Advertisement

Communications in Mathematical Physics

, Volume 95, Issue 3, pp 257–276 | Cite as

Eigenvalue inequalities for fermions in gauge theories

  • Cumrun Vafa
  • Edward Witten
Article

Abstract

We show that QCD with a sufficient number of fermions of zero bare mass has physical massless particles. That result also follows from triangle anomalies, so only our method is novel. Our method involves proving special cases of recently conjectured paramagnetic inequalities for fermions. The proofs are simple applications of the Atiyah-Patodi-Singer theorem on spectral flow.

Keywords

Neural Network Statistical Physic Complex System Gauge Theory Nonlinear Dynamics 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tomboulis, E.T.: Permanent confinement in four-dimensional non-abelian lattice gauge theory. Phys. Rev. Lett.50, 88 (1983)Google Scholar
  2. 2.
    Weingarten, D.: Mass inequalities for quantum chromodynamics. Phys. Rev. Lett.51, 1830 (1983)Google Scholar
  3. 3.
    Vafa, C., Witten, E.: Restrictions on symmetry breaking in vector-like gauge theories. Nucl. Phys. B234, 173 (1984)Google Scholar
  4. 4.
    Witten, E.: Some inequalities among hadron masses. Phys. Rev. Lett.51, 2351 (1983)Google Scholar
  5. 5.
    Nussinov, S.: Baryon-meson mass inequality. Phys. Rev. Lett.51, 1081 (1983); Mass inequalities in quantum chromodynamics52, 966 (1984)Google Scholar
  6. 6.
    Tomboulis, E.T., Yaffe, L.: Princeton preprint (1983)Google Scholar
  7. 7.
    Hogreve, H., Schrader, R., Seiler, R.: A conjecture on the spinor functional determinant. Nucl. Phys. B142, 525 (1978);Google Scholar
  8. 7a.
    Brydges, D., Fröhlich, J., Seiler, E.: On the construction of quantized gauge fields. I. General results. Ann. Phys.121, 227 (1979);Google Scholar
  9. 7b.
    Avron, J., Simon, B.: A counterexample to the paramagnetic conjecture. Phys. Lett.75A, 41 (1979)Google Scholar
  10. 8.
    't Hooft, G.: In: Recent developments in gauge theories, 't Hooft, G. et al. (eds.). New York: Plenum Press 1980Google Scholar
  11. 9.
    Asorey, M., Mitter, P.K.: Regularized, continuum Yang-Mills process and Feynman-Kac functional integral. Commun. Math. Phys.80, 43 (1981)Google Scholar
  12. 10.
    Banks, T., Casher, A.: Chiral symmetry breaking in confining theories. Nucl. Phys. B169, 102 (1980)Google Scholar
  13. 10a.
    Kogut, J., Stone, M., Wyld, H.W., Gibbs, W.R., Shigemitsu, J., Shenker, S.H., Sinclair, D.K.: Deconfinement and chiral symmetry restoration at finite temperature in SU(2) and SU(3) gauge theories. Phys. Rev. Lett.50, 393 (1983);Google Scholar
  14. 10b.
    Scales of chiral symmetry breaking in quantum chromodynamics. Phys. Rev. Lett.48, 1140 (1982)Google Scholar
  15. 11.
    Atiyah, M.F., Patodi, V., Singer, I.: Math. Proc. Camb. Philos. Soc.79, 71 (1976)Google Scholar
  16. 12.
    Callan, C.G., Dashen, R., Gross, D.J.: Toward a theory of the strong interactions. Phys. Rev. D17, 2717 (1978)Google Scholar
  17. 12a.
    Kiskis, J.: Fermion zero modes and level crossing. Phys. Rev.D18, 3690 (1978)Google Scholar
  18. 13.
    Atiyah, M.F., Singer, I.M.: Ann. Math.87, 485, 546, (1968);93, 1, 119, 139 (1971)Google Scholar
  19. 13a.
    Atiyah, M.F., Segal, G.B.: Ann. Math.87, 531 (1968)Google Scholar
  20. 14.
    Gromov, M., Lawson, H.B., Jr.: Ann. Math.111, 209 (1980)Google Scholar
  21. 15.
    Witten, E.: J. Diff. Geom.17, 661 (1982); Alvarez-Gaumé, L., Ginsparg, P.: The topological meaning of non-Abelian anomalies. Harvard preprint (1983)Google Scholar
  22. 16.
    Kato, T.: Israel J. Math.13, 135 (1972)Google Scholar
  23. 17.
    Feynman, R.P.: The qualitative behavior of Yang-Mills theory in 2+1 dimensions. Nucl. Phys. B188, 479 (1981)Google Scholar
  24. 17a.
    Singer, I.M.: Phys. Scripta24, 817 (1981)Google Scholar
  25. 18.
    Jackiw, R., Templeton, S.: How super-renormalizable interactions cure their infrared divergences. Phys. Rev. D23, 2291 (1981)Google Scholar
  26. 18a.
    Schönfeld, J.: A mass term for three-dimensional gauge fields. Nucl. Phys. B185, 157 (1981); Deser, S., Jackiw, R., Templeton, S.: Three-dimensional massive gauge theories. Phys. Rev. Lett.48, 975 (1982); Topologically massive gauge theories, Ann. Phys. (N.Y.)140, 372 (1982)Google Scholar
  27. 19.
    Appelquist, T., Pisarski, R.D.: High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics. Phys. Rev. D23, 2305 (1981)Google Scholar
  28. 20.
    Affleck, I., Harvey, J., Witten, E.: Instantions and (super-)symmetry breaking In (2+1) dimensions. Nucl. Phys. B206, 413 (1982)Google Scholar
  29. 21.
    Redlich, N.: Gauge noninvariance and parity nonconservation of three-dimensional fermions, Phys. Rev. Lett.52, 18 (1984)Google Scholar
  30. 21a.
    Alvarez-Gaumé, L., Witten, E.: Gravitational anomalics, Nucl. Phys. B234, 269 (1984)Google Scholar
  31. 22.
    Pisarski, R.D.: University of California preprint (1984)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Cumrun Vafa
    • 1
  • Edward Witten
    • 1
    • 2
  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrincetonUSA
  2. 2.Institute for Advanced StudyPrincetonUSA

Personalised recommendations