2 1. INTRODUCTION phenomena occurring at energies less than Λ, we can compute any quantity we would like in terms of the effective action Seff[Λ]. If Λ Λ, then the energy Λ effective field theory can be deduced from knowledge of the energy Λ effective field theory. This leads to an equation expressing the scale Λ effective action Seff[Λ ] in terms of the scale Λ effective action Seff[Λ]. This equation is called the renormalization group equation. If we do have a continuum quantum field theory (whatever that is!) we should, in particular, have a low-energy effective field theory for every energy. This leads to our definition : a continuum quantum field theory is a sequence of low-energy effective actions Seff[Λ], for all Λ ∞, which are related by the renormalization group flow. In addition, we require that the Seff[Λ] satisfy a locality axiom, which says that the effective actions Seff[Λ] become more and more local as Λ → ∞. This definition aims to be as parsimonious as possible. The only as- sumptions I am making about the nature of quantum field theory are the following: (1) The action principle: physics at every energy scale is described by a Lagrangian, according to Feynman’s sum-over-histories philosophy. (2) Locality: in the limit as energy scales go to infinity, interactions between fields occur at points. 1.2. In this book, I develop complete foundations for perturbative quan- tum field theory in Riemannian signature, on any manifold, using this defi- nition. The first significant theorem I prove is an existence result: there are as many quantum field theories, using this definition, as there are Lagrangians. Let me state this theorem more precisely. Throughout the book, I will treat as a formal parameter all quantities will be formal power series in . Setting to zero amounts to passing to the classical limit. Let us fix a classical action functional Scl on some space of fields E , which is assumed to be the space of global sections of a vector bundle on a manifold M 1 . Let T (n) (E , Scl) be the space of quantizations of the classical theory that are defined modulo n+1 . Then, Theorem 1.2.1. T (n+1) (E , Scl) → T (n) (E , Scl) is a torsor for the abelian group of Lagrangians under addition (modulo those Lagrangians which are a total derivative). Thus, any quantization defined to order n in can be lifted to a quan- tization defined to order n + 1 in , but there is no canonical lift any two lifts differ by the addition of a Lagrangian. 1 The classical action needs to satisfy some non-degeneracy conditions

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