5. LOCALITY 13 4. A Wilsonian definition of a quantum field theory Any detector one could imagine has some finite resolution, and so only probes some low-energy effective theory, described by some Seff[Λ]. How- ever, one could imagine building detectors of arbitrarily high (but finite) resolution, and so one could imagine probing Seff[Λ] for arbitrarily high (but finite) Λ. As is usual in physics, one should only consider those objects which can in principle be observed. Thus, one should say that all aspects of a quantum field theory are encoded in its various low-energy effective theories. Let us make this into a (rough) definition. A more precise version of this definition is given later in this introduction a completely precise version is given in the body of the book. Definition 4.0.1. A (continuum) quantum field theory is: (1) An effective action Seff[Λ] : C∞(M)[0,Λ] → R[[ ]] for all Λ ∈ (0, ∞). More precisely, Seff[Λ] should be a formal power series both in the field φ ∈ C∞(M) [0,Λ] and in the variable . (2) Modulo , each Seff[Λ] must be of the form Seff[Λ](φ) = − 1 2 M φ D φ + cubic and higher terms. where D is the positive-definite Laplacian. (If we want to consider a massive scalar field theory, we can replace D by D +m2). (3) If Λ Λ, Seff[Λ ] is determined from Seff[Λ] by the renormaliza- tion group equation (which makes sense in the formal power series setting). (4) The effective actions Seff[Λ] satisfy a locality axiom, which we will sketch below. Earlier I described several different versions of the renormalization group equation one based on the world-line formulation of quantum field theory, and one defined by considering arbitrary parametrices for the Laplacian. One gets an equivalent definition of quantum field theory using either of these versions of the renormalization group flow. 5. Locality Locality is one of the fundamental principles of quantum field theory. Roughly, locality says that any interaction between fundamental particles occurs at a point. Two particles at different points of space-time cannot spontaneously affect each other. They can only interact through the medium of other particles. The locality requirement thus excludes any “spooky ac- tion at a distance”.

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