CHAPTER 2 Theories, Lagrangians and counterterms 1. Introduction In this chapter, we will make precise the definition of quantum field theory we sketched in Chapter 1. Then, we will show the main theorem: Theorem A. Let T (n) (M) denote the space of scalar field theories on a manifold M, defined modulo n+1 . Then T (n+1) (M) → T (n) (M) is (in a canonical way) a principal bundle for the space of local action functionals. Further, T (0) (M) is canonically isomorphic to the space of local action functionals which are at least cubic. This theorem has a less natural formulation, depending on an additional choice, that of a renormalization scheme. A renormalization scheme is an object of a “motivic” nature, defined in Section 9. Theorem B. The choice of a renormalization scheme leads to a section of each principal bundle T (n+1) (M) → T (n) (M), and thus to an isomor- phism between the space of theories and the space of local action functionals of the form ∑ iSi, where S0 is at least cubic. 1.1. Let me summarize the contents of this chapter. The first few sections explain, in a leisurely fashion, the version of the renormalization group flow we use throughout this book. Sections 2 and 4 introduce the heat kernel version of high-energy cut-off we will use through- out the book. Section 3 contains a general discussion of Feynman graphs, and explains how certain finite dimensional integrals can be written as sums over graphs. Section 5 explains why infinities appear in the naive func- tional integral formulation of quantum field theory. Section 6 shows how the weights attached to Feynman graphs in functional integrals can be in- terpreted geometrically, as integrals over spaces of maps from graphs to a manifold. In Section 7, we finally get to the precise definition of a quantum field theory and the statement of the main theorem. Section 8 gives a variant of this definition which doesn’t rely on the heat kernel, but instead works with an arbitrary parametrix for the Laplacian. This variant definition is equiv- alent to the one based on the heat kernel. Section 9 introduces the concept of renormalization scheme, and shows how the choice of renormalization scheme allows one to extract the singular part of the weights attached to 31 http://dx.doi.org/10.1090/surv/170/02

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