3 research outputs found
The status of Quantum Geometry in the dynamical sector of Loop Quantum Cosmology
This letter is motivated by the recent papers by Dittrich and Thiemann and,
respectively, by Rovelli discussing the status of Quantum Geometry in the
dynamical sector of Loop Quantum Gravity. Since the papers consider model
examples, we also study the issue in the case of an example, namely on the Loop
Quantum Cosmology model of space-isotropic universe. We derive the
Rovelli-Thiemann-Ditrich partial observables corresponding to the quantum
geometry operators of LQC in both Hilbert spaces: the kinematical one and,
respectively, the physical Hilbert space of solutions to the quantum
constraints. We find, that Quantum Geometry can be used to characterize the
physical solutions, and the operators of quantum geometry preserve many of
their kinematical properties.Comment: Latex, 12 page
Closed FRW model in Loop Quantum Cosmology
The basic idea of the LQC applies to every spatially homogeneous cosmological
model, however only the spatially flat (so called ) case has been
understood in detail in the literature thus far. In the closed (so called: k=1)
case certain technical difficulties have been the obstacle that stopped the
development. In this work the difficulties are overcome, and a new LQC model of
the spatially closed, homogeneous, isotropic universe is constructed. The
topology of the spacelike section of the universe is assumed to be that of
SU(2) or SO(3). Surprisingly, according to the results achieved in this work,
the two cases can be distinguished from each other just by the local properties
of the quantum geometry of the universe. The quantum hamiltonian operator of
the gravitational field takes the form of a difference operator, where the
elementary step is the quantum of the 3-volume derived in the flat case by
Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are
studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself
is not an eigenvalue, the eigenvectors form a basis. An estimate on the
dimension of the spectral projection on any finite interval is provided.Comment: 19 pages, latex, no figures, high quality, nea