Dirac-Coulomb System Research Articles

Asymptotics of the radiation field for the massless Dirac–Coulomb system

We consider the long-time behavior of solutions to the massless Dirac equation coupled to a Coulomb potential. For nice enough initial data, we find a joint asymptotic expansion for solutions near the null and future infinities and characterize explicitly the decay rates seen in the expansion.This paper can be viewed as a successor to previous work on asymptotic expansions for the radiation field [4,5,2]. The key new elements are propagation estimates near the singularity of the potential, building on work of the first author with Wunsch [3] and an explicit calculation with hypergeometric functions to determine the rates of decay.

Vacuum charge and current densities in the supercritical two-dimensional Dirac–Coulomb system in a magnetic field with an axial-vector potential

Vacuum charge and current densities in the supercritical two-dimensional Dirac–Coulomb system in a magnetic field with an axial-vector potential

Vacuum charge and current densities in the supercritical two-dimensional Dirac-Coulomb system in a magnetic field with an axial-vector potential

Для планарной системы Дирака-Кулона со сверхкритическим протяженным аксиально-симметричным кулоновским источником с зарядом $Z > Z_{cr,1}$ и радиусом $R_0$ в присутствии магнитного поля с аксиальным векторным потенциалом рассмотрены непертубативные эффекты поляризации вакуума в закритической области. Исследовано поведение вакуумных плотностей заряда $\rho_{\mathrm{\scriptscriptstyle VP}}(\vec r )$ и тока $\vec{j}_\mathrm{{\scriptscriptstyle VP}}(\vec r )$. Основное внимание уделено расходимостям теории, соответствующей перенормировке и сходимости парциальных рядов для $\rho_\mathrm{{\scriptscriptstyle VP}}(\vec r )$ и $\vec{j}_\mathrm{{\scriptscriptstyle VP}}(\vec r )$. Подчеркнуто, что, в отличие от вакуумной плотности заряда, для вычисления вакуумной плотности тока в присутствии локализованного при $R_1>R_0$ внешнего магнитного поля необходимо учитывать парциальные каналы с большими значениями третьей проекции полного углового момента $|m_j|$. Показано, что в присутствии закритического кулоновского источника наведенное магнитное поле при определенных значениях параметров внешнего векторного потенциала способно усиливать исходное магнитное поле.

Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac-Coulomb System for Z > Zcr: Vacuum Polarization Effects

For a planar Dirac-Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge Z > Zcr,1and radius R0, we consider essentially nonperturbative vacuum polarization effects in the overcritical region. Using results obtained in our preceding paper for the induced charge density \({\rho _{{\rm{VP}}}}(\vec r)\), we thoroughly consider the calculation of the vacuum energy eVPbased on the renormalization, the convergence of the partial expansion for \({\rho _{{\rm{VP}}}}(\vec r)\), and the behavior of the integral induced charge QVPin the overcritical region. In particular, we show that the renormalization using the fermionic loop with two external lines turns out to be a universal technique, which eliminates the divergence of the theory in the purely perturbative and essentially nonperturbative modes for \({\rho _{{\rm{VP}}}}(\vec r)\) and eVP. The most significant result is that for Z ≫ Zcr,1in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z reaching large negative values; its behavior is estimated from below (in absolute value) as ∼ −|ηeffZ3|/R0. We also study the dependence of the effect of the decrease in eVPon the cutoff of the Coulomb asymptotics of the external field at different scales R1 gt; R0and R1 ≫ R0.

Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac–Coulomb System with Z > Zcr: Vacuum Charge Density

For a planar Dirac–Coulomb system with a supercritical axially symmetric Coulomb source with the charge Z > Zcr,1 and radius R0, we consider essentially nonperturbative vacuum-polarization effects. Based on a special combination of analytic methods, computer algebra, and numerical calculations used in our previous papers to study analogous effects in the one-dimensional “hydrogen atom,” we study the behavior of both the vacuum density ρVP(r) and the total induced charge and also the vacuum-polarization energy EVP. We mainly focus on divergences of the theory and the corresponding renormalization, on the convergence of partial series for ρVP(r) and ɛVP, on the integer-valuedness of the total induced charge, and on the behavior of the vacuum energy in the overcritical region. In particular, we show that the renormalization via the fermion loop with two external legs turns out to be a universal method, which removes the divergence of the theory in the purely perturbative and essentially nonperturbative modes for ρVP and ɛVP. The most important result is that for Z ≫ Zcr,1 in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z, which reaches large negative values and whose behavior is estimated from below (in absolute value) as ~ −|ηeffZ3|/R0. We also study the dependence of polarization effects on the cutoff of the Coulomb asymptotic form of the external field. We show that screening the asymptotic value significantly changes the structure and properties of the first partial channels with mj = ±1/2,±3/2. We consider the nonperturbative calculation technique and the behavior of the induced density and the integral induced charge QVP in the overcritical region in detail.

