Klein-Gordonization: mapping superintegrable quantum mechanics to resonant spacetimes

Abstract

We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and Anti-de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in Anti-de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of differential geometry. As an illustration, we explicitly demonstrate how to apply this procedure to superintegrable Rosochatius systems, resulting in a large family of spacetimes with resonant spectra for massless wave equations.

v2: minor improvements, to appear in PRD

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Cited by in corpus (11)

• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Disformal Transition of a Black Hole to a Wormhole in Scalar-Tensor Horndeski Theory
• Supersymmetric hyperbolic Calogero-Sutherland models by gauging
• Complex plane representations and stationary states in cubic and quintic resonant systems
• Energy returns in global AdS_4
• Fully resonant scalars on asymptotically AdS wormholes
• ${\mathbb{CP}}^N$-Rosochatius system, superintegrability, supersymmetry
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Spectroscopy instead of scattering: particle experimentation in AdS spacetime
• A superintegrable quantum field theory
• Superintegrable deformations of oscillator and Coulomb systems