In this paper Quantum Mechanics with Fundamental Length is chosen as Quantum Mechanics at Planck's scale. This is possible due to the presence in the theory of General Uncertainty Relations. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation whereas so far commutators have been deformed. The density matrix obtained by deformation of quantum-mechanical density one is named throughout this paper density pro-matrix. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed approach allows to describe dynamics. In particular, the explicit form of deformed Liouville's equation and the deformed Shrödinger's picture are given. Some implications of obtained results are discussed. In particular, the problem of singularity, the hypothesis of cosmic censorship, a possible improvement of the definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that obtained results allow to deduce in a simple * Phone (+375) 172 883438; e-mail: a.shalyt@mail.ru; alexm@hep.by † Phone (+375) 172 883438; e-mail: suarez@hep.by, jsuarez@mail.tut.by 1 and natural way the Bekenstein-Hawking's formula for black hole entropy in semiclassical approximation.

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. The authors are of the opinion that the approach proposed may lead to proof of these relations. To this end, the statistical mechanics deformation at Planck scale. 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 statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Plank's. The associated deformation of a canonical Gibbs distribution is given explicitly.

Quantum Mechanics of the Early Universe is considered as deformation of a well-known Quantum Mechanics. Similar to previous works of the author, the principal approach is based on deformation of the density matrix with concurrent development of the wave function deformation in the respective Schrödinger picture, the associated deformation parameter being interpreted as a new small parameter. It is demonstrated that the existence of black holes in the suggested approach in the end twice causes nonunitary transitions resulting in the unitarity. In parallel this problem is considered in other terms: entropy density, Heisenberg algebra deformation terms, respective deformations of Statistical Mechanics, -all showing the identity of the basic results. From this an explicit solution for Hawking's informaion paradox has been derived.

Abstract:The present work is a continuation of the previous papers written by the author on the subject. In terms of the measurability (or measurable quantities) notion introduced in a minimal length theory, first the consideration is given to a quantum theory in the momentum representation. The same terms are used to consider the Markov gravity model that here illustrates the general approach to studies of gravity in terms of measurable quantities.

Abstract. It is demonstrated that entropy and its density play a significant role in solving the problem of the vacuum energy density (cosmological constant) in the Universe and hence the dark energy problem. Taking this in mind, two most popular models for dark energy -Holographic Dark Energy Model and Agegraphic Dark Energy Model -are analyzed. It is shown that the fundamental quantities in the first of these models may be expressed in terms of a new small parameter. Besides, the results obtained on the uncertainty relation of the pair "cosmological constant -volume of space-time", where the cosmological constant is a dynamic quantity, are reconsidered and generalized up to the Generalized Uncertainty Relation (GUP).