From Polchinski's String Theory:
Point-particle example
Let us consider an example, the point particles. Expanding out the condensed notation above, the local symmetry is coordinate reparameterization $\delta \tau(\tau)$, so the index $\alpha$ just becomes $\tau$ and a basis of infinitesimal transformations is $\delta_{\tau_1} \tau(\tau) = \delta (\tau - \tau_{1})$. These acts on the fields as $$\delta_{\tau_1} X^{\mu}(\tau) = \delta(\tau - \tau_1)\partial_{\tau}X^{\mu}(\tau), \quad \delta_{\tau_1} e(\tau) = - \partial_{\tau}[\delta(\tau - \tau_1)e(\tau)]. \tag{4.2.19}$$
Why does the transformation act differently when applied to $X$ and when applied to $e$?
