A covariant polarization operator for s=1 particles

An operator corresponding to the 4-vector operator byMichel andWightman is given for the Duffin Kemmer Petiau equation.     

Interaction quasipotential for spinor and scalar particles

The two-time Green functions and corresponding quasipotentials for the system of two relativistic particles with spins 0 and 1/2, interacting through exchange of a massless vector boson and a massive scalar boson, are calculated. The calculations are performed using the covariant single-time method of Logunov and Tavkhelidze in the second order of perturbation theory. The dependence of these quantities on the total energy of the system is given. It is shown that, despite a nonlocal form of the quasipotentials, the three-dimensional equations for the wavefunctions can be reduced to the one-dimensional equations using the partial wave decomposition.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 57–61, February, 1989.In conclusion, the authors express their gratitude to A. A. Afonin, E. A. Dei, V. I. Savrin, and N. B. Skachkov for helpful discussions.     

A covariant kinetic equation for a self-gravitating system

Recently, Israel and Kandrup have developed a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. One by-product of that approach has been the formulation of an approximate kinetic equation for the evolution of a self-gravitating system, valid in an “impulse” or “weak coupling” approximation in the limit that radiative effects may be neglected. This paper exploits the theory of random functions to present a much simpler derivation of that approximate kinetic equation.     

A duality theorem for the automorphism group of a covariant system

In this paper we examine the covariant representation theory of a covariant system (A, G) introduced by Doplicher, Kastler and Robinson. (A is aC*-algebra andG is a locally compact group of automorphisms ofA.) We define the concept of left tensor product of two covariant representations. Loosely stated, our duality theorem says thatG is canonically isomorphic to the set of bounded operator valued maps on the set of covariant representations of the covariant system (A, G) which preserve direct sums, unitary equivalence and left tensor products. We further show that the enveloping von Neumann algebraA(A, G) of the covariant system (A, G) admits a (not necessarily injective) comultiplicationd such that (A(A, G),d) is a Hopf von Neumann algebra. The intrinsic group of this Hopf von Neumann algebra is canonically isomorphic and (strong operator topology) homeomorphic toG.     

Gauge covariant formulation of the generating operator. 1. The Zakharov-Shabat system

A gauge covariant formulation of the generating operator Λ, related to the Zakharov-Shabat system L is proposed. The operator Λ̃, corresponding to $\text{L}\text{?}\text{= ψ}{}_0{}^{\mathrm{?1}}\text{Lψ}{}_0$ in the pole gauge is explicitly calculated. Thus the unified approach to the higher nonlinear Schrödinger equations, based on Λ is automatically reformulated with Λ̃ for the higher Heisenberg ferromagnet equations.     

Covariant interaction operators for a composite system ofN scalar particles

The explicit form of relativistic, energy-dependent interaction operators for a composite system of N scalar particles in the second order of perturbation theory is obtained within the framework of the simultaneous covariant approach of quantum field theory on the basis of the retarding part of two-time Green's functions. Explicitly covariant equations for the wave function of the bound state and the conditions for its normalization are obtained. Frantsiska Skorina State University, Gomel'. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 17–24, August, 1997.     

Solving a relativistic quasipotential equation for a superposition of a nonlocal separable and a local quasipotential

Within the relativistic quasipotential approach to quantum field theory, a method is developed for solving a finite-difference quasipotential equation for the case where a total quasipotential describing the interaction of two relativistic spinless particles of unequal masses is a superposition of a nonlocal separable and a local quasipotential. The cases are investigated where the local component of the total interaction—it is assumed to be known—either admits or does not admit the existence of bound states. This makes it possible to obtain an exact expression for the increment of the phase shift, to determine the conditions of the existence of bound states, and to give a generalization of the Levinson theorem.     

A new approach to obtaining positive-definite Wigner operator for two entangled particles with different masses

Based on our previously proposed Wigner operator in entangled form,we introduce the generalized Wigner operator for two entangled particles with different masses,which is expected to be positive-definite.This approach is able to convert the generalized Wigner operator into a pure state so that the positivity can be ensured.The technique of integration within an ordered product of operators is used in the discussion.     

Approximate diagonalization of the energy operator of a system of interacting particles

The present work considers a method of diagonalization of the energy operator of a system of valence electrons in a simple model of a semiconductor, and in a simple model of a metal. The results obtained are used in the theory of superconductivity and in the theory of superfluidity.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 43–47, December, 1971.     

The multiloop covariant tadpole operator and amplitudes for the bosonic string

An explicit expression is given for the covariant tadpole operator with an arbitrary number of loops, including the ghost zero mode contribution. Its integrand is recast in geometrical terms and is recognized to give the g-vacuum for the bosonic string recently discussed in the literature. By joining the g-tadpole to an N-reggeon vertex the general orientable dual amplitude with g loops is obtained.     

Gauge covariant theory of the generating operator. I

A gauge covariant formulation of the generating operator (-operator) theory for the Zakharov-Shabat system is proposed. The operator , corresponding to the gauge equivalent system in the pole gauge is explicitly calculated. Thus the unified approach to the nonlinear Schrödinger-type equations based on is automatically reformulated with the help of for the Heisenberg ferromagnet-type equations. Consequently, it is established that the conserved densities for the Heisenberg-ferromagnet-type equations are polynomial inS(x) and itsx-derivatives. Special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge-equivalent equations. It is shown that the geometrical properties of the conjugated operator * are gaugeindependent.     

The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.     

Approximately relativistic center-of-mass variables for a system of two interacting particles

A canonical transformation is found that enables one in the first quasirelativistic approximation (which takes into account orders c^–2) to separate the internal motion in a classical system of two particles from its motion as a whole in the case of arbitrary interaction between the particles.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 49–53, October, 1980.     

Asymptotic behavior of the quasipotential wave function of a two-particle system

The asymptotic behavior of relativistic wave functions of two-particle systems, consisting of spin 0 and 1/2 particles, is found with the help of integral equations derived within the framework of a covariant single-time approach in quantum field theory.Gomel'skii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 102–105, June, 1993.     

Tensor representations of conformal algebra and conformally covariant operator product expansion

A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O(4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O(4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n-point functions from causality are described.     

Collision operator for relativistic electrons in a cold gas of atomic particles

The collision operator of relativistic electrons with a cold gas of atomic particles is derived consistently taking into account elastic interactions, excitation of electron shells, and ionization. The creation of secondary electrons is described accurately. In the range of energies exceeding the binding energy of atomic electrons, the operator implicates only the angular scattering by nuclei and the ionization integral that automatically allows for scattering by atomic electrons. The collision operator used earlier for studying the kinetics of avalanches of relativistic runaway electrons is analyzed. A more exact operator derived in the present study is simpler in form and saves time in computer calculations.