Distinguished Self-Adjoint Extension of the Two-Body Dirac Operator with Coulomb Interaction

Deckert, Dirk-André; Oelker, Martin

doi:10.1007/s00023-019-00802-6

Cited by 13 publications

(12 citation statements)

References 30 publications

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“…This is due to the lack of a consistent many body Dirac theory, in contrast with the many body Schrödinger theory so successful in the non relativistic regime. It is not even clear, to give a basic example, the behavior of the system composed by two Dirac particles interacting via a Coulomb potential (see [15] and the recent preprint [14]); for this elementary two particle system it is widely believed but not proved that the essential spectrum is given by the whole real line and there are no eigenvalues. A possible and perhaps unavoidable way out of the difficulties caused by the spectral obstruction to a many body Dirac theory consists in resorting on Quantum Electrodynamics to obtain an effective theory.…”

Section: Introductionmentioning

confidence: 99%

A Dirac field interacting with point nuclear dynamics

2019

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The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent onebody Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in H σ with σ ∈ [0, 3 2 [. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities.

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Section: Introductionmentioning

confidence: 99%

A Dirac field interacting with point nuclear dynamics

2019

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“…Under appropriate circumstances, it is clear that (1) defines an interacting dynamics (see e.g. [1] and references therein). However, (1) is not Lorentz invariant.…”

Section: Introductionmentioning

confidence: 99%

“…ψ can be considered a generalization of the single-time wave function ϕ in the Schrödinger picture, as in Eq. (1). The relation of ψ to ϕ is straightforwardly given by ϕpt, x 1 , x 2 q " ψppt, x 1 q, pt, x 2 qq.…”

Section: Introductionmentioning

confidence: 99%

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Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions

Lienert,

Nöth

2019

Preprint

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Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function ψpx 1 , x 2 q with x 1 , x 2 P R 4 is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half space, meaning that Minkowski spacetime is cut off before t " 0. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at t " 0 can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in 1+3 dimensions.

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“…Finally, let us mention another interesting related topic, where the question of distinguished self-adjoint realizations arises: 2-body Dirac-Coulomb Hamiltonians. Their mathematical study was undertaken in [4]. Even though the physical significance of these Hamiltonians is not very clear, they are widely used in quantum chemistry.…”

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confidence: 99%

Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

Dereziński,

Ruba

2021

Preprint

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We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form D ω,λ :=. We describe their closed realizations in the sense of the Hilbert space L 2 (R + , C 2 ), allowing for complex values of the parameters λ, ω. In physical situations, λ is proportional to the electric charge and ω is related to the angular momentum. We focus on realizations of D ω,λ homogeneous of degree −1. They can be organized in a single holomorphic family of closed operators parametrized by a certain 2-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator.It is well-known that D ω,λ arise after separation of variables of the Dirac-Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension.Our work is mainly motivated by a large literature devoted to distinguished selfadjoint realizations of Dirac-Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac-Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

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Distinguished Self-Adjoint Extension of the Two-Body Dirac Operator with Coulomb Interaction

Cited by 13 publications

References 30 publications

A Dirac field interacting with point nuclear dynamics

A Dirac field interacting with point nuclear dynamics

Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions

Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

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