The Concept of the Renormalization Group
In the opening chapter we introduce the renormalization group (RG) and associated concepts in a general form in order that the complications of particular applications do not obscure the simplicity of the ideas. There are several forms of the RG but our a
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The Concept of the Renormalization Group
In the opening chapter we introduce the renormalization group (RG) and associated concepts in a general form in order that the complications of particular applications do not obscure the simplicity of the ideas. There are several forms of the RG but our approach is the one pioneered by Wilson and for which he won the Physics Nobel Prize in 1982 (Wilson 1983). This approach follows from a remarkably simple and intuitive idea and yields a very powerful way to think about quantum field theories (QFT).
1.1 Effective Theories The key thread running through the RG is the way that phenomena on different distance, or equivalently energy/momentum, scales relate to one another. The basic idea is that if we want to describe phenomena on length scales down to μ−1 ,1 then we should be able to use a set of variables defined at the length scale μ−1 . For example, hadrons and mesons are built from quarks. As long as we consider processes which occur at low enough energies then the description is best couched in terms of hadrons and mesons. However, when we start to consider processes at higher energies, i.e. shorter distances, then the quark degrees-of-freedom cannot be ignored. At low energies, the point is not that the quarks do not matter, but the only way that they do matter is to set various coupling constants and masses in the effective theory where the manifest degrees-of-freedom are the hadrons and mesons. The notion of an effective theory will be fundamental to our discussion. The idea is that the description of the physical world on distance scales >μ−1 is most efficiently described by a theory where the degrees-of-freedom are defined around the scale μ−1 . In this case there are no unnecessary degrees-of-freedom and the description is in some sense optimal. The effective theory will usually break down in some way for length scales smaller than μ−1 . At these smaller scales, a new effective theory 1
In the following μ is a momentum scale.
T. J. Hollowood, Renormalization Group and Fixed Points, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-36312-2_1, © The Author(s) 2013
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1 The Concept of the Renormalization Group
is called for, containing new degrees-of-freedom. The important point is that the parameters of the first effective theory (coupling constants and masses, etc.) will be fixed by properties of the more basic underlying theory. So there will be a set of matching conditions at momentum μ between the two layers of description. The basic underlying assumption here is the intuitive notion of a separation of scales. The important point is that to make a successful effective theory one must identify the physically relevant variables at the scale in which one is interested and then understand how these variables interact. This will involve various “couplings” which could in principle be calculated from first principles by using the underlying microscopic theory. As long as there are only a few effective variables and a few couplings, these details can be fixed exp
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