CHAPTER 1 Introduction 1. Overview Quantum field theory has been wildly successful as a framework for the study of high-energy particle physics. In addition, the ideas and techniques of quantum field theory have had a profound influence on the development of mathematics. There is no broad consensus in the mathematics community, however, as to what quantum field theory actually is. This book develops another point of view on perturbative quantum field theory, based on a novel axiomatic formulation. Most axiomatic formulations of quantum field theory in the literature start from the Hamiltonian formulation of field theory. Thus, the Segal (Seg99) axioms for field theory propose that one assigns a Hilbert space of states to a closed Riemannian manifold of dimension d − 1, and a unitary operator between Hilbert spaces to a d-dimensional manifold with boundary. In the case when the d- dimensional manifold is of the form M × [0,t], we should view the corresponding operator as time evolution. The Haag-Kastler (Haa92) axioms also start from the Hamiltonian for- mulation, but in a slightly different way. They take as the primary object not the Hilbert space, but rather a C algebra, which will act on a vacuum Hilbert space. I believe that the Lagrangian formulation of quantum field theory, using Feynman’s sum over histories, is more fundamental. The axiomatic frame- work developed in this book is based on the Lagrangian formalism, and on the ideas of low-energy effective field theory developed by Kadanoff (Kad66), Wilson (Wil71), Polchinski (Pol84) and others. 1.1. The idea of the definition of quantum field theory I use is very simple. Let us assume that we are limited, by the power of our detectors, to studying physical phenomena that occur below a certain energy, say Λ. The part of physics that is visible to a detector of resolution Λ we will call the low-energy effective field theory. This low-energy effective field theory is succinctly encoded by the energy Λ version of the Lagrangian, which is called the low-energy effective action Seff[Λ]. The notorious infinities of quantum field theory only occur if we con- sider phenomena of arbitrarily high energy. Thus, if we restrict attention to 1 http://dx.doi.org/10.1090/surv/170/01

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