Classical gauge fixing in quantum field theory

Quantization and gauge fixing commute: if one fixes algebraic gauge invariances at the classical level by elimination of classical fields, the modified transformation rules of the remaining classical fields lead to the same quantum action as obtained by solving the field equations of the corresponding nonpropagating ghosts. This is shown for simple supergravity with symmetric vierbein fields in detail. The structure functions are modified but do not depend on auxiliary fields, the four-ghost couplings are unchanged, but new vertices in the two-ghost sector are due to elimination of the Lorentz ghosts and the Becchi-Rouet-Stora-Tyutin transformation laws of the remaining ghosts are modified.