Skip to main content
SpringerLink
Log in

The Quantum Theory of the Lorentzian Fermionic Differential Forms

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the quantum theory of the Lorentzian fermionic differential forms and the corresponding bispinor quantum fields, which are expansion coefficients of the forms in the bispinor basis of Becher and Joos. We describe the canonical quantization procedure for the bispinor gauge theory in terms of its Dirac spinor constituents in detail and derive the corresponding Feynman rules and also all possible mass terms for massive fermions in the bispinor gauge theory. We classify the solutions by a scalar spin quantum number, a number with no analogue in the standard gauge theory and in the Standard Model. The possible mass terms correspond to combinations of scalar spin-zero singlets and scalar spin-1/2 doublets in the generation space. We describe the connection between the Lorentz spin of bispinors and the scalar spin of bispinor constituents.

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. D. Iwanenko and L. Landau, “Zur Theorie des magnetischen Elektrons: I,” Z. Phys., 48, 340–348 (1928).

    Article  ADS  Google Scholar 

  2. P. A. M. Dirac, “The quantum theory of the electron,” Proc. Roy. Soc. London Ser. A, 117, 610–624 (1928).

    Article  ADS  Google Scholar 

  3. E. Kahler, “Der innere Differentialkalkül,” Rend. Mat. Appl., 21, 425–523 (1962).

    MathSciNet  MATH  Google Scholar 

  4. W. Graf, “Differential forms as spinors,” Ann. Inst. H. PoincaréSect. A, n.s., 29, 85–109 (1978).

    MathSciNet  MATH  Google Scholar 

  5. I. M. Benn and R. W. Tucker, “A generation model based on Kähler fermions,” Phys. Lett. B, 119, 348–350 (1982).

    Article  ADS  Google Scholar 

  6. T. Banks, Y. Dothan, and D. Horn, “Geometric fermions,” Phys. Lett. B, 117, 413–417 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  7. P. Becher and H. Joos, “The Dirac—Kähler equation and fermions on the lattice,” Z. Phys. C, 15, 343–365 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  8. B. Holdom, “Gauged fermions from tensor fields,” Nucl. Phys. B, 233, 413–432 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  9. A. N. Jourjine, “Space-time Dirac—Kahler spinors,” Phys. Rev. D, 35, 757–758 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  10. I. M. Benn and R. W. Tucker, “Fermions without spinors,” Commun. Math. Phys., 89, 341–362 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  11. D. D. Ivanenko, Yu. N. Obukhov, and S. N. Solodukhin, “On antisymmetric tensor representation of the Dirac equation,” ICTP Preprint IC/85/2, ICTP, Trieste (1985).

    Google Scholar 

  12. J. Kato, N. Kawamoto, and A. Miyake, “N=4 twisted superspace from Dirac—Kähler twist and off-shell SUSY invariant actions in four dimensions,” Nucl. Phys. B, 721, 229–286 (2005); arXiv:hep-th/0502119v1 (2005).

    Article  ADS  Google Scholar 

  13. K. Nagata and Y.-S. Wu, “Twisted supersymmetric invariant formulation of Chern—Simons gauge theory on a lattice,” Phys. Rev. D, 78, 065002 (2008); arXiv:0803.4339v4 [hep-lat] (2008).

    Article  ADS  Google Scholar 

  14. F. Bruckmann, S. Catterall, and M. de Kok, “Critique of the link approach to exact lattice supersymmetry,” Phys. Rev. D, 75, 045016 (2007); arXiv:hep-lat/0611001v1 (2006).

    Article  ADS  Google Scholar 

  15. S. Arianos, A. D’Adda, N. Kawamoto, and J. Saito, “Lattice supersymmetry in 1D with two supercharges,” PoS (LATTICE 2007), 42, 259 (2007); arXiv:0710.0487v1 [hep-lat] (2007).

    MATH  Google Scholar 

  16. K. Nagata, “On the continuum and lattice formulations of N=4 D=3 twisted super Yang—Mills,” JHEP, 0801, 041 (2008); arXiv:0710.5689v2 [hep-th] (2007); “Exact lattice supersymmetry at large N,” JHEP, 0810, 036 (2008); arXiv:0805.4235v2 [hep-lat] (2008).

    Article  ADS  MathSciNet  Google Scholar 

  17. H. Echigoya and T. Miyazaki, “De Rham—Kodaira’s theorem and dual gauge transformations,” arXiv:hep-th/0011263v1 (2000).

  18. Y.-G. Miao, R. Manvelyana, and H. J. W. Muller-Kirsten, “Self-duality beyond chiral p-form actions,” Phys. Lett. B, 482, 264–270 (2000); arXiv:hep-th/0002060v2 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  19. B. de Wit and M. van Zalk, “Supergravity and M-theory,” Gen. Rel. Grav., 41, 757–784 (2009); arXiv:0901.4519v1 [hep-th] (2009).

    Article  ADS  MathSciNet  Google Scholar 

  20. T. Kobayashi and S. Yokoyama, “Gravitational waves from p-form inflation,” JCAP, 2009, 004 (2009); arXiv:0903.2769v2 [astro-ph.CO] (2009).

    Article  Google Scholar 

  21. A. N. Jourjine, “Mass mixing, the fourth generation, and the kinematic Higgs mechanism,” Phys. Lett. B, 693, 149–154 (2010); arXiv:1005.3593v4 [hep-ph] (2010).

    Article  ADS  Google Scholar 

  22. A. N. Jourjine, “The spectrum of the 4-generation Dirac—Kähler extension of the SM,” Phys. Lett. B, 695, 482–488 (2011); arXiv:1011.0382v3 [hep-ph] (2010).

    Article  ADS  MathSciNet  Google Scholar 

  23. A. N. Jourjine, “Scalar spin of elementary fermions,” Phys. Lett. B, 728, 347–357 (2014); arXiv:1307.2694v2 [hep-ph] (2013).

    Article  ADS  Google Scholar 

  24. P. Salgado and S. Salgado, “Extended gauge theory and gauged free differential algebras,” Nucl. Phys. B, 926, 179–199 (2018); arXiv:1702.07819v4 [hep-th] (2017).

    Article  ADS  MathSciNet  Google Scholar 

  25. A. Jourjine, “Extended gauge theory, bi-spinors, and scalar supersymmetry,” arXiv:1706.01269v3 [hep-th] (2017).

  26. C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York (1980).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FG Center for Theoretical Physics, Dresden, Germany

    A. Jourjine

Authors
  1. A. Jourjine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Jourjine.

Additional information

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 202, No. 2, pp. 207–242, February, 2020.

Conflicts of interest

The author declares no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jourjine, A. The Quantum Theory of the Lorentzian Fermionic Differential Forms. Theor Math Phys 202, 183–213 (2020). https://doi.org/10.1134/S004057792002004X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S004057792002004X

Keywords

Search

Navigation