Advertisement

Minimal realization of ℓ-conformal Galilei algebra, Pais-Uhlenbeck oscillators and their deformation

  • Sergey Krivonos
  • Olaf LechtenfeldEmail author
  • Alexander Sorin
Open Access
Regular Article - Theoretical Physics

Abstract

We present the minimal realization of the ℓ-conformal Galilei group in 2+1 dimensions on a single complex field. The simplest Lagrangians yield the complex PaisUhlenbeck oscillator equations. We introduce a minimal deformation of the ℓ = 1/2 conformal Galilei (a.k.a. Schrödinger) algebra and construct the corresponding invariant actions. Based on a new realization of the d = 1 conformal group, we find a massive extension of the near-horizon Kerr-dS/AdS metric.

Keywords

Conformal and W Symmetry Space-Time Symmetries Black Holes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    P. Havas and J. Plebanski, Conformal extensions of the Galilei group and their relation to the Schrödinger group, J. Math. Phys. 19 (1978) 482.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Henkel, Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems, Phys. Rev. Lett. 78 (1997) 1940 [cond-mat/9610174] [INSPIRE].
  3. [3]
    M. Henkel, Phenomenology of local scale invariance: From conformal invariance to dynamical scaling, Nucl. Phys. B 641 (2002) 405 [hep-th/0205256] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Negro, M.A. del Olmo and A. Rodriguez-Marco, Nonrelativistic conformal groups, J. Math. Phys. 38 (1997) 3786.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J.A. de Azcarraga, F.J. Herranz, J.C. Perez Bueno and M. Santander, Central extensions of the quasiorthogonal Lie algebras, J. Phys. A 31 (1998) 1373 [q-alg/9612021] [INSPIRE].
  6. [6]
    Y.-H. Gao, Symmetries, matrices and de Sitter gravity, Conf. Proc. C 0208124 (2002) 271 [hep-th/0107067] [INSPIRE].Google Scholar
  7. [7]
    A. Galajinsky and I. Masterov, Remarks on l-conformal extension of the Newton-Hooke algebra, Phys. Lett. B 702 (2011) 265 [arXiv:1104.5115] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    K. Andrzejewski, A. Galajinsky, J. Gonera and I. Masterov, Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator, Nucl. Phys. B 885 (2014) 150 [arXiv:1402.1297] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Masterov, Dynamical realizations of N = 1 l-conformal Galilei superalgebra, J. Math. Phys. 55 (2014) 102901 [arXiv:1407.1438] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Galajinsky and I. Masterov, Dynamical realizations of l-conformal Newton-Hooke group, Phys. Lett. B 723 (2013) 190 [arXiv:1303.3419] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Fedoruk, E. Ivanov and J. Lukierski, Galilean Conformal Mechanics from Nonlinear Realizations, Phys. Rev. D 83 (2011) 085013 [arXiv:1101.1658] [INSPIRE].ADSGoogle Scholar
  12. [12]
    D. Martelli and Y. Tachikawa, Comments on Galilean conformal field theories and their geometric realization, JHEP 05 (2010) 091 [arXiv:0903.5184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    K. Andrzejewski, Hamiltonian formalisms and symmetries of the Pais-Uhlenbeck oscillator, Nucl. Phys. B 889 (2014) 333 [arXiv:1410.0479] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    I. Masterov, Remark on higher-derivative mechanics with l-conformal Galilei symmetry, J. Math. Phys. 57 (2016) 092901 [arXiv:1607.02693] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
  16. [16]
    C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
  17. [17]
    D.V. Volkov, Phenomenological lagrangians, Sov. J. Part. Nucl. 4 (1973) 3.MathSciNetGoogle Scholar
  18. [18]
    V.I. Ogievetsky, Nonlinear realizations of internal and space-time symmetries, in: Proceedings of the Xth Winter School of Theoretical Physics in Karpacz, Vol.1, (1974), p. 117.Google Scholar
  19. [19]
    E.A. Ivanov and V.I. Ogievetsky, The Inverse Higgs Phenomenon in Nonlinear Realizations, Teor. Mat. Fiz. 25 (1975) 164 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Pais and G.E. Uhlenbeck, On field theories with nonlocalized action, Phys. Rev. 79 (1950) 145 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D. Chernyavsky and A. Galajinsky, Ricci-flat spacetimes with l-conformal Galilei symmetry, Phys. Lett. B 754 (2016) 249 [arXiv:1512.06226] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    A. Galajinsky and K. Orekhov, On the near horizon rotating black hole geometries with NUT charges, Eur. Phys. J. C 76 (2016) 477 [arXiv:1604.08056] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    D. Chernyavsky, Coset spaces and Einstein manifolds with l-conformal Galilei symmetry, Nucl. Phys. B 911 (2016) 471 [arXiv:1606.08224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Galajinsky, Conformal mechanics in Newton-Hooke spacetime, Nucl. Phys. B 832 (2010) 586 [arXiv:1002.2290] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    S. Bellucci and S. Krivonos, Potentials in N = 4 superconformal mechanics, Phys. Rev. D 80 (2009) 065022 [arXiv:0905.4633] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205 [Teor. Mat. Fiz. 65 (1985) 347] [INSPIRE].
  27. [27]
    A.M. Polyakov, Gauge Transformations and Diffeomorphisms, Int. J. Mod. Phys. A 5 (1990) 833 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys. 139 (1991) 71 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J.M. Bardeen and G.T. Horowitz, The Extreme Kerr throat geometry: A vacuum analog of AdS 2 × S 2, Phys. Rev. D 60 (1999) 104030 [hep-th/9905099] [INSPIRE].ADSMathSciNetGoogle Scholar
  30. [30]
    U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta 46 (1973) 191 [INSPIRE].Google Scholar
  31. [31]
    U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation., Helv. Phys. Acta 45 (1972) 802 [INSPIRE].MathSciNetGoogle Scholar
  32. [32]
    B. Bina and M. Günaydin, Real forms of nonlinear superconformal and quasisuperconformal algebras and their unified realization, Nucl. Phys. B 502 (1997) 713 [hep-th/9703188] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    E. Ivanov, S. Krivonos and A.S. Sorin, N = 2 super-W 3(2) algebra in superfields, Mod. Phys. Lett. A 10 (1995) 2439 [hep-th/9505142] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sergey Krivonos
    • 1
  • Olaf Lechtenfeld
    • 2
    Email author
  • Alexander Sorin
    • 1
    • 3
    • 4
  1. 1.Bogoliubov Laboratory of Theoretical Physics, JINRDubnaRussia
  2. 2.Institut für Theoretische Physik and Riemann Center for Geometry and PhysicsLeibniz Universität HannoverHannoverGermany
  3. 3.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia
  4. 4.Dubna International UniversityDubnaRussia

Personalised recommendations