Closed-form solutions of the Wheeler-DeWitt equation in a scalar-vector field cosmological model by Lie symmetries

TL;DR

This paper uses Lie symmetries to determine specific forms of functions in a scalar-vector cosmological model, enabling explicit solutions to the Wheeler-DeWitt equation and revealing conserved quantities.

Contribution

It introduces a method to identify the form of coupling and potential functions in quantum cosmology using Lie symmetry analysis, leading to exact solutions.

Findings

Explicit forms of coupling and potential functions derived

Exact solutions to the Wheeler-DeWitt equation obtained

Conservation laws constructed from symmetries

Abstract

We apply as selection rule to determine the unknown functions of a cosmological model the existence of Lie point symmetries for the Wheeler-DeWitt equation of quantum gravity. Our cosmological setting consists of a flat Friedmann-Robertson-Walker metric having the scale factor $a(t)$, a scalar field with potential function $V(\phi)$ minimally coupled to gravity and a vector field of its kinetic energy is coupled with the scalar field by a coupling function $f(\phi)$. Then, the Lie symmetries of this dynamical system are investigated by utilizing the behavior of the corresponding minisuperspace under the infinitesimal generator of the desired symmetries. It is shown that by applying the Lie symmetry condition the form of the coupling function and also the scalar field potential function may be explicitly determined so that we are able to solve the Wheeler-DeWitt equation. Finally, we show how we can use the Lie symmetries in order to construct conservation laws and exact solutions for the field equations.