Fermion pair creation by a strong Coulomb field in 2+1 dimensions

The creation of charged fermion pairs by a strong external Coulomb field in a space with two dimensions is investigated. Exact solutions to the Dirac equation are found for the Coulomb external field in 2+1 dimensions. The equation for determining the critical charge is obtained and is numerically solved for a simplified model. The critical charge for 2+1 dimensions is much less than the critical charge for the similar model with 3+1 dimensions. The influence of the vacuum polarization on the critical charge is studied in the one-loop approximation to the (2+1)-dimensional quantum electrodynamics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 2, pp. 277–287, August, 1998.     

Spontaneous fermion creation in the coulomb field and Aharonov-Bohm potential in 2+1 dimensions

We find exact solutions of the Dirac equation and the fermion energy spectrum in the Coulomb (vector and scalar) potential and Aharonov-Bohm potential in 2+1 dimensions taking the particle spin into account. We describe the fermion spin using the two-component Dirac equation with the additional (spin) parameter introduced by Hagen. We consider the effect of creation of fermion pairs from the vacuum by a strong Coulomb field in the Aharonov-Bohm potential in 2+1 dimensions. We obtain transcendental equations implicitly determining the electron energy spectrum near the boundary of the lower energy continuum and the critical charge. We numerically solve the equation for the critical charge. We show that for relatively weak magnetic flows, the critical charge decreases (compared with the case with no magnetic field) if the energy of interaction of the electron spin magnetic moment with the magnetic field is negative and increases if this energy is positive. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 250–262, February, 2009.     

Zero-mass fermions in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions

We consider the motion of a relativistic charged zero-mass fermion in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions. With these singular external potentials, we construct one-parameter self-adjoint Dirac Hamiltonians classified by self-adjoint boundary conditions. We show that if the so-called effective charge becomes overcritical, then virtual (quasistationary) bound states occur. The wave functions corresponding to these states have large amplitudes near the Coulomb center, and their energy spectrum is quasidiscrete and consists of a number of broadened levels of a width related to the inverse lifetime of the quasistationary state. We derive equations for the quasidiscrete spectra and quasistationary state lifetimes and solve these equations in physically interesting cases. We study the so-called local densities of state, which can be assessed in physical experiments, as functions of the energy and the problem parameters, investigating these densities both analytically and graphically.     

Fermion bound states in the Aharonov-Bohm field in 2+1 dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in 2+1 dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions describe the behavior of relativistic fermions in the field of the cosmic string in 3+1 dimensions, our results can presumably be used to describe fermions in the cosmic string field.     

Discrete spectra of the Dirac Hamiltonian in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions

We find all self-adjoint Dirac Hamiltonians in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions with the fermion spin taken into account. We obtain implicit equations for the spectra and construct eigenfunctions for all self-adjoint Dirac Hamiltonians in the indicated external fields. We find explicit solutions of the equations for the spectra in some cases.     

Induced vacuum charge of massless fermions in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions

We study the vacuum polarization of zero-mass charged fermions in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions. For this, we construct the Green’s function of the two-dimensional Dirac equation in the considered field configuration and use it to find the density of the induced vacuum charge in so-called subcritical and supercritical regions. The Green’s function is represented in regular and singular (in the source) solutions of the Dirac radial equation for a charged fermion in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions and satisfies self-adjoint boundary conditions at the source. In the supercritical region, the Green’s function has a discontinuity related to the presence of singularities on the nonphysical sheet of the complex plane of “energy,” which are caused by the appearance of an infinite number of quasistationary states with negative energies. Ultimately, this situation represents the neutral vacuum instability. On the boundary of the supercritical region, the induced vacuum charge is independent of the self-adjoint extension. We hope that the obtained results will contribute to a better understanding of important problems in quantum electrodynamics and will also be applicable to the problem of screening the Coulomb impurity due to vacuum polarization in graphene with the effects associated with taking the electron spin into account.     

Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

Polarization of vacuum by an external magnetic field in the (2+1)-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the (2+1)-dimensional quantum electrodynamics (QED ₂₊₁) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogenous magnetic field with a nonzero chemical potential is derived in the one-loopQED ₂₊₁ approximation. The contribution of the induced Chern—Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of n relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern—Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 132–151, October, 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.     

