Relativistic particle phase-space action ($I=\int dt[p_m{\dot{x}}^m-\frac{e}{2}(p^2+M^2)]$)

The worldline einbein imposes the relativistic mass-shell condition. With the Minkowski metric of signature $(-,+,\dots ,+)$, its first-class constraint is $\phi =(p^2+M^2)/2$.

An einbein specifies a one-dimensional metric tensor along a worldline. In the relativistic particle phase-space action it is the Lagrange multiplier for the mass-shell condition and transforms as $\delta e=\dot{ϵ}$ under the corresponding canonical gauge transformation. After eliminating the momentum, its geometric interpretation depends on the normalization of the particle action.

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 $M\int edt$.