- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Quantum gravity and the Coulomb potential

Viqar Husain1,2,*, Jorma Louko3,†, and Oliver Winkler1,2,‡

  • 1Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada
  • 2Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, N2L 2Y5 ON, Canada
  • 3School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom

  • *husain@math.unb.ca
  • †jorma.louko@nottingham.ac.uk
  • ‡owinkler@perimeterinstitute.ca
PDF

Phys. Rev. D 76, 084002 – Published 3 October, 2007

DOI: https://doi.org/10.1103/PhysRevD.76.084002

Abstract

We apply a singularity-resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on R with the singular −1/|x| potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite-difference approximation of the Schrödinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at x=0, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized.

View figure in article

Article Text

References (26)

  1. B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
  2. J. A. Wheeler, in Battelle Rencontres 1967, edited by C. DeWitt and J. A. Wheeler (Benjamin, New York, 1968).
  3. T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cambridge, England, 2007).
  4. A. Ashtekar, S. Fairhurst, and J. Willis, Classical Quantum Gravity 20, 1031 (2003).
  5. T. Thiemann, Classical Quantum Gravity 15, 839 (1998).
  6. M. Bojowald, Phys. Rev. D 64, 084018 (2001).
  7. V. Husain and O. Winkler, Phys. Rev. D 69, 084016 (2004).
  8. V. Husain and O. Winkler, Classical Quantum Gravity 22, L127 (2005).
  9. J. Brunnemann and T. Thiemann, Classical Quantum Gravity 23, 1395 (2006).
  10. J. Brunnemann and T. Thiemann, Classical Quantum Gravity 23, 1429 (2006).
  11. J. M. Velhinho, Classical Quantum Gravity 24, 3745 (2007).
  12. C. Corduneau, Almost Periodic Functions (Interscience, New York, 1968).
  13. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis (Academic, New York, 1963), Vol. 1.
  14. H. Halvorson, Stud. Hist. Phil. Mod. Phys. 35, 45 (2004).
  15. K. Fredenhagen and F. Reszewski, Classical Quantum Gravity 23, 6577 (2006).
  16. A. Corichi, T. Vukasinac, and J. A. Zapata, arXiv:0704.0007 [Phys. Rev. D (to be published)].
  17. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), 2nd ed.
  18. C. J. Fewster, arXiv:hep-th/9305102.
  19. S. Elaydi, An Introduction to Difference Equations (Springer, New York, 2005), 3rd ed.
  20. A. Gordon, C. Jirauschek, and F. X. Kärtner, Phys. Rev. A 73, 042505 (2006).
  21. A. Ashtekar and M. Bojowald, Classical Quantum Gravity 23, 391 (2006).
  22. P. Hájíček, Nucl. Phys. B603, 555 (2001).
  23. G. T. Horowitz and D. Marolf, Phys. Rev. D 52, 5670 (1995).
  24. R. C. Helling and G. Policastro, arXiv:hep-th/0409182.
  25. J. Lewandowski and D. Marolf, Int. J. Mod. Phys. D 7, 299 (1998).
  26. R. Gambini, J. Lewandowski, D. Marolf, and J. Pullin, Int. J. Mod. Phys. D 7, 97 (1998).

Outline

Information

Phys. Rev. D 76, 084002– Published 3 October, 2007
  • Received 18 July 2007

© 2007 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation