Solutions of the Maxwell equations describing the spectrum of hydrogen

We obtain a new class of solutions of the Maxwell equations describing the spectrum of hydrogen. We prove that, instead of the quantum-mechanical Dirac equation, the ordinary classical Maxwell equations can be applied to the solution of many problems in atomic and nuclear physics. Institute of Electron Physics, Ukrainian Academy of Sciences, Uzhgorod. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 958–969, July, 1997.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

The Selfconsistent Pauli Equation

 We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model.     

The Selfconsistent Pauli Equation

 We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model. (Received 22 May 2000)     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

Chirality-violating condensates in QCD and their connection with zero-mode solutions of quark dirac equations

We demonstrate that chirality-violating condensates in massless QCD arise entirely from zero-mode solutions of the Dirac equation in arbitrary gluon fields. We propose a model in which the zero-mode solutions are the ones for quarks moving in the instanton field and calculate the quark condensate magnetic susceptibilities χ of dimension three and κ and ξ of dimension five based on this model. The good correspondence of the values of χ, κ, and ξ obtained using this approach with the values found from the hadronic spectrum is a serious argument that instantons are the only source of chirality-violating condensates in QCD. We discuss the temperature dependence of the quark condensate and show that the phase transition corresponding to the temperature dependence α(T) of the quark condensate as an order parameter is a crossover-type transition.     

Schr?dinger and Dirac particles in quasi-one-dimensional systems with a Coulomb interaction

We consider specific features and principal distinctions in the behavior of the energy spectra of Schr?dinger and Dirac particles in the regularized ??Coulomb?? potential V_??(z) = ?q/(|z|+??) as functions of the cutoff parameter ?? in 1+1 dimensions. We show that the discrete spectrum becomes a quasiperiodic function of ?? for ?? ? 1 in such a one-dimensional ??hydrogen atom?? in the relativistic case. This effect is nonanalytically dependent on the coupling constant and has no nonrelativistic analogue in this case. This property of the Dirac spectral problem explicitly demonstrates the presence of a physically informative energy spectrum for an arbitrarily small ?? > 0, but also the absence of a regular limit transition ?? ?? 0 for all nonzero q. We also show that the three-dimensional Coulomb problem has a similar property of quasiperiodicity with respect to the cutoff parameter for q = Z?? > 1, i.e., in the case where the domain of the Dirac Hamiltonian with the nonregularized potential must be especially refined by specifying boundary conditions as r ?? 0 or by using other methods.     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.      

Kepler problem in a constant-curvature space

We algebraically derive the spectrum of a hydrogen atom in a space with constant curvature. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 317–326, May, 2008.     

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag–Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.      

Maxwell equations in complex form of Majorana–Oppenheimer,solutions with cylindric symmetry in Riemann <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript> and Lobachevsky <Emphasis Type="Italic">H</Emphasis><Subscript>3</Subscript> spaces

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space-time in accordance with the tetrad recipe of Tetrode–Weyl–Fock–Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for the Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.     

On Bers Generating Functions for First Order Systems of Mathematical Physics

Considering one of the fundamental notions of Bers’ theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the examples the Maxwell system for an inhomogeneous medium is considered, and as one of the consequences of the introduced approach we find a relation between the time-dependent one-dimensional Maxwell system and hyperbolic pseudoanalytic functions and obtain an infinite system of solutions of the Maxwell system. Other considered examples are the system describing force-free magnetic fields and the Dirac system from relativistic quantum mechanics.     

Bosonic symmetries of the massless Dirac equation

The results of spin 1 symmetries of massless Dirac equation [21] are proved completely in the space of 4-component Dirac spinors on the basis of unitary operator in this space connecting this equation with the Maxwell equations containing gradient-like sources. Nonlocal representations of conformal group are found, which generate the transformations leaving the massless Dirac equation being invariant. The Maxwell equations with gradient-like sources are proved to be invariant with respect to fermionic representations of Poincaré and conformal groups and to be the kind of Maxwell equations with maximally symmetrical properties. Brief consideration of an application of these equations in physics is discussed.     

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

Asymptotic expression for the correlation function of twisted fields in the two-dimensional Dirac model on a lattice

A determinant representation is obtained for the correlation function of twisted fields in the two-dimensional Dirac model on a lattice. These fields are determined by twisted boundary conditions for the Dirac fermions. The asymptotic expression is calculated for the correlation function at large distances (the vacuum expectation of the twisted field) at the critical point and in the scaling region. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2 pp. 329–346, Nember, 1999.     

Extended Dirac factorization: ‘multi-mass’ special cases in clifford form

The hypercomplex approach to Dirac physics is here summarized. The new mass term is shown to give a ψ wave, in the extension of Dirac’s Klein-Gordon factorization (when the usual Dirac mass part equals zero), that is completely within the ‘complexified’ Hamilton-Pauli sub-algebraE. This may help with the physical interpretation of this new mass, with its five possible parts. New, spin 1/2 quanta also seem possible. These are all variations on Dirac’s factorization, where mass is generalized to several parts (multi-parts) or, instead, each new mass part, of the five, can be taken one at a time in nature, for individual wave equations. Lorentz symmetry is also naturally extended from 6 parameters to 8 parameters, and this has sweeping metaphysical implications. Dirac algebra is doubled also in a natural way using quaternions.     

Fermionic entropy in BTZ space time

The thermodynamics of Dirac field is discussed in the backgrounds of 3 dimensional Banados-Teitelboim-Zanelli space time. The Dirac equation is solved under “quasi-periodic” boundary condition and the exact solution is obtained, from which the corresponding free energy and Fermionic entropy are calculated Project supported by the National Natural Science Foundation of China (Grant No. 9873013).     

Weyl laws on open manifolds

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is purely discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume. The author has been partially supported by the Research and Training Network HPRN-CT-1999-00118 “Geometric Analysis” funded by the European Commission.