A supersymmetric model of scalar and spinor fields in a closed isotropic universe

In this talk, a closed isotropic universe with scalar and spinor fields is considered within the framework of the extended phase space approach. This approach implies the derivation of the Schrödinger equation for the wave function of the universe from the path integral with the Faddeev–Popov effective action, including gauge fixing and ghost terms, instead of the Wheeler–DeWitt equation. In the path integral, we use a mixed representation: a coordinate representation for gravitational variables (the lapse function and scale factor) and the so-called holomorphic representation for scalar and spinor fields. In the first step, the scalar field is assumed to be conformal. It enables us to find exact solutions to the Schrödinger equation for special gauge conditions. We consider a set of supersymmetric multiplets of scalar and spinor fields to cancel vacuum divergences and to determine vacuum energy in a closed universe. In the next step, the scalar field is slightly non-conformal. In a time-dependent gravitational field, it gives rise to scalar particle production. In its turn, it acts as a perturbation and results in transitions between quantum states in the Early Universe. Making use of the perturbation theory technique, it is possible to compute probabilities of transitions between states with different energy values.