Physical time and other conceptual issues of quantum gravity on the example of loop quantum cosmology

Several conceptual aspects of quantum gravity (QG) are studied on the example of the homogeneous isotropic loop quantum cosmology (LQC) model. In particular: (i) the proper time of the comoving observers is shown to be a quantum operator and a quantum spacetime metric tensor operator is derived. (ii) Solutions of the quantum scalar constraint for two different choices of the lapse function are compared and contrasted. In particular it is shown that in the case of a model with massless scalar field and cosmological constant Lambda, the physical Hilbert spaces constructed for two choices of lapse are the same for Lambda < 0 while they are significantly different for Lambda > 0. (iii) The mechanism of the singularity avoidance is analyzed via detailed studies of an energy density operator, whose essential spectrum was shown to be an interval [0, rho(cr)], where rho(cr) approximate to 0.41 rho(Pl). (iv) The relation between the kinematical and the physical quantum geometry is discussed on the level of relation between observables.