Conformal bridge in a cosmic string background

Abstract

Hidden symmetries of non-relativistic so(2, 1) ∼ = sl(2, R) invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the "geometrical parameter" α, determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at α = q/k with q, k = 1, 2, . . . , which for q > 1 are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of α; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when α is an integer, that reveals a quantum anomaly.

Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background

“…The dynamics of test particles in rigidly rotating spacetime backgrounds is essentially affected by the appearance of gravitoelectromagnetic fields at the classical and quantum levels. For special values of the parameters of such and similar systems, classical dynamics of test particles can be completely integrable due to appearance of hidden symmetries, which also reveal themselves in peculiar properties of the corresponding quantum systems [5,6,10,11,12,17,39]. This section aims to investigate such effects for geodesic motion in a rotating conical background.…”

“…For a given rational value (3.21) of γ, the true integrals L (ǫ),± α,s 1 ,s 2 and the dynamical quantities J (δ),± α,s 1 ,s 2 generate a non-linear algebra, which in the Euclidean isotropic case α = 1, γ = 0 reduces to the linear sp(4, R) algebra [17,39]. As in the Euclidean case, the transformation γ → 1/γ changes the system…”

“…When R 1 = R 2 , the path goes through the origin, however this case corresponds to the "falling to the center", which is excluded from our analysis. Using this information, one can draw the trajectory of the particle in the (x 1 , x 2 ) plane when α = q by following the geometric arguments similar to those for the free particle in conical background of Ref [17]. In this context, the complex integrals b…”

“…n (z) to be the generalized Laguerre polynomials. For more details on application of this technique to other models, see [17,39,29]. Here, we just take into account that the Hamiltonian of our conical ERIHO system is a linear combination of the integrals J (α) 0 and L (α) , the quantum versions of which are simultaneously diagonalized by wave functions (4.17).…”

“…Since the structure of this operator is a linear combination of the Hamiltonian of the isotropic harmonic oscillator in the cone [17] and the angular momentum operator, it is easy to show that the eigenstates correspond to (4.17), and the spectrum is…”