70 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

(3) The functional I[P ] has smooth first derivative. Recall, as ex-

plained in Section 8, that this means the following. There is a

continuous linear map

C∞(M)

→

O(C∞(M

))[[ ]]

φ →

dI[P ]

dφ

.

Saying that I[Φ has smooth first derivative means that this map

extends to a continuous linear map

D(M) →

O(C∞(M

))[[ ]]

where D(M) is the space of distributions on M.

In order to verify these properties, it is convenient to choose a renormaliza-

tion scheme, so that we can write

I[L] = lim

ε→0

W

(

P (ε, L),I −

Ict(ε)

)

.

Now let us choose a cut-off function Ψ ∈

C∞(M

× M) which is 1 in a

neighbourhood of the diagonal, and 0 outside a small neighbourhood of the

diagonal. Then, ΨP (0,L) is a parametrix, which agrees with P (0,L) near

the diagonal. Thus, we have

I[ΨP (0,L)] = lim

ε→0

W

(

ΨP (ε, L),I −

Ict(ε)

)

.

Since I and

Ict(ε)

are local, if we choose the function Ψ to be supported

in a very small neighbourhood of the diagonal then we can ensure that

Ii,k[ΨP (0,L)] is supported within an arbitrarily small neighbourhood of the

small diagonal in M

k.

From this and from the identity

I[P ] = W (P − ΨP (0,L),I[ΨP (0,L)])

we can check that, by choosing the parametrix P to have support very close

to the diagonal, we can ensure that Ii,k[P ] has support arbitrarily close to

the small diagonal in M

k.

The point is that the combinatorial formulae for

W (Φ,J) (where Φ ∈

C∞(M 2)

and J ∈

O(C∞(M 2))+[[

]]) allows one to

control the support of Wi,k (Φ,J) in terms of the support of J and that of

Φ.

This shows that I[P ] satisfies the first two properties we want to ver-

ify. It remains to check that I[P ] has smooth first derivative. This follows

from the fact that all local functionals have smooth first derivative, and

that the property of having smooth first derivative is preserved under the

renormalization group flow.

Now, we need to prove the converse. This follows from the following

lemma.