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MATHEMATICAL STRUCTURES OF THE UNIVERSE

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79. 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.

80. 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.

81. Zdzisław A. Golda
Density Perturbations in Open Models of Early Universe with Positive Cosmological Constant
Acta Phys. Pol. , vol. B36, pp. 2149-2164 (2005).
[abstract] [journal]

Abstract:
The analytical solutions of the density perturbation equation in the Friedman--Lema\^{\i}tre--Robertson--Walker (FLRW) open cosmological models with radiation and positive cosmological constant are provided. The perturbations are of two types: the first propagating as acoustical waves, and the second of non-wave nature. It is shown that there occurs dispersion on curvature and cosmological constant for acoustical perturbations. The wave solutions have anomalous dispersion.

82. Zdzisław A. Golda, Andrzej Woszczyna, Karolina Zawada
Canonical gauge-invariant variables
Acta Phys. Pol., B , vol. B36, p. 2133 (2005).
[abstract] [preprint] [journal]

Abstract:
Under an appropriate change of the perturbation variable Lifshitz-Khalatnikov propagation equations for the scalar perturbation reduce to d'Alembert equation. The change of variables is based on the Darboux transform

83. Leszek M. Sokołowski
Elementy Kosmologii
(ZamKor 2005) ISBN: 83-88830-39-2 [opis]

Abstract:

84. G. Siemieniec-Ozieblo, A. Woszczyna
Acoustic instabilities at the transition from the radiation-dominated to the matter-dominated universe
Astron. Astrophys. , vol. 419, pp. 801-810 (2004).
[abstract] [preprint] [journal]

Abstract:
The transition from acoustic noise in the radiation-dominated universe to the density structures in the matter dominated epoch is considered. The initial state is a stochastic field of sound waves moving in different directions. The construction of the initial state is compatible with the hyperbolic type of propagation equation for density perturbations, and parallel to the theory of stochastic background of gravitational waves. Instantaneous transition between the cosmological epochs is assumed, and Darmois-Israel joining conditions are applied to match solutions for sound waves with growing or decaying modes at the decoupling. As a result a substantial amplification of the low scale structures is obtained.

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