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Иваненко, Д.; Курдгелаидзе, Д. Ф.

doi:10.1007/bf00818558

Cited by 9 publications

(6 citation statements)

References 2 publications

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“…Nonlinear selfcouplings of the spinor fields may arise as a consequence of the geometrical structure of the space-time and, more precisely, because of the existence of torsion. As soon as 1938, Ivanenko [2][3][4] showed that a relativistic theory imposes in some cases a fourth order selfcoupling. In 1950 Weyl [5] proved that, if the affine and the metric properties of the space-time are taken as independent, the spinor field obeys either a linear equation in a space with torsion or a nonlinear one in a Reimannian space.…”

Section: Introductionmentioning

confidence: 99%

Spinor field in a Bianchi type-I universe: Regular solutions

2001

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Self-consistent solutions to the nonlinear spinor field equations in general relativity are studied for the case of Bianchi type-I ͑BI͒ space-time. It is shown that, for some special type of nonlinearity the model provides a regular solution, but this singularity-free solution is attained at the cost of breaking the dominant energy condition in the Hawking-Penrose theorem. It is also shown that the introduction of a ⌳ term in the Lagrangian generates oscillations of the BI model, which is not the case in the absence of a ⌳ term. Moreover, for the linear spinor field, the ⌳ term provides oscillatory solutions, which are regular everywhere, without violating the dominant energy condition.

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Section: Introductionmentioning

confidence: 99%

Spinor field in a Bianchi type-I universe: Regular solutions

2001

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“…The first to propose and study models for the description of this phenomenon were Ivanenko [97], Weyl [169] and Heisenberg [90]. Therefore, we do not consider any external potential (but possibly a self-generated one).…”

Section: Nonlinear Dirac Equations For a Free Particlementioning

confidence: 99%

“…The Hartree-Fock energy E HF appearing in (108) is the same as (97), but with D c replaced by −∆/2 and C 4 by C 2 . It can be proved [65] that the critical points constructed in Theorem 23 all satisfy the assumptions of Theorem 25 for any fixed j.…”

Section: Theorem 25 (Nonrelativistic Limit Of the Dirac-fock Equationsmentioning

confidence: 99%

Variational methods in relativistic quantum mechanics

Esteban

Lewin

Séré

2008

Bull. Amer. Math. Soc.

114

104

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This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems.In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow us to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein-Gordon-Dirac equations.The second part is devoted to the presentation of min-max principles allowing us to characterize and compute the eigenvalues of linear Dirac operators with an external potential in the gap of their essential spectrum. Many consequences of these min-max characterizations are presented, among them are new kinds of Hardy-like inequalities and a stable algorithm to compute the eigenvalues.In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit.In the last part, we present a more involved relativistic model from Quantum Electrodynamics in which the behavior of the vacuum is taken into account, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

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“…Stationary solutions of such equations represent the state of a localized particle which can propagate without changing its shape. The first to propose and study models for the description of this phenomenon were Ivanenko [97], Weyl [169] and Heisenberg [90]. We refer to Rañada [138] for a very interesting review on the historical background of this kind of models.…”

Section: Nonlinear Dirac Equations For a Free Particlementioning

confidence: 99%

Variational methods in relativistic quantum mechanics

Esteban,

Lewin,

séré

2007

Preprint

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This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems.In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein-Gordon-Dirac equations.The second part is devoted to the presentation of min-max principles allowing to characterize and compute the eigenvalues of linear Dirac operators with an external potential, in the gap of their essential spectrum. Many consequences of these min-max characterizations are presented, among them a new kind of Hardy-like inequalities and a stable algorithm to compute the eigenvalues.In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers, lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit.In the last part, we present a more involved relativistic model from Quantum Electrodynamics in which the behavior of the vacuum is taken into account, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

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Quark stars

Cited by 9 publications

References 2 publications

Spinor field in a Bianchi type-I universe: Regular solutions

Spinor field in a Bianchi type-I universe: Regular solutions

Variational methods in relativistic quantum mechanics

Variational methods in relativistic quantum mechanics

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