Perturbation method for particle-like solutions of the Einstein–Dirac–Maxwell equations

Nodari, Simona Rota

doi:10.1016/j.crma.2010.06.003

Cited by 8 publications

(6 citation statements)

References 5 publications

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“…We can not complete this quick review of NLD models in high-energy physics without mentioning the recent work of [121] (see also [114]), where a variant of the NLD is applied to the study of neutrino oscillations. Related systems are the Dirac-Maxwell system [23,41,60,79,166], the Einstein-Dirac system [140,155], and Einstein-Dirac-Maxwell system [141]. In quantum chemistry, the Dirac-Hartree-Fock model [63,64,105] takes into account the fermionic properties of electrons (describing the exchange interaction, which is a fundamental effect of purely quantum nature) and is used for accurate computation of the electronic energy [134,164]; this model has also started to receive mathematical attention [63][64][65].…”

Section: Introductionmentioning

confidence: 99%

Solitary Waves in the Nonlinear Dirac Equation

Cuevas–Maraver

Boussaïd

Comech

et al. 2018

Understanding Complex Systems

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In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schrödinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT -symmetric variant of the model.

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Section: Introductionmentioning

confidence: 99%

Solitary Waves in the Nonlinear Dirac Equation

Cuevas–Maraver

Boussaïd

Comech

et al. 2018

Understanding Complex Systems

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show abstract

“…To prove Theorem 1.1, we construct solitary wave solutions by deforming the solutions to the nonrelativistic limit (represented by the Choquard equation) via the implicit function theorem. Such a method was employed in [Oun00, Gua08,BC17a] for the nonlinear Dirac equation and in [RN10a,Stu10,RN10b] for Einstein-Dirac and Einstein-Dirac-Maxwell systems.…”

Section: To Vladimir Georgiev On the Occasion Of His 60th Birthday 1 ...mentioning

confidence: 99%

Small amplitude solitary waves in the Dirac-Maxwell system

Comech¹,

Stuart²

2018

Communications on Pure &Amp; Applied Analysis

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We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form φ(x, ω)e −iωt , with ω ∈ (−m, ω * ) for some ω * > −m. The solutions satisfy φ( · , ω) ∈ H 1 (R 3 , C 4 ), and are small amplitude in the sense that φ( · , ω) 2The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation. To Vladimir Georgiev on the occasion of his 60th birthdayWe consider the system of Dirac-Maxwell equations, where the electron, described by the standard "linear" Dirac equation, interacts with its own electromagnetic field which is in turn required to obey the Maxwell equations:with the charge-current density J µ = (ρ, J) ∈ R × R 3 generated by the spinor field itself:Above, ρ and J are the charge and current, respectively. We denoteψ = (γ 0 ψ) * = ψ * γ 0 , with ψ * the hermitian conjugate of ψ ∈ C 4 . The charge is denoted by q (so that for the electron q < 0); the fine structure constant is the dimensionless coupling constant α ≡ q 2 c ≈ 1/137. We choose the units so that = c = 1. We have written the Maxwell equations using the Lorentz gauge condition ∂ µ A µ = 0. The Dirac γ-matrices satisfy the anticommutation relations {γ µ , γ ν } = 2g µν , 0 ≤ µ, ν ≤ 3,

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“…Subsequent work has extended this analysis to charged fermions [6], the addition of non-Abelian gauge fields [7], proofs of existence [8][9][10], and consideration of the Newtonian limit [11]. Comparison between the fermionic and bosonic cases is presented in [12,13], while axisymmetric solutions corresponding to single fermion states have recently been found [14].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear effects in the excited states of many-fermion Einstein-Dirac solitons

et al. 2021

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We present an analysis of excited-state solutions for a gravitationally localized system consisting of a filled shell of high-angular-momentum fermions, using the Einstein-Dirac formalism introduced by Finster, Smoller, and Yau [Phys. Rev. D 59, 104020 (1999). We show that, even when the particle number is relatively low (N f ≥ 6), the increased nonlinearity in the system causes a significant deviation in behavior from the two-fermion case. Excited-state solutions can no longer be uniquely identified by the value of their central redshift, with this multiplicity producing distortions in the characteristic spiraling forms of the mass-radius relations. We discuss the connection between this effect and the internal structure of solutions in the relativistic regime.

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Perturbation method for particle-like solutions of the Einstein–Dirac–Maxwell equations

Cited by 8 publications

References 5 publications

Solitary Waves in the Nonlinear Dirac Equation

Solitary Waves in the Nonlinear Dirac Equation

Small amplitude solitary waves in the Dirac-Maxwell system

Nonlinear effects in the excited states of many-fermion Einstein-Dirac solitons

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