The Dirac Equation as One Fourth-Order Equation for One Function: A General, Manifestly Covariant Form

  • Andrey Akhmeteli
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series (STEAM)


Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also manifestly relativistically covariant and should be useful in applications of the Dirac equation.



The author is grateful to V.G. Bagrov, A.V. Gavrilin, A.Yu. Kamenshchik, P.W. Morgan, nightlight, R. Sverdlov, and H. Westman for their interest in this work and valuable remarks.


  1. 1.
    Ahuja, R., Blomqvist, A., Larsson, P., Pyykkö, P., Zaleski-Ejgierd, P.: Phys. Rev. Lett. 106, 018301 (2011)Google Scholar
  2. 2.
    Akhmeteli, A.: J. Math. Phys. 52, 082303 (2011)Google Scholar
  3. 3.
    Akhmeteli, A.: Int. J. Quantum Inf. 9(Suppl.), 17 (2011)Google Scholar
  4. 4.
    Akhmeteli, A.: Eur. Phys. J. C 73, 2371 (2013)Google Scholar
  5. 5.
    Bagrov, V.G., Gitman, D.: The Dirac Equation and Its Solutions. Walter de Gruyter GmbH, Berlin/Boston (2014)Google Scholar
  6. 6.
    Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, 3rd edn. Wiley, New York (1980)Google Scholar
  7. 7.
    Bogolubov, N.N., Logunov, A.A., Todorov, I.T.: Introduction to Axiomatic Quantum Field Theory. W. A. Benjamin, Inc., Reading, MA (1975)Google Scholar
  8. 8.
    Dirac, P.A.M.: Proc. R. Soc. Lond. A117, 610 (1928)Google Scholar
  9. 9.
    Feynman, R.P., Gell-Mann, M.: Phys. Rev. 109, 193 (1958)Google Scholar
  10. 10.
    Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)Google Scholar
  11. 11.
    Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Company, Evanston (1961)Google Scholar
  12. 12.
    Wilczek, F.: Int. J. Mod. Phys. A 19(Suppl.), 45 (2004)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Andrey Akhmeteli
    • 1
  1. 1.LTASolid Inc.HoustonUSA

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