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Nf , each transforming in the fundamental representation of the gauge group G whose generators T a satisfy the commutation relations [T a , T b ] = f abc T c . The Lagrangian density is given by Á2 1 a a 1 À F F @ Aa þ@ D þ 4 2 Nf X A þ ðiD = À mA Þ A LQCD ¼ À ½42 A¼1 a where F = @ Aa À @ Aa þ f abc Ab Ac and D = @ þ a a ieR(T )A , with R the pertinent representation of G a (R(T a )bc = fbc for the adjoint representation and a a R(T ) = T for the fundamental representation).

A universal formulation of all symmetries which can be derived from the field equation in classical field theory is the ‘‘master Ward identity’’ (which presupposes (N3) and (N4)) (Boas and Du¨tsch 2002, Du¨tsch and Fredenhagen 2003); see next section. The problem of perturbative renormalization is to construct a solution of all these normalization conditions. Epstein and Glaser have constructed the solutions of (N0)–(N3). Recently, the conditions (N4) and (AWI) have been included. The master Ward identity cannot always be fulfilled, the obstructions being the famous ‘‘anomalies’’ of perturbative quantum field theory.

The extension of this property when the external momenta are continued to physical on-shell values is difficult. The Kinoshita–Lee–Nauenberg theorem states that, as a consequence of unitarity, transition probabilities in a theory involving massless particles are finite when the sum over all degenerate states (initial and final) is taken. This is true order by order in perturbation theory in bare quantities or if minimal subtraction renormalization is used (to avoid infrared or mass singularities in the renormalization constants).