Density Matrix Deformations in Quantum and Statistical Mechanics at Planck-Scale

In this work the Quantum and Statistical Mechanics of the Early Universe, i.e. at Planck scale, is considered as a deformation of the well-known theories. In so doing the primary object under deformation in both cases is the density matrix. It is demonstrated that in construction of the deformed quantum mechanical and statistical density matrices referred to as density pro-matrices there is a complete analog. The principal difference lies in a nature of the deformation parameter that is associated with the fundamental length in the first case and with a maximum temperature in the second case. Consideration is also given to some direct consequences, specifically the use of the explicitly specified exponential ansatz in the derivation from the basic principles of a semiclassical Bekenstein-Hawking formula for the black hole entropy and of the high-temperature complement to the canonical Gibbs distribution