Geometry of the isotropic oscillator driven by the conformal mode

Galajinsky, Anton

doi:10.1140/epjc/s10052-018-5568-8

Cited by 10 publications

(14 citation statements)

References 17 publications

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“…The scalar charged particle in the monopole background that we considered is subjected to a central potential V (r) = α 2mr 2 + mω 2 2 r 2 , which is a three-dimensional analog of the AFF model's potential, and therefore the system posseses the conformal Newton-Hooke symmetry [54,55,56,57]. For coupling constant α = ν 2 , trajectories are closed only for some particular initial conditions.…”

Section: Discussion and Outlookmentioning

confidence: 99%

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Hidden symmetry and (super)conformal mechanics in a monopole background

Inzunza

Plyushchay

Wipf

2020

J. High Energ. Phys.

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We study classical and quantum hidden symmetries of a particle with electric charge e in the background of a Dirac monopole of magnetic charge g subjected to an additional central potential V (r) = U (r) + (eg) 2 /2mr 2 with U (r) = 1 2 mω 2 r 2 , similar to that in the one-dimensional conformal mechanics model of de Alfaro, Fubini and Furlan (AFF). By means of a non-unitary conformal bridge transformation, we establish a relation of the quantum states and of all symmetries of the system with those of the system without harmonic trap, U (r) = 0. Introducing spin degrees of freedom via a very special spin-orbit coupling, we construct the osp(2|2) superconformal extension of the system with unbroken N = 2 Poincaré supersymmetry and show that two different superconformal extensions of the one-dimensional AFF model with unbroken and spontaneously broken supersymmetry have a common origin. We also show a universal relationship between the dynamics of a Euclidean particle in an arbitrary central potential U (r) and the dynamics of a charged particle in a monopole background subjected to the potential V (r).To solve the vector equation in (2.4) in the case J 2 > ν 2 , which we will assume in what follows, we decompose n into the component parallel to the angular momentum and the orthogonal component,whereĴ is the unit vector in the direction of J . Since the parallel component n is constant, we conclude that the orthogonal component n ⊥ (t) has constant length and thus describes a circle in the plane orthogonal to J , n ⊥ (t) = n ⊥ (0) cos ϕ(t) +Ĵ × n ⊥ (0) sin ϕ(t) .(2.15) Here, θ, ϕ and ψ are the Euler angles, 0 ≤ ϕ, ψ < 2π, 0 ≤ θ < π, and J i J i = K a K a = J 2 . The common eigenstates of J 2 , J 3 and K 3 satisfying relations

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Section: Discussion and Outlookmentioning

confidence: 99%

“…In this subsection we derive the conformal Newton-Hooke symmetry [54,55,56,57] for the system (2.1) and compare it with the conformal symmetry of the model with ω = 0 studied in [27].…”

Section: Conformal Newton-hooke Symmetrymentioning

confidence: 99%

Hidden symmetry and (super)conformal mechanics in a monopole background

Inzunza

Plyushchay

Wipf

2020

J. High Energ. Phys.

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“…(3.11) Integrals H g , D and K of the 'regularized' AFF system generate the Newton-Hooke symmetry [55,56,10]…”

Section: Aff Conformal Mechanics Modelmentioning

confidence: 99%

“…We show that at half-integer ν, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.constant g = ν(ν + 1) ≥ −1/4 1 . Its non-relativistic conformal symmetry and supersymmetric extensions [5,6,7,8,9,10] find a variety of interesting applications including the particles dynamics in black hole backgrounds [11,12,13,14,15,16], cosmology [17,18,19], non-relativistic AdS/CFT correspondence [20,21,22,23,24], QCD confinement problem [25,26], and physics of Bose-Einstein condensates [27,28].On the other hand, the time-dependent Schrödinger equation for the AFF conformal mechanics model reveals a discrete Klein four-group symmetry generated by transformation of the parameter ν → −ν − 1, and by the spatial Wick rotation x → ix accompanied by the time reflection t → −t. In the picture of the stationary Schrödinger equation the time reflection transforms into the change of the eigenvalue's sign E → −E.…”

