7. A DEFINITION OF A QUANTUM FIELD THEORY 53 Then, the correlation function for the effective interaction I[L] is defined by EI[L](f1, . . . , fn) = γ∈Γn 1 Aut(γ) n−g(γ)−1 Cγ(I[L])(f1,...,fn). Unlike the heuristic graphical expansion we gave for the correlation functions of the φ3 theory, this expression is well-defined. We should interpret this expansion as saying that we can compute the correlation functions from the effective interaction I[L] by allowing particles to propagate in the usual way, and to interact by I[L] except that in between any two interactions, particles must travel for a proper time of at least L. This accounts for the fact that edges which join to internal vertices are labelled by P (L, ∞). If we have a collection {I[L] | L ∈ (0, ∞)} of effective interactions, then the renormalization group equation is equivalent to the statement that all the correlation functions constructed from I[L] using the prescription given above are independent of L. 7. A definition of a quantum field theory 7.1. Now we have some preliminary definitions and an understanding of why the terms in the graphical expansion of a functional integral diverge. This book will describe a method for renormalizing these functional integrals to yield a finite answer. This section will give a formal definition of a quantum field theory, based on Wilson’s philosophy of the effective action and a precise statement of the main theorem, which says roughly that there’s a bijection between theories and Lagrangians. Definition 7.1.1. A local action functional I ∈ O(C∞(M )) is a func- tional which arises as an integral of some Lagrangian. More precisely, if we Taylor expand I as I = ∑ k Ik where Ik(λa) = λkI k (a) (so that Ik is homogeneous of degree k of the variable a ∈ C∞(M)), then Ik must be of the form Ik(a) = s j=1 M D1,j(a) · · · Dk,j(a) where Di,j are arbitrary differential operators on M. Let Oloc(C∞(M )) ⊂ O(C∞(M )) be the subspace of local action function- als. As before, let O+ loc (C∞(M))[[ ]] ⊂ Oloc(C∞(M ))[[ ]] be the subspace of those local action functionals which are at least cubic modulo .

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