abstract

Several physical problems such as the “twin paradox” in curved spacetimes have a purely geometrical nature and are reduced to studying properties of bundles of timelike geodesics. The paper is a general introduction to systematic investigations of the geodesic structure of physically relevant spacetimes. These are focussed on the search of locally maximal timelike geodesics. The method is based on determining conjugate points on chosen geodesic curves. The method presented here is effective at least in the case of radial and circular geodesics in static spherically symmetric spacetimes. Our approach shows that even in Schwarzschild spacetime (as well as in other static spherically symmetric ones), one can find a new unexpected geometrical feature: each stable circular orbit contains besides the obvious set of conjugate points two other sequences of conjugate points. The obvious limitations of the approach arise from one’s inability to solve involved ordinary differential equations and the recent progress in the field allows one to increase the range of metrics and types of geodesic curves tractable by this method.

abstract

We study the trapping problem associated with a random walk process that takes place in deterministic multiplex networks. To this end, we consider the Average Trapping Time (ATT) and explore the properties of this system by adjusting the coupling strength \(\lambda \). We get the analytical expression of ATT with the help of the properties of block matrix, and apply it to two types of deterministic multiplex networks. We find that the ATT in our examples presents a minimum with the change of \(\lambda \) and that the emergence of the minimum under some special initial conditions has a potential relationship with the structural difference of the two graphs in the multiplex network. Our results provide a potential way to control the trapping time in multiplex networks.

abstract

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanying the procedure of the semiclassical quantization in polygon billiards. Unfolding rational polygon billiards (RPB) into corresponding Riemann surfaces (RS) periodic structures of the latter are demonstrated with \(2g\) independent periods on the respective multitori with \(g\) as their genuses. However, it is the two-dimensional real space of the real linear combinations of these periods which is used for quantizing RPBs. A class of doubly rational polygon billiards (DRPB) is distinguished for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is shown that semiclassical quantization of both the classical momenta and the energy spectra are completely determined by the periodic structure of the corresponding RSs. Each RS can be reduced to elementary polygon patterns (EPP) as its basic periodic elements. Each such EPP can be glued to a torus of genus \(g\). Semiclassical wave functions (SWF) are then constructed on EPPs. The SWFs for DRPBs appear to be exact and have forms of coherent sums of plane waves. They satisfy well defined boundary conditions — the Dirichlet, the Neumann or the mixed ones. Not every mixing of such conditions is allowed however. A respective incompleteness of SWFs provided by the method used in the paper is discussed. Dense families of DRPBs are used for approximate semiclassical quantization of RPBs. General rational polygons are quantized being approximated by DRPBs. An extension of the formalism to irrational polygons is described as well. The semiclassical approximations constructed in the paper are controlled by a general criteria of the eigenvalue theory. A relation between the superscar solutions and SWFs constructed in the paper is also discussed.

abstract

Discussed are some classical and quantization problems of test affinely-rigid bodies moving in Riemannian spaces. We investigate the systems with potential energies for which the variables can be separated. The special case of constant curvature two-dimensional spaces is discussed. Some explicit solutions are found using the Sommerfeld polynomial method.

abstract

Possible low-\(|t|\) structures in the differential cross section of \(pp\) elastic scattering at the LHC are predicted. It is argued that the change of the slope of the elastic cross section near \(t=-0.1\) GeV\(^2\) has the same origin as that observed in 1972 at the ISR, both related to the \(4m_{\pi }^2\) branch point in the \(|t|\)-channel of the scattering amplitude. Apart from that structure, tiny oscillations at small \(|t|\) may be present on the cone at low \(|t|\).

abstract

In this work, we have analyzed the dynamics of the model of a free particle over a 2-sphere in a noncommutative (NC) phase-space. Besides, we have shown that the solution of the equations of motion allows one to show the equivalence between the movement of the particle upon a 2-sphere and the one described by a central field. We have considered the effective force felt by the particle as being caused by the curvature of the space. We have analyzed the NC Poisson algebra of classical observables in order to obtain the NC corrections to Newton’s second law analogous to the one caused by a central field. We have also discussed the relation between affine connection and Dirac brackets, as they describe the proper evolution of the model over the surface of constraints in the Lagrangian and Hamiltonian formalisms, respectively. As an application, we have treated the so-called Zitterbewegung of the Dirac electron. Since it is assumed to be an observable effect, then we have traced its physical origin by assuming that the electron has an internal structure.