External Potentials and Ehrenfest Relations in Lagrangian Field Theories

This paper develops a general method to construct Ehrenfest-like relations for Lagrangian field theories when an external, coordinate-dependent scalar potential is applied. To do so, we derive continuity equations in which the spatial and temporal derivatives of the potential can be interpreted as a source of field momentum and field energy, respectively. For a non-relativistic Schr\"odinger field theory, these continuity equations yield Ehrenfest's theorem for energy, linear momentum, and angular momentum. We then derive a relativistic counterpart for these relations using complex Klein-Gordon fields coupled with an electric potential.