Kl0

∈

C∞(M

× M) denote the heat kernel for D; thus,

y∈M

Kl0(x,

y)φ(y) =

e−l Dφ

(x)

for all φ ∈

C∞(M).

We can write

Kl0

in terms of a basis of eigenvalues for D as

Kl0

=

e−lλi

ei ⊗ ei.

Let

Kl =

e−lm2

Kl0

be the kernel for the operator

e−l(D

+m2).

Then, the propagator P can be

written as

P =

∞

l=0

Kldl.

For ε, L ∈ [0, ∞], let

P (ε, L) =

L

l=ε

Kldl.