By Olivier Piguet, Silvio P. Sorella
This e-book presents a pedagogical and self-contained creation to the algebraic approach to renormalization in perturbative quantum box concept. this technique is predicated on common theorems of renormalization, particularly at the Quantum motion precept. It permits us to regard the issues of the renormalizability and the anomalies of versions with neighborhood or international symmetries when it comes to the algebraic houses of classical box polynomials. numerous examples (e.g. topological types) are thought of in a few element. one of many major merits 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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Extra info for Algebraic renormalization: perturbative renormalization, symmetries and anomalies
74) Differentiating this extended Slavnov-Taylor identity with respect to X yields 0 O--~Z(J, q) - S Z ( J , q) = 0. 72) we wanted. 15), acquires this x-dependence through the action of the BRS operator on a. Thus the problem of the gauge independence is reduced to the problem of implementing the extended Slavnov-Taylor identity to all orders. 73) form a BRS doublet (see Def. 7 in Chap. 5). Prop. 8 applies and shows that the cohomology does not depend on a and X. Hence no other obstruction than the gauge a n o m a l y - assumed to be absent - c a n occur.
2Au and c are chosen real, ~, pU and a imaginary. ),0, I7. = y+~0. With this choice the action as well as the Slavnov-Taylor operator are real. 40 4. 1. Ghost numbers and dimensions. s Ghost number Dimension Au 1 0 0 1 ¢ c 5 0 3/2 1 -1 0 2 B pu y a 0 2 -1 3 -1 5/2 -2 4 Remarks. 1. 8) belongs to the class of the linear, covariant gauges discussed by 't Hooft . Besides of them there is a wide class of linear, noncovariant gauges [70, 71], for example the axial gauge  and the light-cone gauge 3.
It acts in the space of local Poincar6 invariant field polynomials, the "local functionals". 41) if and only if A ' - A" = bz~ , with z~ a local functional of dimension 4. 37). An anomaly is a nonvanishing cohomology class. A detailed discussion of these aspects will be given in Chap. 5. 42) for ,4 a local functional of dimension 4 and of ghost number 1. 30). ¢(x) =: icaSa¢(X), and by relying on the proof, given in Subsect. 1, that such an invariance can be maintained to all orders. 45) that the breaking A of the Slavnov-Taylor identity is rigid invariant: ]/VaA = O.
Algebraic renormalization: perturbative renormalization, symmetries and anomalies by Olivier Piguet, Silvio P. Sorella