Few-Body Systems

, Volume 49, Issue 1–4, pp 129–147 | Cite as

Mini Review of Poincaré Invariant Quantum Theory

  • W. N. PolyzouEmail author
  • Ch. Elster
  • W. Glöckle
  • J. Golak
  • Y. Huang
  • H. Kamada
  • R. Skibiński
  • H. Witała


We review the construction and applications of exactly Poincaré invariant quantum mechanical models of few-degree of freedom systems. We discuss the construction of dynamical representations of the Poincaré group on few-particle Hilbert spaces, the relation to quantum field theory, the formulation of cluster properties, and practical considerations related to the construction of realistic interactions and the solution of the dynamical equations. Selected applications illustrate the utility of this approach.


Unitary Transformation Wigner Function Cluster Property Mass Operator Faddeev Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wigner E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149 (1939)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics (1980)Google Scholar
  3. 3.
    Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Osterwalder K., Schrader R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83 (1973)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Dirac P.A.M.: Forms of relativistic dynamics. Rev. Mod. Phys. 21, 392 (1949)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Bakamjian B., Thomas L.H.: Relativistic particle dynamics. Phys. Rev. 92, 1300 (1953)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Coester F.: Scattering theory for relativistic particles. Helv. Phys. Acta 38, 7 (1965)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Sokolov S.N.: Theory of relativistic direct interactions. Dokl. Akad. Nauk SSSR 233, 575 (1977)MathSciNetGoogle Scholar
  9. 9.
    Coester F., Polyzou W.N.: Relativistic quantum mechanics of particles with direct interactions. Phys. Rev. D 26, 1348 (1982)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Polyzou W.N.: Relativistic two-body models. Ann. Phys. 193, 367 (1989)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Polyzou W.N.: Relativistic quantum mechanics—particle production and cluster properties. Phys. Rev. C 68, 015202 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Keister B.D., Polyzou W.N.: Relativistic Hamiltonian dynamics in nuclear and particle physics. Adv. Nucl. Phys. 20, 225 (1991)Google Scholar
  13. 13.
    Bakker B.L.G., Kondratyuk L.A., Terentev M.V.: On the formulation of two-body and three-body relativistic equations employing light-front dynamics. Nucl. Phys. B 158, 497 (1979)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Grach I.L., Kondratyuk L.A.: Electromagnetic form-factors of deuteron in relativistic dynamics. Two nucleon and six quark components. Sov. J. Nucl. Phys. 39, 198 (1984)Google Scholar
  15. 15.
    Chung P.L., Polyzou W.N., Coester F., Keister B.D.: Hamiltonian light front dynamics of elastic electron deuteron scattering. Phys. Rev. C 37, 2000 (1988)ADSCrossRefGoogle Scholar
  16. 16.
    Chung P.L., Coester F., Polyzou W.N.: Charge form-factors of quark model pions. Phys. Lett. B 205, 545 (1988)ADSCrossRefGoogle Scholar
  17. 17.
    Cardarelli F., Pace E., Salme G., Simula S.: Nucleon and pion electromagnetic form-factors in a light front constituent quark model. Phys. Lett. B 357, 267 (1995)ADSCrossRefGoogle Scholar
  18. 18.
    Polyzou W.N., Glöckle W.: Scaling for deuteron structure functions in relativistic light-front models. Phys. Rev. C 53, 3111 (1996)ADSCrossRefGoogle Scholar
  19. 19.
    Krutov A.F.: Electroweak properties of light mesons in the relativistic model of constituent quarks. Phys. Atom. Nucl. 60, 1305 (1997)ADSGoogle Scholar
  20. 20.
    Allen T.W., Klink W.H., Polyzou W.N.: Comparison of relativistic nucleon-nucleon interactions. Phys. Rev. C 63, 034002 (2001)ADSCrossRefGoogle Scholar
  21. 21.
    Wagenbrunn R.F., Boffi S., Klink W., Plessas W., Radici M.: Covariant nucleon electromagnetic form factors from the goldstone-boson exchange quark model. Phys. Lett. B 511, 33 (2001)ADSCrossRefGoogle Scholar
  22. 22.
    Julia-Diaz B., Riska D.O., Coester F.: Baryon form factors of relativistic constituent-quark models. Phys. Rev. C 69, 035212 (2004)ADSCrossRefGoogle Scholar
  23. 23.
    Sengbusch E., Polyzou W.N.: Pointlike constituent quarks and scattering equivalences. Phys. Rev. C 70, 058201 (2004)ADSCrossRefGoogle Scholar
  24. 24.
