18 1. INTRODUCTION This expression is the renormalization group flow from infinite energy to energy Λ. This is an infinite dimensional integral, as the field φH has un- bounded energy. This functional integral is renormalized using the technique of coun- terterms. This involves first introducing a regulating parameter r into the functional integral, which tames the singularities arising in the Feynman graph expansion. One choice would be to take the regularized functional integral to be an integral only over the finite dimensional space of fields φ ∈ C∞(M) (Λ,r] . Sending r → ∞ recovers the original integral. This limit won’t exist, but one renormalizes this limit by introducing counterterms. Counterterms are functionals SCT (r, φ) of both r and the field φ, such that the limit lim r→∞ φH∈C∞(M)(Λ,r] exp 1 S(φL + φH) − 1 SCT (r, φL + φH) exists. These counterterms are local, and are uniquely defined once one chooses a renormalization scheme. The effective action Seff[Λ] is then defined by this limit: Seff[Λ](φL) = lim r→∞ φH∈C∞(M)(Λ,r] exp 1 S(φL + φH) − 1 SCT (r, φL + φH) 6.2. In practise, we don’t use the energy-scale regulator r but rather a length-scale regulator ε. The reason is the same as before: it is easier to construct local theories using the length-scale regulator than the energy- scale regulator. In what follows, I will ignore this rather technical point to make the following discussion completely accurate, the reader should replace the energy-scale regulator r by the length-scale regulator we will use later. The counterterms SCT are constructed by a simple inductive procedure, and are local action functionals of the field φ ∈ C∞(M). Once we have chosen such a renormalization scheme, we find a set of counterterms SCT (r, φ) for any local action functional S. These countert- erms are uniquely determined by the requirements that firstly, the r → ∞ limit above exists, and secondly, they are purely singular as a function of the regulating parameter r. 6.3. What we see from this is that the bijection between theories and local action functionals is not canonical, but depends on the choice of a renormalization scheme. Thus, theorem A is the most natural formulation: there is no natural bijection between theories and local action functionals. Theorem A implies that the space of theories is an infinite dimensional manifold, modelled on the topological vector space of R[[ ]]-valued local action functionals on C∞(M).

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