Theory of Spontaneous Symmetry Breakdown and the Appearance of Massless Particles

  • Y. Frishman
  • A. Katz
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 3/1966)


Broken symmetry solutions appeared in field theory when Nambu and Jona-Lasinio [1] tried to fashion a theory of strongly interacting particles analogous to the Bardeen, Cooper and Schrieffer [2] theory of superconductivity. The mass of the baryons was supposed to be completely dynamical — a result of spontaneous symmetry breakdown. The mass of the pions (analogs of the collective excitations of superconductivity) turned out to be zero. Soon afterwards Goldstone [3] conjectured that massless particles appear whenever spontaneous symmetry breakdown occurs. Arguments leading to this conclusion were later presented by Goldstone, Salam and Weinberg [4].


Symmetry Breaking Mass Splitting Break Symmetry Massless Particle Heisenberg Picture 


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References and Footnotes

  1. 1.
    Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961)ADSCrossRefGoogle Scholar
  2. 2.
    J. Bardeen, L. N. Cooper and S. R. Schrieffer, Phys. Rev. 108, 1175 (1957)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    J. Goldstone, Nuovo Cim. 19, 154 (1961)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    S. A. Bludman and A. Klein, Phys. Rev. 131, 2364 (1963)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    G. Jona-Lasinio, Nuovo Cim. 34, 1790 (1964)CrossRefGoogle Scholar
  7. 7.
    A. Klein and B. W. Lee, Phys. Rev. Lett. 12, 266 (1964)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    P. W. Higgs, Phys. Lett. 12, 132 (1964)ADSGoogle Scholar
  9. 9.
    G. S. Guralnik, Phys. Rev. Lett. 13, 295 (1964)CrossRefGoogle Scholar
  10. 10.
    J. Lopuszanski and H. Reeh, Phys. Rev. 140, B926 (1965)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    W. Gilbert, Phys. Rev. Lett., 12, 713 (1964)CrossRefGoogle Scholar
  12. 12.
    Y. Frishman and A. Katz, to be publishedGoogle Scholar
  13. 13.
    R. Haag, Nuovo Cim. 25, 287 (1962)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    R. F. Streater, Proc. Roy. Soc. 287, 510 (1965)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    D. Kastler, D. W. Robinson and A. Swieca, preprint (Univ. of Illinois)Google Scholar
  16. 16.
    J. Schwinger, Phys. Rev. Lett. 3, 296 (1959)CrossRefGoogle Scholar
  17. 17.
    K. Johnson, Nuclear Physics, 25, 431 (1961)ADSMATHCrossRefGoogle Scholar
  18. 18.
    K. Johnson, Phys. Lett. 5, 253 (1963)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964)ADSCrossRefGoogle Scholar
  20. 20.
    R. V. Lange, Phys. Rev. Lett. 14, 3 (1965)ADSMATHCrossRefGoogle Scholar
  21. 21.
    R. F. Streater, Phys. Rev. Lett. 15, 475 (1965)CrossRefGoogle Scholar
  22. 22.
    N. Fuchs (Phys. Rev. Lett. 15, 911 (1965)) gets, as a result of assuming a difference in the two point functions of two fields as spontaneous, that there are “states 101> with momentum and energy zero which however are not Lorentz invariant. His treatment of these 101> appears incorrect because of the points raised in ref.[4] sec. IIIGoogle Scholar
  23. 23.
    A. Katz: “Is the Heisenberg picture better than the Schrödinger picture?”preprint.Google Scholar
  24. 24.
    The Hamiltonian of the Heisenberg ferromagnet is a spin-spin interaction \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 % da9iabgkHiTmaaqafabaGaamOsamaaBaaaleaacaWGPbGaamOAaaqa % baaabaGaamyAaiaadQgaaeqaniabggHiLdGcceWGtbGbaSaadaWgaa % WcbaGaamyAaaqabaGcceWGtbGbaSaadaWgaaWcbaGaamOAaaqabaaa % aa!4377! H = - \sum\limits_{ij} {J_{ij} } \vec S_i \vec S_j \) Si being the spin operator associated with the i-th point of the lattice, and Jij are positive, usually assumed to be a constant J > 0 for nearest neighbours and zero otherwise. (See W. Heisenberg, Z. Physik, 49, 619 (1928)). The ground state of this Hamiltonian is obviously a state with all spins pointing in the same direction and as such not rotationally invariant. The fact that the spectrum contains massless excitations is demonstrated in almost every book on solid state physics.Google Scholar
  25. 25.
    M. Baker and S. L. Glashow, Phys. Rev. 128, 2462 (1962)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Th. A. J. Maris, V. E. Herscovitz and G. Jacob, Nuovo Cim. 34, 946 (1964); a second paper by the same authors is to be published in Nuovo Cim.Google Scholar
  27. 27.
    R. Arnowitt and S. Deser, Phys. Lett. 13, 256 (1964); Phys. Rev. 138, B712 (1965)MathSciNetGoogle Scholar
  28. 28.
    G. S. Guralnik, Preprint (University of Rochester).Google Scholar
  29. 29.
    A. Katz and Y. Frishman: “Massless Particles rather than spurious States following broken Symmetry”, to be published in Nuovo Cimento.Google Scholar
  30. 30.
    Y. Frishman and A. Katz, Phys. Rev. Lett. 16, 370 (1966)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Wien 1966

Authors and Affiliations

  • Y. Frishman
    • 1
  • A. Katz
    • 1
  1. 1.Department of PhysicsWeizmann InstituteRehovothIsrael

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