Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 029, 12 pages      math-ph/0602066      http://dx.doi.org/10.3842/SIGMA.2006.029

Large-j Expansion Method for Two-Body Dirac Equation

Askold Duviryak
Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine, 1 Svientsitskii Str., Lviv, 79011 Ukraine

Received December 01, 2005, in final form February 15, 2006; Published online February 28, 2006

Abstract
By using symmetry properties, the two-body Dirac equation in coordinate representation is reduced to the coupled pair of radial second-order differential equations. Then the large-j expansion technique is used to solve a bound state problem. Linear-plus-Coulomb potentials of different spin structure are examined in order to describe the asymptotic degeneracy and fine splitting of light meson spectra.

Key words: Breit equation, two body Dirac equation, large-N expansion, Regge trajectories.

pdf (248 kb)   ps (166 kb)   tex (17 kb)

References

  1. Breit G., The effect of retardation on the interaction of two electrons, Phys. Rev., 1929, V.34, 553-573.
  2. Barut A.O., Komy S., Derivation of nonperturbative reativistic two-body equation from the action principle in quantum electrodynamics, Fortschr. Phys., 1985, V.33, 309-318.
  3. Barut A.O., Ünal N., A new approach to bound-state quantum electrodynamics, Phys. A, 1987, V.142, 467-487.
  4. Grandy W.T.Jr., Relativistic quantum mechanics of leptons and fields, Dordrecht - Boston - London, Kluwer Academic Publishers, 1991.
  5. Darewych J.W.. Di Leo L., Two-fermion Dirac-like eigenstates of the Coulomb QED Hamiltonian, J. Phys. A: Math. Gen., 1996, V.29, 6817-6841.
  6. Darewych J.W., Few-particle eigenstates in the Yukawa model, Condensed Matter Physics, 1998, V.1, N 3(15), 593-604.
  7. Darewych J.W., Duviryak A., Exact few-particle eigenstates in partially reduced QED, Phys. Rev. A, 2002, V.66, 032102, 20 pages, nucl-th/0204006.
  8. Duviryak A., Darewych J.W., Variational wave equations of two fermions interacting via scalar, pseudoscalar, vector, pseudovector and tensor fields, Cent. Eur. J. Phys., 2005, V.3, N 3, 1-17.
  9. Fushchich W.I., Nikitin A.G., On the new constants of motion for two- and three-particle equations, J. Phys. A: Math. Gen., 1990, V.23, L533-L535.
  10. Nikitin A.G., Fushchich W.I., Non-Lie integrals of the motion for particles of arbitrary spin and for systems of interacting particles, Teor. Mat. Fiz., 1991, V.88, 406-515 (English transl.: Theor. Math. Phys., 1991, V.88, 960-967).
  11. Simenog I.V., Turovsky A.I., A relativistic model of the two-nucleon problem with direct interaction, Ukraïn. Fiz. Zh., 2001, V.46, 391-401 (in Ukrainian).
  12. Simenog I.V., Turovsky A.I., The model of deuteron in Dirac-Breit approach with direct interaction, J. Phys. Studies, 2004, V.8, 23-34 (in Ukrainian).
  13. Krolikowski W., Relativistic radial equations for 2 spin-1/2 particles with a static interaction, Acta Phys. Polon. B, 1976, V.7, 485-496.
  14. Childers R.W., Effective Hamiltonians for generalized Breit interactions in QCD, Phys. Rev. D, 1987, V.36, 606-614.
  15. Brayshaw D.D., Relativistic description of quarkonium, Phys. Rev. D, 1987, V.36, 1465-1478.
  16. Tsibidis G.D., Quark-antiquark bound states and the Breit equation, Acta Phys. Polon. B, 2004, V.35, 2329-2366, hep-ph/0007143.
  17. Khelashvili A.A., Radial quasipotential equation for a fermion and antifermion and infinitely rising central potentials, Teor. Mat. Fiz., 1982, V.51, 201-210 (English transl.: Theor. Math. Phys., 1982, V.51, 447-453).
  18. Crater H.W., Wong C.W. and Wong C.-Y., Singularity-free Breit equation from constraint two-body Dirac equations, Internat. J. Modern Phys. E, 1996, V.5, 589-615, hep-ph/9603402.
  19. Mlodinov L.D., Shatz M.P., Solving the Schrödinger equation with use of 1/N perturbation theory, J. Math. Phys., 1984, V.25, 943-950.
  20. Imbo T., Pagnamenta A. And Sukhatme U., Energy eigenstates of spherically symmetric potentials using the shifted 1/N expansion, Phys. Rev. D, 1984, V.29, 1669-1681.
  21. Vakarchuk I.O., The 1/N-expansion in quantum mechanics. High-order approximations, J. Phys. Studies 2002, V.6, 46-54.
  22. Mustafa O., Barakat T., Nonrelativistic shifted-l expansion technique for three- and two-dimensional Schrödinger equation, Commun. Theor. Phys., 1997, V.28, 257-264, math-ph/9910040.
  23. Mustafa O., Barakat T., Relativistic shifted-l expansion technique for Dirac and Klein-Gordon equations, Commun. Theor. Phys., 1998, V.29, 587-594, math-ph/9910039.
  24. Todorov I.T., Quasipotential equation correspondong to the relativistic eiconal approximation, Phys. Rev. D, 1971, V.3, 2351-2356.
  25. Rizov V.A., Sazdian H., Todorov I.T., On the relativistic quantum mechanics of two interacting spinless particles, Ann. of Phys. (NY), 1985, V.165, 59-97.
  26. Duviryak A., Heuristic models of two-fermion relativistic systems with field-type interaction, J. Phys. G, 2002, V.28, 2795-2809, nucl-th/0206048.
  27. Lucha W., Schoberl F.F., Gromes D., Bound states of quarks, Phys. Rep., 1991, V.200, Issue 4, 127-240.
  28. Eddington A.S., The charge of an electron, R. Soc. Lond. Proc. Ser. A, 1929, V.122, N 789, 358-369.
  29. Gaunt J.A., The triplets of Helium, Philos. Trans. R. Soc. Lond. Ser. A, 1929, V.228, 151-196.
    Gaunt J.A., The triplets of Helium, R. Soc. Lond. Proc. Ser. A, 1929, V.122, N 790, 513-532.
  30. Salpeter E.E., Mass corrections to the fine structure of hydrogen-like atoms, Phys. Rev., 1952, V.87, 328-343.
  31. Faustov R.N., The proton structure and hyperfine splitting of hydrogen energy levels, Nucl. Phys., 1966, V.75, 669-681.
  32. Khelashvili A.A., Quasipotential equation for the system of two particles with spin 1/2, Communications of the Joint Institute for Nuclear Physics, P2-4327, Dubna, 1969 (in Russian).
  33. Long C., Robson D., Bound states of a relativistic quark confined by a vector potential, Phys. Rev. D, 1983, V.27, 644-646.
  34. Baric N., Jena S.N., Lorentz structure vs relativistic consistency of an effective power-law potential model for quark-antiquark systems, Phys. Rev. D, 1982, V.26, 2420-2429.
  35. Haysak I., Lengyel V., Shpenik A., Challupka S., Salak M., Quark masses in the relativistic analytic model, Ukraïn. Fiz. Zh., 1996, V.41, 370-372 (in Ukrainian).

Previous article   Next article   Contents of Volume 2 (2006)