Zeitschrift für Physik C Particles and Fields

, Volume 15, Issue 4, pp 343–365 | Cite as

The Dirac-Kähler equation and fermions on the lattice

  • P. Becher
  • H. Joos
Article

Abstract

The geometrical description of spinor fields by E. Kähler is used to formulate a consistent lattice approximation of fermions. The relation to free simple Dirac fields as well as to Susskind's description of lattice fermions is clarified. The first steps towards a quantized interacting theory are given. The correspondence between the calculus of differential forms and concepts of algebraic topology is shown to be a useful method for a completely analogous treatment of the problems in the continuum and on the lattice.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K.G. Wilson: Phys. Rev.D10, 2445 (1974)Google Scholar
  2. 2.
    M. Creutz: Phys. Rev. Lett.45, 313 (1980); B. Berg: Phys. Lett.97B, 401 (1980); G. Bhanot, C. Rebbi: Nucl. Phys.B189, 469 (1981); K. Ishikawa, G. Schierhólz., M. Teper: Phys. Lett.110B, 399 (1982)Google Scholar
  3. 3.
    H. Hamber, G. Parisi: Phys. Rev. Lett.47, 1972 (1981); E. Marinari, G. Parisi, C. Rebbi: Phys. Rev. Lett.47, 1795 (1981); N. Kawamoto: Nucl. Phys.B190, 617 (1981); Niels Bohr Institute preprint NBI-HE-82-4 (1982); D. Weingarten: Phys. Lett.109B, 57 (1982); A. Hasenfratz, Z. Kunszt, P. Hasenfratz, C.B. Lang: CERN preprint, Ref. TH. 3220-CERN (1981)Google Scholar
  4. 4.
    For a review of this problem see: L.H. Karsten, J. Smit: Nucl. Phys.B183, 103 (1981)Google Scholar
  5. 5.
    K.G. Wilson: In: New phenomena in subnuclear physics. A. Zichichi ed. (“Erice 1975”), New York: Plenum 1977. P.H. Ginsparg, K.G. Wilson: Harvard Univ. Preprint HUTP-81/A060 (1981)Google Scholar
  6. 6.
    L. Susskind: Phys. Rev.D16, 3031 (1977); H.S. Sharatchandra H.J. Thun, P. Weisz: Nucl. Phys.B192, 205 (1981)Google Scholar
  7. 7.
    S.D. Drell, M. Weinstein, S. Yankielowicz: Phys. Rev.D14, 487, 1627 (1976)Google Scholar
  8. 8.
    J.M. Rabin: Phys. Rev.D24, 3218 (1981); In a recent preprint (Yale University YTP81-29), J.M. Rabin followed a suggestion of [12] and discussed the spectrum problem in the framework of the Dirac-Kähler equationGoogle Scholar
  9. 9.
    C.N. Yang: Phys. Rev. Lett.33, 445 (1974); M.E. Maver, W. Drechsler: Fiber bundle techniques in gauge theories. Lecture Notes in Physics, 67. Berlin, Heidelberg, New York: Springer 1977; G. Mack: Fortschr. Phys.29, 135 (1981)Google Scholar
  10. 10.
    W. Graf: Ann. Inst. H. Poincaré. Sect.A29, 85 (1978)Google Scholar
  11. 11.
    E. Kähler: Rend. Mat. Ser. V,21, 425 (1962)Google Scholar
  12. 12.
    P. Becher, Phys. Lett.104B, 221 (1981)Google Scholar
  13. 13.
    I.M. Singer, J.A. Thorpe: Lecture notes on elementary topology and geometry. Glenview, ILL: Scott, Foresman 1967 P.S. Aleksandrov: Combinatorial topology, vol. 1–3. Rochester, N.Y. Graylock 1956/57 and Albany, N.Y. (1960)Google Scholar
  14. 14.
    P.A.M. Dirac: Proc. Royal Soc. LondonA117, 610 (1928);A118, 351 (1928)Google Scholar
  15. 15.
