arXiv subjects
Publications and source records attributed to Wlodzimierz Piechocki.
We examine spectrum of the physical volume operator within the non-standard loop quantum cosmology. The spectrum is discrete with equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semi-classical level which may be detected in astro-cosmo observations.
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.
We examine the nature of the cosmological Big Bounce transition within the loop geometry underlying loop quantum cosmology at classical and quantum levels. Our canonical quantization method is an alternative to the standard loop quantum cosmology. An evolution parameter we use has a clear interpretation. Our method opens the door for analyses of spectra of physical observables like the energy density and the volume operator. We find that one cannot determine the energy scale specific to the Big Bounce by making use of the loop geometry without an extra input from observational cosmology.
We examine the nature of the cosmological big-bounce (BB) transition within the loop geometry underlying Loop Quantum Cosmology (LQC) at classical and quantum levels. Our canonical quantization method is an alternative to the standard LQC. Our method opens the door for analyzes of spectra of physical observables like energy density and volume operator. We find that one cannot determine the energy scale specific to BB by making use of the loop geometry without an extra input from observational cosmology.
We analyze classical theory of a membrane propagating in a singular background spacetime. The algebra of the first-class constraints of the system defines the membrane dynamics. A membrane winding uniformly around compact dimension of embedding spacetime is described by two constraints, which are interpreted in terms of world-sheet diffeomorhisms. The system is equivalent to a closed bosonic string propagating in a curved spacetime. Our results may be used for finding a quantum theory of a membrane in the compactified Milne space.
We present analytical results on the propagation of a classical string in non-zero modes through the singularity of the compactified Milne space. We restrict our analysis to a string winding around the compact dimension of spacetime. The compact dimension undergoes contraction to a point followed by re-expansion. We demonstrate that the classical dynamics of the string in excited states is non-singular in the entire spacetime.
Our results concern the transition of a quantum string through the singularity of the compactified Milne (CM) space. We restrict our analysis to the string winding around the compact dimension (CD) of spacetime. The CD undergoes contraction to a point followed by re-expansion. We demonstrate that both classical and quantum dynamics of considered string are well defined. Most of presently available calculations strongly suggest that the singularity of a time dependent orbifold is useless as a model of the cosmological singularity. We believe that our results bring, to some extent, this claim into question.
We examine the transition of a particle across the singularity of the compactified Milne (CM) space. Quantization of the phase space of a particle and testing the quantum stability of its dynamics are consistent to one another. One type of transition of a quantum particle is described by a quantum state that is continuous at the singularity. It indicates the existence of a deterministic link between the propagation of a particle before and after crossing the singularity. Regularization of the CM space leads to the dynamics similar to the dynamics in the de Sitter space. The CM space is a promising model to describe the cosmological singularity deserving further investigation by making use of strings and membranes.
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.
We compare the respective efficiencies of three quantization methods (group theoretical, coherent state and geometric) by quantizing the dynamics of a free massive particle in two-dimensional de Sitter space. For each case we consider the realization of the principal series representation of $SO_0(1,2) $ group and its two-fold covering SU(1,1). We demonstrate that standard technique for finding an irreducible representation within the geometric quantization scheme fails. For consistency we recall our earlier results concerning the other two methods, make some improvements and generalizations.
Evolution of winding strings in spacetimes with cycles whose proper lengths depend on time is examined. It was established earlier that extended objects wrapping the shrinking dimension in compactified Milne spacetime enjoy classically nonsingular evolution. Extensions of this observation to other spacetimes are discussed.
We present a coherent state quantization of the dynamics of a relativistic test particle on a one-sheet hyperboloid embedded in a three-dimensional Minkowski space. The group SO_0(1,2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO_0(1,2) seems to be a new result.
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.
We consider classical and quantum dynamics of a free particle in de Sitter's space-times with different topologies to see what happens to space-time singularities of removable type in quantum theory. We find analytic solution of the classical dynamics. The quantum dynamics is solved by finding an essentially self-adjoint representation of the algebra of observables integrable to the unitary representations of the symmetry group of each considered gravitational system. The dynamics of a massless particle is obtained in the zero-mass limit of the massive case. Our results indicate that taking account of global properties of space-time enables quantization of particle dynamics in all considered cases.
We examine the dependence of quantization on global properties of a classical system. Quantization based on local properties may lead to ambiguities and inconsistency between local and global symmetries of a quantum system. Our quantization method based on global characteristics has sound foundation. Presented results give insight into the nature of removable type singularity of spacetime at the quantum level.
We compare classical and quantum dynamics of a particle in the de Sitter spacetimes with different topologies to show that the result of quantization strongly depends on global properties of a classical system. We present essentially self-adjoint representations of the algebra of observables for each system. Quantization based on global properties of a classical system accounts properly its symmetries.
We analyze particle dynamics on $N$ dimensional one-sheet hyperboloid embedded in $N+1$ dimensional Minkowski space. The dynamical integrals constructed by $SO_\uparrow (1,N)$ symmetry of spacetime are used for the gauge-invariant Hamiltonian reduction. The physical phase-space parametrizes the set of all classical trajectories on the hyperboloid. In quantum case the operator ordering problem for the symmetry generators is solved by transformation to asymptotic variables. Canonical quantization leads to unitary irreducible representation of $SO_\uparrow (1,N)$ group on Hilbert space $L^2(S^{N-1})$.
General relativity is unable to determine the topology of the Universe. We propose to apply quantum approach. Quantization of dynamics of a test particle is sensitive to the spacetime topology. Presented results for a particle in de Sitter spacetimes favor a finite universe.