By Olivier Piguet, Silvio P. Sorella

This ebook offers a pedagogical and self-contained advent to the algebraic approach to renormalization in perturbative quantum box concept. this system relies 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 international 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 merits of this system, past its simplicity, is its nice energy, simply because no specific subtraction or regularization scheme maintaining the symmetries of the matter is a priori required.

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1. 1 Tree Approximation, Gauge Fixing and BRS Invariance 39 This means that the Lagrange multiplier dependent terms of the action are prescribed to be: Tr f d4x (BOA + 2aB2 ) . The special case with a = 0 is called the Landau gauge. The gauge transformations are replaced by the BRS transformations sA t, = Ot,c + i[c, A~,] = D~,c , s¢ --iaaTa~), SC = ic 2 8~) = ieTac a , 1 ( 8 c a _.. 9) ), sB=O, which are nilpotent: s 2 = O. 10) The BRS transformations of some of the fields being nonlinear, one h a s t o introduce BRS invariant external fields p', Y and a coupled to them.

But it very often happens that the subtraction procedure breaks these identities. In such a case the question is: is it possible to adjust some of the counterterms which have been left free after the subtractions, in such a way that the Ward identities are recovered? , of a breaking of the symmetry through the radiative corrections. Let us show that there is no anomaly in the present case. The proof is recursive. 37) we shall show that it holds at the next order, provided the free counterterms of order n which are not U(1)-invariant are suitably chosen.

L(Xn-1)Tai,-,J'~¢Jn(Xn)¢i-+ l(xn+l)" "¢iN(XN)> (0) __ O. 10) n=l Such relations between Green functions, due to a symmetry, are called Ward identities. 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 .