By Olivier Piguet, Silvio P. Sorella
This booklet offers a pedagogical and self-contained creation to the algebraic approach to renormalization in perturbative quantum box thought. this technique relies on normal 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 worldwide symmetries when it comes to the algebraic houses of classical box polynomials. numerous examples (e.g. topological versions) 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 protecting the symmetries of the matter is a priori required.
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Additional resources for Algebraic renormalization: perturbative renormalization, symmetries and anomalies
Hence no other obstruction than the gauge a n o m a l y - assumed to be absent - c a n occur. 55), then the anomaly coefficient r at its lowest nonvanishing order is independent from the gauge parameter a since r A belonging to the extended BRS cohomology cannot depend on a. 76) 54 with 4. 13), 0 oo F = S r F . 6 Gauge Invariance Together with Rigid Invariance Up to the present point, the BRS formalism has been applied either for a gauge symmetry, in this chapter, or very briefly for a rigid symmetry in Sect.
They are fixed in terms of the parameters of the action S(~-1), which was assumed to be known. 2. It is clear that, at the order n we are considering, we are still free to add to the action any invariant counterterm of dimension bounded by four. 32): f d%~ /d4x O~O¢fl , , /d4x (~599) 2 . 45) (Pl • • •P4) - g. p~=,2, (pl+pj)2=~,~(~3) 3. The free invariant counterterms are in one to one correspondence with the terms of the classical action. This means that they correspond to a renormalization of the initial parameters of the theory.
6) Dr, ¢ = (0 t, - i A ~ T o . ) ¢ . 3)) for the fermions forbids any mass term. The gauge fixing is done in the BR8 way [16, 65]. One introduces Faddeev-Popov ghost and antighost fields c(x) and a(x), and the Lautrup-Nakanishi  field B ( x ) . The latter plays the role of a Lagrange multiplier for the gauge condition. These three fields are Lie algebra valued: c(x) = ca(x)ra, etc. The components of c and a. are anticommuting. One chooses a covariant linear gauge condition, which one can express by the functional identity 5S 6-B = O~*A~* + a B .