Abstract
Hidden symmetries of non-relativistic \( \mathfrak{so}\left(2,1\right)\cong \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) \) 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Cariglia, Hidden Symmetries of Dynamics in Classical and Quantum Physics, Rev. Mod. Phys. 86 (2014) 1283 [arXiv:1411.1262] [INSPIRE].
J. de Boer, F. Harmsze and T. Tjin, Nonlinear finite W symmetries and applications in elementary systems, Phys. Rept. 272 (1996) 139 [hep-th/9503161] [INSPIRE].
J. Beckers, Y. Brihaye and N. Debergh, On realizations of nonlinear Lie algebras by differential operators and some physical applications, J. Phys. A 32 (1999) 2791 [hep-th/9803253] [INSPIRE].
W. Pauli, Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik, Z. Phys. 36 (1926) 336 [INSPIRE].
D. M. Fradkin, Three-dimensional isotropic harmonic oscillator and SU3, Am. J. Phys. 33 (1965) 207.
M. S. Plyushchay and A. Wipf, Particle in a self-dual dyon background: hidden free nature, and exotic superconformal symmetry, Phys. Rev. D 89 (2014) 045017 [arXiv:1311.2195] [INSPIRE].
L. Inzunza, M. S. Plyushchay and A. Wipf, Hidden symmetry and (super)conformal mechanics in a monopole background, JHEP 04 (2020) 028 [arXiv:2002.04341] [INSPIRE].
S. Wojciechowski, Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983) 279.
V. B. Kuznetsov, Hidden symmetry of the quantum Calogero-Moser system, Phys. Lett. A 218 (1996) 212 [solv-int/9509001] [INSPIRE].
S. W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
R. Britto-Pacumio, J. Michelson, A. Strominger and A. Volovich, Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces, NATO Sci. Ser. C 556 (2000) 255 [hep-th/9911066] [INSPIRE].
P. Claus, M. Derix, R. Kallosh, J. Kumar, P. K. Townsend and A. Van Proeyen, Black holes and superconformal mechanics, Phys. Rev. Lett. 81 (1998) 4553 [hep-th/9804177] [INSPIRE].
J. A. de Azcarraga, J. M. Izquierdo, J. C. Perez Bueno and P. K. Townsend, Superconformal mechanics and nonlinear realizations, Phys. Rev. D 59 (1999) 084015 [hep-th/9810230] [INSPIRE].
G. W. Gibbons and P. K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [INSPIRE].
V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
A. Galajinsky, O. Lechtenfeld and K. Polovnikov, \( \mathcal{N} \) = 4 superconformal Calogero models, JHEP 11 (2007) 008 [arXiv:0708.1075] [INSPIRE].
N. Kozyrev, S. Krivonos, O. Lechtenfeld and A. Sutulin, SU(2|1) supersymmetric mechanics on curved spaces, JHEP 05 (2018) 175 [arXiv:1712.09898] [INSPIRE].
B. Carter, Axisymmetric Black Hole Has Only Two Degrees of Freedom, Phys. Rev. Lett. 26 (1971) 331 [INSPIRE].
G. W. Gibbons, R. H. Rietdijk and J. W. van Holten, SUSY in the sky, Nucl. Phys. B 404 (1993) 42 [hep-th/9303112] [INSPIRE].
M. Cariglia, Quantum mechanics of Yano tensors: Dirac equation in curved spacetime, Class. Quant. Grav. 21 (2004) 1051 [hep-th/0305153] [INSPIRE].
V. P. Frolov and D. Kubiznak, Hidden Symmetries of Higher Dimensional Rotating Black Holes, Phys. Rev. Lett. 98 (2007) 011101 [gr-qc/0605058] [INSPIRE].
V. P. Frolov and D. Kubiznak, Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables, Class. Quant. Grav. 25 (2008) 154005 [arXiv:0802.0322] [INSPIRE].
V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
G. ’t Hooft, Nonperturbative Two Particle Scattering Amplitudes in (2 + 1)-Dimensional Quantum Gravity, Commun. Math. Phys. 117 (1988) 685 [INSPIRE].
S. Deser and R. Jackiw, Classical and Quantum Scattering on a Cone, Commun. Math. Phys. 118 (1988) 495 [INSPIRE].
B. S. Kay and U. M. Studer, Boundary conditions for quantum mechanics on cones and fields around cosmic strings, Commun. Math. Phys. 139 (1991) 103 [INSPIRE].
C. Furtado and F. Moraes, Harmonic oscillator interacting with conical singularities, J. Phys. A 33 (2000) 5513 [INSPIRE].
J. L. A. Coelho and R. L. P. G. Amaral, Coulomb and quantum oscillator problems in conical spaces with arbitrary dimensions, J. Phys. A 35 (2002) 5255 [gr-qc/0111114] [INSPIRE].
