By Olivier Piguet, Silvio P. Sorella

This booklet presents a pedagogical and self-contained advent to the algebraic approach to renormalization in perturbative quantum box idea. this technique is predicated on common theorems of renormalization, specifically at the Quantum motion precept. It permits us to regard the issues of the renormalizability and the anomalies of versions with neighborhood or worldwide symmetries by way of the algebraic houses of classical box polynomials. a number of examples (e.g. topological types) are thought of in a few element. one of many major benefits of this technique, past its simplicity, is its nice energy, simply because no specific subtraction or regularization scheme holding the symmetries of the matter is a priori required.

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3. , by certain identities which the Green functions of the theory must obey. The aim of the renormalization procedure is to prove the renormalizability of a given symmetry of the corresponding classical theory. This means that one has to construct a theory of Green functions fulfilling perturbatively, to all orders, the Ward identities which express this symmetry. We shall explain in this chapter how to write the Ward identities in the functional formalism, beginning with the classical theory - equivalent to the tree approximation of the quantum t h e o r y - and then we shall proceed to the extension to all orders of perturbation theory, for very simple models with rigid symmetry.

6. 2. 55) that in theories not involving the tensor ~ p ~ (for example chromodynamics), or the tensor dabc (for example a SU(2) gauge theory) there is no room for an anomaly. 4 Stability We have seen in the last section how noninvariant counterterms have in general to be included in the action, order by order, for the theory to remain BRS invariant. We have also seen that there may be an obstruction to this programme, namely the gauge anomaly. The anomaly will be assumed throughout this section to be absent.

51) yields the Callan-Symanzik equation [57, 62] (~na~ + ~0~ + ~0~ - ~N) r = 2~n~(~ + 5 ) ~ . r. /3 expresses the renormalization of the coupling constant following a change of scale, whereas 7 expresses the renormalization of the field amplitude. Remark. 55) -(1) where the superscript (1) means one-loop graph contributions. 3 Renormalization Group Equation Whereas the Callan-Symanzik equation describes the response of the theory to a change of scale, as we have seen in Sect. f. 45) - defining the parametrization of the theory.

### Algebraic renormalization: perturbative renormalization, symmetries and anomalies by Olivier Piguet, Silvio P. Sorella

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