68 2. THEORIES, LAGRANGIANS AND COUNTERTERMS functionals – implies that this quantity is local. Thus, let Ii,k = Ii,k[L] − W R i,k ⎛ ⎝P (0,L), (r,s) (i,k) r Ir,s⎠ ⎞ . Then, W R a,b ⎛ ⎝P (0,L), (r,s)≤(i,k) r Ir,s⎠ ⎞ = Ia,b[L] for all (a, b) ≤ (i, k), and we can continue the induction. 11.1. Recall that we defined T (n) to be the space of theories defined modulo n+1 , and T (∞) to be the space of theories. Thus, we have shown that the choice of renormalization scheme sets up a bijection between T (∞) and functionals I ∈ O+ loc (C∞(M))[[ ]], and between T (n) and functionals I ∈ O+ loc (C∞(M))[ ]/ n+1 . The more fundamental statement of the theorem on the bijection be- tween theories and local action functionals is that the map T (n+1) → T (n) makes T (n+1) into a principal bundle for the group Oloc(C∞(M )), in a canonical way (independent of any arbitrary choices, such as that of a renor- malization scheme). In this subsection we will use the bijection constructed above to prove this statement. The bijection between theories and Lagrangians shows that the map T (n+1) → T (n) is surjective this is the only place the bijection is used. To show that T (n+1) → T (n) is a principal bundle, suppose that {I[L]}, {J[L]} are two theories which are defined modulo n+2 and which agree modulo n+1 . Let I0[L] ∈ T (0) be the classical theory corresponding to both I[L] and J[L]. Let us consider the tangent space to T (0) at I[L], which includes infinitesimal deformations of classical theories which do not have to be at least cubic. More precisely, let T I0[L] T (0) be the set of H[L] ∈ O(C∞(M )) such that I0[L] + δH[L] satisfies the classical renormalization group equation modulo δ2, I0,i[L] + δHi[L] = W0,i P (ε, L), I0,j[ε] + δHj[ε] modulo δ2, and which satisfy the usual locality axiom, that H[L] has a small L asymp- totic expansion in terms of local action functionals. The bijection between classical theories and local action functionals is canonical, independent of the choice of renormalization scheme. This is true even if we include non-cubic terms in our effective interaction, as long as these non-cubic terms are accompanied by nilpotent parameters. Thus, we have a canonical isomorphism of vector spaces T I0[L] T (0) ∼ Oloc(C∞(M )).

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