We consider (0+1) and (1+1) dimensional Yukawa theory in various scalar field backgrounds, which are solving classical equations of motion: $\ddot��_{cl} = 0$ or $\Box ��_{cl} = 0$, correspondingly. The (0+1)--dimensional theory we solve exactly. In (1+1)--dimensions we consider background fields of the form $��_{cl} = E\, t$ and $��_{cl} = E\, x$, 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.

54 pages, 12 figures. v3: mistake in the section 4 is corrected, new appendix is added