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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.

Quantum entropy of a black hole is commonly considered as a formula for the communal entropy representing a series, where the major term is coincident with Bekenstein-Hawking entropy in a semiclassical approximation, whereas other terms are its quantum corrections. This paper presents the development of a new approach to a quantum theory close to the singularity (Early Universe), whose side product is the application to a quantum theory of black holes the calculations of entropy including. The principal method of this paper is deformation of a quantum-mechanical density matrix in the Early Universe. And a quantum mechanics of the Early Universe is considered as Quantum Mechanics with Fundamental Length (QMFL), the associated deformed density matrix being referred to as a density pro-matrix. 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

Besides, in _{min}_{p}^{′} ^{′} as an arbitrary constant without giving it any definite value. Equation (1) is identified as the Generalized Uncertainty Relations in Quantum Mechanics.

The inequality (1) is quadratic in △_{p}_{min}_{p}

As demonstrated in [_{min}

Any system in QMFL is described by a density pro-matrix of the form

0 <

The vectors |

_{i}

_{i}_{i}_{i}_{i}

For every operator

Finally, in order that our definition 1 be in agreement with the result of ([

According to point 5, _{α}_{α}_{µ}, p_{ν}_{α}

It should be noted that:

The above limit covers both Quantum and Classical Mechanics. Indeed, since ^{3}^{2}, we obtain:

(

(

As a matter of fact, the deformation parameter _{min}_{min}_{m}_{i}_{n}

We consider possible solutions for (7). For instance, one of the solutions of (7), at least to the first order in _{i}^{∗}(_{pl}

In [_{α}_{α}_{1}_{2} ≤ 1

_{2} by the observer who is at a scale corresponding to the deformation parameter _{1}. Note that with this approach the diagonal element _{α}

For the initial (approximately pure) state

Using the exponential ansatz(9),we obtain:

So increase in the entropy density for an external observer at the large-scale limit is 1/4. Note that increase of the entropy density (information loss) for the observer that is crossing the horizon of the black hole’s events and moving with the information flow to singularity will be smaller:

Now we consider the general Information Problem. Note that with the classical Quantum Mechanics (QM) the entropy density matrix

Of course in this case any conservation of information is impossible as these theories are based on different concepts of entropy. Simply saying, it is incorrect to compare the entropy interpretations of two different theories (QM and QMFL, where this notion is originally differently understood. So the chain above must be symmetrized by accompaniment of the arrow on the left ,so in an ordinary situation we have a chain:

So it’s more correct to compare entropy close to the initial and final (Black hole) singularities. In other words, it is necessary to take into account not only the state, where information disappears, but also that whence it appears. The question arises, whether the information is lost in this case for every separate observer. For the event under consideration this question sounds as follows: are the entropy densities S(in) and S(out) equal for every separate observer? It will be shown that in all conceivable cases they are equal.

For the observer in the large-scale limit (producing measurements in the semiclassical approximation) _{1} = 0

S(

S(

So

For the observer moving together with the information flow in the general situation we have the chain:

This case is a special case of 2), when we do not come out of the early Universe considering the processes with the participation of black mini-holes only. In this case the originally specified chain becomes shorter by one Section:

It should be noted that in terms of deformation the Liouville’s equation (

In the last few years Quantum Mechanics of black holes has been studied under the assumption that GUR are valid [_{p}^{−33}

In connection with this remark of J.Bekenstein [

An approach proposed in [

As demonstrated in [_{2} to 1/4.

In works of J.Bekenstein, [

This necessitates mentioning of the recent findings of R.Bousso [

Also it should be noted that the approach proposed in [

Qualitative analysis performed in this work reveals that the Information Loss Problem in black holes with the canonical problem statement [

This paper presents certain results pertinent to the application of the above methods in a Quantum Theory of Black Holes. Further investigations are still required in this respect, specifically for the complete derivation of a semiclassical Bekenstein-Hawking formula for the Black Hole entropy, since in

As indicated in papers [_{0},_{1},... or the definition of additional members in the exponent ”destroying” _{0},_{1},... [_{0} = −3_{1} and _{2}. Fixing one of them, e.g. _{1}, it is possible to expand the series in terms of _{2} parameter and to obtain the quantum corrections to the main result as more and more higher-order terms of this series. In the process, (13) is a partial case of the approach to _{1} = 0 and _{2} close to 1/4.

Thus, in this paper it is demonstrated that the developed approach to study a quantum theory of the Early Universe - density matrix deformation at Planck’s scales - leads to a new method of studying the black hole entropy its quantum aspects including. Despite the fact that quite a number of problems require further investigation, the proposed approach seems a worthy contribution: first, it allows for the direct calculations of entropy and, second, the method enables its determination proceeding from the basic principles with the use of the density matrix. Actually, we deal with a substantial modification of the conventional density matrix known from the quantum mechanics - deformation at Planck’s scales or minimum length scales. Moreover, within this approach it is possible to arrive at a very simple derivation and physical interpretation for the Bekenstein-Hawking formula of black hole entropy in a semi-classical approximation. Note that the proposed approach enables one to study the information problem of the Universe proceeding from the basic principles and two types of the existing quantum mechanics only: QM that describes nature at the well known scales and QMFL at Planck’s scales. The author is of the opinion that further development of this approach will allow to research the information problem in greater detail. Besides, it is related to other methods, specifically to the holographic principle, as the entropy density matrix studied in this work is related to the two-dimensional objects.

To conclude, it should be noted that an important problem of the extremal black holes was beyond the scope of this paper. In the last decade this problem has, however, attracted much attention in connection with a string theory and quantum gravitation(e.g., [