Worldline gauge-orbit determinant (${\Delta }_{\mathrm{FP}}=\det {}^{\prime }{\partial }_t$)

On a unit interval with gauge parameters vanishing at its ends, $e=s+\dot{ϵ}$ separates the worldline einbein into its invariant average $s$ and a canonical gauge transformation. The Jacobian for nonconstant modes is $\det {\partial }_t$ between the appropriate boundary-condition spaces. On a circle its prime excludes the constant gauge-parameter zero mode. This Faddeev-Popov determinant can be represented by anticommuting $b,c$ fields with an action proportional to $\int b\dot{c}dt$.