Spinor fields in the theory of relativity and a generalization of Heisenberg's nonlinear spinor equation

Some years ago it was shown that the nonlinear term of Heisenberg's spinor equation can be derived by torsion of the Minkowski space (Cartan space). This result is applied in the investigations of this paper. As the Heisenberg equation does not show any connection with recent phenomenological theories in high energy physics, like the parton or quark model, the problems of the metric of space-time are discussed from the aspect of fundamental axioms of topology (Hausdorff space). It will be shown that Feynman's relativistic parton theory can be derived by means of a quantised de Sitter space, where the constant curvature can assume only discrete values. It is also possible to derive the Dirac equation from the same mathematical considerations. A nonlinear spinor equation will be formulated which contains the parton theory and the nonlinear term of the Heisenberg equation as different approaches in the theory of elementary particles.     

Mass operator for wave scattering from a slightly random surface

This paper deals with the mass operator representing multiple-scattering effects in the theory of wave scattering from a slightly random surface. By means of the stochastic-functional approach, a recurrence equation for the mass operator is obtained in the form of an iterative integral. However, its solution oscillates in a non-physical manner against the number of iterations. Next, the recurrence equation may be regarded as a nonlinear integral equation, when the number of iterations goes to infinity. An analytical solution of the nonlinear integral equation is presented for a special case in which the roughness spectrum is the Dirac delta function. Then, the nonlinear integral equation is solved numerically for the Gaussian roughness spectrum by iteration, starting from such an analytical solution. It is shown that only a few iterations are required to obtain the mass operator, even when the correlation distance is small. Effects of the mass operators on the coherent reflection coefficient and the incoherent scattering cross section are calculated and shown in figures.     

Resonance conditions for the spectral function and single-hole states in finite nuclei

An equation for the spectroscopic amplitudes in finite nuclei is determined both in the discrete and in the continuous spectrum of the residual nucleus hamiltonian. Its structure, involving the hole mass operator, is investigated. In the continuous spectrum, the spectroscopic amplitude equation leads to a precise definition of the hole decay width (related to the average value of the antihermitian part of the hole mass operator). Besides, one deduces a formula which describes the resonant behaviour of the spectral function in terms of the hole decay widths. Finally, the resonance conditions are investigated. Close to sharp resonances (included the ones of the discrete spectrum), one interpretes the spectroscopic amplitudes as single-hole wave functions which satisfy a Schrödinger-like equation where the hermitian part of the hole mass operator plays the role of an effective potential.     

Elastic scattering of acoustic phonons on point mass defects

The relaxation of homogeneous states of long-wave acoustic phonon gas scattered by point mass defects in transversely—isotropic media is studied. The spectrum of the suitable collision operator of the Boltzmann-Peierls equation is investigated. It consists of a continuous part and several discrete eigenvalues. Both continuous and discrete part of the spectrum depend on the values of components of the elastic constant tensor. For some values of elastic constants the continuous part splits up into two separate intervals and some of the discrete eigenvalues appear in the gap. The number of discrete eigenvalues and their arrangement are also affected by elastic properties of medium.     

Quantum-like Interpretation of Dirac Wave Equation in?Robertson-Walker Geometry

The Dirac wave equation is separated in the Robertson-Walker metric. The resulting radial equation is interpreted as a one dimensional quantum-like equation that is explicitly solved. There results that the energy spectrum, that is determined in the flat, open and closed universe, is independent of the mass of the particle. Moreover it is the same of the massless neutrino case previously studied. In the closed metric case the discrete positive spectrum is asymptotically determined. The separation of the energy levels is however very far from being experimentally tested.     

Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons

We develop a mean-field theory of the dynamics of a nonequilibrium Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity. The spectrum of elementary excitations around the stationary state is analytically studied by means of a generalized Gross-Pitaevskii equation. A diffusive behavior of the Goldstone mode is found in the spatially homogeneous case and new features are predicted for the Josephson effect in a two-well geometry.     

Conformal invariance and the Higgs boson mass in the Weinberg-Salam theory with gravitational mechanism of instability

The assumption that the Higgs scalar field equation is conformally invariant leads to new features of the unified gauge theories including classical gravitation. Both the self-consistent approach and the external curved space-time method are discussed here. The purpose is to compute the upper and lower bounds on the mass of the stable Higgs particle. Also an attempt to obtain a discrete mass spectrum at classical level was made.     

Self-consistent many-body theory for the standard basis operator Green's functions

We propose a self-consistent many-body theory for the standard basis operator Green's functions and obtain an exact Dyson-type matrix equation for the interacting many-level systems. A zeroth order approximation, which neglects all the damping effects, is investigated in detail for the anisotropic Heisenberg model, the isotropic quadrupolar system and the Hubbard model. In the case of the anisotropic Heisenberg ferromagnet with both exchange and single-ion anisotropy the low-temperature renormalization of the spin-waves for the uniaxial ordering agrees with the Bloch-Dyson theory. For the spin-1 easy-plane ferromagnet, the critical parameters for the phase transition at zero temperature are determined and compared with other theories. The elementary excitation spectrum of the spin-1 isotropic quadrupolar system is calculated and compared with the random phase approximation and Callen-like decoupling schemes. Finally, the theory is applied to the study of the single-particle excitation spectrum of the Hubbard model.     