C. C. Barros Jr., Quantum mechanics in curved space-time, Eur. Phys. J. C 42 (2005) 119 [physics/0409064] [INSPIRE].
G. De A. Marques, V. B. Bezerra and S. G. Fernandes, Exact solution of the Dirac equation for a Coulomb and scalar potentials in the gravitational field of a cosmic string, Phys. Lett. A 341 (2005) 39 [INSPIRE].
K. Kowalski and J. Rembieliński, On the dynamics of a particle on a cone, Annals of Physics 329 (2013) 146 [arXiv:1304.4412] [INSPIRE].
F. M. Andrade and E. O. Silva, Effects of spin on the dynamics of the 2D Dirac oscillator in the magnetic cosmic string background, Eur. Phys. J. C 74 (2014) 3187 [arXiv:1403.4113] [INSPIRE].
M. Hosseinpour, F. M. Andrade, E. O. Silva and H. Hassanabadi, Scattering and bound states for the Hulthén potential in a cosmic string background, Eur. Phys. J. C 77 (2017) 270 [Erratum ibid. 77 (2017) 373] [arXiv:1608.03558] [INSPIRE].
F. Ahmed, Relativistic quantum dynamics of spin-0 massive charged particle in the presence of external fields in 4D curved space-time with a cosmic string, Eur. Phys. J. Plus 135 (2020) 108 [arXiv:1910.12700] [INSPIRE].
T. W. B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387 [INSPIRE].
A. Vilenkin, Gravitational Field of Vacuum Domain Walls and Strings, Phys. Rev. D 23 (1981) 852 [INSPIRE].
A. Vilenkin, Cosmic Strings and Domain Walls, Phys. Rept. 121 (1985) 263 [INSPIRE].
J. S. Dowker, Quantum Field Theory on a Cone, J. Phys. A 10 (1977) 115 [INSPIRE].
M. Visser, Traversable wormholes: Some simple examples, Phys. Rev. D 39 (1989) 3182 [arXiv:0809.0907] [INSPIRE].
J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford and G. A. Landis, Natural wormholes as gravitational lenses, Phys. Rev. D 51 (1995) 3117 [astro-ph/9409051] [INSPIRE].
M. O. Katanaev and I. V. Volovich, Theory of defects in solids and three-dimensional gravity, Annals Phys. 216 (1992) 1 [INSPIRE].
G. E. Volovik, The universe in a helium droplet Oxford Science Publications (2003) [DOI].
N. S. Manton, Five Vortex Equations, J. Phys. A 50 (2017) 125403 [arXiv:1612.06710] [INSPIRE].
D. D. Sokolov and A. A. Starobinsky, On the structure of curvature tensor on conical singularities, Dokl. Akad. Nauk 234 (1977) 1043 [Sov. Phys. Dokl. 22 (1977) 312].
M. Aryal, L. H. Ford and A. Vilenkin, Cosmic Strings and Black Holes, Phys. Rev. D 34 (1986) 2263 [INSPIRE].
C. Furtado, B. G. C. da Cunha, F. Moraes, E. R. Bezerra de Mello and V. B. Bezerra, Landau levels in the presence of disclinations, Phys. Lett. A 195 (1994) 90 [INSPIRE].
S. N. Solodukhin, The Conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].
M. G. Germano, V. B. Bezerra and E. R. Bezerra de Mello, Gravitational effects due to a cosmic string in Schwarzschild space-time, Class. Quant. Grav. 13 (1996) 2663 [INSPIRE].
E. R. B. de Mello and A. A. Saharian, Vacuum polarization induced by a cosmic string in anti-de Sitter spacetime, J. Phys. A 45 (2012) 115002 [arXiv:1110.2129] [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta 46 (1973) 191 [INSPIRE].
P. D. Alvarez, J. Gomis, K. Kamimura and M. S. Plyushchay, (2 + 1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase, Annals Phys. 322 (2007) 1556 [hep-th/0702014] [INSPIRE].
A. Galajinsky, Conformal mechanics in Newton-Hooke spacetime, Nucl. Phys. B 832 (2010) 586 [arXiv:1002.2290] [INSPIRE].
K. Andrzejewski, Conformal Newton-Hooke algebras, Niederer’s transformation and Pais-Uhlenbeck oscillator, Phys. Lett. B 738 (2014) 405 [arXiv:1409.3926] [INSPIRE].
L. Inzunza, M. S. Plyushchay and A. Wipf, Conformal bridge between asymptotic freedom and confinement, Phys. Rev. D 101 (2020) 105019 [arXiv:1912.11752] [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation., Helv. Phys. Acta 45 (1972) 802 [INSPIRE].
