5. LOCALITY 15 This problem, however, is an artifact of the particular form of the renor- malization group equation we are using. The notion of “energy” is very non-local: high-energy eigenvalues of the Laplacian are spread out all over the manifold. Things work much better if we use the version of the renor- malization group flow based on length rather than energy. The length-based version of the renormalization group flow was sketched earlier. It will be described in detail in Chapter 2, and used throughout the rest of the book. This length scale version of the renormalization group equation is essen- tially equivalent to the version based on energy, in the following sense: Any solution to the length-scale RGE can be translated into a solution to the energy-scale RGE and conversely3. Under this transformation, large length will correspond (roughly) to low energy, and vice-versa. The great advantage of working with length scales, however, is that one can make sense of locality. Unlike the energy-scale renormalization group flow, the length-scale renormalization group flow diffuses from local to non- local. We have seen earlier that it is more convenient to describe the length- scale version of an effective field theory by an effective interaction Ieff[L] rather than by an effective action. If the length-scale L effective interaction Ieff[L] is close to being local, then Ieff[L + ε] is slightly less local, and so on. As L 0, we approach more “fundamental” interactions. Thus, the locality axiom should say that Ieff[L] becomes more and more local as L 0. Thus, one can correct the tentative definition asymptotic locality to the following: Definition 5.1.2 (Asymptotic locality). A collection of low-energy ef- fective actions Ieff[L] satisfying the length-scale version of the renormaliza- tion group equation is asymptotically local if there exists a small L asymp- totic expansion of the form Seff[L](φ) fi(L)Θi(φ) where the Θi are local action functionals. (The actual L 0 limit will not exist, in general). Because solutions to the length scale and energy scale RGEs are in bi- jection, this definition applies to solutions to the energy scale RGE as well. We can now update our definition of quantum field theory: Definition 5.1.3. A (continuum) quantum field theory is: (1) An effective action Seff[Λ] : C∞(M)[0,Λ] R[[ ]] 3 The converse requires some growth conditions on the energy-scale effective actions Seff [Λ].
Previous Page Next Page