Theory of relativistic systems

Main goal of research in this field consists in the construction of the theory of relativistic systems which implies a development and application of theoretical methods of the description of relativistic particles at those conditions and in those regimes where relativistic effects are essential.

Physical systems which fall under consideration are both macroscopic as well as microscopic objects with intrinsic structure, in particular, systems of gravitating bodies, atoms and their nuclei, hadrons (within quark models) etc.

The scientist who inspired and arranged the development of this research field in the Ukraine was Roman Gaida (1928-1998 рр.), the doctore of physical and mathematical sciences, professor, a full member of the Shevchenko Scientific Society in Lviv, Head of the Physical Committee of the Shevchenko Sci. Soc., a member of the American Mathematical Society. In the seventies he joined scientists who worked at different institutions but engaged in related scientific themes in the field of the relativistic direct interaction theory which is an unconventional approach to the description of relativistic particle systems without use of a field as a mediator of interaction (Yu. Kluchkovsky, V. Tretyak, A. Duviryak, Yu. Yaremko, V. Shpytko).

Valuable results in the development of the classical and quantum relativistic dynamics have been derived in the Institute. Apart a principal substantiation of the possibility of consistent Poincaré-invariant description of a wide class of realistic direct interactions, they include a number of methods for the description of relativistic effects in the systems of interacting particles. A formulation of geometric concept of the forms of relativistic dynamics has been performed; on this ground a consistent scheme of the Lagrangian description of particle systems with the use of higher derivatives of covariant (physical) coordinates has been developed, a relation of this description with nonlocal Lagrangians typical for the Fokker type formalism has been studied; a transition to the Hamiltonian description and various realizations of a quantization is realized. Relativistic rules for a composition of interactions within the Lagrangian and Hamiltonian descriptions have been derived, the theory of invertible nonpoint contact transformations with application to the Wheeler-Feynman electrodynamics has been constructed. By means of the introduction of relativistic center-of-mass variables a class of integrable two-particle models has been considerably expanded. New exactly integrable models of relativistic two- and N-particle systems have been invented within the frontal and isotropic forms of dynamics and within the Fokker’s formalism. Thorough study of weekly-relativistic approximations for various phenomenological and field interactions in the classical, quantum and statistical approaches was carried out.

This completed approach to the description of relativistic particle systems gave one a possibility to avoid problems which are typical for field-theoretical descriptions of interactions (non-renormalizability, the problem of bound states etc.) and to obtain important physical results. Among them, those are noteworthy: deriving higher-order quasi-relativistic corrections for the particle systems with electromagnetic and gravitational interactions, as well as exact expressions for interactions of arbitrary tensor dimension; a calculation of spectra of those two-particle systems and stating their consistency with the results of the quantum electrodynamics, the chromodynamics and the general relativity; the construction, on the base of the effective field theory with higher derivatives, the Poincaré-invariant potential two-quark model which, contrary to other known approaches, reproduces well characteristic features of spectra of both light and heavy mesons; the calculation of quasi-relativistic corrections to thermodynamical functions of weekly-imperfect gas and an exactly solvable relativistic generalization of the model of rigid rods, as well as deriving important relations for statistical mechanics of pointlike charges.

Recently, research themes in the field of relativistic dynamics got considerably enlarged and now include quantum field-theoretical models, the models of classical massless fields in spacetime continua of different dimensions, and the relativistic hydrodynamics.

It was proposed new approach for deriving relativistic wave equations for a system of particles (bosons, fermions) with account of relativistic retardation effects. There were taken for its base appropriate results derived earlier for the Fokker type integrals within the direct interaction theory. The method has been applied to a wide class of interactions and approved perturbatively and numerically for physically interesting examples. Preliminary results for field systems have been obtained in the example of generalized Yukawa model. For the case of vector (electromagnetic) and scalar interactions the derived equations are in agreement with ones known in literature.

A necessity of solving quantum-mechanical problem with a strong coupling has led to the elaboration of non-perturbative and pseudo-perturbative methods, such as expansions in 1/N (dimension inversed) or 1/L (angular momentum inversed). The application of these methods to relativistic problems was limited due to insufficient maturity of the methods as well as due to a complexity of problems. It is proposed in the Institute a scheme of non-perturbative analysis of two-particle Dirac equations, which arise in effective field theories, by the use of radial and algebraic reduction. In this way a set of exactly solvable examples, which are integrable extensions of known in the literature Dirac oscillators, has been found. On the ground of two-particle Dirac equation it was constructed the potential model of light mesons which reproduces well an experimentally observable behavior of Regge trajectories.