11. PROOF OF THE MAIN THEOREM 67 Since Wi,k ⎛ ⎝P (L, L ),W (i,k) ⎛ ⎝P (ε, L),I − (r,s) (i,k) r Ir,s CT (ε)⎠⎠ ⎞⎞ is non-singular, the last equation reduces to Ii,k CT (L , ε) = Singε Wi,k ⎛ ⎝P (ε, L),I − (r,s) (i,k) r Ir,s CT (ε)⎠ ⎞ = Ii,k CT (L, ε) as desired. Thus, ICT i,k (L, ε) is independent of L, and can be written as ICT i,k (ε) and we can continue the induction. 11. Proof of the main theorem Let I ∈ O+ loc (C∞(M))[[ ]] be a local action functional. What we have shown so far allows us to construct the corresponding theory. Let W R (P (0,L),I) = lim ε→0 W ( P (ε, L),I − ICT (ε) ) be the renormalized version of the renormalization group flow from scale 0 to scale L applied to I. The theory associated to I has scale L effective interaction W R (P (0,L),I). It is easy to see that this satisfies all the axioms of a perturbative quantum field theory, as given in Definition 7. The locality axiom of the definition of perturbative quantum field theory follows from Theorem 9.5.1. We need to show the converse: that to each theory there is a correspond- ing local action functional. This is a simple induction. Let {I[L]} denote a collection of effective interactions defining a theory. Let us assume, by induction on the lexicographic ordering as before, that we have constructed local action functionals Ir,s for (r, s) (i, k) such that Wa,b R ⎛ ⎝P (0,L), (r,s) (i,k) r Ir,s⎠ ⎞ = Ia,b[L] for all (a, b) (i, k). Then, the infinitesimal renormalization group equation implies that W R i,k ⎛ ⎝P (0,L), (r,s) (i,k) r Ir,s⎠ ⎞ − Ii,k[L] is independent of L. The locality axiom for the theory {I[L]} – which says that each Ii,k[L] has a small L asymptotic expansion in terms of local action

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