 Solution

For a timelike worldline, $-{\eta }_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }>0$. Its square root is a Lorentz scalar, andThus the first term is the Lorentz-invariant free-particle action $-m{c}^2\int d\tau$. Written as a line element, it is also invariant under a change of the worldline parameter $\lambda$.

The interaction is the scalar line integralUnder an electromagnetic gauge transformationit changes only byThis endpoint term does not affect a fixed-endpoint variational problem. Hence both Lorentz invariance and gauge invariance select the displayed relativistic charged-particle action, with the expected free-particle and minimal-coupling terms.