By Olivier Piguet, Silvio P. Sorella
This booklet offers a pedagogical and self-contained advent to the algebraic approach to renormalization in perturbative quantum box concept. this technique relies on common 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 houses of classical box polynomials. a number of examples (e.g. topological types) are thought of in a few aspect. one of many major merits of this system, past its simplicity, is its nice energy, simply because no specific subtraction or regularization scheme holding the symmetries of the matter is a priori required.
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Additional info for Algebraic renormalization: perturbative renormalization, symmetries and anomalies
One chooses a covariant linear gauge condition, which one can express by the functional identity 5S 6-B = O~*A~* + a B . 8) 1According to the rules established in Sect. 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 _..
4Since we shall be working in a space of functionals obeying the ghost equation, it is convenient to write these transformation laws in terms of the shifted external field t~~. 44 4. 37) where/~ is an integrated local field polynomial of dimension 4, (that this is a solution is the consequence of the nilpotency of b). 32) to the next order. 20). It should therefore obey the constraints ~/X = 0. 24). 34)). , according to the degree of homogeneity in the fields B and ~: z~=y~z~, with Afz~=n/~. 34).
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. In such a case one speaks of a multiplicatively renormalizable theory. In more physical terms, this property expresses the stability of the theory upon quantum fluctuations. 2 The Callan-Symanzik Equation In the classical theory the scale invariance is broken by the mass term of the action.
Algebraic renormalization: perturbative renormalization, symmetries and anomalies by Olivier Piguet, Silvio P. Sorella