The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism

Bialynicki‐Birula, Iwo; Bialynicka‐Birula, Zofia

doi:10.1088/1751-8113/46/5/053001

Cited by 137 publications

(159 citation statements)

References 60 publications

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“…The d = 3 ultrarelativistic operator can be given a physical interpretation within the photon wave mechanics framework, [9], see also [1,16]. As well it may serve as a natural approximation of the "true" generator in the quasirelativistic quantum mechanics of nearly massless particles.…”

Section: Motivationmentioning

confidence: 99%

Ultrarelativistic bound states in the spherical well

Żaba

Garbaczewski

2016

Journal of Mathematical Physics

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We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−∆) 1/2 , whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into nonoverlapping, orbitally labelled E (k,l) series. For each orbital label l = 0, 1, 2, ... the label k = 1, 2, ... enumerates consecutive l-th series eigenvalues. Each of them is 2l + 1-degenerate. The l = 0 eigenvalues series E (k,0) are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E (k,0) (d = 3) = E 2k (d = 1). Likewise, the eigenfunctions ψ (k,0) (d = 3) and ψ 2k (d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).

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Section: Motivationmentioning

confidence: 99%

Ultrarelativistic bound states in the spherical well

Żaba

Garbaczewski

2016

Journal of Mathematical Physics

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“…Thus we exchange

\begin{matrix} true \\ right {Ξ^{c}}_{a} & left \to Ξ_{c a} \\ right D^{n} & left \to D_{n} \\ right {\bar{L}}_{f n} & left \to {\bar{L}_{f}}^{n} \\ right {\bar{L}}_{r n} & left \to {\bar{L}_{r}}^{n} \\ right H^{c} & left \to H_{c} . \end{matrix}

In this way we describe the Riemann–Silberstein vector in the Hermitian Kaluza–Klein Theory in an electromagnetic case (see Ref. )

F_{c}

and the second vector

G_{c}

.…”

Section: Conclusion and Prospects For Further Researchmentioning

confidence: 99%

The Nonsymmetric Kaluza-Klein Theory and Modern Physics A Novel Approach

Kalinowski

2015

Fortschr. Phys.

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We consider in the paper the Nonsymmetric Kaluza-Klein Theory finding a condition for a color confinement in the theory. We consider also a Kerner-Wong-Kopczyński equation in this theory. The Nonsymmetric Kaluza-Klein Theory with a spontaneous symmetry breaking and Higgs' mechanism is examined. We find a mass spectrum for a broken gauge bosons and Higgs' particles. We derive a generalization of Kerner-Wong-Kopczyński equation in the presence of Higgs' field. A new term in the equation is a generalization of a Lorentz force term for a Higgs' field. We consider also a bosonic part of GSW (Glashow-Salam-Weinberg) model in our theory, getting masses for W , Z bosons and for a Higgs' boson agreed with an experiment. We consider Kerner-Wong-Kopczyński equation in GSW model obtaining some additional charges coupled to Higgs' field.

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“…Silberstein noticed that such a vector made its first appearance in the lectures on partial differential equations by Riemann edited and published in 1901 by Weber [4]. The name the Riemann-Silberstein vectors was introduced by Bialynicki-Birula in his works on quantum electrodynamics in [5,6], and we will use this name as the most reasonable for our formulation. After Silberstein's work the same idea has been remarked upon during the 20th century.…”

Section: Introductionmentioning

confidence: 99%

Utilization of Riemann-Silberstein Vectors in Electromagnetics

Belkovich¹,

Kogan²

2016

PIER B

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Abstract-Electromagnetic field modal expansion is traditionally an effective technique for solving Maxwell's Equations for numerous high-frequency engineering problems. In this paper, an alternative form of electromagnetic field representation is described. It is based on the Riemann-Silberstein vectors, which are a linear combination of the electric and magnetic field vectors. Utilizing such combination in homogeneous space, Maxwell's Equations are converted into a system of two independent equations. Under these circumstances, each vector describes the total electromagnetic field of an ideal circular polarization. Electromagnetic fields are simply expressed in the form of the Riemann-Silberstein vectors using the helical coordinate system and special functions, which form a set of generalized spherical harmonics. The new representation of vector spherical harmonics differs in simplicity and symmetry while having a more physically apparent expression. The amount of computational work is reduced due to the initial independence of the Riemann-Silberstein vectors. The purpose of this paper is to show the efficiency of a new approach that is based on Riemann-Silberstein vector field representation and spherical wave expansion.

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The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism

Cited by 137 publications

References 60 publications

Ultrarelativistic bound states in the spherical well

Ultrarelativistic bound states in the spherical well

The Nonsymmetric Kaluza-Klein Theory and Modern Physics A Novel Approach

Utilization of Riemann-Silberstein Vectors in Electromagnetics

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