C. R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].
A. O. Barut, Conformal group → Schrödinger group → dynamical group — the maximal kinematical group of the massive Schrödinger particle, Helv. Phys. Acta 46 (1973) 496.
C. Leiva and M. S. Plyushchay, Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence, Annals Phys. 307 (2003) 372 [hep-th/0301244] [INSPIRE].
M. Henkel and J. Unterberger, Schrödinger invariance and space-time symmetries, Nucl. Phys. B 660 (2003) 407 [hep-th/0302187] [INSPIRE].
D. T. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
M. S. Plyushchay, Quantization of the classical SL(2, ℝ) system and representations of SL(2, ℝ) group, J. Math. Phys. 34 (1993) 3954 [INSPIRE].
V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [INSPIRE].
P. A. M. Dirac, Forms of Relativistic Dynamics, Rev. Mod. Phys. 21 (1949) 392 [INSPIRE].
F. Correa, V. Jakubsky and M. S. Plyushchay, P T -symmetric invisible defects and confluent Darboux-Crum transformations, Phys. Rev. A 92 (2015) 023839 [arXiv:1506.00991] [INSPIRE].
L. Inzunza and M. S. Plyushchay, Klein four-group and Darboux duality in conformal mechanics, Phys. Rev. D 99 (2019) 125016 [arXiv:1902.00538] [INSPIRE].
M. de Crombrugghe and V. Rittenberg, Supersymmetric Quantum Mechanics, Annals Phys. 151 (1983) 99 [INSPIRE].
M. S. Plyushchay, Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions, Annals Phys. 245 (1996) 339 [hep-th/9601116] [INSPIRE].
M. Plyushchay, Hidden nonlinear supersymmetries in pure parabosonic systems, Int. J. Mod. Phys. A 15 (2000) 3679 [hep-th/9903130] [INSPIRE].
F. Correa and M. S. Plyushchay, Hidden supersymmetry in quantum bosonic systems, Annals Phys. 322 (2007) 2493 [hep-th/0605104] [INSPIRE].
V. Jakubsky, L.-M. Nieto and M. S. Plyushchay, The origin of the hidden supersymmetry, Phys. Lett. B 692 (2010) 51 [arXiv:1004.5489] [INSPIRE].
R. Bonezzi, O. Corradini, E. Latini and A. Waldron, Quantum Mechanics and Hidden Superconformal Symmetry, Phys. Rev. D 96 (2017) 126005 [arXiv:1709.10135] [INSPIRE].
L. Inzunza and M. S. Plyushchay, Hidden superconformal symmetry: Where does it come from?, Phys. Rev. D 97 (2018) 045002 [arXiv:1711.00616] [INSPIRE].
M. Crampin, Hidden symmetries and killing tensors, Rept. Math. Phys. 20 (1984) 31.
M. S. Plyushchay, The Model of relativistic particle with torsion, Nucl. Phys. B 362 (1991) 54 [INSPIRE].
P. A. Horvathy and M. S. Plyushchay, Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane, JHEP 06 (2002) 033 [hep-th/0201228] [INSPIRE].
P. A. Horvathy and M. S. Plyushchay, Anyon wave equations and the noncommutative plane, Phys. Lett. B 595 (2004) 547 [hep-th/0404137] [INSPIRE].
J. M. Leinaas and J. Myrheim, On the theory of identical particles, Nuovo Cim. B 37 (1977) 1.
J. M. Leinaas and J. Myrheim, Intermediate statistics for vortices in superfluid films, Phys. Rev. B 37 (1988) 9286 [INSPIRE].
R. MacKenzie and F. Wilczek, Peculiar Spin and Statistics in Two Space Dimensions, Int. J. Mod. Phys. A 3 (1988) 2827 [INSPIRE].
C. M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
A. Mostafazadeh, PseudoHermiticity versus PT symmetry. The necessary condition for the reality of the spectrum, J. Math. Phys. 43 (2002) 205 [math-ph/0107001] [INSPIRE].
C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].
A. Fring, PT-symmetric deformations of integrable models, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120046 [arXiv:1204.2291] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, From PT-symmetric quantum mechanics to conformal field theory, Pramana 73 (2009) 217 [arXiv:0906.1130] [INSPIRE].
J. Mateos Guilarte and M. S. Plyushchay, Perfectly invisible PT-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry, JHEP 12 (2017) 061 [arXiv:1710.00356] [INSPIRE].
J. Mateos Guilarte and M. S. Plyushchay, Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal mechanics systems, JHEP 01 (2019) 194 [arXiv:1806.08740] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2012.04613
Rights and permissions
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.
About this article
Cite this article
Inzunza, L., Plyushchay, M.S. Conformal bridge in a cosmic string background. J. High Energ. Phys. 2021, 165 (2021). https://doi.org/10.1007/JHEP05(2021)165
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)165