Invariant solutions of the Wheeler-DeWitt equation in Hybrid Gravity
Symmetries play an important role in physics and, in particular, the Noether symmetry is a useful tool to select models motivated at a fundamental level, and it is useful in finding exact solutions for a specific Lagrangian. In this work, we consider the application of point symmetries in the recently proposed hybrid metric-Palatini gravitational theory in order to select the f(R) function and to find analytical solutions of the field equations and of the Wheeler-DeWitt (WDW) equation in quantum cosmology. We show that in order to find new nonlinear f(R) theories which are integrable, we have to apply conformal transformations in the Lagrangian. In this context, we explore a conformal transformation of the form dτ = N(a)dt. For the conformal transformation we found the f(R) function where the field equations admit Noether symmetries. We transform the field equation by use of the normal coordinates to simplify the dynamical system and write the exact solutions. Furthermore, we have analyzed the quantization and written the WDW equation. We determined the Lie point symmetries of the WDW equations and applied the Lie invariants in order to find invariant solutions of the WDW equations.