Breaking \(so(4)\) Symmetry Without Degeneracy Lift

  • M. Kirchbach
  • A. Pallares Rivera
  • F. de J. Rosales Aldape
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 111)


We consider on S R 3 the quantum motion of a scalar particle of mass m, perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, , the 3D orbital angular momentum, and another, an external scale introducing continuous parameter, B. We show that a loss of the geometric hyper-spherical so(4) symmetry of the free motion can occur that leaves intact the unperturbed \(\mathcal{N}^{2}\)-fold degeneracy patterns, with \(\mathcal{N} = (\ell+n + 1)\) and n denoting the nodes of the wave function. Our point is that although the number of degenerate states for any \(\mathcal{N}\) matches dimensionality of an irreducible so(4) representation space, the corresponding set of wave functions do not transform irreducibly under any so(4). Indeed, in expanding the Scarf I wave functions in the basis of properly identified so(4) representation functions, we find power series in the perturbation parameter, B, where 4D angular momenta \(K \in [\ell,\mathcal{N}- 1]\) contribute up to the order \(\mathcal{O}\left (\frac{2mR^{2}B} {\hslash ^{2}} \right )^{\mathcal{N}-1-K}\). In this fashion, we work out an explicit example on a symmetry breakdown by external scales that retains the degeneracy. The scheme extends to so(d + 2) for any d.


Orbital Angular Momentum Conformal Symmetry Jacobi Polynomial Casimir Operator External Scale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



One of us (M.K.) thanks the organizers of the “Lie Theory and Its Applications In Physics” 2013 conference in Varna for their hospitality and efforts. We are indebted to Dr. Jose Antonio Vallejo for helpful comments and Jose Limon Castillo for assistance in computer matters.


  1. 1.
    Abramovic, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Tables. Dover Publications, New York (1972)Google Scholar
  2. 2.
    Beringer, J. et al. (Particle Data Group): Review of particle physics. Phys. Rev. D 86, 010001 (2012)Google Scholar
  3. 3.
    Deur, A., Burker, V., Chen, J.P., Korsch, W.: Determination of the effective strong coupling constant alpha(s, g(1)) (Q**2) from CLAS spin structure function data. Phys. Lett. B665, 349–351 (2008)CrossRefGoogle Scholar
  4. 4.
    Fock, V.A.: Zur Theorie des Wasserstoffatoms. Z. Phys. 98, 145–154 (1935)Google Scholar
  5. 5.
    Hands, S., Hollowood, T.J., Myers, J.C.: QCD with chemical potential in a small hyperspherical box. JHEP 1007, 086 (2010)CrossRefGoogle Scholar
  6. 6.
    Kalnins, E., Miller, W., Pogosyan, G.S.: The Coulomb-oscillator relation on n-dimensional spheres. Phys. At. Nucl. 65, 1086–1094 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Kirchbach, M., Compean, C.: Conformal symmetry and light flavor baryon spectra. Phys. Rev. D82, 034008 (2010)Google Scholar
  8. 8.
    Levai, G.: Solvable potentials associated with su(1,1) algebras:a systematic study. J. Phys. A Math. Gen. 27, 3809–3828 (1994)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Lüscher, M., Mack, G.: Global conformal invariance in quantum field theory. Commun. Math. Phys. 41, 203–234 (1975)CrossRefGoogle Scholar
  10. 10.
    Witten, E.: Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2, 233–291 (1998)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • M. Kirchbach
    • 1
  • A. Pallares Rivera
    • 1
  • F. de J. Rosales Aldape
    • 1
  1. 1.Institute of PhysicsAutonomous University at San Luis PotosiSan Luis PotosiMexico

Personalised recommendations