Knotted solutions, from electromagnetism to fluid dynamics

Alves, Daniel W. F.; Hoyos, Carlos; Năstase, Horaţiu; Sonnenschein, Jacob

doi:10.1142/s0217751x17502001

Cited by 67 publications

(21 citation statements)

References 50 publications

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“…The topological charge associated with solutions equivalent to maps

ϕ : S^{3} \to S^{2}

is the Hopf charge. For a complex scalar ϕ, consider the “field strength”

F_{i j} = \partial_{i} A_{j} - \partial_{j} A_{i} = \frac{1}{4 π i} \frac{\partial_{i} ϕ \partial_{j} ϕ^{*} - \partial_{i} ϕ^{*} \partial_{j} ϕ}{{false(1 + {false| ϕ false|}^{2} false)}^{2}},

coming from the “gauge field”

A_{i} = \frac{- 1}{8 π i} \frac{\partial_{i} ln ϕ - \partial_{i} ln ϕ^{*}}{1 + {| ϕ |}^{2}} .

Then the Hopf index (charge) is (see [] for more details)

\begin{matrix} true \\ right N & = & left \int_{S^{3}} F \land A \\ = & left \int_{S^{3}} \frac{d^{3} x}{32 π^{2}} ε^{i j k} \frac{false(\partial_{i} prefixln ϕ - \partial_{i} prefixln ϕ^{*} false) false(\partial_{i} ϕ \partial_{j} ϕ^{*} - \partial_{i} ϕ^{*} \partial_{j} ϕ false)}{{false(1 + false| ϕ false| false)}^{3}} . \end{matrix}

A related quantity is obtained by projecting ϕ (defined on

C^{*}

) onto a vector

\overset{n}{true}

on S 3 in Euclidean coordinates. The map is done using the standard stereographic projection

…”

Section: Other Null Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Năstase

Sonnenschein

2018

Fortschritte der Physik

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Knotted solutions to electromagnetism are investigated as an independent subsector of the theory. We write down a Lagrangian and a Hamiltonian formulation of Bateman's construction for the knotted electromagnetic solutions. We introduce a general definition of the null condition and generalize the construction of Maxwell's theory to massless free complex scalar, its dual two form field, and to a massless DBI scalar. We set up the framework for quantizing the theory both in a path integral approach, as well as the canonical Dirac method for a constrained system. We make several observations about the semi-classical quantization of systems of null configurations. IntroductionElectromagnetism is a free (non-selfinteracting) theory so, according to standard lore, we wouldn't expect topologically nontrivial solutions. Indeed, solitons are usually found in interacting theories, as in the case of water solitons, which started the field, with John Scott Russel's observation of a solitonic wave in a canal in Scotland. Sometimes there is a topological reason for the existence and stability of a soliton, which is the case for "kinks" in 1+1 dimensional scalar theories, vortices in 2+1 dimensional gauge theories, or monopoles in 3+1 dimesional gauge theories, for instance.But the existence of a topological constraint, of a fixed topological number, turns out to be possible even in a free theory like Maxwell electromagnetism without sources. Thus it was realized rather late that there exist solutions with a nonzero Hopf index, or "Hopfions," and the explicit solutions were written only in [1,2] by Rañada, after the early work by Trautman in [3]. The standard Hopfion solution is null in the sense of the Riemann-Silberstein (RS) vector F = E + i B, i.e. F 2 = 0, corresponding to E 2 = B 2 and E · B = 0, but there are also partially null solutions, as we will explain in the following.

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“…The topological charge associated with solutions equivalent to maps

ϕ : S^{3} \to S^{2}

is the Hopf charge. For a complex scalar ϕ, consider the “field strength”

F_{i j} = \partial_{i} A_{j} - \partial_{j} A_{i} = \frac{1}{4 π i} \frac{\partial_{i} ϕ \partial_{j} ϕ^{*} - \partial_{i} ϕ^{*} \partial_{j} ϕ}{{false(1 + {false| ϕ false|}^{2} false)}^{2}},

coming from the “gauge field”

A_{i} = \frac{- 1}{8 π i} \frac{\partial_{i} ln ϕ - \partial_{i} ln ϕ^{*}}{1 + {| ϕ |}^{2}} .

