In the present work the approach - density matrix deformation - earlier developed by the author to study a quantum theory of the Early Universe (Planck's scales) is applied to study a quantum theory of black holes. On this basis the author investigates the information paradox problem, entropy of the black hole remainders after evaporation, and consistency with the holographic principle. The possibility for application of the proposed approach to the calculation of quantum entropy of a black hole is considered.

The principal objective of this paper is to show that deformation of the General Relativity within the scope of both the Generalized Uncertainty Principle (UV-cutoff) and the Extended Uncertainty Principle (IR-cutoff) in the cases when the corresponding gravitational equations have a thermodynamic interpretation may be studied in terms of a small parameter introduced in previous works of the author. From this viewpoint the parameter is though to be universal. Consideration is given to the possibility for extension of the obtained results to more general cases. Possible generalization of the uncertainty relation for the pair (cosmological constant, space-time volume), where the cosmological constant is regarded as a dynamic quantity at high and low energies, is analyzed. This paper is devoted to the 70-th Anniversary of Professor Nikolai Maksimovich Shumeiko, Director of the Belarusian Particles and High Energy Physics National Centre.

It is demonstrated that entropy and its density play a significant role in solving the problem of the vacuum energy density (cosmological constant) of 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 analysed. It is shown that the fundamental quantities in the first of these models may be expressed in terms of a new small dimensionless parameter that is naturally occurring in High Energy Gravitational Thermodynamics and Gravitational Holography (UV-limit). On this basis, the possibility of a new approach to the problem of Quantum Gravity is discussed. 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.

It is demonstrated that provided a theory involves a minimal length, this theory must be free from such infinitesimal quantities as infinitely small variations in surface of the holographic screen, its volume, and entropy. The corresponding infinitesimal quantities in this case must be replaced by the minimal variations possiblefinite quantities dependent on the existent energies. As a result, the initial low-energy theory (quantum theory or general relativity) inevitably must be replaced by a minimal length theory that gives very close results but operates with absolutely other mathematical apparatus.

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.