v 2 2 2 N ov 2 00 5 The Quantum and Statistical Theory of Early Universe and Its Possible Applications to Cosmology Alex

In this paper a new approach to investigation of Quantum and Statistical Mechanics of the Early Universe (Planck scale)-density matrix deformation-is proposed. The deformation is understood as an extension of a particular theory by inclusion of one or several additional parameters in such a way that the initial theory appears in the limiting transition. In the first part of this chapter Quantum Mechanics of the Early Universe is treated as a Quantum Mechanics with Fundamental Length considering the fact that different approaches to quantum gravitation exhibited in the Early Universe were inevitably leading to the notion of fundamental length on the order of Plancks. Besides, this is possible due to the involvement in this theory of the Generalized Uncertainty Relations. And Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach as compared to the previous ones is the fact that here the density matrix is subjected to deformation, rather than commutators or (that is the same) Heisenberg's Algebra. In this chapter the density matrix obtained by deformation of the quantum-mechanical one is referred to as a density pro-matrix. Within our * Phone (+375) 172 883438; 1 approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure associated with this interpretation. The proposed approach allows for description of dynamics, in particular, the explicit form of deformed Liouville equation and the deformed Shrodinger's picture. Some implications of the obtained results are discussed including the singularity problem, hypothesis of cosmic censorship, possible improvement of the definition for statistical en-tropy. It is shown that owing to the obtained results one is enabled to deduce in a simple and natural way the Bekenstein-Hawking formula for black hole entropy in a semiclassical approximation. In the second part of the chapter it is demonstrated that Statistical Mechanics of the Early Universe is a deformation of the conventional Statistical Mechanics. The statistical-mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum-mechanics and statistical density matrices at Plank scale (i.e. density pro-matrices). It is shown that an ordinary …