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confidence: 99%

Klein four-group and Darboux duality in conformal mechanics

Inzunza

Plyushchay

2019

Phys. Rev. D

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We study the Klein four-group (K 4 ) symmetry of the time-dependent Schrödinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant g = ν(ν + 1) ≥ −1/4. We show that it undergoes a complete or partial (at half-integer ν) breaking on eigenstates of the system, and is the automorphism of the osp(2, 2) superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of N = 2 Poincaré supersymmetry. We exploit the K 4 symmetry and its relation with the conformal symmetry to construct the dual Darboux transformations which generate spectrally shifted pairs of the rationally deformed AFF models. Two distinct pairs of intertwining operators originated from Darboux duality allow us to construct complete sets of the spectrum generating ladder operators that identify specific finite-gap structure of a deformed system and generate three distinct related versions of nonlinearly deformed sl(2, R) algebra as its symmetry. We show that at half-integer ν, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.constant g = ν(ν + 1) ≥ −1/4 1 . Its non-relativistic conformal symmetry and supersymmetric extensions [5,6,7,8,9,10] find a variety of interesting applications including the particles dynamics in black hole backgrounds [11,12,13,14,15,16], cosmology [17,18,19], non-relativistic AdS/CFT correspondence [20,21,22,23,24], QCD confinement problem [25,26], and physics of Bose-Einstein condensates [27,28].On the other hand, the time-dependent Schrödinger equation for the AFF conformal mechanics model reveals a discrete Klein four-group symmetry generated by transformation of the parameter ν → −ν − 1, and by the spatial Wick rotation x → ix accompanied by the time reflection t → −t. In the picture of the stationary Schrödinger equation the time reflection transforms into the change of the eigenvalue's sign E → −E. The discrete symmetry K 4 , however, turns out to be completely broken at the level of the quantum states when ν is not a half-integer number : application of the group generators to physical eigenstates produces formal eigenstates which do not satisfy the necessary boundary conditions. In the case of half-integer values of the parameter ν the K 4 discrete symmetry breaks partially, and transformation ν → −ν − 1, as we shall see, turns out to be a true symmetry nontrivially realized on the spectrum of the system. In the physics of anyons, where the AFF model is used to generate the transmutation of statistics, halfinteger values of ν correspond to the two-particle system of identical fermions [29,30,31]. In the context of the problem we consider here, even though the new solutions with arbitrary value of ν generated by transformations of the discrete group are not acceptable from the physical point of view, the analogs of such non-physical states in other q...

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“…Recent studies of dynamical realizations of the non-relativistic conformal algebras [1,2] revealed an interesting extension of the 1d conformal mechanics by de Alfaro, Fubini, and Furlan [3]. It describes a particle parametrized by the coordinates x i , i = 1, 2, 3, which moves along an ellipse and undergoes periods of accelerated/decelerated motion controlled by the conformal mode ρ(t)…”

Section: Introductionmentioning

confidence: 99%

A variant of Schwarzian mechanics

Galajinsky

2018

Nuclear Physics B

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The Schwarzian derivative is invariant under SL(2, R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2, R)-invariant 1d mechanics or the Schwarzian mechanics for short. In this note, we consider the simplest variant which results from setting the Schwarzian derivative to be equal to a dimensionful coupling constant. It is shown that the corresponding dynamical system in general undergoes stable evolution but for one fixed point solution which is only locally stable. Conserved charges associated with the SL(2, R)-symmetry transformations are constructed and a Hamiltonian formulation reproducing them is proposed. An embedding of the Schwarzian mechanics into a larger dynamical system associated with the geodesics of a Brinkmann-like metric obeying the Einstein equations is constructed.

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Geometry of the isotropic oscillator driven by the conformal mode

Cited by 10 publications

References 17 publications

Hidden symmetry and (super)conformal mechanics in a monopole background

Hidden symmetry and (super)conformal mechanics in a monopole background

Klein four-group and Darboux duality in conformal mechanics

A variant of Schwarzian mechanics

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