A study of the problem of time in a Friedmann quantum cosmology

After reviewing Dirac's canonical quantization program and the canonical quantization of General Relativity, we study the problem of time in the context of a quantum minisuperspace cosmological model: a Friedmann-Lemaître-Robertson-Walker spacetime coupled minimally to a scalar field. We explore different methods to include time and evolution in our formalism. We begin by discussing the possibility to identify a dynamical time variable before quantization. Such a time variable is constructed as a function of the phase space variables and leads to a multiple choice problem for the evolution of our quantum system. We then explore the connection between the Born-Oppenheimer (BO) approach to the problem of time and gauge fixing. We find that by choosing a particular gauge we can recover the Born-Oppenheimer approach ansatz both in the classical and in the quantum theory. In the latter, the result of the BO approach is recovered by performing a phase transformation in the Wheeler-DeWitt equation and requiring that the resulting Schrödinger-like equation is unitary