We consider ($0+1$) and ($1+1$) dimensional Yukawa theory in various scalar field backgrounds, which are solving classical equations of motion: ${\ddot{\varphi }}_{\mathrm{cl}}=0$ or $\square {\varphi }_{\mathrm{cl}}=0$, correspondingly. The ($0+1$)–dimensional theory we solve exactly. In ($1+1$)–dimensions we consider background fields of the form ${\varphi }_{\mathrm{cl}}=Et$ and ${\varphi }_{\mathrm{cl}}=Ex$, 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.

• Received 18 November 2019

DOI:https://doi.org/10.1103/PhysRevD.101.025005

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Published by the American Physical Society