42 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

We can contract this tensor with the element of E

⊗2E(γ)

given by the tensor

product of the propagators; the result of this contraction is

wγ(P, I) ∈ Hom(E

⊗T (γ),

R).

Thus, one can define

W (P, I) =

γ

1

|Aut(γ)|

wγ(P, I) ∈

O+(E

)[[ ]]

exactly as before.

The interpretation in terms of differential operators works in this situa-

tion too. As in the finite dimensional situation, we can define an order two

differential operator

∂P : O(E ) → O(E ).

On the direct factor

Hom(E

⊗n,

K)Sn =

Symn

E

∨

of O(E ), the operator ∂P comes from the map

Hom(E

⊗n,

K) → Hom(E

⊗n−2,

K)

given by contracting with the tensor P ∈ E ⊗2.

Then,

W (P, I) = log {exp( ∂P ) exp(I/ )}

as before.

4. Sharp and smooth cut-offs

4.1. Let us return to our scalar field theory, whose action is of the form

S(φ) = −

1

2

φ, (D

+m2)φ

+ I(φ).

The propagator P is the kernel for the operator (D

+m2)−1.

There are

several natural ways to write this propagator. Let us pick a basis {ei}

of

C∞(M)

consisting of orthonormal eigenvectors of D, with eigenvalues

λi ∈ R≥0. Then,

P =

i

1

λi 2 + m2

ei ⊗ ei.

There are natural cut-off propagators, where we only sum over some of the

eigenvalues. For a subset U ⊂ R≥0, let

PU =

i such that λi∈U

1

λi

2

+ m2

ei ⊗ ei.

Note that, unlike the full propagator P , the cut-off propagator PU is a

smooth function on M × M as long as U is a bounded subset of R≥0.