Relativistic charged-particle action

For metric signature $(-,+,+,+)$, a particle of mass $m$ and charge $q$ coupled to a four-potential has the reparametrization-invariant action

Under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\chi$, the interaction changes byFixed-endpoint equations of motion are therefore gauge invariant.

Varying the relativistic charged-particle action with fixed endpoints givesor the equivalent raised-index equation.