# Deformed Density Matrix and Quantum Entropy of the Black Hole

## Abstract

**:**

**03.65**;

**05.70**

## 1 Introduction. Deformed Density Matrix in QMFL

_{min}is a must, as follows from the generalized uncertainty relations [8,9,10,11,12] and not only [13]. Then, as noted earlier(e.g., see [2]), the fundamental length may be included into quantum mechanics by the use of the density matrix deformation. Recall the main features of the associated construction. We begin with the Generalized Uncertainty Relations (GUR) [8]:

_{p}is the Planck’s length: and α

^{′}> 0 is a constant. In [9] it was shown that this constant may be chosen equal to 1. However, here we will use α

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

_{p}. As reviewed previously in [13], the fundamental length appears quite naturally at Planck scale, being related to the quantum-gravitational effects.Let us consider equation (3) in some detail. Squaring both its sides, we obtain

_{min}∼ L

_{p}.

_{min}is going up. In the notation system used for , where x is the scale for the fundamental deformation parameter.

**Definition 1.**(Quantum Mechanics with Fundamental Length)

- 0 < α ≤ 1/4.
- The vectors |i > form a full orthonormal system.
- ω
_{i}(α) ≥ 0, and for all i the finite limit exists. - Sp[ρ(α)] = ∑
_{i}ω_{i}(α) < 1, ∑_{i}ω_{i}= 1. - For every operator B and any α there is a mean operator B depending on α:

**Definition 1.**, some explanatory remarks are needed. Of course, any theory may be associated with a number of deformations. In case under consideration the deformation is ”minimal” as only the probabilities are deformed rather than the state vectors. This is essential for the external form of the density pro-matrix, and also for points 2 and 3 in

**Definition 1.**. This suggests point 5 of the Definition: deformation of the average values of the operators. And point 4 follows directly from point 3,(6) and remark before this limiting transition. Finally, limitation on the parameter is inferred from the relation

_{α}= Sp[ρ(α)]. Therefore for any scalar quantity f we have < f >

_{α}= f Sp[ρ(α)]. In particular, the mean value < [x

_{µ}, p

_{ν}] >

_{α}is equal to

- The above limit covers both Quantum and Classical Mechanics. Indeed, since Għ/c
^{3}x^{2}, we obtain:- (ħ ≠ 0, x → ∞) ⇒ (α → 0) for QM;
- (ħ → 0, x → ∞) ⇒ (α → 0) for Classical Mechanics;

- As a matter of fact, the deformation parameter α should assume the value 0 < α ≤ 1. As seen from (8), however, Sp[ρ(α)] is well defined only for 0 < α ≤ 1/4. That is if x = il
_{min}and i ≥ 2, then there is no any problem. At the point of x = l_{min}there is a singularity related to the complex values following from Sp[ρ(α)] , i.e. to the impossibility of obtaining a diagonalized density pro-matrix at this point over the field of real numbers. For this reason definition 1 has no sense at the point x = l_{m}_{i}_{n}. - We consider possible solutions for (7). For instance, one of the solutions of (7), at least to the first order in α, is
_{i}> 0 are independent of α and their sum is equal to 1. In this way Sp[ρ^{∗}(α)] = exp(−α). We can easily verify that_{pl}is the Planck momentum. When present in the matrix elements, exp(−α) can damp the contribution of great momenta in a perturbation theory.

## 2 Entropy Density Matrix and Information Loss Problem

_{α}generalizing the ordinary statistical entropy:

_{α}means of the entropy density on a unit minimum area depending on the scale. In fact a more general concept accepts the form of the entropy density matrix [4],[5],[7]:

_{1}, α

_{2}≤ 1/4.

_{2}by the observer who is at a scale corresponding to the deformation parameter α

_{1}. Note that with this approach the diagonal element S

_{α}= ${S}_{\alpha}^{\alpha}$,of the described matrix is the density of entropy measured by the observer who is at the same scale as the measured object associated with the deformation parameter α. In [2] Section 6 such a construction was used implicitly in derivation of the semiclassical Bekenstein-Hawking formula for the Black Hole entropy:

- a)
- For the initial (approximately pure) state
- b)
- Using the exponential ansatz(9),we obtain:

- 1)
- For the observer in the large-scale limit (producing measurements in the semiclassical approximation) α
_{1}= 0S(in) = (Origin singularity)S(out) = (Singularity in Black Hole)So S(in) = S(out) = . Consequently, the initial and final densities of entropy are equal and there is no any information loss. - 2)
- For the observer moving together with the information flow in the general situation we have the chain:
- 3)
- 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:(Early Universe, origin singularity, QMFL, density pro-matrix)→ (Black Mini-Hole, singularity, QMFL, density pro-matrix),

## 3 Entropy Bounds, Entropy Density and Holographic Principle

_{p}. However, J.Bekenstein in [25] has credited such an approach as problematic, since then the objects with dimensions on the order of the Planck length ∼ 10

^{−33}cm should have very great entropy thus making problems in regard to the entropy bounds of the black hole remnants [26].

- An approach proposed in [34],[4] and in the present paper gives a deeper insight into the cause of high entropy for Planck’s black hole remnants, namely: high entropy density that by this approach at Planck scales takes place for every fixed observer including that on a customary scale, i.e. on α ≈ 0. In [4] using the exponential ansatz (Section 3) it has been demonstrated how this density can increase in the vicinity of the singularities withAs demonstrated in [34],[4], increase in the entropy density will be realized also for the observer moving together with the information flow: , though to a lesser extent than in the first case. Obviously, provided the existing solutions for (7) are different from the exponential ansatz, the entropy density for them will be increasing as compared to with a tendency of α
_{2}to 1/4. - In works of J.Bekenstein, [26] in particular, a ”universal entropy bound” has been used [27]:
- This necessitates mentioning of the recent findings of R.Bousso [28],[29], who has derived the Bekenstein’s ”universal entropy bound” for a weakly gravitating matter system, and among other things in flat space, from the covariant entropy bound [30] associated with the holographic principle of Hooft-Susskind [31],[32],[33].

## 4 Quantum corrections to black hole entropy. Heuristic approach

_{0},a

_{1},... or the definition of additional members in the exponent ”destroying” a

_{0},a

_{1},... [4]. It is easy to check that close to the singularity α = 1/4 the exponential ansatz gives a

_{0}= −3/2, being coincident with the logarithmic correction factor for the black hole entropy [39]. However, by the proposed approach - density matrix deformation at Planck’s scales - the quantum entropy receives a wider and more productive interpretation due to the notion of entropy density matrix introduced in [4],[5],[7],[34] and Section 3. Indeed, the value may be considered as a series of two variables α

_{1}and α

_{2}. Fixing one of them, e.g. α

_{1}, it is possible to expand the series in terms of α

_{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 α

_{1}= 0 and α

_{2}close to 1/4.

## 5 Conclusion

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Shalyt-Margolin, A.E.
Deformed Density Matrix and Quantum Entropy of the Black Hole. *Entropy* **2006**, *8*, 31-43.
https://doi.org/10.3390/e8010031

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Shalyt-Margolin AE.
Deformed Density Matrix and Quantum Entropy of the Black Hole. *Entropy*. 2006; 8(1):31-43.
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**Chicago/Turabian Style**

Shalyt-Margolin, A. E.
2006. "Deformed Density Matrix and Quantum Entropy of the Black Hole" *Entropy* 8, no. 1: 31-43.
https://doi.org/10.3390/e8010031