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
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