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 K0 l ∈ C∞(M × M) denote the heat kernel for D thus, y∈M Kl0(x, y)φ(y) = e−l D φ (x) for all φ ∈ C∞(M). We can write K0 l in terms of a basis of eigenvalues for D as Kl0 = e−lλiei ⊗ ei. Let Kl = e−lm2K0 l 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.
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