67. L. Pysiak, M. Heller, Z. Odrzygóźdź, W. Sasin
Observables in a Noncommutative Approach to the Unification of Quanta and Gravity: A Finite Model
General Relativity and Gravitation, vol. 37, pp. 541-555 (2005).

Abstract:

68. Marek Demianski, Zdzislaw A. Golda, Andrzej Woszczyna
Evolution of density perturbations in a realistic universe
Gen. Rel. Grav., vol. 37, pp. 2063-2082 (2005).
[abstract] [preprint] [journal]

Abstract:
Prompted by the recent more precise determination of the basic cosmological parameters and growing evidence that the matter-energy content of the universe is now dominated by dark energy and dark matter we present the general solution of the equation that describes the evolution of density perturbations in the linear approximation. It turns out that as in the standard CDM model the density perturbations grow very slowly during the radiation dominated epoch and their amplitude increases by a factor of about 4000 in the matter and later dark energy dominated epoch of expansion of the universe.

69. Marek Szydlowski, Adam Krawiec, Wojciech Czaja
Phantom cosmology as a simple model with dynamical complexity
Phys. Rev., vol. E72, p. 036221 (2005).
[abstract] [preprint] [journal]

Abstract:
We study the Friedman-Robertson-Walker model with phantom fields modeled in terms of scalar fields. We apply the Ziglin theory of integrability and find that the flat model is nonintegrable. Then we cannot expect to determine simple analytical solutions of the Einstein equations. We demonstrate that there is only a discrete set of parameters where this model is integrable. For comparison we describe the phantoms fields in terms of the barotropic equation of state. It is shown that in contrast to the phantoms modeled as scalar fields, the dynamics is always integrable and phase portraits are contracted. In this case we find the duality relation.

70. Marek Szydlowski, Wojciech Czaja
Modified Friedmann cosmologies -Theory and observations
Annals Phys. , vol. 320, pp. 261-281 (2005).
[abstract] [preprint] [journal]

Abstract:
This paper shows the results of the investigation of a class of Cardassian scenarios of the evolution of the universe in the formulation of the qualitative theory of dynamical systems. That theory allowed us to analyze all solutions for all possible initial conditions on the phase plane. In the Cardassian models we find that big-rip singularities are present as a typical behavior in the future if $n < 0$. We were also able to find some exact solution for the flat Cardassian models as well as a duality relation. In turn for the statistical analysis of SNIa data, without any priors on the matter content in the model, we obtain that the big-rip scenario is favored. The potential function for the Hamiltonian description of dynamics was reconstructed from the SNIa data (inverse dynamical problem).

71. Michael Heller, Leszek Pysiak and Wiesłw Sasin
Noncommutative Dynamics of Random Operators
International Journal of Theoretical Physics, vol. 44, pp. 619-628 (2005).
[abstract] [preprint] [journal] [download]

Abstract:
We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.

72. Michael Heller, Leszek Pysiak, and Wiesław Sasin
Noncommutative unification of general relativity and quantum mechanics
J. Math. Phys., vol. 46, p. 122501 (2005).
[abstract] [preprint] [journal]

Abstract:
We present a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry is developed in terms of a noncommutative algebra A which is defined on a transformation groupoid Γ given by the action of a noncompact group G on the total space E of a principal fiber bundle over space-time M. The case is important since to obtain physical effects predicted by the model we should assume that G is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalue equation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model “collapses” to the usual quantum mechanics.