Deformed Density Matrix and Quantum Entropy of the Black Hole
Abstract
:1 Introduction. Deformed Density Matrix in QMFL
- 0 < α ≤ 1/4.
- The vectors |i > form a full orthonormal system.
- ωi(α) ≥ 0, and for all i the finite limit
exists.
- Sp[ρ(α)] = ∑i ωi(α) < 1, ∑i ωi = 1.
- For every operator B and any α there is a mean operator B depending on α:
- The above limit covers both Quantum and Classical Mechanics. Indeed, since
Għ/c3x2, 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 = ilmin and i ≥ 2, then there is no any problem. At the point of x = lmin 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 = lmin.
- We consider possible solutions for (7). For instance, one of the solutions of (7), at least to the first order in α, is
, where ppl 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
- 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) α1 = 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:
= S. Here S is the ordinary entropy at quantum mechanics(QM), but S(in) = S(out) =
,value considered in QMFL. So in this case the initial and final densities of entropy are equal without any loss of information.
- 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),
= S disappears at the corresponding chain of the entropy density associated with the large-scale consideration:
the density of entropy is preserved. Actually this event was mentioned in [2],where from the basic principles it has been found that black mini-holes do not radiate, just in agreement with the results of other authors [21,22,23,24]. As a result, it’s possible to write briefly
3 Entropy Bounds, Entropy Density and Holographic Principle
- 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 α2 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
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
Shalyt-Margolin AE. Deformed Density Matrix and Quantum Entropy of the Black Hole. Entropy. 2006; 8(1):31-43. https://doi.org/10.3390/e8010031
Chicago/Turabian StyleShalyt-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
APA StyleShalyt-Margolin, A. E. (2006). Deformed Density Matrix and Quantum Entropy of the Black Hole. Entropy, 8(1), 31-43. https://doi.org/10.3390/e8010031