Skip to main content
SpringerLink
Log in

New representation of Dirac-Yang-Mills equations

  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

With the help of the apparatus of tensors with values in the Atiyah-Kähler algebra (the apparatus of lambda tensors), a new representation for the relativistic equation of field theory is proposed. The circle of questions under consideration is standard for investigations concerning the theory of the Dirac equation (in this paper, we treat no problems dealing with second quantization).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. G. Marchuk, “Technique of Tensors with Values in the Atiyah-Kähler Algebra in the New Representation of Dirac-Yang-Mills Equations,” Russ. J. Math. Phys. 13(3), 299–314 (2006).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. M. Riesz, “Sur certaines notions fondamentales en théorie quantique relativiste,” in C. R. Dixième Congrés Math. Scandinaves 1946, pp. 123–148 (Jul. Gjellerups Forlag, Copenhagen, 1947); Reprinted in: L. Gårding, L. Hërmander, Eds., Marcel Riesz, Collected Papers (Springer, Berlin, 1988), pp. 814–832.

    Google Scholar 

  3. E. Kähler, “Der innere Differentialkalkul,” Rend. Mat. Appl. (Roma) Ser. 5 21, 425–523 (1962).

    Google Scholar 

  4. D. Hestenes, Space-Time Algebra (Gordon and Beach, New York, 1967).

    Google Scholar 

  5. C. Møller, “On the Crisis in the Theory of Gravitation and a Possible Solution,” Mat. Fys. Medd. Danske Vid. Selsk. 39, 13 (1978).

    Google Scholar 

  6. M. Atiyah, Vector Fields on Manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200 (Westdeutscher Verlag, Cologne, 1970).

    MATH  Google Scholar 

  7. N. G. Marchuk, “Dirac γ-Equation, Classical Gauge Fields, and Clifford Algebra,” Adv. Appl. Clifford Algebras 8(1), 181–225 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  8. N. G. Marchuk, “Dirac Equation in Riemannian Space without Tetrads,” Nuovo Cimento Soc. Ital. Fis. B 115(11), 1267–1301 (2000).

    MathSciNet  ADS  Google Scholar 

  9. N. G. Marchuk, “Dirac-Type Tensor Equations,” Nuovo Cimento Soc. Ital. Fis. B 116(10), 1225–1248 (2001).

    MathSciNet  ADS  Google Scholar 

  10. N. G. Marchuk, “A Concept of Dirac-Type Tensor Equations,” Nuovo Cimento Soc. Ital. Fis. B 117(12), 1357–1388 (2002).

    MathSciNet  ADS  Google Scholar 

  11. N. G. Marchuk, “Coordinate-Free Form of the Dirac Equation for an Electron,” Dokl. Akad. Nauk 392(3), 318–321 (2003).

    MathSciNet  Google Scholar 

  12. D. Iwanenko and L. Landau, “Zur Theorie des magnetischen Elektrons,” Z. Phys. A 48(5–6), 340–348 (1928).

    Google Scholar 

  13. P. Becher, “Dirac Fermions on the Lattice — a Local Approach Without Spectrum Degeneracy,” Phys. Lett. B 104(3), 221–225 (1981).

    Article  ADS  Google Scholar 

  14. J. M. Rabin, “Homology Theory of Lattice Fermion Doubling,” Nuclear Phys. B 201(2), 315–332 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  15. P. Becher and H. Joos, “The Dirac-Kähler Equation and Fermions on the Lattice,” Z. Phys. C15, 343–365 (1982).

    MathSciNet  ADS  Google Scholar 

  16. T. Banks, Y. Dothan, and D. Horn, “Geometric Fermions,” Phys. Lett. B 117, 413 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  17. M. Reuter, “Symplectic Dirac-Kähler Fields,” J. Math. Phys. 40, 5593–5640 (1999).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. N. Kawamoto, T. Tsukioka, and H. Umetsu, “Generalized Gauge Theories and Weinberg-Salam Model with Dirac-Kähler Fermions,” Internat. J. Modern. Phys. A 16, 3867–3896 (2001).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. I. Kanamori and N. Kawamoto, “Dirac-Kähler Fermion from Clifford Product with Noncommutative Differential Form on a Lattice,” Internat. J. Modern. Phys. A 19, 695–736 (2004).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. V. de Beauce, S. Sen, and J. C. Sexton, “Chiral Dirac Fermions on the Lattice Using Geometric Discretisation,” Nuclear Phys. B Proc. Suppl. 129–130, 468–470 (2004).

    Google Scholar 

  21. S. I. Kruglov, “Dirac-Kähler Equation,” Internat. J. Theoret. Phys. 41(4), 653–687 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Green, J. Schwarz, and E. Witten, Superstring Theory, Vols. 1,2 (Cambridge Univ. Press, 1988; Russian transl. by Mir, Moscow, 1990).

  23. I. M. Benn and R. W. Tucker, An Introduction to Spinors and Geometry with Applications in Physics (Adam Hilger, Ltd., Bristol, 1987).

    Google Scholar 

  24. P. Lounesto, Clifford Algebras and Spinors (Cambridge Univ. Press, 1997, 2001).

  25. M. Nakahara, Geometry, Topology and Physics (Institute of Physics Publishing, Bristol and Philadelphia, 1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V. A. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina ul. 8, Moscow, 119991, Russia

    N. G. Marchuk

Authors
  1. N. G. Marchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported by the Program for Supporting Leading Scientific Schools under grant no. NSh-6705.2006.1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marchuk, N.G. New representation of Dirac-Yang-Mills equations. Russ. J. Math. Phys. 13, 397–413 (2006). https://doi.org/10.1134/S1061920806040030

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920806040030

Keywords

Search

Navigation