    The description of Fermi fields in terms of geometric algebras was attempted by M. Schönberg: An. Acad. Brasil Ci.29, 473 (1957);30, 1, 117, 259, 429 (1958); Suppl. Nuovo CimentoVI, Ser. X, 356 (1957)Google Scholar
  16. 16.
    W. Pauli: Die allgemeinen Prinzipien der Wellenmechanik. Handbuch der Physik 24 Berlin: Springer 1933Google Scholar
  17. 17.
    M. Atiyah: Vector fields on manifolds. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Natur-, Ingenieur-und Gesellschaftswissenschaften, Helt 200 (1970)Google Scholar
  18. 18.
    E.P. Wigner: Group theory and its application to the quantum mechanics of atomic spectra. Expanded and improved edition, p. 210ff. New York, London: Academic Press 1959Google Scholar
  19. 19.
    Textvooks on quantum field theory; f.i.: C. Itzykson, J.-B. Zuber: Quantum field theory. New York: McGraw-Hill 1980Google Scholar
  20. 20.
    L. Susskind:Ref. [6]; T. Banks et al.: Phys. Rev.D15, 1111 (1977)Google Scholar
  21. 21.
    C.C. Chevalley: The algebraic theory of spinors. Morningside Heights, N.Y., Columbia, Univ. Press 1954Google Scholar
  22. 22.
    B.L. van der Waerden, Group theory and quantum mechanics. Berlin, Heidelberg, New York, Springer 1974Google Scholar
  23. 23.
    H. Weyl: The theory of groups and quantum mechanics Chap. IV, 14. New York: Dover 1931; R. Haag, D. Kastler: J. Math. Phys.5, 848 (1964)Google Scholar
  24. 24.
    T. Nakano: Progr. Theor. Phys.21, 241 (1959)Google Scholar
  25. 25.
    M.E. Mayer, et al.:Ref. [9]; W. Greub, S. Halperin, R. Vanstone: Connections, curvature, and cohomology, vol. I. New York: Academic Press 1972Google Scholar
  26. 26.
    K. Osterwalder, R. Schrader: Helv. Phys. Acta46, 277 (1973)Google Scholar
  27. 27.
    R.P. Feynman, A.R. Hibbs: Quantum mechanics and path integrals. New York: McGraw-Hill 1965Google Scholar
  28. 28.
    F.A. Berezin: The method of second quantization. New York, London: Academic Press 1966Google Scholar
  29. 29.
    J.-P. Serre: Ann. Math.54, 425 (1951); P.J. Hilton, S. Wylie: Homology theory, Cambridge: Cambridge University Press 1960Google Scholar
  30. 30.
    A. Chodos, J.B. Healy: Nucl. Phys.B127, 426 (1977); H.S. Sharatchandra, et al.: Ref [6]Google Scholar
  31. 31.
    H.S. Sharatchandra, et al.:Ref. [6]Google Scholar
  32. 32.
    A similar statement in the Hamiltonian approach has recently been made by A. Dhar, R. Shankar: Tata Institute Bombay preprint TIFR/TH/82-4 (1982)Google Scholar
  33. 33.
    Some references on this much disputed problem are: H.B. Nielsen, M. Ninomiya: Phys. Lett.105B, 219 (1981): L. Karsten, et al.: Ref. [4]; P.H. Ginsparg, et al.: Ref. [5]; W. Kerler: Phys. Rev.D23, 2384 (1981)Google Scholar
  34. 34.
    N.H. Kuiper: In: The CHERN-Symposium 1979. W.-Y. Hsiang, et al., eds.. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  35. 35.
    F. Flanders: Differential forms with applications to the physical sciences. New York: Academic Press 1963Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • P. Becher
    • 1
  • H. Joos
    • 2
  1. 1.Physikalisches Institut der UniversitätWürzburgFederal Republic of Germany
  2. 2.Deutsches Elektronen-Synchrotron DESYHamburgFederal Republic of Germany

Personalised recommendations