Essentially nonperturbative vacuum polarization effects in a two-dimensional Dirac-Coulomb system with $Z>Z_\mathrm{cr}$: Vacuum charge density

Для планарной системы Дирака-Кулона со сверхкритическим протяженным аксиально-симметричным кулоновским источником с зарядом $Z>Z_\mathrm{cr,1}$ и радиусом $R_0$ рассмотрены существенно непертурбативные эффекты поляризации вакуума. На основе специальной комбинации аналитических методов, компьютерной алгебры и численных расчетов, использованной в предыдущих работах авторов для изучения аналогичных эффектов в одномерном "атоме водорода", исследованы поведение вакуумной плотности $\rho_{{\textrm{VP}}}(\vec r )$ и полного индуцированного заряда, а также вакуумная энергия поляризации $\mathcal{E}_{_\textrm{VP}}$. Основное внимание уделяется расходимостям теории и соответствующей перенормировке, сходимости парциальных рядов для $\rho_{\textrm{VP}}(\vec r )$ и $\mathcal{E}_{\textrm{VP}}$, целочисленности полного индуцированного заряда и поведению вакуумной энергии в закритической области. В частности, показано, что перенормировка через фермионную петлю с двумя внешними линиями оказывается универсальным приемом, который устраняет расходимость теории как в чисто пертурбативном, так и в существенно непертурбативном режимах как для $\rho_{\textrm{VP}}$, так и для $\mathcal{E}_{\textrm{VP}}$. Наиболее значимый результат состоит в том, что в такой системе при $Z \gg Z_\mathrm{cr,1}$ вакуумная энергия становится быстро убывающей функцией заряда источника $Z$, достигающей больших отрицательных значений, поведение которой оценивается снизу (по модулю) как $\sim-|\eta_\mathrm{eff}Z^3| /R_0$. Исследована также зависимость эффектов поляризации от обрезания кулоновской асимптотики внешнего поля. Показано, что экранировка асимптотики существенно изменяет структуру и свойства первых парциальных каналов с $m_j=\pm 1/2,\pm 3/2$. Подробно рассматриваются техника непертурбативного вычисления и поведение индуцированной плотности и интегрального наведенного заряда $Q_{\textrm{VP}}$ в закритической области.

Casimir (vacuum) energy in planar QED with strong coupling

The essentially non-perturbative vacuum polarization effects, caused by an extended external supercritical Coulomb source, are explored for a planar Dirac-Coulomb (DC) system with strong coupling (similar to graphene and graphene-based heterostructures). Taking account of results, obtained in Ref. [1] for the induced charge density ρVP(r→), in the present paper the evaluation of the Casimir (vacuum) energy EVP is presented. The main result is that for a wide range of the system parameters in the overcritical region EVP turns out to be a rapidly decreasing negative function ∼−Z3∕R0 with Z, R0 being the charge and the size of the external source. By an explicit calculation the possibility for complete screening of the electrostatic reflection self-energy of the external source by such polarization effects for Z ≫ Zcr,1 is demonstrated. The dependence of the Casimir energy on the screening of the Coulomb asymptotics of the external source at some R1 > R0 is also explored in detail, and some peculiar effects in the partial channels with the lowest rotational numbers mj = ±1∕2, ±3∕2 in the screened case are also discussed.

Essentially non-perturbative and peculiar polarization effects in planar QED with strong coupling

The essentially non-perturbative polarization effects are considered for a planar supercritical Dirac-Coulomb system with strong coupling (similar to graphene and graphene-based heterostructures) in terms of induced charge density ρVP(r→). The main attention is paid to the renormalization, convergence of the partial expansion and the behavior of ρVP(r→) and the integral induced charge QVP in the overcritical region. The dependence of the induced density on the screening of the Coulomb asymptotics of the external source is also explored in detail. Some peculiar effects in the discrete spectrum with the lowest rotational numbers mj = ±1∕2, ±3∕2 in the screened case are detected and their possible role in the transition through corresponding Zcr is also discussed.

Nonperturbative vacuum polarization effects in two-dimensional supercritical Dirac–Coulomb system II. Vacuum energy

Nonperturbative vacuum polarization effects are explored for a supercritical Dirac–Coulomb system with [Formula: see text] in 2+1[Formula: see text]D, based on the original combination of analytical methods, computer algebra and numerical calculations, proposed recently in Refs. 1–3. Both the vacuum charge density [Formula: see text] and vacuum energy [Formula: see text] are considered. Due to a lot of details of calculation the whole work is divided into two parts I and II. Taking account of results, obtained in the part I4 for [Formula: see text], in the present part II, the evaluation of the vacuum energy [Formula: see text] is investigated with emphasis on the renormalization and convergence of the partial expansion for [Formula: see text]. It is shown that the renormalization via fermionic loop turns out to be the universal tool, which removes the divergence of the theory both in the purely perturbative and essentially nonperturbative regimes of the vacuum polarization. The main result of calculation is that for a wide range of the system parameters in the overcritical region [Formula: see text] turns out to be a rapidly decreasing function [Formula: see text] with [Formula: see text] and [Formula: see text] being the size of the external Coulomb source. To the end the similarity in calculations of [Formula: see text] in 2+1 and 3+1[Formula: see text]D is discussed, and qualitative arguments are presented in favor of the possibility for complete screening of the classical electrostatic energy of the Coulomb source by the vacuum polarization effects for [Formula: see text] in 3+1[Formula: see text]D.

