Gnatenko K. Effect of space quantization on the properties of classical and quantum systems

The work is devoted to studies of influence of features of space structure on the Planck scale on the properties of quantum and classical systems. Significant increasing of interest to such studies in the last years is caused by the development of the String Theory and Quantum Gravity. In the work, the theory of quantum space constructed on the basis of the idea that the ordinary commutation relations for operators of coordinates and momenta can be deformed was studied. Noncommutative algebras of canonical type, Lie type, and nonlinear deformed algebras were considered. Properties of the kinetic energy of a macroscopic body, independence of coordinates of mass, the weak equivalence principle were recovered in a space which is characterized by nonrelativistic Snyder algebra and in the space with deformed Kempf algebra. We obtained that in the case when parameters of coordinate noncommutativity are proportional inversely to mass and parameters of momentum noncommutativity are proportional to mass the noncommutative coordinates do not depend on mass and can be considered as kinematic variables, the properties of the kinetic energy are preserved, the problem of description of motion of a composite system is solved (the total momentum can be introduced as integrals of motion, the motion of the center-of-mass is independent of the relative motion), the weak equivalence principle is recovered, the trajectory of motion of free particle does not depend on mass. The motion of the Sun-Earth-Moon system was studied in noncommutative phase space of canonical type and the corrections to the Eӧtvӧs-parameter for Earth and Moon caused by noncommutativity were analyzed. On the basis of studies of perihelion shift of the Mercury planet with taking into account features of description of motion of a macroscopic body in noncommutative phase space the upper bound for the parameter of momentum of noncommutativity which at least 10 orders less than results presented in literature was obtained. The problem of violation of the time reversal and rotational symmetries was studied in noncommutative phase space of canonical type. We found that because of noninvariance of noncommutative algebra upon time reversal the transformation of coordinates and momenta upon time reversal depend on their representation, period of the circular motion depends on its direction. Noncommutative algebra which is rotationally-invariant, time reversal invariant and equivalent to the noncommutative algebra of canonical type was constructed. It was found that idea to relate parameters of noncommutativity with mass is also important for solving the problem of description of motion of many-particle system and problem of violation of the equivalence principle in rotationally-invariant noncommutative phase space. The spectrum of system of interacting harmonic oscillators (symmetric network of harmonic oscillators, harmonic oscillator chain) was obtained in the noncommutative phase space with rotational symmetry. It was shown that noncommutativity of coordinates and noncommutativity of momenta affect on the frequencies of the systems. It was found that noncommutativity of coordinates better appears in spectrum of atoms with large reduced mass, the influence of momentum noncommutativity is bigger in the case of atoms with small reduced mass. The influence of noncommutativity on the spectrum of the hydrogen atom and exotic atoms (muonic hydrogen, antiprotonic helium) was found and analyzed. On the basis of comparison of obtained results with experimental ones the upper bounds for the parameter of coordinate noncommutativity and parameter of momentum noncommutativity were found. It was concluded that studies of antiprotonic helium open good possibilities for estimation of the minimal length. The problem of description of motion of macroscopic body and the problem of violation of the equivalence principle in a space with noncommutative algebra of Lie type was solved due to relation of parameters of noncommutativity with mass. Therefore we concluded that the idea of relation of parameters with mass opens possibility to build theory of quantum space with preserved fundamental laws and principles. Importance of this idea is justified by the number of deformed algebras and number of results that can be obtained due to its consideration. It was found that zeros of the correlation functions of Bose gas in a space with minimal length and minimal momentum (q-deformed Bose gas) are related with Fisher zeros of partition function. Complex temperature is caused by q-deformation and evolution of correlation function. Similar relation of zeros of partition function with zeros of correlation functions was found for the interacting Bose gas and for spin systems in the ordinary space. The obtained results open new possibilities for experimental observation of zeros of partition function, which have fundamental importance in statistical physics.