= Faddeev-Popov gauge-orbit identity
{c}
{title2=$1=\Delta_{\rm FP}[A]\int\mathcal D\omega\,\delta(G[A^\omega]-f)$}
Suppose a <gauge fixing> condition intersects each <gauge orbit> once in the local region under consideration, and its linearized <Faddeev-Popov operator> has no residual zero modes after boundary conditions are imposed. The <functional Jacobian> for changing from the <gauge transformation> parameter to the gauge condition gives the displayed identity. Inserting it in a <gauge-invariant> <functional integral> allows the integration along the <gauge orbit> to factor out and cancel the formal gauge-group volume. In a perturbative neighborhood the Jacobian's sign or phase is fixed; a field-independent determinant factor is absorbed into normalization. The <Gribov ambiguity> prevents treating this local identity as a globally unique gauge choice without further qualifications.
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