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Published online by Cambridge University Press: 31 October 2009
Developing the theory of renormalization in order to extend renormalization from primitive divergences to all diagrams and demonstrate preservation of global and local symmetries.
Introduction
Chapter 4 opened the topic of renormalization, indicating the need for counterterms to balance divergent contributions to the scalar self-energy. There we also noted that interactions of mass dimension greater than four would generate an infinite tower of counterterms. Chapter 10 provided a foundation for renormalization in the theory of Green functions, and gave examples of the use of renormalization conditions in the computation of first-order corrections. Chapter 19 discussed the renormalization of the primitive divergences of QED in detail, introducing the use of a photon mass as an infrared regulator. In this chapter, we extend the theoretical foundation of renormalization and in particular justify the approach taken in Chapter 19.
Sections 20.1 and 20.2 present the renormalization procedure of Bogoliubov, Parasiuk, Hepp, and Zimmermann (BPHZ). This procedure is effectively the counterterm renormalization technique expressed in terms of Taylor series. It extends the renormalization techniques of the last chapter to all diagrams. Section 20.3 concludes the first unit of the chapter with some renormalization theorems which show that BPHZ renormalization preserves global symmetries.
At this point in the chapter, we still have no guarantee that renormalization will preserve a local symmetry.
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