    Coester F., Polyzou W.N.: Charge form factors of quark-model pions. Phys. Rev. C 71, 028202 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    Huang Y., Polyzou W.N.: Exchange current contributions in null-plane quantum models of elastic electron deuteron scattering. Phys. Rev. C 80, 025503 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    Arrington J., Coester F., Holt R.J., Lee T.S.H.: Neutron structure functions. J. Phys. G 36, 025005 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    Desplanques B.: RQM description of the charge form factor of the pion and its asymptotic behavior. Eur. Phys. J. A 42, 219 (2009)ADSCrossRefGoogle Scholar
  28. 28.
    Glöckle W., Lee T.S.H., Coester F.: Relativistic effects in three-body bound states. Phys. Rev. C 33, 709 (1986)ADSCrossRefGoogle Scholar
  29. 29.
    Kamada H. et al.: Lorentz boosted nucleon-nucleon T-matrix and the triton binding energy. Mod. Phys. Lett. A 24, 804 (2009)ADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Fuda M.G., Zhang Y.: Light front dynamics of one boson exchange models of the two-nucleon system. Phys. Rev. C 51, 23 (1995)ADSCrossRefGoogle Scholar
  31. 31.
    Lin T., Elster C., Polyzou W.N., Glöckle W.: First order relativistic three-body scattering. Phys. Rev. C 76, 014010 (2007)ADSCrossRefGoogle Scholar
  32. 32.
    Lin T., Elster C., Polyzou W.N., Glöckle W.: Relativistic effects in exclusive pd breakup scattering at intermediate energies. Phys. Lett. B 660, 345 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    Lin T., Elster C., Polyzou W.N., Witała H., Glöckle W.: Poincaré invariant three-body scattering at intermediate energies. Phys. Rev. C 78, 024002 (2008)ADSCrossRefGoogle Scholar
  34. 34.
    Witała H. et al.: Relativity and the low energy nd Ay puzzle. Phys. Rev. C 77, 034004 (2008)ADSCrossRefGoogle Scholar
  35. 35.
    Witała H. et al.: Relativistic effects in 3N reactions. Mod. Phys. Lett. A 24, 871 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Fuda M.G., Bulut F.: Three-particle model of the pion-nucleon system. Phys. Rev. C 80, 024002 (2009)ADSCrossRefGoogle Scholar
  37. 37.
    Newton T.D., Wigner E.P.: Localized states for elementary systems. Rev. Mod. Phys. 21, 400 (1949)ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Melosh H.J.: Quarks: currents and constituents. Phys. Rev. D 9, 1095 (1974)ADSCrossRefGoogle Scholar
  39. 39.
    Joos H.: On the representation theory of inhomogeneous Lorentz groups as the foundation of quantum mechanical kinematics. Fortschr. Physik 10, 109 (1962)Google Scholar
  40. 40.
    Moussa, P., Stora, R.: In: Brittin, W.E., Barut, A.O. (eds.) Lectures in Theoretical Physics, vol. VIIA, p. 248. The University of Colorado Press (1965)Google Scholar
  41. 41.
    Ekstein H.: Equivalent Hamiltonians in scattering theory. Phys. Rev. 117, 1590 (1960)MathSciNetADSzbMATHCrossRefGoogle Scholar
  42. 42.
    Kato T.: Perturbation Theory for Linear Operators. Spinger-Verlag, Berlin (1966)zbMATHGoogle Scholar
  43. 43.
    Chandler C., Gibson A.: Invariance principle for modified wave operators. Indiana J. Math. 25, 443 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Baumgärtel H., Wollenberg M.: Mathematical Scattering Theory. Spinger-Verlag, Berlin (1983)Google Scholar
  45. 45.
    Haag R.: Quantum field theories with composite particles and asymptotic conditions. Phys. Rev. 112, 669 (1958)MathSciNetADSzbMATHCrossRefGoogle Scholar
  46. 46.
    Ruelle V.D.: On the asymptotic condition in quantum field theory. Helv. Phys. Acta. 35, 147 (1962)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Coester F., Pieper S.C., Serduke F.J.D.: Relativistic effects in phenomenological nucleon-nucleon potentials and nuclear matter. Phys. Rev. C 11, 1 (1975)ADSCrossRefGoogle Scholar
  48. 48.
    Wiringa R.B., Stoks V.G.J., Schiavilla R.: An accurate nucleon-nucleon potential with charge independence breaking. Phys. Rev. C 51, 38 (1995)ADSCrossRefGoogle Scholar
  49. 49.
    Machleidt R.: The high-precision, charge-dependent Bonn nucleon-nucleon potential (CD-Bonn). Phys. Rev. C 63, 024001 (2001)ADSCrossRefGoogle Scholar
  50. 50.
