By Olivier Piguet, Silvio P. Sorella
This ebook offers a pedagogical and self-contained creation to the algebraic approach to renormalization in perturbative quantum box idea. this technique is predicated on basic 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 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 keeping the symmetries of the matter is a priori required.
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Extra info for Algebraic renormalization: perturbative renormalization, symmetries and anomalies
1. 43) in the language of differential forms (the ~2's, there, are the duals of the Q's). 53) ~)abcd ) re,vp~CaO~ abcf~v p ~ Ab AcAd = d " AbA~ + 1 2 ~' p " +BRS variation + total derivative, with r being some calculable coefficient depending on the parameters of the theory. 55) -- r~vpa d4x CaOIz Aabc~YAPAa . . . b"c + 12 AVAPAa ~b~c~d " Remarks. 1. The actual presence of an anomaly depends on the nonvanishing of its coefficient r. 4 Stability 49 in the one-loop approximation (triangle graph).
We shall restrict the discussion here to Yang-Mills theories in four-dimensional space-time. Beyond their obvious physical relevance , these theories have played an eminent historical role. It was indeed in the search for a rigorous renormalization scheme for these theories that Becchi, Rouet and Stora [15, 16] in 1974 discovered BRS invariance and demonstrated its role in guarantying the unitarity of the S-matrix and the gauge independence of the physical observables, on the one hand, and set up the general iterative construction needed in the absence of an invariant regularization.
2 T h e G a u g e A n o m a l y a n d t h e D e s c e n t E q u a t i o n s The last step is to solve the consistency condition (where we have replaced b by s since both operators coincide on A and c): sA(A, c) = O . 50) This is nothing else than the Wess-Zumino consistency condition , written in the BRS formalism. 51) we infer from the consistency condition that the BRS variation of the integrand must be a total derivative: sQ1 - O Q~~ . ] means antisymmetry in #, ~, .... The argument can be repeated and leads us to the descent equations [83, 84] 6The superscript indicates the ghost number.
Algebraic renormalization: perturbative renormalization, symmetries and anomalies by Olivier Piguet, Silvio P. Sorella