Vectons in a Coulomb field

The problem is solved within the framework of an approach developed by the author. The author believes this approach to be free of deficiencies characteristic of theories for particles with unit and larger spins. We derive integrals of motion, separate variables, derive radial-state equations, and then solve them. We find coupled particle states; these are vector atoms that may be observed in nature. They possess a discrete energy spectrum similar to that of electrons in hydrogen atoms as seen in Dirac theory. We also note cases of particle repulsion from the center of fields, zero rest mass, and elementary free motion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 107–111, November, 1991.     

杨诺赛波动方程作为原子核或元粒子理论的数学形式

本文指出杨诺赛波动方程作为电子的个体理论的困难;指出这一类型波动方程可以作为原子核系综理论的数学形式,也可以作为元粒子系综理论的数学形式。本文并讨论了这一类型波动方程的第二级量子化问题。     

Radiative mass generation in perturbation theory and the renormalization group

The mechanism of dynamical mass generation is investigated in perturbation theory for the spontaneously broken supersymmetry model of O'Raifeartaigh. The generated mass obeys an homogeneous renormalization group equation. The compatibility of the perturbation solution with the exact solutions spectrum of the renormalization group equation is shown.     

Continuum Field Approximation for the η Pairing Mechanism of a Hubbard Model Within the Framework of a Mean-Field Theory

A quantum field theory for the η pairing mechanism of a Hubbard model within the mean-field framework is presented in this letter. It is found that the energy spectrum has two separate branches, and the eigenfunction has a plane-wave type. This result differs from the discrete lattice case in momentum space.     

Linear and nonlinear vibrations and waves in optical or acoustic superlattices (photonic or phonon crystals)

A model of a dielectric or an elastic superlattice is proposed which describes quite simply the frequency spectrum of electromagnetic or acoustic waves. The frequency band spectrum of a one-dimensional lattice consists of minibands, which narrow down with increasing frequency (so that the forbidden bands in the spectrum broaden with increasing frequency). An elementary analysis of the spectrum of a one-dimensional lattice reveals the presence of many forbidden frequency bands in this case as well. It is shown that dynamic equations for superlattices can be generalized to the nonlinear case, leading to equations of the type of the nonlinear Schrödinger equation for the lattice. Soliton excitations are described and the particle-like dynamics of solitons is demonstrated. Local vibrations near point defects of different complexity in superlattices are studied and graphically illustrated. The existence of Bloch oscillations of a wave packet in a superlattice in a homogeneous external field is discussed.     

On the Discrete Spectrum of the Nonstationary Schrödinger Equation and Multipole Lumps of the Kadomtsev–Petviashvili I Equation

The discrete spectrum of the nonstationary Schr?dinger equation and localized solutions of the Kadomtsev–Petviashvili-I (KPI) equation are studied via the inverse scattering transform. It is shown that there exist infinitely many real and rationally decaying potentials which correspond to a discrete spectrum whose related eigenfunctions have multiple poles in the spectral parameter. An index or winding number is asssociated with each of these solutions. The resulting localized solutions of KPI behave as collection of individual humps with nonuniform dynamics. Received: 30 September 1998 / Accepted: 30 March 1999     

Local Fields Without Restrictions on the Spectrum of 4-Momentum Operator and Relativistic Lindblad Equation

Quantum theory of Lorentz invariant local scalar fields without restrictions on 4-momentum spectrum is considered. The mass spectrum may be both discrete and continues and the square of mass as well as the energy may be positive or negative. One may assume the existence of such fields only if they interact with ordinary fields very weakly. Generalization of Kallen-Lehmann representation for propagators of these fields is found. The considered generalized fields may violate CPT-invariance. Restrictions on mass-spectrum of CPT-violating fields are found. Local fields that annihilate vacuum state and violate CPT-invariance are constructed in this scope. Correct local relativistic generalization of Lindblad equation for density matrix is written for such fields. This generalization is particularly needed to describe the evolution of quantum system and measurement process in a unique way. Difficulties arising when the field annihilating the vacuum interacts with ordinary fields are discussed.     

On the generation of solitons and breathers in the modified Korteweg-de Vries equation

We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.     

Spinor Structure and Matter Spectrum

Classification of relativistic wave equations is given on the ground of interlocking representations of the Lorentz group. A system of interlocking representations is associated with a system of eigenvector subspaces of the energy operator. Such a correspondence allows one to define matter spectrum, where the each level of this spectrum presents a some state of elementary particle. An elementary particle is understood as a superposition of state vectors in nonseparable Hilbert space. Classification of indecomposable systems of relativistic wave equations is produced for bosonic and fermionic fields on an equal footing (including Dirac and Maxwell equations). All these fields are equivalent levels of matter spectrum, which differ from each other by the value of mass and spin. It is shown that a spectrum of the energy operator, corresponding to a given matter level, is non-degenerate for the fields of type (l, 0) ⊕ (0, l), where l is a spin value, whereas for arbitrary spin chains we have degenerate spectrum. Energy spectra of the stability levels (electron and proton states) of the matter spectrum are studied in detail. It is shown that these stability levels have a nature of threshold scales of the fractal structure associated with the system of interlocking representations of the Lorentz group.