Abstract
Dynamics of small-amplitude perturbations in the global anti-de Sitter (AdS) spacetime is restricted by selection rules that forbid effective energy transfer between certain sets of normal modes. The selection rules arise algebraically because some integrals of products of AdS mode functions vanish. Here, we reveal the relation of these selection rules to AdS isometries. The formulation we discover through this systematic approach is both simpler and stronger than what has been reported previously. In addition to the selection rule considerations, we develop a number of useful representations for the global AdS mode functions, with connections to algebraic structures of the Higgs oscillator, a superintegrable system describing a particle on a sphere in an inverse cosine-squared potential, where the AdS isometries play the role of a spectrum-generating algebra.
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References
G.M. Chechin and V.P. Sakhnenko, Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results, Physica D 117 (1998) 43.
P.W. Higgs, Dynamical symmetries in a spherical geometry 1, J. Phys. A 12 (1979) 309.
H.I. Leemon, Dynamical symmetries in a spherical geometry. 2, J. Phys. A 12 (1979) 489 [INSPIRE].
P. Bizon and A. Rostworowski, On weakly turbulent instability of Anti-de Sitter space, Phys. Rev. Lett. 107 (2011) 031102 [arXiv:1104.3702] [INSPIRE].
M. Maliborski and A. Rostworowski, Lecture notes on turbulent instability of Anti-de Sitter spacetime, Int. J. Mod. Phys. A 28 (2013) 1340020 [arXiv:1308.1235] [INSPIRE].
P. Bizon, Is AdS stable?, Gen. Rel. Grav. 46 (2014) 1724 [arXiv:1312.5544] [INSPIRE].
B. Craps and O. Evnin, AdS (in)stability: an analytic approach, arXiv:1510.07836 [INSPIRE].
B. Craps, O. Evnin and J. Vanhoof, Renormalization group, secular term resummation and AdS (in)stability, JHEP 10 (2014) 048 [arXiv:1407.6273] [INSPIRE].
B. Craps, O. Evnin and J. Vanhoof, Renormalization, averaging, conservation laws and AdS (in)stability, JHEP 01 (2015) 108 [arXiv:1412.3249] [INSPIRE].
A. Buchel, S.R. Green, L. Lehner and S.L. Liebling, Conserved quantities and dual turbulent cascades in Antide Sitter spacetime, Phys. Rev. D 91 (2015) 064026 [arXiv:1412.4761] [INSPIRE].
P. Basu, C. Krishnan and A. Saurabh, A stochasticity threshold in holography and the instability of AdS, Int. J. Mod. Phys. A 30 (2015) 1550128 [arXiv:1408.0624] [INSPIRE].
P. Basu, C. Krishnan and P.N. Bala Subramanian, AdS (in)stability: lessons from the scalar field, Phys. Lett. B 746 (2015) 261 [arXiv:1501.07499] [INSPIRE].
I.-S. Yang, Missing top of the AdS resonance structure, Phys. Rev. D 91 (2015) 065011 [arXiv:1501.00998] [INSPIRE].
O.J.C. Dias, G.T. Horowitz and J.E. Santos, Gravitational turbulent instability of Anti-de Sitter space, Class. Quant. Grav. 29 (2012) 194002 [arXiv:1109.1825] [INSPIRE].
G.T. Horowitz and J.E. Santos, Geons and the instability of Anti-de Sitter spacetime, arXiv:1408.5906 [INSPIRE].
O. Evnin and C. Krishnan, A hidden symmetry of AdS resonances, Phys. Rev. D 91 (2015) 126010 [arXiv:1502.03749] [INSPIRE].
M. Lakshmanan and K. Eswaran, Quantum dynamics of a solvable nonlinear chiral model, J. Phys. A 8 (1975) 1658 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective conformal theory and the flat-space limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].
G.S. Pogosyan, A.N. Sissakian, S.I. Vinitsky, Interbasis sphere-cylinder expansions for the oscillator in the three-dimensional space of constant positive curvature, in Frontiers of fundamental physics, M. Barone and F. Selleri eds., Springer, Germany (1994).
J. Kaplan, Lectures on AdS/CFT from the bottom up, http://www.pha.jhu.edu/~jaredk/AdSCFTCourseNotesPublic.pdf.
M. Hamermesh, Group theory and its application to physical problems, Dover, U.S.A. (1989).
A.L. Besse, Manifolds all of whose geodesics are closed, Springer, Germany (1978).
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS geometry of conformal blocks, arXiv:1508.00501 [INSPIRE].
