20 1. INTRODUCTION 7.2. As our definition of renormalizability only makes sense on Rn, we will now restrict to considering scalar field theories on Rn. We want to mea- sure Seff[Λ] as Λ ∞, after we have changed units. Define RGl(Seff[Λ]) by RGl(Seff [Λ])(φ) = Seff[l−2Λ](R l (φ)) Thus, RGl(Seff [Λ]) is the effective action Seff[l2Λ], but measured in units that have been rescaled by l. We can use the map RGl to implement precisely the definition of renor- malizability suggested above. Definition 7.2.1. A theory {Seff[Λ]} is renormalizable if RGl(Seff[Λ]) grows at most logarithmically as l 0. 7.3. It turns out that the map RGl defines a flow on the space of theories. Lemma 7.3.1. If {Seff[Λ]} satisfies the renormalization group equation, then so does {RGl(Seff[Λ]}. Thus, sending {Seff[Λ]} {RGl(Seff[Λ])} defines a flow on the space of theories: this is the local renormalization group flow. Recall that the choice of a renormalization scheme leads to a bijection between the space of theories and Lagrangians. Under this bijection, the local renormalization group flow acts on the space of Lagrangians. The con- stants appearing in a Lagrangian (the coupling constants) become functions of l the dependence of the coupling constants on the parameter l is called the β function. Renormalizability means these coupling constants have at most logarithmic growth in l. The local renormalization group flow RGl, as l 0, can be interpreted geometrically as focusing on smaller and smaller regions of space-time, while always using units appropriate to the size of the region one is considering. In energy terms, applying RGl as l 0 amounts to focusing on phenomena of higher and higher energy. The logarithmic growth condition thus says the theory doesn’t break down completely when we probe high-energy phenomena. If the effective actions displayed polynomial growth, for instance, then one would find that the perturbative description of the theory wouldn’t make sense at high en- ergy, because the terms in the perturbative expansion would increase with the energy. 7.4. The definition of renormalizability given above can be viewed as a perturbative approximation to an ideal non-perturbative definition. Definition 7.4.1 (Ideal definition). A non-perturbative theory is renor- malizable if, as we flow the theory under RGl and let l 0, we converge to a fixed point.
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