4. SHARP AND SMOOTH CUT-OFFS 43
By analogy with the case of finite-dimensional integrals, we have the
W (P, I) (a) = log
I(φ + a)
Both sides of this equation are ill-defined. The propagator P is not a smooth
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 ))[[ ]]:
I(φ + a) .
Both sides of this identity are well-defined; the propagator P[Λ
is a smooth
function on M × M, so that W
is well-defined. The right hand
side is a finite dimensional integral.
The equation (†) says that the map
))[[ ]] →
I → W
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
× M) denote the heat kernel for D; thus,
for all φ ∈
We can write
in terms of a basis of eigenvalues for D as
ei ⊗ ei.
be the kernel for the operator
Then, the propagator P can be
For ε, L ∈ [0, ∞], let
P (ε, L) =