We consider (0+1) and (1+1) dimensional Yukawa theory in various scalar field backgrounds, which are solving classical equations of motion:φ cl = 0 or φ cl = 0, correspondingly. The (0+1)-dimensional theory we solve exactly. In (1+1)-dimensions we consider background fields of the form φ cl = E t and φ 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. * akhmedov@itep.ru †

The Sachdev–Ye–Kitaev model and two-dimensional dilaton gravity have recently been attracting increasing attention of the high-energy and condensed-matter physics communities. The success of these models is due to their remarkable properties. Following the original papers, we broadly discuss the properties of these models, including the diagram technique in the limit of a large number of degrees of freedom, the emergence of conformal symmetry in the infrared limit, effective action, four-point functions, and chaos. We also briefly discuss some recent results in this field. On the one hand, we attempt to be maximally rigorous, which means considering all the details and gaps in the argument; on the other hand, we believe that this review can be suitable for those who are not familiar with the relevant models.