Quantization in background scalar fields

We consider (0 + 1) and (1 + 1) dimensional Yukawa theory in various scalar field backgrounds, which are solving classical equations of motion: (phi) over dot(cl) = 0 or square phi(cl )= 0, correspondingly. The (0 + 1)-dimensional theory we solve exactly. In (1 + 1)-dimensions we consider background fields of the form phi(cl) = Et and phi(cl) = Ex, which are inspired by the constant electric field. Here E is a constant. We study the backreaction problem by various methods, including the dynamics of a coherent state. We also calculate loop corrections to the correlation functions in the theory using the Schwinger-Keldysh diagrammatic technique.