Neutral Fermion Having Electric and Magnetic Moments in an External Electromagnetic Field

We show that in 2+1 dimensions, the Dirac equation for a neutral fermion possessing electric and magnetic dipole moments in an external electromagnetic field reduces to the Dirac equation for a charged fermion in a external field characterized by a certain 3-pseudo-vector potential. The effective charge of the neutral fermion is determined by its dipole moments. The effects of coupling electric and magnetic moments of the neutral fermion to the external electromagnetic field seem to be inseparable in physical experiments of any type. We find an exact solution of the Dirac equation for a massive neutral fermion with electric and magnetic dipole moments in a external plane-wave electromagnetic field. We derive expressions for the fermionic vacuum current induced by neutral fermions in the presence of external electromagnetic fields.     

Three-dimensional Gross-Neveu model in external chromomagnetic fields at finite temperature

A (2+1)-dimensional four-fermion theory at finite temperature in external chromomagnetic fields that model a vacuum gluon condensate is considered. The critical properties of the theory are investigated, and the dependence of the fermion mass on the external field is also found.Institute of High Energy Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 391–401, December, 1994.     

A (2+1)-Dimensional Gauge Model with Electrically Charged Fermions

We consider a (2+1)-dimensional gauge theory with a nonzero fermion density and an initial Chern–Simons topological term, whose Lorentz invariance is spontaneously broken in a certain Lorentz reference frame by the generation of a constant homogenous magnetic field. We propose interpreting the number =±1, which characterizes the two nonequivalent representations of Dirac matrices in 2+1 dimensions, as a quantum number that explicitly describes the spin of the fermion. In particular, this interpretation allows determining the vacuum state of the model in a constant homogenous magnetic field as the state whose fermion and spin numbers are equal to zero.     

QED2+1 radiation effects in a strong magnetic field

We develop the eigenfunction method for the Dirac operator in a background magnetic field in the (2+1)-dimensional quantum electrodynamics (QED₂₊₁). In the eigenfunction repressentation, we find the exact solutions and the Green's functions of the Dirac equation in a strong constant homogeneous magnetic field in 2+1 dimensions. In the one-loop QED₂₊₁ approximation, we derive the effective Lagrangian, the density of vacuum fermions induced by the field, and the electron mass operator in a homogeneous background magnetic field. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 412–423, December, 1999.     

Neutral Fermion with a Magnetic Moment in External Electromagnetic Fields

The interaction between a massive neutral fermion with a static (spin) magnetic dipole moment and an external electromagnetic field is described by the Dirac–Pauli equation. Exact solutions of this equation are obtained along with the corresponding energy spectrum for an axially symmetric external magnetic field and for some centrally symmetric electric fields. It is shown that the spin–orbital interaction of a neutral fermion with a magnetic moment determines both the characteristic properties of the quantum states and the fermion energy spectrum. It is found that (1) the discrete energy spectrum of a neutral fermion depends on the projection of the fermion spin on a certain quantization axis, (2) the ground energy level of a fermion in these electric fields as well as the energy levels of all bound states with a fixed value of the quantum number characterizing the projection of the fermion spin in the electric field E = er is degenerate and the degeneration order is countably infinite, and (3) the energy spectra of neutral fermions and antifermions with spin magnetic moments are symmetric in centrally symmetric fields. Bound states of a neutral fermion with a magnetic moment in an external electric field do exist even if the Dirac–Pauli equation does not explicitly contain the term with the fermion mass. In addition, in centrally symmetric electric fields, there exist a countably infinite set of pairs of isolated charge-conjugate zero-energy solutions of the Dirac–Pauli equation.     

Kadomtsev–Petviashvili Equation Revisited and Integrability in 4 + 2 and 3 + 1

The Cauchy problem of Kadomtsev–Petviashvili I (KPI) was reduced to a nonlocal Riemann–Hilbert (RH) problem by the author and Ablowitz in 1983. This formulation was based on the introduction of two spectral functions (nonlinear Fourier transforms, FTs). This formalism was improved by Boiti et al. [ 1 ], where it was shown that the earlier nonlocal RH problem can be formulated in terms of a single spectral function (nonlinear FT). A different formalism was presented by Zhou [ 2 ], where the Cauchy problem was rigorously solved in terms of a linear integral equation involving a nonanalytic eigenfunction. Here, we first revisit the above results and then review some recent results about the derivation of integrable generalizations of KP in 4 + 2 (i.e., in four spatial and two temporal dimensions), as well as in 3 + 1 (i.e., in three spatial and one temporal dimensions).     

