Topological Electromagnetism with Hidden Nonlinearity

Rañada, Antonio Fernández; Trueba, José L.

doi:10.1002/0471231495.ch2

Cited by 19 publications

(38 citation statements)

References 33 publications

(69 reference statements)

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“…The Hopf fibration can also be used in the construction of finite-energy radiative solutions to Maxwell's equations and linearized Einstein's equations [8]. Some examples are Ranada's null electromagnetic (EM) hopfion [9,10] and its generalization to torus knots [1,11,12].…”

Section: Introductionmentioning

confidence: 99%

Constructing a class of topological solitons in magnetohydrodynamics

et al. 2014

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We present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, n t and n p , respectively. The special case of n t = 1 and n p = 1 corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked exactly once, and form the surfaces of nested tori. We show that for n t ∈ Z + and n p = 1, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity, and we estimate the soliton lifetime in such a medium as a function of the toroidal winding number.

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Section: Introductionmentioning

confidence: 99%

Constructing a class of topological solitons in magnetohydrodynamics

et al. 2014

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“…In order to find these fields we use the method described in [12] to find electromagnetic knots (although the electromagnetic field that we will use in the present work is not a Rañada electromagnetic knot). Let φ(r), θ(r), two complex scalar fields such that can be considered as maps φ, θ : S 3 → S 2 after identifying the physical space R 3 with S 3 and the complex plane with S 2 .…”

Section: Construction and Resultsmentioning

confidence: 99%

Exchange of helicity in a knotted electromagnetic field

Arrayás

Trueba

2011

Annalen der Physik

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There are solutions of Maxwell equations in vacuum in which the magnetic and the electric lines have a nontrivial topology. This behaviour has physical consequences since it is related to classical expressions indicating aspects of the photon content of the electromagnetic field. In this work we present for the first time an exact solution of Maxwell equations in vacuum, having non trivial topology, in which there is an exchange of helicity between the electric and magnetic part of such field. We calculate the temporal evolution of the magnetic and electric helicities, and explain the exchange of helicity making use of the Chern-Simon form. We also have found and explained that, as time goes to infinity, both helicities reach the same value and the exchange between the magnetic and electric part of the field stops.

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“…The two scalars are assumed to have only one value at infinity, which is equivalent to compactifying the three-space into the sphere S 3 . This implies that they can be interpreted (via stereographic projection) as two maps S 3 → S 2 , which can be classified in homotopy classes and, as such, characterized by the value of the Hopf index n. It can be shown that the two scalars have the same Hopf index and that the magnetic (resp.…”

Section: The Topological Model Of Electromagnetismmentioning

confidence: 99%

“…In Section 8 of Ref. 3, it is argued that this interpretation is plausible since the quantum vacuum is dielectric but paramagnetic, so that it changes the electron charge from e 0 to e (<e 0 ) and the monopole charge from g 0 = e 0 /c to g(> g 0 ). Intriguingly, it turns out that α 0 is close to the estimated value of α s at the unification scale.…”

Section: Three Quantization Conditionsmentioning

confidence: 99%

“…In the TME, this field strength corresponds to the scalar φ (which gives a map S 3 → S 2 ). (3) Given the configuration of the magnetic lines of this solenoid, it is impossible for φ to be regular in the entire sphere S 3 . However, we may consider the 3-space as S 2 × R and require that φ be regular for the induced map S 2 → S 2 , the first S 2 being the plane (x, y), and the second the complex plane, both completed with the point at infinity).…”

Section: Flux Quantization In An Infinite Solenoidmentioning

confidence: 99%

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Topological Quantization of the Magnetic Flux

Rañada

Trueba

2006

Found Phys

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THE TOPOLOGICAL MODEL OF ELECTROMAGNETISMThis section summarizes the basic elements of the a topological model of electromagnetism (TME from now on) previously proposed by one of us, (1) which is locally equivalent to Maxwell's standard theory but implies furthermore some topological quantization conditions with intriguing physical implications (2)−(5) ( (3) is a review of the results obtained up to 2001). The TME makes use of two fundamental complex scalar fields (φ, θ ), their level curves being the magnetic and electric lines, respectively, so that each one of these lines is labelled by a particular value of the corresponding scalar. It turns out that the set of magnetic and electric lines has very curious and interesting topological properties.The two scalars are assumed to have only one value at infinity, which is equivalent to compactifying the three-space into the sphere S 3 . This implies that they can be interpreted (via stereographic projection) as two maps S 3 → S 2 , which can be classified in homotopy classes and, as such,

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Topological Electromagnetism with Hidden Nonlinearity

Cited by 19 publications

References 33 publications

Constructing a class of topological solitons in magnetohydrodynamics

Constructing a class of topological solitons in magnetohydrodynamics

Exchange of helicity in a knotted electromagnetic field

Topological Quantization of the Magnetic Flux

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