By Olivier Piguet, Silvio P. Sorella

This publication offers a pedagogical and self-contained advent to the algebraic approach to renormalization in perturbative quantum box conception. this system is predicated on normal theorems of renormalization, particularly at the Quantum motion precept. It permits us to regard the issues of the renormalizability and the anomalies of types with neighborhood or worldwide symmetries by way of the algebraic homes of classical box polynomials. numerous examples (e.g. topological versions) are thought of in a few aspect. one of many major merits of this technique, past its simplicity, is its nice strength, simply because no specific subtraction or regularization scheme protecting the symmetries of the matter is a priori required.

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**Extra info for Algebraic renormalization: perturbative renormalization, symmetries and anomalies**

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42) yields O = bA = xTr d% \-~c + iac4 + terms independent of a = ixTr f d 4 x ([c, a]c3+ ac 4) + terms independent of a = -iz fd z oc + terms independent of a hence x = 0, and A is independent of a. , a . 48) is the b-variation of -2iyf d4xfZT~¢c~ up to IF-independent terms. 41) (and of Y as well). At last the most general dependence in t3 is given by A ~ Tr f d 4 x ~ ' R , ( A , c) + terms independent of t3, Y, a R , ( A , c) = zO, cc + tcO, c + u m , c2 + vcA, c + wc2m, . The consistency condition writes 0 = bA = Tt / d4x 5S _ ~ b R .

11) Sext(¢, P) --- i f dxpieaRai to the original action S(¢). , tree graph, approximation. We would like to derive Ward identities as in the linear case. 1) because of the external field piece. 4). 14) and the total action is now invariant: 5F (°) = 0. The latter invariance can be written in a functional form as S ( F (°)) = f dx 6F (°) 6F (°) 6p i 6¢i 1 hc%be c OF ~ (°) = 0 . 15) 1The typical example is that of the BRS symmetry of the nonabelian gauge theories. It will be studied in Chap. 4. 24 3.

59). A quantum extension of this basis may be written in the form of a set of insertions generated by the action on the vertex functional of four independent s y m m e t r i c differential operators. , which fulfills the conditions 52 4. 64) ~-~ ( v r ) = 0 , G ( v r) = 0 . 60)) 0 ZU, g(J Ale, AfA, Aft. 59). Hence from Prop. 1 of Chap. 3 they constitute a basis for the quantum invariant insertions of dimension 4 and ghost number 0. The expansion of #O~,Fin this basis yields the Callan-Symanzik or renormalization group equation s c r := ( , o .