Quantum Mechanics at Planck's Scale and Density Matrix

Abstract

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. 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. Within our approach two main features of Quantum Mechanics are conserved: the probabilistic interpretation of the theory and the well-known measuring procedure corresponding to that interpretation. The proposed 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 definition of statistical entropy and the problem of information loss in black holes are considered. It is shown that obtained results allow to deduce in a simple * Phone (+375) 172 883438; e-mail: a.shalyt@mail.ru; alexm@hep.by † Phone (+375) 172 883438; e-mail: suarez@hep.by, jsuarez@mail.tut.by 1 and natural way the Bekenstein-Hawking's formula for black hole entropy in semiclassical approximation.

The Universe as a Nonuniform Lattice in Finite-Volume Hypercube I: Fundamental Definitions and Particular Features

“…Now it is obvious that in the latter the notion of the fundamental (minimum) length l min ∼ l p is a requisite [4], where l p is the Plancks length. It has been demonstrated [9]- [13] that on retention of a well-known measuring procedure the density matrix becomes dependent on the additional parameter α = l 2 min /x 2 , where x is the measuring scale. In this way the density matrix is subjected to the deformation procedure (e.g.…”

“…It is no use to enumerate all the evident implications and applications of Definition 1., better refer to [12], [13]. Nevertheless, it is clear that for α → 0 the above limit covers both the Classical or Quantum Mechanics depending on → 0 or not.…”

“…T max [13]- [15]that is associated with the same time point t * min = 2t min . For QMFL this has been noted in the previous section.…”

“…The first approach suffers from two serious disadvantages: 1) the deformation parameter is a dimensional variable κ with a dimension of mass [6]; 2) in the limiting transition to QM this parameter goes to infinity and fluctuations of other values are hardly sensitive to it. The second approach is devoid of such limitations as in this approach the deformation parameter is represented by the dimensionless quantity α = l 2 min /x 2 , where x is the measuring scale and the variation interval α is finite 0 < α ≤ 1/4 [9]- [13]. Moreover, it gives a key to the solution of particular problems: an extra term in Liouville equation in the processes associated with black holes [11]- [13]; singularities and cosmic censorship [12], [13];derivation of a semiclassical Bekenstein-Hawking formula for the black hole entropy and some others [13], [16].…”

“…The second approach is devoid of such limitations as in this approach the deformation parameter is represented by the dimensionless quantity α = l 2 min /x 2 , where x is the measuring scale and the variation interval α is finite 0 < α ≤ 1/4 [9]- [13]. Moreover, it gives a key to the solution of particular problems: an extra term in Liouville equation in the processes associated with black holes [11]- [13]; singularities and cosmic censorship [12], [13];derivation of a semiclassical Bekenstein-Hawking formula for the black hole entropy and some others [13], [16]. In [16]- [18]it has been shown that within the developing paradigm there is a possibility for the solution of Hawkings information Paradox problem [19]- [21]and α may be interpreted as a new small parameter of quantum theory.…”