    Keister B.D., Polyzou W.N.: Quantitative relativistic effects in the three-nucleon problem. Phys. Rev. C 73, 014005 (2006)ADSCrossRefGoogle Scholar
  51. 51.
    Kamada H., Glöckle W.: Realistic two-nucleon potentials for the relativistic two-nucleon Schroedinger equation. Phys. Lett. B 655, 119 (2007)ADSCrossRefGoogle Scholar
  52. 52.
    Miller G.A., Schwenk A.: Resonant relativistic corrections and the A y problem. Phys. Rev. C 76, 024001 (2007)ADSCrossRefGoogle Scholar
  53. 53.
    Cub J. et al.: Analyzing power for elasticn-d scattering at 13 MeV. Few-Body Syst. 6, 151 (1989)ADSCrossRefGoogle Scholar
  54. 54.
    Tornow W. et al.: Convergent theory for effective interaction in nuclei. Phys. Lett. B 257, 273 (1991)ADSCrossRefGoogle Scholar
  55. 55.
    Fachruddin I., Elster C., Glöckle W.: Nucleon-nucleon scattering in a three dimensional approach. Phys. Rev. C 62, 044002 (2000)ADSCrossRefGoogle Scholar
  56. 56.
    Liu H., Elster C., Glöckle W.: Three-body scattering at intermediate energies. Phys. Rev. C 72, 054003 (2005)ADSCrossRefGoogle Scholar
  57. 57.
    Malfliet R.A., Tjon J.A.: Solution of the Faddeev equations for the triton problem using local two-particle interactions. Nucl. Phys. A 127, 161 (1969)ADSCrossRefGoogle Scholar
  58. 58.
    Punjabi V. et al.: 2H(p,2p)n at 508 MeV: recoil momenta ≤ 200 MeV/c. Phys. Rev. C 38, 2728 (1988)ADSCrossRefGoogle Scholar
  59. 59.
    Lomon, E.L.: Effect of revised R n measurements on extended Gari-Krumpelmann model fits to nucleon electromagnetic form factors (2006). arxiv:nucl-th/0609020v2Google Scholar
  60. 60.
    Budd, H., Bodek, A., Arrington, J.: Modeling quasi-elastic form factors for electron and neutrino scattering (2003). arXiv:hep-ex/0308005v2Google Scholar
  61. 61.
    Bradford R., Bodek A., Budd H., Arrington J.: Nucl. Phys. Proc. Suppl. 159, 127 (2006)ADSCrossRefGoogle Scholar
  62. 62.
    Kelly J.J.: Simple parametrization of nucleon form factors. Phys. Rev. C 70(6), 068202 (2004)ADSCrossRefGoogle Scholar
  63. 63.
    Bijker R., Iachello F.: Reanalysis of the nucleon spacelike and timelike electromagnetic form factors in a two-component model. Phys. Rev. C 69(6), 068201 (2004)ADSCrossRefGoogle Scholar
  64. 64.
    Riska D.O.: Exchange currents. Phys. Rep. 181, 207 (1989)ADSCrossRefGoogle Scholar
  65. 65.
    Buchanan C.D., Yearian M.R.: Elastic electron-deuteron scattering and possible meson-exchange effects. Phys. Rev. Lett. 15(7), 303 (1965)ADSCrossRefGoogle Scholar
  66. 66.
    Elias J.E. et al.: Measurements of elastic electron-deuteron scattering at high momentum transfers. Phys. Rev. 177(5), 2075 (1969)ADSCrossRefGoogle Scholar
  67. 67.
    Benaksas D., Drickey D., Frèrejacque D.: Deuteron electromagnetic form factors for 3 F −2 < q 2 < 6 F −2. Phys. Rev. 148(4), 1327 (1966)ADSCrossRefGoogle Scholar
  68. 68.
    Arnold R.G., Chertok B.T., Dally E.B., Grigorian A., Jordan C.L., Schütz W.P., Zdarko R., Martin F., Mecking B.A.: Measurement of the electron-deuteron elastic-scattering cross section in the range 0.8 ≤ q2 ≤ 6 GeV 2. Phys. Rev. Lett. 35(12), 776 (1975)ADSCrossRefGoogle Scholar
  69. 69.
    Platchkov S. et al.: Deuteron A(Q 2) structure function and the neutron electric form-factor. Nucl. Phys. A 510, 740 (1990)ADSCrossRefGoogle Scholar
  70. 70.