Nonperturbative vacuum polarization effects in two-dimensional supercritical Dirac–Coulomb system I. Vacuum charge density

Based on the original combination of analytical methods, computer algebra tools and numerical calculations, proposed recently in Refs. 1–3, the nonperturbative vacuum polarization effects in the 2+1[Formula: see text]D supercritical Dirac–Coulomb system with [Formula: see text] are explored. Both the vacuum charge density [Formula: see text] and vacuum energy [Formula: see text] are considered. The main result of the work is that in the overcritical region [Formula: see text] turns out to be a rapidly decreasing function [Formula: see text] with [Formula: see text] and [Formula: see text] being the size of the external Coulomb source. Due to a lot of details of calculation the whole work is divided into two parts I and II. In the present part I, we consider the evaluation and behavior of the vacuum density [Formula: see text], which further is used in part II for evaluation of the vacuum energy, with emphasis on the renormalization, convergence of the partial expansion for [Formula: see text] and behavior of the integral induced charge [Formula: see text] in the overcritical region.

Nonperturbative effects of vacuum polarization for a quasi-one-dimensional Dirac–Coulomb system with Z > Z cr

Major nonperturbative properties of the vacuum charge density ρ VP (x) and vacuum-polarization energy ξ VP are considered for an overcritical Coulomb source with Z > Z cr in 1 + 1 D. We demonstrate that, for a broad range of external-field parameters, vacuum energy may significantly deviate from the perturbative quadratic increase and even decrease towards deeply negative values due to vacuum polarization in the overcritical region.

Vacuum energy of one-dimensional supercritical Dirac–Coulomb system

Nonperturbative vacuum polarization effects are explored for a supercritical Coulomb source with [Formula: see text] in 1[Formula: see text]+[Formula: see text]1D. Both the vacuum charge density [Formula: see text] and vacuum energy [Formula: see text] are considered. It is shown that in the overcritical region the behavior of vacuum energy could be significantly different from perturbative quadratic growth up to decrease reaching large negative values.

Polarons as stable solitary wave solutions to the Dirac–Coulomb system

We consider solitary wave solutions to the Dirac–Coulomb system both from physical and mathematical points of view. Fermions interacting with gravity in the Newtonian limit are described by the model of Dirac fermions with the Coulomb attraction. This model also appears in certain condensed matter systems with emergent Dirac fermions interacting via optical phonons. In this model, the classical soliton solutions of equations of motion describe the physical objects that may be called polarons, in analogy to the solutions of the Choquard equation. We develop analytical methods for the Dirac–Coulomb system, showing that the no-node gap solitons for sufficiently small values of charge are linearly (spectrally) stable.

Algebraic Approach to the Fixed Point Structure of the Quantum Mechanical Dirac -Coulomb System

Abstract It is shown that a non-perturbative ß like function can be obtained for a Dirac-Coulomb system with both vector and scalar couplings using the properties of 0(2,1) algebra and the tilting operator mechanism.

Energy and beta -function solutions to relativistic Hamiltonians with Coulombic and linear potentials.

It is argued that quantum-mechanical \ensuremath{\beta} functions can be derived consistently by using renormalization length scales implied by the quasiclassical minimization of Hamiltonian forms. Our renormalization-group results for the Dirac-Coulomb system can then be related analytically to ones obtained in ladder quantum electrodynamics in four space-time dimensions. Involving spinless relativistic two-body Hamiltonians, one can present exact energy- and \ensuremath{\beta}-function solutions for the Lorentz scalar linear potential. Inverse-square, other linear, and mixed linear plus Coulomb potentials can also be treated. The exact energy of this last potential has been discussed in conjunction with an alternative \ensuremath{\beta} function.

On the fixed point structure of the quantum mechanical Dirac-Coulomb system

Abstract Proofs are given that β-like functions for hamiltonian systems can be established by using the minimum location of the quantum-mechanical 1/ N method as a renormalization length scale. Then the typical cubic square-root form characterizing the nonperturbative β-function in QED can also be derived in terms of a related Dirac-Coulomb system with both vector and scalar couplings, now within the complementary region of subcritical vector coupling.

Supersymmetry in second-order relativistic equations for the hydrogen atom

The factorisation method for constructing eigenstates and eigenvalues of the Schrodinger equation, based on the algebra of supersymmetry, is applied to second-order relativistic equations such as Kramers' equation for the Dirac-Coulomb system (hydrogen atom). The well known helicity degeneracy, for example, of the 2s1/2 and 2p1/2 levels, which is broken by the Lamb shift, in thus associated with supersymmetry. A novel form of supersymmetry is found when l=0: two interpenetrating ladders, founded on different and non-degenerate 'ground' states, coexist. One ladder, corresponding to a deeply bound ground state, has no counterpart in the physical hydrogen spectrum. Analogous results are obtained for the Klein-Gordon-Coulomb system in one and three dimensions. Eigenstates and eigenvalues for the one-dimensional Dirac-Coulomb system are found by projection.