Renormalization and the UV Problem
In this chapter the UV divergences are discussed and shown to be removed by renormalization. We proceed as in Chap. 9. In Sect. 10.1 the problem and its solution will be explained in detail for simple but typical examples in low orders. Then the general formulation of the renormalization procedure will be stated in Sect. 10.2 and shown to satisfy the renormalization conditions. In these two first sections we work with the p-space form of the graph rules, because the renormalization conditions have been formulated in Sects. 8.1 and 8.2 for that space, and because the rules are easier to describe and motivate there. But the proof that these rules indeed remove the UV divergences is simpler in x-space. Also, the proof that renormalization does not destroy the W-properties needs to be carried out partly in x-space. We therefore describe the x-space form of renormalization in Sect. 10.3 and give the necessary proofs, as far as feasible at this stage. The final complete verifications that all is as it should be can only be given in the next chapter, after the IR problem for Wightman functions has been solved.
KeywordsExternal Line Loop Integral Cross Line Finite Part Mass Renormalization
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