The two quantum symmetries associated with a classical symmetry

Given a classical local symmetry, it is shown how to build two associated quantum symmetries s and s. These generalized BRS and anti-BRS symmetries interchange classical and ghost fields. A direct consequence of the closure and Jacobi identities of the classical algebra are the complete nilpotency relations s² = ss+ss = s² = 0. This nilpotency allows one to derive very simply the correct quantum lagrangian from the classical one without recourse to the Faddeev-Popov construction. A simple formal proof is given, showing that s and s are preserved by renormalization.