Proper-time modulus ($s$)

The constant mode of the worldline einbein is not removed by gauge transformations whose parameters vanish at the ends of a fixed interval. This Schwinger proper-time parameter remains integrated over after gauge fixing. For $\phi =(p^2+M^2)/2$, the geometric on-shell proper length is $M\int edt$.

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 $\int b\dot{c}dt$.