Renormalization: general theory (in metric ( ; +; +; +))

Quantum eld theories (QFTs) provide a natural framework for quantum theories that obey the principles of special relativity. Among their most striking features are ultra-violet divergences, which at rst sight invalidate the existence of the theories. The divergences arise from Fourier modes of very high wave number, and hence from the structure of the theories at very short distances. In the very restricted class of theories called \renormalizable", the divergences may be removed by a singular redenition of the parameters of the theory. This is the process of renormalization, that denes a QFT as a nontrivial limit of a theory with a UV cut-o. A very important QFT is the Standard Model, an accurate and successful theory for all the known interactions except gravity. Calculations using renormalization and related methods are vital to the theory’s success. The basic idea of renormalization predates QFT. Suppose we treat an observed electron as a combination of a bare electron of mass m0 and the associated classical electromagnetic eld down to a radius a. The observed mass of the electron is its bare mass plus the energy in the eld (divided by c 2 ). The eld energy is substantial, e.g., 0:7 MeV when a = 10 15 m, and it diverges when a ! 0. The observed mass, 0:5 MeV, is the sum of the large (or innite) eld contribution compensated by a negative and large (or innite) bare mass. This calculation needs replacing by a more correct version for short distances, of course, but it remains a good motivation. In this article, I review the theory of renormalization in its classic form, as applied to weak-coupling perturbation theory, or Feynman graphs. It is this method, rather than the Wilsonian approach reviewed elsewhere in this volume, that is typically used in practice for perturbative calculations in the Standard Model, especially its QCD part. Much of the emphasis is on weak-coupling perturbation theory, where there are well-known algorithmic rules for performing calculations and renormalization. Applications | see the article on QCD and connemen t for some important non-trivial examples | involve further related results, such as the operator product expansion, factorization theorems, and the renormalization group,