2004, Vol.7, No.2, pp.129-139

In this paper an approach is developed which allows to propose Quantum Mechanics with Fundamental Length as the theory to describe quantum-mechanical behavior of nature at Planck's scale (that is at the Early Universe). Such approach seems to be logically supported, since in all known methods for researching quantum-gravitational effects, somehow or other, a fundamental length of the Planck-scale order appears. Quantum Mechanics with Fundamental Length is obtained by deformation of Quantum Mechanics. The novelty of the presented approach is in carrying out a deformation of density matrix, instead of deforming commutators (in other words, a deformation of Heisenberg's algebra) as it was done up to now. In our approach two fundamental features of Quantum Mechanics are conserved. Namely, the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation (at the moment in first approximation). Some dynamical aspects of the theory are discussed. An explicit form of deformed Liouville equation is given. Some implications of obtained results are analyzed. In particular, the problem of singularity and the hypothesis of cosmic censorship are tried. The Density-entropy concept is introduced, improving the definition of statistical entropy. Density of entropy is used to deal with the problem of information loss in black holes. Bekenstein-Hawking's formula for black hole entropy is deduced from the first principles.
Key words: fundamental length, general uncertainty relations, density pro-matrix, deformed Liouville's equation, density of entropy, Bekenstein-Hawking formula

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