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.
