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Generalized uncertainty relations, fundamental length, density matrix and new view on the black holes information paradox

Alexander Shalyt-Margolin, A. Ya. Tregubovich

Published Jul 18, 2002• 3 citations
OpenAlex

Abstract

It was shown that if in Quantum Theory a fundamental length exists, then in this case there is not pure states. It was noted, that theory satisfying General Uncertainty Relations (GUR) are related to the mentioned above theories. Moreover, the density matrix at the Planck scale can not be defined in the usual way, because in this case the density matrix trace is strongly less than 1. Then density matrix must be changed by its progenitrix or as we call it density pro-matrix, which at low energy limit turns to the density matrix. This transition corresponds to non-unitary one from Quantum Mechanics with GUR to Quantum mechanics. Below the implications of obtained results are enumerated. A new view on the Black Holes Information Paradox are discussed.

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It was shown that if in Quantum Theory a fundamental length exists, then in this case there is not pure states. It was noted, that theory satisfying General Uncertainty Relations (GUR) are related to the mentioned above theories. Moreover, the density matrix at the Planck scale can not be defined in the usual way, because in this case the density matrix trace is strongly less than 1. Then density matrix must be changed by its progenitrix or as we call it density pro-matrix, which at low energy limit turns to the density matrix. This transition corresponds to non-unitary one from Quantum Mechanics with GUR to Quantum mechanics. Below the implications of obtained results are enumerated. A new view on the Black Holes Information Paradox are discussed.
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 (GUR). 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, which at low energy limit turns to the density matrix. This transition corresponds to non-unitary one from Quantum Mechanics with GUR to Quantum mechanics. Below the implications of obtained results are enumerated.New view on the Black Holes Information Paradox are discussed
It was shown that if in Quantum Theory a fundamental length exists and a well-known measurement procedure is used, then the density matrix at the Planck scale cannot be defined in the usual way, because in this case density matrix trace is strongly less than one. Density matrix must be changed by a progenitrix or as we call it throughout this paper, density pro-matrix. This pro-matrix is a deformed density matrix, which at low energy limit turns to usual one. Below the explicit form of the deformation is described. Implications of obtained results are summarized as well as their application to the interpretation of Information Paradox on the Black Holes.
information paradox problem 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.
In this paper Quantum Mechanics with Fundamental Length is chosen as the theory for describing the early Universe. This is possible due to the presence in the theory of General Uncertainty Relations from which unavoidable it follows that in nature a fundamental length exits. Here Quantum Mechanics with Fundamental Length is obtained as a deformation of Quantum Mechanics. The distinguishing feature of the proposed in this paper approach in comparison with previous ones, lies on the fact that here density matrix subjects to deformation as well as so far commutators had been deformed. The deformed density matrix mentioned above, 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 exact predefined measuring procedure corresponding to that interpretation. The proposed here 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 statistical entropy definition and the problem of information loss in black holes are considered.
Thirty years ago, John Preskill concluded "that the information loss paradox may well presage a revolution in fundamental physics" and mused that "Conceivably, the puzzle of black hole evaporation portends a scientific revolution as sweeping as that that led to the formulation of quantum theory in the early 20th century." Many still agree with this assessment. On the other hand, it seems to me the "paradox" has little to do with the physical world but rather, at best, simply points out the possible inconsistency of two, already disparate, theories (mathematical models) of nature, general relativity and quantum mechanics, with virtually no conceivable observational consequences. The information paradox hinges on the concepts of a pure quantum state, the unitarity of quantum mechanics, and Hawking's semi-classical calculation of black hole evaporation. I used the qualifier "at best" above because, for me, the concept of a quantum state is far more restrictive than required by the paradox while unitarity is not a property of nature but rather of a mathematical model and is already violated by the process of making a measurement. Furthermore, the semi-classical calculation of Hawking is surely of limited applicability.
It is noted that in the quantum theory at Planck’s scales (Early Universe), with introduction of the fundamental length, the measuring procedure is changed, leading to the density-matrix deformation. This deformation is described explicitly, and its inferences may be used for the determination of pure and mixed states in the associated quantum mechanics, for Hawking’s Problem solution in case of the coherence loss on black hole evaporation, and in calculations of the black hole quantum entropy.