The aim of this section is to give a description of the Faddeev–Popov procedure for gauge field theories first introduced by Faddeev and Popov in the case of Yang–Mills theories [FP]. It yields a device to compute formally functional integrals over configurations of fields, the integrand being invariant under the action of the gauge group. This invariance leads to an infinite expression and the Faddeev–Popov procedure gives a prescription for “factoring out” the infinite volume of the gauge group. More precisely, one considers the integration of a G-invariant function on a principal fiber bundle with structure group G, picking out a well chosen set of representatives (a local section of the bundle) of the orbits on which the integration will actually be done. The Faddeev–Popov procedure then tells us how to relate the integral along this section with the integral on the whole orbit space. The presentation that follows is close to [Pa2], [Pa3]. Further discussions concerning the interpretation of the Faddeev–Popov procedure were presented in [AP2], [AP3].

Going from an integral on the configuration space to an integral on a section gives rise to a Jacobian determinant, the Faddeev–Popov determinant; it is, up to a finite-dimensional determinant which depends on the choice of the slice, entirely determined by the geometric data, namely, the fiber bundle P → P/G equipped with a Riemannian structure.