Quantum particle on hyperboloid

We present quantization of particle dynamics on one-sheet hyperboloid embedded in three dimensional Minkowski space. Taking account of all global symmetries enables unique quantization. Making use of topology of canonical variables not only simplifies calculations but also gives proper framework for analysis.Comment: 7 pages, no figures, revtex

Simple model of big-crunch/big-bang transition

We present classical and quantum dynamics of a test particle in the compactified Milne space. Background spacetime includes one compact space dimension undergoing contraction to a point followed by expansion. Quantization consists in finding a self-adjoint representation of the algebra of particle observables. Our model offers some insight into the nature of the cosmic singularity.Comment: 17 pages, no figures, RevTeX4, accepted for publication in Class. Quantum Gra

Bianchi I model in terms of nonstandard loop quantum cosmology: Quantum dynamics

We analyze the quantum Bianchi I model in the setting of the nonstandard loop quantum cosmology. Elementary observables are used to quantize the volume operator. The spectrum of the volume operator is bounded from below and discrete. The discreteness may imply a foamy structure of spacetime at semiclassical level. The results are described in terms of a free parameter specifying loop geometry to be determined in astro-cosmo observations. An evolution of the quantum model is generated by the so-called true Hamiltonian, which enables an introduction of a time parameter valued in the set of all real numbers.Comment: 18 pages, version accepted for publication by Class. Quant. Gra

Examination of the nature of the Bianchi type cosmological singularities

We present quantum (and classical) Bianchi I model, with free massless scalar field, of the Universe. Our model may be treated as the simplest prototype of the quantum BKL (Belinskii-Khalatnikov-Lifshitz) scenario. The quantization is done by making use of the nonstandard Loop Quantum Cosmology (LQC). Since the method is quite new, we present in details its motivation and the formalism. To make the nonstandard method easily understandable, we include its application to the FRW model. In the nonstandard LQC, we first solve the Hamiltonian constraint of the theory at the classical level and find elementary observables. Physical compound observables are defined in terms of elementary ones. We find that classical Big Bang singularity is replaced by quantum Big Bounce transition due to modification of classical theory by holonomy around a loop with finite size. The energy density of matter fields at the Big Bounce depends on a free parameter {\lambda}, which value is expected to be determined from future cosmological observations. The phase space is divided into two distinct regions: Kasner-like and Kasner-unlike. We use the elementary observables to quantize volume and directional volume operators in both cases. Spectra of these operators are bounded from below and discrete, and depend on {\lambda}. The discreteness may imply a foamy structure of spacetime at semiclassical level. At the quantum level an evolution of the model is generated by the so-called true Hamiltonian. This enables introducing a time parameter valued in the set of all real numbers.Comment: This is my PhD Thesis, about 80 page

Turning Big Bang into Big Bounce: I. Classical Dynamics

The big bounce (BB) transition within a flat Friedmann-Robertson-Walker model is analyzed in the setting of loop geometry underlying the loop cosmology. We solve the constraint of the theory at the classical level to identify physical phase space and find the Lie algebra of the Dirac observables. We express energy density of matter and geometrical functions in terms of the observables. It is the modification of classical theory by the loop geometry that is responsible for BB. The classical energy scale specific to BB depends on a parameter that should be fixed either by cosmological data or determined theoretically at quantum level, otherwise the energy scale stays unknown.Comment: 14 pages, 1 figure, version accepted for publication in Physical Review