Quasi-exact solution of the problem of relativistic bound states in the (1+1)-dimensional case

We investigate the problem of bound states for bosons and fermions in the framework of the relativistic configurational representation with the kinetic part of the Hamiltonian containing purely imaginary finite shift operators e^±ihd/dx instead of differential operators. For local (quasi)potentials of the type of a rectangular potential well in the (1+1)-dimensional case, we elaborate effective methods for solving the problem analytically that allow finding the spectrum and investigating the properties of wave functions in a wide parameter range. We show that the properties of these relativistic bound states differ essentially from those of the corresponding solutions of the Schrödinger and Dirac equations in a static external potential of the same form in a number of fundamental aspects both at the level of wave functions and of the energy spectrum structure. In particular, competition between ? and the potential parameters arises, as a result of which these distinctions are retained at low-lying levels in a sufficiently deep potential well for ? ? 1 and the boson and fermion energy spectra become identical.     

Traveling-Wave Solutions of the Calogero-Degasperis-Fokas Equation in 2+1 Dimensions

Soliton solutions are among the more interesting solutions of the (2+1)-dimensional integrable Calogero-Degasperis-Fokas (CDF) equation. We previously derived a complete group classiffication for the CDF equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. The solutions consequently exhibit a rich variety of qualitative behaviors. Choosing the arbitrary functions appropriately, we exhibit solitary waves and bound states.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 44–55, July, 2005.     

Peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus

We consider the peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus and discuss the long history of an incorrect interpretation of this problem in the case of a pointlike nucleus and its current correct solution. We consider the spectral problem in the case of a regularized Coulomb potential. For some special regularizations, we derive an exact equation for the point spectrum in the energy interval (-m,m) and find some of its solutions numerically. We also derive an exact equation for charges yielding bound states with the energy E = -m; some call them supercritical charges. We show the existence of an infinite number of such charges. Their existence does not mean that the oneparticle relativistic quantum mechanics based on the Dirac Hamiltonian with the Coulomb field of such charges is mathematically inconsistent, although it is physically unacceptable because the spectrum of the Hamiltonian is unbounded from below. The question of constructing a consistent nonperturbative second-quantized theory remains open, and the consequences of the existence of supercritical charges from the standpoint of the possibility of constructing such a theory also remain unclear.     

Spectral and dispersion properties of pair soliton states in a quasi-one-dimensional anisotropic antiferromagnet withS=1/2

By solution of the Schrödinger equation in the continuum approximation, it is shown analytically that there exist excited eigenstates of the quasi-one-dimensional Ising antiferromagnet with spinS=1/2 in the form of spatially localized quantum states. Computer modeling of a discrete model of interacting solitons with allowance for the symmetry of the solutions gives eigenvalues of the Sturm sequence that differ from the solutions of the continuum approximation. The spectral and dispersion properties of the nonlinear bound states of lowest energy and the selection rules in resonance transitions in an external magnetic field applied parallel to and perpendicular to the axis of magnetic anisotropy are calculated.L. V. Kirenski Physics Institute, Siberian Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 112–119, April, 1992.     

Nonconservation of the fermion number and the limiting density of fermionic matter (two-dimensional gauge model)

Conclusions Thus, at the present time there are two possible ways for instability to develop in gauge theories at high fermion density. In the four-dimensional Abelian model considered in [1] there is ultimately formed an anomalous state characterized by zero density of the real fermions, zero scalar condensate, and large gauge field condensate. In the two-dimensional model considered in the present paper, the effects of the complicated vacuum structure have the consequence that the system undergoes a transition to a normal state with low fermion density above a topologically nontrivial vacuum, this transition being accompanied by nonconservation of the fermion number. It is of undoubted interest to clarify which of these possibilities is realized in realistic non-Abelian four-dimensional theories (for example, in the standard model of electroweak interactions), i. e., to consider the existence of stable anomalous states in such theories. This question will be considered in later papers.Institute of Nuclear Research, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 68, No. 1, pp. 3–17, July, 1986.