7. RENORMALIZABILITY 19 7. Renormalizability We have seen that the space of theories is an infinite dimensional man- ifold, modelled on the space of R[[ ]]-valued local action functionals on C∞(M). A physicist would find this unsatisfactory. Because the space of theories is infinite dimensional, to specify a particular theory, it would take an infinite number of experiments. Thus, we can’t make any predictions. We need to find a natural finite-dimensional submanifold of the space of all theories, consisting of “well-behaved” theories. These well-behaved theories will be called renormalizable. 7.1. An old-fashioned viewpoint is the following: A local action functional (or Lagrangian) is renormalizable if it has only finitely many counterterms: SCT (r) = finite fi(r)SiCT In general, this definition picks out a finite dimensional subspace of the in- finite dimensional space of theories. However, it is not natural: the spe- cific counterterms will depend on the choice of renormalization scheme, and therefore this definition may depend on the choice of renormalization scheme. More fundamentally, any definitions one makes should be directly in terms of the only physical quantities one can measure, namely the low-energy effective actions Seff[Λ]. Thus, we would like a definition of renormalizabil- ity using only the Seff[Λ]. The following is the basic idea of the definition we suggest, following Wilson and others. Definition 7.1.1 (Rough definition). A theory, defined by effective ac- tions Seff[Λ], is renormalizable if the Seff[Λ] don’t grow too fast as Λ → ∞. However, we must measure Seff[Λ] in units appropriate to energy scale Λ. For instance, if Seff[1] is measured in joules, then Seff[103] should be measured in kilo-joules, and so on. However, this change of units only makes sense on Rn. Since we can identify energy with length−2, changing the units of energy amounts to rescaling Rn. In addition, the field φ ∈ C∞(Rn) can have its own energy (which should be thought of as giving the target of the map φ : Rn → R some weight). Once we incorporate both of these factors, the procedure of changing units (in a scalar field theory) is implemented by the map Rl : C∞(Rn) → C∞(Rn) φ(x) → l1−n/2φ(l−1x).

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