54 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Thus, local action functionals are the same as Lagrangians modulo those Lagrangians which are a total derivative. Definition 7.1.2. A perturbative quantum field theory, with space of fields C∞(M) and kinetic action − 1 2 φ, (D +m2)φ , is given by a set of effective interactions I[L] ∈ O+(C∞(M ))[[ ]] for all L ∈ (0, ∞], such that (1) The renormalization group equation I[L] = W (P (ε, L),I[ε]) is satisfied, for all ε, L ∈ (0, ∞]. (2) For each i, k, there is a small L asymptotic expansion Ii,k[L] r∈Z≥0 gr(L)Φr where gr(L) ∈ C∞((0, ∞)L) and Φr ∈ Oloc(C∞(M )). Let T (∞) denote the set of perturbative quantum field theories, and let T (n) denote the set of theories defined modulo n+1 . Thus, T (∞) = lim ←− T (n) . Let me explain more precisely what I mean by saying there is a small L asymptotic expansion Ii,k[L] j∈Z≥0 gr(L)Φr. Without loss of generality, we can require that the local action functionals Φr appearing here are homogeneous of degree k in the field a. Then, the statement that there is such an asymptotic expansion means that there is a non-decreasing sequence dR ∈ Z, tending to infinity, such that for all R, and for all fields a ∈ C∞(M), lim L→0 L−dR Ii,k[L](a) − R r=0 gr(L)Φr(a) = 0. In other words, we are asking that the asymptotic expansion exists in the weak topology on Hom(C∞(M)⊗k, R)S k . The main theorem of this chapter (in the case of scalar field theories) is the following. Theorem A. Let T (n) denote the set of perturbative quantum field the- ories defined modulo n+1 . Then T (n+1) is, in a canonical way, a principal bundle over T (n) for the abelian group Oloc(C∞(M )) of local action func- tionals on M. Further, T (0) is canonically isomorphic to the space O+ loc (C∞(M)) of local action functionals which are at least cubic. There is a variant of this theorem, which states that there is a bijection between theories and Lagrangians once we choose a renormalization scheme, which is a way to extract the singular part of certain functions of one vari- able. The concept of renormalization scheme will be discussed in Section 9

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