Then the Hopf index (charge) is (see [] for more details)

\begin{matrix} true \\ right N & = & left \int_{S^{3}} F \land A \\ = & left \int_{S^{3}} \frac{d^{3} x}{32 π^{2}} ε^{i j k} \frac{false(\partial_{i} prefixln ϕ - \partial_{i} prefixln ϕ^{*} false) false(\partial_{i} ϕ \partial_{j} ϕ^{*} - \partial_{i} ϕ^{*} \partial_{j} ϕ false)}{{false(1 + false| ϕ false| false)}^{3}} . \end{matrix}

A related quantity is obtained by projecting ϕ (defined on

C^{*}

) onto a vector

\overset{n}{true}

on S 3 in Euclidean coordinates. The map is done using the standard stereographic projection

…”

Section: Other Null Systemsmentioning

confidence: 99%

“…Note that a plane electromagnetic wave is also null, and one can apply such a procedure on it as well. In [], a connection of null electromagnetism with fluid dynamics was used in order to explore other ways of finding solutions in both theories.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Năstase

Sonnenschein

2018

Fortschritte der Physik

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“…The study of the topological solutions to Maxwell's equations in vacuum, firstly proposed by Trautman and Rañada in [1][2][3], has revealed so far a rich interplay between physical systems and mathematical structures which was previously unexpected in the realm of classical electrodynamics and classical field theory [4]. Since then, the subject of the topological electromagnetic fields has gain momentum with very interesting problems investigated recently, such as the existence of topological solutions of the Einstein-Maxwell theory [5][6][7][8] and of the non-linear electrodynamics [9][10][11][12][13][14]. Also, it has been shown that there are interesting mathematical structures that can be associated to the physical systems a e-mail: adina.crisan@mep.utcluj.ro b e-mail: ionvancea@ufrrj.br (corresponding author) with topological electromagnetic fields and play an important role in their dynamics, such as twistors [15], fibrations [16] and rational functions [17,18] (see for recent reviews [19][20][21]).…”

Section: Introductionmentioning

confidence: 99%

Finsler geometries from topological electromagnetism

Crișan

Vancea

2020

Eur. Phys. J. C

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We analyse the Finsler geometries of the kinematic space of spinless and spinning electrically charged particles in an external Rañada field. We consider the most general actions that are invariant under the Lorentz, electromagnetic gauge and reparametrization transformations. The Finsler geometries form a set parametrized by the gauge fields in each case. We give a simple method to calculate the fundamental objects of the Finsler geometry of the kinematic space of a particle in a generic electromagnetic field. Then we apply this method to calculate the geodesic equations of the spinless and spinning particles. Also, we show that the electromagnetic duality in the Rañada background induces a simple dual map in the set of Finsler geometries. The duality map has a simple interpretation in terms of an electrically charged particle that interacts with the electromagnetic potential and a magnetically charged particle that interacts with the dual magnetoelectric potential. We exemplify the action of the duality map by calculating the dual geodesic equation.

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“…More recent studies have focused on other features of the electromagnetic fields in terms of field lines: in [8,9] the authors studied the connection between the knotted and linked solutions, the dynamics of the electric charges in this background and of the knotted fields were investigated in [11]- [14] and the topological quantization was discussed in [15]- [18]. The generalization of the above solutions to the torus knot topology was given in [19]- [21] and field lines and Hopf solutions were constructed in the nonlinear electromagnetism in [22]- [24]. The importance of the electromagnetic fields in terms of field lines is emphasised by a large range of phenomena in which they seem to be present: in fluid physics [23,24], atmospheric physics [25], liquid crystals [26], plasma physics [27], optical vortices [28,29] and superconductivity [30].…”

Section: Introductionmentioning

confidence: 99%

“…The generalization of the above solutions to the torus knot topology was given in [19]- [21] and field lines and Hopf solutions were constructed in the nonlinear electromagnetism in [22]- [24]. The importance of the electromagnetic fields in terms of field lines is emphasised by a large range of phenomena in which they seem to be present: in fluid physics [23,24], atmospheric physics [25], liquid crystals [26], plasma physics [27], optical vortices [28,29] and superconductivity [30]. (For a review of the subject and some of its applications see [31] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On the existence of the field line solutions of the Einstein–Maxwell equations

Vancea

2018

Int. J. Geom. Methods Mod. Phys.

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The main result of this paper is the proof that there are local electric and magnetic field configurations expressed in terms of field lines on an arbitrary hyperbolic manifold. This electromagnetic field is described by (dual) solutions of the Maxwell's equations of the Einstein-Maxwell theory. These solutions have the following important properties: i) they are general, in the sense that the knot solutions are particular cases of them and ii) they reduce to the electromagnetic fields in the field line representation in the flat spacetime. Also, we discuss briefly the real representation of these electromagnetic configurations and write down the corresponding Einstein equations.

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Knotted solutions, from electromagnetism to fluid dynamics

Cited by 67 publications

References 50 publications

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Lagrangian Formulation, Generalizations and Quantization of Null Maxwell's Knots

Finsler geometries from topological electromagnetism

On the existence of the field line solutions of the Einstein–Maxwell equations

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