Relativistic scattering theory of charged spinless particles

In the context of a relativistic quantum mechanics we discuss the scattering of two and three charged spinless particles. The corresponding transition operators are shown to satisfy four-dimensional Lippmann-Schwinger and eight-dimensional Faddeev-type equations, respectively. A simplified model of two particles with Coulomb interaction can be solved exactly. We calculate: (i) The partial waveS-matrix from which we extract the bound state spectrum. The latter agrees with a fourth-order result of Schwinger, (ii) The full scattering amplitude which in the weakfield limit coincides with the expression derived by Fried et al. from eikonalized QED.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia, May 27–June 1, 1985.Based on part of the Ph D. thesis by M. Hannemann, Mainz, 1985.     

Relativistic quantum theory without quantized fields

Peculiarities of symmetrical quantum systems are considered with the aid of the Mackey's induced representations theory. The four-dimensional coordinate representation of the relativistic quantum mechanics suggested by Stueckelberg in 1941 is rederived, reinterpreted and generalized for an arbitrary spin. Then it is applied to introduce the causal propagator as a particleantiparticle transition amplitude without consideration of a field equation. Finally the theory of relativistic quantum particles interaction is reformulated without an appeal to the concept of quantized fields.     

Relativistic wave mechanics of spinless particles in a curved space-time

Starting from the Klein-Gordon equation, the single-particle approximation for a reiativistic scalar particle in the presence of external electromagnetic and gravitational fields is performed. The nonrelativistic limit is obtained by a Foldy-Wouthuysen transformation on a Schrödinger-type equation. The results are then compared with those obtained in classical mechanics.     

Relativistic particles on quantum space-time

We discuss alternatives to the usual quantization of a relativistic particle which result in discrete spectra for position and time operators.     

Relativistic corrections to the Lagrangian for interacting charged particles

We give the Lagrangian of a system of moving charged particles up to the fourth approximation in 1/c neglecting dipole radiation of the system. In this case the appearance of the electromagnetic waves (quadrupole radiation) by moving charges occurs in the fifth approximation in 1/c.     

Relativistic particle dynamics in external gravitational fields

Using the Riemann metric for event space, which leads to Newtonian mechanics at nonrelativistic velocities and not necessarily weak gravitational fields, the dynamics of relativistic particles in external gravitational fields are considered. Trajectories, laws of motion, and equations of light rays are found in homogeneous and Newtonian fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 56–61, October, 1977.In conclusion the authors thank Yu. G. Pavlenko for his evaluation of the results and valuable advice.     

Relativistic physics from quantum theory

An investigation of the invariance of quantum theory under the complex group reveals a natural origin of relativistic physics from quantum theory. Once such an origin of relativity is accepted, quantum limitations on the applicability of standard relativistic theory also become evident.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.     

Charged particles in external fields

The second-quantized Dirac and Klein-Gordon equations with external fields are solved. It is shown that the interpolating field is local and satisfies the Yang-Feldman equations. The Capri-Wightman approach and the Friedrichs-Segal approach are shown to lead to the same unitaryS-operator. The evolution operator andS-operator are studied. A divergence-free perturbation expansion of theS-operator is derived and the connection with the Feynman-Dyson series is established.     

Simultaneous geometrization of classical and quantum mechanics of spinless particles

It is shown that classical and quantum equations of motion of a relativistic spinless particle (the Lorentz and Klein-Gordon equations) allow for a geometrization on the same manifold ₄. A classical particle on ₄ is described as a free particle ( p=0), while the quantum particle, as a free wave ( s=0).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 70–74, September, 1990.     

Relativistic quantum particle in a homogeneous external field

The model of the relativistic quantum particle in a homogeneous external field is proposed. This model is realized in the one-dimensional relativistic configurational x-space and is described by the finite-difference equation. The momentum p-space in our case is the one-dimensional Lobachevsky space. We have found the wave functions and propagator for the model under study in both x- and p-representations.     

Spectral method in quantum theory of channeling phenomena of fast charged particles in crystals

We show the possibility of using the spectral method to determine the quantum characteristics of interaction between fast charged particles and crystal when the channeling occurs. Basing on this method, we have developed the procedure for calculation of particles transversal motions eigenlevels for plane channeling. We considered origin of new levels with increasing of particle's energy and zone structure forming from these levels with increasing number of potential wells. We discuss also the possibility of using the spectral method in number of other problems of quantum theory of channeling.     

The quantum theory of the radiation from charged polarized fermions moving in crossed fields

The radiation of a charged Dirac particle, moving at relativistic velocity in orthogonal electric and magnetic fields (E H), is examined here. The formulas obtained are applicable for arbitrary orientation of the particle spin.     

Nonlinear quantum theory of interaction of charged particles and monochromatic radiation in a medium

We study the quantum theory of nonlinear interaction of charged particles and a given field of plane-transverse electromagnetic radiation in a medium. Using the exact solution of the generalized Lamé equation, we find the nonlinear solution of the Mathieu equation to which the relativistic quantum equation of particle motion in the field of a monochromatic wave in the medium reduces if one ignores the spin-spin interaction (the Klein-Gordon equation).We study the stability of solutions of the generalized Lamé equation and find a class of bounded solutions corresponding to the wave function of the particle. On the basis of this solution we establish that the particle states in a stimulated Cherenkov process form bands. Depending on the wave intensity and polarization, such a band structure describes both bound particle-wave states (capture) and states in the continuous spectrum. It is obvious that in a plasma there can be no such bands, since bound states of a particle with a transverse wave whose phase velocity v _ph is higher than c are impossible in this case. The method developed in the paper can be applied to a broad class of problems reducible to the solution of the Mathieu equation. Zh. éksp. Teor. Fiz. 113, 43–57 (January 1998)     

Bose condensation in supercritical external fields charged condensates

This paper extends a previous one which was applicable only to short range interactions. We study the relativistic field theory of a charged spin-zero boson field in the presence of the Coulomb field of a prescribed (nuclear) charge distribution. It is shown that for a sufficiently intense field the ground state is unstable against the formation of a Bose-Einstein condensate of negatively charged bosons, positively charged bosons escaping the system. When the effects of weak interaction are included, the instability occurs in a weaker field and positrons are emitted. A consistent quantum theory is formulated after the Coulomb interaction of the bosons is included. Properties of the condensate are examined in the limit of large condensate density, in a mean field approximation, which is also studied numerically. Possible implications concerning the existence of abnormally bound nuclei are presented.     

Quantum theory of particles with spin zero and one half in external fields

The unitary (pseudo unitary) time-evolution operator for a particle with spin half (zero) in an external time-dependent electromagnetic (scalar) field is used to generate a Bogoliubov automorphism on the algebra of the free in field. For the case of an electric external field (scalar field) a finite expression for _out is given and theS-matrix constructed. The latter is unitary and implements the Bogoliubov automorphism. Theorems by Shale and Stinespring are rederived.Supported in part by the U.S. Atomic Energy Commission under Contract No. AT-30-1-3829.