8 1. INTRODUCTION where the low-energy field φL is in C∞(M)≤Λ . This is a finite dimensional integral, and so (under mild conditions) is well defined as formal power series in . This equation is called the renormalization group equation. It says that if Λ Λ, then Seff[Λ ] is obtained from Seff[Λ] by averaging over fluctuations of the low-energy field φL ∈ C∞(M)≤Λ with energy between Λ and Λ. 3.3. Recall that in the naive functional-integral point of view, there is supposed to be a measure on the space C∞(M) of the form eS(φ)/ dφ, where dφ refers to the (non-existent) Lebesgue measure on the vector space C∞(M), and S(φ) is a function of the field φ. It is natural to ask what role the “original” action S plays in the Wilso- nian low-energy picture. The answer is that S is supposed to be the “energy infinity effective action”. The low energy effective action Seff[Λ] is supposed to be obtained from S by integrating out all fields of energy greater than Λ, that is Seff[Λ](φL) = log φH∈C∞(M)(Λ,∞) exp 1 S(φL + φH) . This is a functional integral over the infinite dimensional space of fields with energy greater than Λ. This integral doesn’t make sense the terms in its Feynman graph expansion are divergent. However, one would not expect this expression to be well-defined. The infinite energy effective action should not be defined one would not expect to have a description of how particles behave at infinite energy. The infinities in the naive functional integral picture arise because the classical action functional S is treated as the infinite energy effective action. 3.4. So far, I have explained how to define a renormalization group equation using the eigenvalues of the Laplacian. This picture is very easy to explain, but it has many disadvantages. The principal disadvantage is that this definition is not local on space-time. Thus, it is diﬃcult to integrate the locality requirements of quantum field theory into this version of the renormalization group flow. In the body of this book, I will use a version of the renormalization group flow that is based on length rather than on energy. A complete account of this will have to wait until Chapter 2, but I will give a brief description here. The version of the renormalization group flow based on length is not derived directly from Feynman’s functional integral formulation of quantum field theory. Instead, it is derived from a different (though ultimately equiv- alent) formulation of quantum field theory, again due to Feynman (Fey50). Let us consider the propagator for a free scalar field φ, with action Sfree(φ) = Sk(φ) = − 1 2 φ(D +m2)φ. This propagator P is defined to be the integral kernel for the inverse of the operator D +m2 appearing in the

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