Relativistic Adiabatic Approximation and Geometric Phase

A relativistic analogue of the quantum adiabatic approximation is developed for Klein-Gordon fields minimally coupled to electromagnetism, gravity and an arbitrary scalar potential. The corresponding adiabatic dynamical and geometrical phases are calculated. The method introduced in this paper avoids the use of an inner product on the space of solutions of the Klein-Gordon equation. Its practical advantages are demonstrated in the analysis of the relativistic Landau level problem and the rotating cosmic string.