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Showing papers by "Jerzy Lewandowski published in 2008"


Journal ArticleDOI
TL;DR: In this article, the APS construction of the quantum Hamiltonian is analyzed under the assumption that the cosmological constant is a constant, and the essential self-adjointness of the operator whose square-root defines in [1] is proved.
Abstract: The flat Friedman?Robertson?Walker (FRW) model coupled to the massless scalar field according to the improved, background scale-independent version of Ashtekar, Paw?owski and Singh [1] is considered. The core of the theory is addressed directly: the APS construction of the quantum Hamiltonian is analyzed under the assumption that the cosmological constant ? ? 0. We prove the essential self-adjointness of the operator whose square-root defines in [1] the quantum Hamiltonian operator and therefore provide the explicit definition. If ? 0 being some constants) plus a trace class operator.

61 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the quantum scalar constraint for two different choices of the lapse function and showed that the physical Hilbert spaces constructed for two choices of lapse are the same for both choices.
Abstract: Several conceptual aspects of quantum gravity are studied on the example of the homogeneous isotropic LQC model. In particular: $(i)$ The proper time of the co-moving observers is showed 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 case of model with masless scalar field and cosmological constant $\Lambda$ the physical Hilbert spaces constructed for two choices of lapse are the same 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 $[0,\rhoc]$, where $\rhoc\approx 0.41\rho_{\Pl}$. $(iv)$ The relation between the kinematical and the physical quantum geometry is discussed on the level of relation between observables.

33 citations


Journal ArticleDOI
TL;DR: In this paper, the authors prove several theorems concerning the con- nection between the local CR embeddability of 3-dimensional CR manifolds and the existence of algebraically special Maxwell and gravitational fields.
Abstract: We prove several theorems concerning the con- nection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR in- variant equations on a 3-dimensional CR manifold defined by the fields. Using the reduced Einstein equations we construct two independent CR functions for the corresponding CR man- ifold. We also point out that the Einstein equations, imposed on spacetimes associated with a 3-dimensional CR manifold, imply that the spacetime metric, after an appropriate rescaling, becomes well defined on a circle bundle over the CR manifold. The circle bundle itself emerges as a consequence of Einstein's equations.

20 citations


Journal ArticleDOI
TL;DR: In this paper, the quantum geometry operators of LQC in both the kinematical and the physical Hilbert spaces of solutions to the quantum constraints are derived, and quantum geometry can be used to characterize the physical solutions, and the operators of quantum geometry preserve many of their kinematics.
Abstract: This paper is motivated by the recent papers by Dittrich and Thiemann and, respectively, Rovelli discussing the status of quantum geometry in the dynamical sector of quantum geometry. 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 a space-isotropic universe. We derive the Rovelli–Thiemann–Dittrich partial observables corresponding to the quantum geometry operators of LQC in both Hilbert spaces: the kinematical one and 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.

10 citations


Journal ArticleDOI
TL;DR: The Ahtekar-Isham C*-algebra known from Loop Quantum Gravity is the algebra of continuous functions on the space of (generalized) connections with a compact structure Lie group as mentioned in this paper.
Abstract: The Ahtekar-Isham C*-algebra known from Loop Quantum Gravity is the algebra of continuous functions on the space of (generalized) connections with a compact structure Lie group. The algebra can be constructed by some inductive techniques from the C*-algebra of continuous functions on the group and a family of graphs embedded in the manifold underlying the connections. We generalize the latter construction replacing the commutative C*-algebra of continuous functions on the group by a non-commutative C*-algebra defining a compact quantum group.

5 citations