We use the vacuum Bianchi I model as an example to investigate the concept of
physical evolution in Loop Quantum Cosmology (LQC) in the absence of the
massless scalar field which has been used so far in the literature as an
internal time. In order to retrieve the system dynamics when no such a suitable
clock field is present, we explore different constructions of families of
unitarily related partial observables. These observables are parameterized,
respectively, by: (i) one of the components of the densitized triad, and (ii)
its conjugate momentum; each of them playing the role of an evolution
parameter. Exploiting the properties of the considered example, we investigate
in detail the domains of applicability of each construction. In both cases the
observables possess a neat physical interpretation only in an approximate
sense. However, whereas in case (i) such interpretation is reasonably accurate
only for a portion of the evolution of the universe, in case (ii) it remains so
during all the evolution (at least in the physically interesting cases). The
constructed families of observables are next used to describe the evolution of
the Bianchi I universe. The performed analysis confirms the robustness of the
bounces, also in absence of matter fields, as well as the preservation of the
semiclassicality through them. The concept of evolution studied here and the
presented construction of observables are applicable to a wide class of models
in LQC, including quantizations of the Bianchi I model obtained with other
prescriptions for the improved dynamics.Comment: RevTex4, 22 pages, 4 figure
