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.
