The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following questions:
How does the variational principle for fermions formally work so that it's mathematically correct. I do not want to know how to derive the Euler-Lagrange equation from the Dirac Lagrangian. I know how to do that. I don't know how one even gets these equations since a solution to the Euler-Lagrange equations will not generally connect these chosen points.
If I have an initial and end point that are not connected by an on-shell path (one that solves the Dirac equation), then how do I calculate the path that minimizes the action?