Interferometry as a space-time process is described. Starting from this viewpoint, a convenient formalism for the phase shifts which arise in particle interferometry is developed. This formalism is based on a covariant form of Hamilton`s action principle and Lagrange`s equations of motion. It will be shown that this Lorentz invariant formalism yields a simple perturbation-theoretic expression for the general phase shift that arises in matter wave interferometry. The resulting formalism will be used to analyse the Sagnac effect, gravitational field measurements, and several Aharonov-Bohm-like topological phase shifts. 16 refs., 4 refs.