A system of equations is found for the torsion tensor. It is proved that the equality of the torsion covector to zero is equivalent to the Born–Infeld electrodynamic equations.

We study the temporal behaviour of small perturbations of static kink-solutions of the sine-Gordon equation on a finite line. Two classes of solutions are found. The first contains all oscillatory modes of the \(N\) kinks present, while the second class describes the scattering of phonons by these kinks. For each solution the oscillation frequency is calculated and. plotted as a function of the length of the line.

In the covariant single-time approach to the quantum field theory the expressions of the one-photon exchange quasipotentials (the kernels of three-dimensional integral equations for the relativistic wave functions) of two-body systems are obtained. The systems of particles with spins (1/2, 1/2), (1/2, 0) and (0, 0) are considered. In the calculations the double-time Green’s functions are used. It is shown that the obtained quasipotentials coincide with the corresponding Feynman amplitudes on the energy-shell.

We present a solution for the Schwinger model: quantum electrodynamics in two dimensions for massless fermions, in the canonical Hamiltonian formalism. The equivalence of the Dirac quantization procedure for constrained systems and the Schwinger’s action principle is established.

We discuss the possibilities of extraction of the “square root” of the Dirac equation within \(N\)-extended supersymmetry for construction of the more fundamental dynamical theory. The “square root” of the Dirac operator can be defined in \(N\)-extended superspace for \(N \lt 2\), but it is impossible, in the framework of the standard demands to the field theory, to build a new dynamical model with it.

The matrix representations of Witten’s and \(B\)-algebras of the field string theory in finite dimensional space of the ghost states are suggested for the case of Virasoro algebra truncated to its SU(1,1) subalgebra. In this case all algebraic operations of Witten’s and \(B\)-algebras are realized in explicit form as some matrix operations in the graded complex vector space. The structure of string action coincides with the universal non-linear cubic matrix form of action for the gauge field theories. These representations lead to matrix conditions of theory invariance which can be used for finding of the explicit form of corresponding operators of the string algebras.

The possibility of the existence of magnetically bound positronium states (superpositronium) is investigated starting from the two body Bethe–Salpeter equation. For the total angular momentum \(j = 0\) quasi bound energy states are found in form of resonances of the cross-section calculated with the use of the phase of the scattering states. The energies are in agreement with the observed kinetic energy of the disintegration electrons or positrons in the heavy ion collision experiments. From the width of the resonances the life-time of the quasi-bound states can be estimated. Really bound superpositronium states could not be found.

The differential and total cross sections of process \(e^+e^- \to W^+_i~W_k\) are calculated in analytical form for the SU(2)\(_{\rm L}\) \(\times \) SU(2)\(_{\rm R}\) \(\times \) U(1) model (\(i\), \(k = 1,2\)).

A deterministic cascade model similar in many ways to the random cascade \(\alpha \)-model is described. This model suggests the relation \(a+b = 2\) for the parameters of the (\(l = 2\)) \(\alpha \)-model. Some predictions for experiment following from this relation are presented.