16 1. INTRODUCTION for all Λ ∈ (0, ∞). More precisely, Seff[Λ] should be a formal power series both in the field φ ∈ C∞(M)[0,Λ] and in the variable . (2) Modulo , each Seff[Λ] must be of the form Seff[Λ](φ) = − 1 2 M φ D φ + cubic and higher terms. where D is the positive-definite Laplacian. (If we want to consider a massive scalar field theory, we can replace D by D +m2). (3) If Λ Λ, Seff[Λ ] is determined from Seff[Λ] by the renormaliza- tion group equation (which makes sense in the formal power series setting). (4) The effective actions Seff[Λ], when translated into a solution to the length-scale version of the RGE, satisfy the asymptotic locality axiom. Since solutions to the energy and length-scale versions of the RGE are equivalent, one can base this definition entirely on the length-scale version of the RGE. We will do this in the body of the book. Earlier we sketched a very general form of the RGE, which uses an arbitrary parametrix to define a “scale” of the theory. In Chapter 2, Section 8, we will give a definition of a quantum field theory based on arbitrary parametrices, and we will show that this definition is equivalent to the one described above. 6. The main theorem Now we are ready to state the first main result of this book. Theorem A. Let T (n) denote the set of theories defined modulo n+1 . Then, T (n+1) is a principal bundle over T (n) for the abelian group of local action functionals S : C∞(M) → R. Recall that a functional S is a local action functional if it is of the form S(φ) = M L (φ) where L is a Lagrangian. The abelian group of local action functionals is the same as that of Lagrangians up to the addition of a Lagrangian which is a total derivative. Choosing a section of each principal bundle T (n+1) → T (n) yields an isomorphism between the space of theories and the space of series in whose coeﬃcients are local action functionals. A variant theorem allows one to get a bijection between theories and local action functionals, once one has made an additional universal (but unnatural) choice, that of a renormalization scheme. A renormalization

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