B. de Wit and I. Herger, Anti-de Sitter supersymmetry, Lect. Notes Phys. 541 (2000) 79 [hep-th/9908005] [INSPIRE].
W. Miller, Jr., S. Post and P. Winternitz, Classical and quantum superintegrability with applications, J. Phys. A 46 (2013) 423001 [arXiv:1309.2694] [INSPIRE].
P. Cordero and J. Daboul, Analysis of the spectrum generating algebra method for obtaining energy spectra, J. Math. Phys. 46 (2005) 053507.
O.F. Gal’bert, Ya.I. Granovskii and A.S. Zhedanov, Dynamical symmetry of anisotropic singular oscillator, Phys. Lett. A 153 (1991) 177.
Ya.I. Granovskii, A.S. Zhedanov and I.M. Lutsenko, Quadratic algebras and dynamics in curved spaces I: oscillator, Teor. Mat. Fiz. 91 (1992) 207.
Ya.I. Granovskii, A.S. Zhedanov and I.M. Lutsenko, Quadratic algebras and dynamics in curved spaces. II: the Kepler problem, Teor. Mat. Fiz. 91 (1992) 396.
A.S. Zhedanov, The ‘Higgs algebra’ as a ‘quantum’ deformation of SU(2), Mod. Phys. Lett. A 7(1992) 507 [INSPIRE].
D. Bonatsos, C. Daskaloyannis and K.D. Kokkotas, Deformed oscillator algebras for two-dimensional quantum superintegrable systems, hep-th/9309088 [INSPIRE].
P. Létourneau and L. Vinet, Quadratic algebras in quantum mechanics, in Symmetries in science VII, Plenum Press (1994).
C. Daskaloyannis, Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems, J. Math. Phys. 42 (2001) 1100 [math-ph/0003017].
D. Ruan, Single mode realizations of the Higgs algebra, quant-ph/0111056.
D. Ruan, Two-boson realizations of the polynomial angular momentum algebra and some applications, J. Math Chem 39 (2006) 417.
D. Ruan, Two boson realizations of the Higgs algebra and some applications, math-ph/0312002 [INSPIRE].
E.G. Kalnins, J.M. Kress, W. Miller Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006) 093501.
C. Quesne, Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions , SIGMA 3 (2007) 067 [arXiv:0705.2577].
V.X. Genest, L. Vinet and A. Zhedanov, The singular and the 2 : 1 anisotropic Dunkl oscillators in the plane, J. Phys. A 46 (2013) 325201, arXiv:1305.2126.
V.X. Genest, L. Vinet and A. Zhedanov, Superintegrability in two dimensions and the Racah-Wilson algebra, Lett. Math. Phys. 104 (2014) 931 [arXiv:1307.5539].
E.G. Kalnins and W. Miller Jr., Quadratic algebra contractions and 2nd order superintegrable systems, arXiv:1401.0830.
T.R. Govindarajan, P. Padmanabhan and T. Shreecharan, Beyond fuzzy spheres, J. Phys. A 43 (2010) 205203 [arXiv:0906.1660] [INSPIRE].
C. Daboul, J. Daboul and P. Slodowy, The dynamical algebra of the hydrogen atom as a twisted loop algebra, talk given at the XX International Colloquium on Group Theoretical Methods in Physics, August 20-26, Osaka, Japan (1994), hep-th/9408080 [INSPIRE].
S.-L. Zhang, Yangian Y (sl(2)) in Coulomb problem, Helv. Phys. Acta 71 (1998) 586 [INSPIRE].
C.M. Bai, M.L. Ge, K. Xue, Further understanding of hydrogen atom: Yangian approach and physical effect, J. Stat. Phys. 102 (2001) 545.
C.M. Bender and A. Turbiner, Analytic continuation of eigenvalue problems, Phys. Lett. A 173 (1993) 442 [INSPIRE].
C.M. Bender, D.W. Hook and S.P. Klevansky, Negative-energy PT-symmetric Hamiltonians, J. Phys. A 45 (2012) 444003 [arXiv:1203.6590] [INSPIRE].
J.A. Omolo, Positive-negative energy partner states and conjugate quantum polynomials for a linear harmonic oscillator, Fund. J. Math. Sci. 2 (2015) 55.
E.D. Rainville, A relation between Jacobi and Laguerre polynomials, Bull. Amer. Math. Soc. 51 (1945) 266.
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Evnin, O., Nivesvivat, R. AdS perturbations, isometries, selection rules and the Higgs oscillator. J. High Energ. Phys. 2016, 151 (2016). https://doi.org/10.1007/JHEP01(2016)151
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DOI: https://doi.org/10.1007/JHEP01(2016)151