The problem of time in quantum cosmology

Abstract: This thesis contains an analysis of the problem of time in quantum cosmology and its application to a cosmological minisuperspace model. In the first part, we introduce the problem of time and the theoretical foundations. In the second part, we focus on a specific minisuperspace universe, analyse it classically, and quantise it using the canonical quantisation method. The chosen model is a flat FLRW universe with a free massless scalar field and a perfect fluid. We extract the Wheeler--DeWitt equation, and calculate its solutions. There are three dynamical variables that may be used as clock parameters, namely a coordinate $t$ conjugated to the perfect fluid mass, the massless scalar field $\varphi$, and $v$, a positive power of the scale factor. We define three quantum theories, each one based on assuming one of the previous dynamical quantities as the clock. This quantisation method is then compared with the Dirac quantisation. We find that, in each quantisation procedure, covariance is broken, leading to inequivalent quantum theories. In the third part, the properties of each theory are analysed. Unitarity of each theory is implemented by adding a boundary condition on the allowed states. Requiring unitarity is what breaks general covariance in the quantum theory. In the fourth part, we study the numerical properties of the wave functions in the three theories, paying special attention to singularity resolution and other divergences from the classical theory. The $t$-clock theory is able to resolve the singularity, the $\varphi$-clock theory presents some non trivial dynamics that can be associated with a resolution of spatial infinity, and the $v$-clock theory does not show significant deviations from the classical theory. In the last part, we expand our analysis in order to include another quantisation method: path integral quantisation, and finally, we conclude.