Nonlinear Phenomena in Complex Systems, 2005, Vol.8, No.1, pp.10-18

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2005, Vol.8, No.1, pp.10-18

The Density Matrix Deformation in Quantum and Statistical Mechanics of the Early Universe and Some of its Applications.
A.E. Shalyt-Margolin and V.I. Strazhev

In this work a new approach to studies in quantum and statistical mechanics of the Early Universe (at Plank scale) is proposed - the density matrix deformation approach. A deformation of a particular theory is understood as its extension through one or several parameters in such a way that the initial theory appears in the limiting transition. The distinguishing feature of the proposed approach in comparison to the previous ones lies in the fact that here a deformed object is the density matrix rather than the commutators as it has been so far in case of quantum mechanics. The deformation parameter is dimensionless in the finite interval. The implications and applications of the above approach are explicitly demonstrated for the solution of the following problems: Liouville equation modifications in processes associated with inflation and black holes, Hawking information paradox by the different approaches (S.Hawking and R.Penrose). It is shown that the developed methods enable one to obtain from the basic principles the Bekenstein-Hawking formula for the black hole entropy in semiclassical approximation.Also it is shown that high entropy for Planck's remnants of black holes appearing in the assumption of the Generalized Uncertainty Relations may be explained within the scope of the density matrix entropy introduced by the first author previously. It is noted that the suggested paradigm is consistent with the Holographic Principle. Because of this, a conjecture is made about the possibility for obtaining the Generalized Uncertainty Relations from the covariant entropy bound at high energies in the same way as R.Bousso has derived Heisenberg's uncertainty principle for the flat space. Deformation in statistical mechanics at Planck scale is constructed in similar way with the statistical density matrix as a primary object. The principal difference is the deformation parameter that is associated with a maximum temperature rather than with the fundamental length like in the first case. Some obvious implications are also presented with the formula for a high-temperature complement to the canonical Gibbs distribution. Besides, a generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature.
Key words: deformed density matrix in Quantum and Statistical Mechanics at Plank's Scale, density of entropy, Holographic Principle, Generalized Uncertainty Relations in Thermodynamics

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