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

Abstract

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

Conformal bridge transformation and PT symmetry

“…The non-relativistic conformal symmetry occurs naturally in a wide variety of physical phenomena, and attracted recently a lot of attention in the context of non-relativistic AdS/CFT correspondence [10,11,12,13], black hole physics [14,15,16], cosmology [17,18,19,20], AdS/CDM correspondence [21,22,23,24] and QCD confinement [25,26], to name a few. The non-relativistic conformal symmetry of the free particle and its generalizations lie in the base of the so-called conformal bridge transformation (CBT) [27,28,29,30] by which the dynamics and symmetries of an asymptomatically free conformally invariant system can be mapped into those of the associated in a certain way harmonically trapped system. This corresponds to the picture described in the Dirac seminal article [31], where different forms of dynamics are studied by choosing, in the general case, a linear combination of the generators of a given symmetry as the Hamiltonian of the system.…”

“…As the transformation is based on the algebraic arguments, it can be applied to systems in any conformallyinvariant space-time and gauge backgrounds. In this way, it was employed to study the dynamics and hidden symmetries in backgrounds of the Dirac monopole [28] and cosmic string [29]. One of the goals of this article is to review how this transformation works and the scope of its applications.…”

“…In fact, it is expected that the possibilities to connect the systems by the CBT expand with increasing the number of dimensions and with the conformally invariant change of the space-time metric. Additionally, some hints on a possible close relationship of the CBT with PTsymmetric systems [33,34,35,36] were indicated in the original works [27,28,29,30]. They are based on the fact that at the quantum level, the transformation is realized by a non-unitary operator Ŝ that transforms a non-Hermitian operator i D, where D is a generator of dilations, into the Hermitian compact generator of the sl(2, R) symmetry which has a real discrete spectrum.…”

Generalized spinning particles on $${\mathcal {S}}^2$$ in accord with the Bianchi classification

“…More recently, the focus of research shifted to the study of superconformal mechanics extended by spin degrees of freedom [2][3][4][5][6][7][8][9][10][11][12]. Such variables typically arise when gauging U (n) isometry of the matrix superfield systems [2] or supersymmetrizing the Euler-type extension of the Calogero model [5].…”

Conformal bridge in a cosmic string background

“…Integrals of this type are higher-order functions of the canonical momenta, and they may generate nonlinear symmetry algebras [2,3]. Some examples of integrals related to hidden symmetries are the Laplace-Runge-Lenz vector for the Kepler-Coulomb problem [4], the Fradkin tensor for the isotropic harmonic oscillator [5], the analogs of these integrals in a monopole background [6,7], the higher-order symmetry generators of the anisotropic harmonic oscillator with commensurable frequencies [2], and the N integrals in involution in the Calogero models of N particles [8,9].…”