Abstract
Hidden symmetries of non-relativistic mathfrak{so}left(2,1right)cong mathfrak{sl}left(2,mathrm{mathbb{R}}right) 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.
Highlights
Hole, which attracted attention in the context of AdS/CFT correspondence [12,13,14,15]
The systems are quantized for arbitrary values of α; the well-defined hidden symmetry operators associated with spectral degeneracies only exist when α is an integer, that reveals a quantum anomaly
Our goal here is to study the influence of the geometrical properties of this background on the dynamics of the systems from the perspective of well-defined integrals of motion in the phase space when considering classical cases, and well-defined symmetry operators for the corresponding quantum versions of the systems
Summary
Cosmic strings are hypothetical one-dimensional topological defects which may have formed in the spontaneous symmetry-breaking phase transition in the expanding Universe [36,37,38,39]. The space-time metric (2.1) with spatial part presented in the form (2.11) looks like the metric of (2+1)-dimensional Minkowski space dS2 = ημνdXμdXν, X0 = ct, ημν = diag (−1, 1, 1) It is conformally invariant under transformations of the conformal SO(3, 2) group, whose classical generators are P μ, Jμν = XμP ν − XνP μ, Kμ = 2Xμ(XP ) − X2P μ and D = XP , where Pμ = ημνP ν are the momenta canonically conjugate to Xμ. The change of coordinates (2.10) and the geodesic analysis presented in section 5 will show that the non-relativistic dynamics in the cosmic string background can be related to the free motion in the Euclidean plane Bearing this in mind, instead of jumping directly to the analysis of the dynamics in the conical geometry, it is appropriate to review some important characteristics related to the motion in R2
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