Study of the Wheeler-DeWitt equation with the Wigner-Weyl transform

The scope of this thesis is studying a new approach to the Wheeler-DeWitt equation, consisting in a Born-Oppenheimer decomposition of the "Universe wave function" and a description of the gravitational sector through the Wigner function. This new approach is more general than the standard one since it can be applied to mixed state which may arise, in the early Universe, as the consequence of the interaction with some hidden sector of the theory. The gravity wave function gets transformed according to Wigner-Weyl, different solutions to the gravity equation are studied and their association with the Wigner function calculated. The form of the Wigner function which solves the gravity equation then affects the form of the matter (inflaton) equation. In the semi-classical limit the matter equation takes the form of a Schrodinger equation, and the time can be defined in it. In the Born-Oppenheimer approach, non-adiabatic next-to-leading order corrections emerge in the gravitational and in the matter equation. Such corrections have a quantum-gravitational origin in this context. The study of the Wigner function indeed is useful to keep track of the quantum effect of gravity, and to better understand their role in the matter-gravity system. The above study is applied in particular for two different sets of initial condition for the gravitational wavefunction: the first one is the Hartle-Hawking initial condition which describe the Universe as a superposition of expanding and contracting solutions, the second one is the Vilenkin initial condition, which describe an expanding Universe. For both cases different approximation methods and procedures are analyzed. The Vilenkin solution, in particular, has been shown to generate quantum-corrections to the definition of time inside the matter equation, which can be described as the presence of an "early-Universe virtual fluid" possibly affecting the slow-roll parameters of and the spectral indices of the primordial spectre.