By Olivier Piguet, Silvio P. Sorella
This e-book presents a pedagogical and self-contained advent to the algebraic approach to renormalization in perturbative quantum box idea. this technique is predicated on basic 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 by way of the algebraic houses of classical box polynomials. numerous examples (e.g. topological versions) are thought of in a few aspect. one of many major merits of this system, past its simplicity, is its nice strength, simply because no specific subtraction or regularization scheme conserving the symmetries of the matter is a priori required.
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Additional info for Algebraic renormalization: perturbative renormalization, symmetries and anomalies
19) with a polynomial Q~(x) of dimension dQ. -coupled with an external field pc(x) of dimension D - dQo, (see Sect. 1) - o n e obtains, still formally: Oa(x) 6¢~(x) " Z - hJ~(X) 6p~(x) = 0. 24) where A~'(x) is a local composite field operator of dimension bounded by (D-d,+dQo). For the vertex functional this becomes 5F 5F = A~'(x) • F . 25) are the main content of the quantum action principle [12, 13, 14] (ref.  concerns the generalization of the BPHZ renormalization of composite field operators and of the quantum action principle to theories where some fields are massless).
F. Prop. 10 in Chap. 5). 52) is worked out in Chap. 5, Subsect. 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.
5 we can then restrict the cohomology problem to the space of rigid invariant local functionals. Remark. Rigid invariance is a simplifying hypothesis, which is not necessary , and anyhow could not be imposed in more general cases, as for example in the case of a theory with spontaneously broken gauge invariance. 46 4. 41), A(A, 42, c, ~, ~, Y, a) ~ A(A, ¢, c) . 47) It has been proved in quite generality [78, 79] that in a gauge theory such as the one considered in the present chapter the cohomology in the sector of ghost number 1 is independent from the external fields.