    Galster S. et al.: Elastic electron-deuteron scattering and the electric neutron form factor at four-momentum transfers 5 fm2 < q2 < 14 fm2. Nucl. Phys. B 32, 221 (1971)ADSCrossRefGoogle Scholar
  71. 71.
    Cramer R., Renkhoff M., Drees J., Ecker U., Jagoda D., Koseck K., Pingel G.R., Remenschnitter B., Ritterskamp A.: Measurement of the magnetic formfactor of the deuteron for Q/sup 2/=0.5 to 1.3 (GeV/c)/sup 2/ by a coincidence experiment. ZPC 29(4), 513 (1985)CrossRefGoogle Scholar
  72. 72.
    Simon G.G., Schmitt C., Walther V.H.: Elastic electric and magnetic e-d scattering at low momentum transfer. Nuclear Phys. A 364, 285 (1985)ADSCrossRefGoogle Scholar
  73. 73.
    Abbott D. et al.: Precise measurement of the deuteron elastic structure function A(Q2). Phys. Rev. Lett. 82(7), 1379 (1999)ADSCrossRefGoogle Scholar
  74. 74.
    Alexa L.C. et al.: Measurements of the deuteron elastic structure function A(Q2) for 0.7 ≤ Q2 ≤ 6.0 (GeV/c)2 at Jefferson laboratory. Phys. Rev. Lett. 82(7), 1374 (1999)ADSCrossRefGoogle Scholar
  75. 75.
    Berard R.W. et al.: Elastic electron deuteron scattering. Phys. Lett. B 47, 355 (1973)ADSCrossRefGoogle Scholar
  76. 76.
    Bosted P.E. et al.: Measurements of the deuteron and proton magnetic form factors at large momentum transfers. Phys. Rev. C 42(1), 38 (1990)ADSCrossRefGoogle Scholar
  77. 77.
    Martin F., Arnold R.G., Chertok B.T., Dally E.B., Grigorian A., Jordan C.L., Schütz W.P., Zdarko R., Mecking B.A.: Measurement of the magnetic structure function of the deuteron at q2 = 1.0 (GeV/c)2. Phys. Rev. Lett. 38(23), 1320 (1977)ADSCrossRefGoogle Scholar
  78. 78.
    Auffret S. et al.: Magnetic form factor of the deuteron. Phys. Rev. Lett. 54(7), 649 (1985)ADSCrossRefGoogle Scholar
  79. 79.
    Dmitriev V.F. et al.: First measurement of the asymmetry in electron scattering by a jet target of polarized deuterium atoms. Phys. Lett. B 157, 143 (1985)ADSCrossRefGoogle Scholar
  80. 80.
    Voitsekhovsky B.B. et al.: Asymmetry in the reaction D(e,eD) at a momentum transfer of 1FM −1 − 1.5FM −1. JETP Lett. 43, 733 (1986)ADSGoogle Scholar
  81. 81.
    Gilman R. et al.: Measurement of tensor analyzing power in electron-deuteron elastic scattering. Phys. Rev. Lett. 65(14), 1733 (1990)ADSCrossRefGoogle Scholar
  82. 82.
    Schulze M.E. et al.: Measurement of the tensor polarization in electron-deuteron elastic scattering. Phys. Rev. Lett. 52(8), 597 (1984). doi: 10.1103/PhysRevLett.52.597 ADSCrossRefGoogle Scholar
  83. 83.
    The I. et al.: Measurement of tensor polarization in elastic electron-deuteron scattering in the momentum-transfer range 3.8 ≤ q ≤ 4.6 fm −1. Phys. Rev. Lett. 67(2), 173 (1991)ADSCrossRefGoogle Scholar
  84. 84.
    Abbott D. et al.: Measurement of tensor polarization in elastic electron-deuteron scattering at large momentum transfer. Phys. Rev. Lett. 84(22), 5053 (2000)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • W. N. Polyzou
    • 1
    Email author
  • Ch. Elster
    • 2
  • W. Glöckle
    • 3
  • J. Golak
    • 4
  • Y. Huang
    • 1
  • H. Kamada
    • 5
  • R. Skibiński
    • 4
  • H. Witała
    • 4
  1. 1.Department of Physics and AstronomyThe University of IowaIowa CityUSA
  2. 2.Institute of Nuclear and Particle Physics, Department of Physics and AstronomyOhio UniversityAthensUSA
  3. 3.Institut für theoretische Physik IIRuhr-Universität BochumBochumGermany
  4. 4.M. Smoluchowski Institute of PhysicsJagiellonian UniversityKrakówPoland
  5. 5.Department of Physics, Faculty of EngineeringKyushu Institute of TechnologyKitakyushuJapan

Personalised recommendations