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On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation

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Abstract

Extended gamma matrix Clifford–Dirac and SO(1,9) algebras in the terms of \(8 \times 8\) matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. Two isomorphic realizations \(\textit{C}\ell ^{\texttt {R}}\)(0,8) and \(\textit{C}\ell ^{\texttt {R}}\)(1,7) are considered. The corresponding gamma matrix representations of 45-dimensional SO(10) and SO(1,9) algebras, which contain standard and additional spin operators, are introduced as well. The SO(10), SO(1,9) and the corresponding \(\textit{C}\ell ^{\texttt {R}}\)(0,8)\(, \textit{C}\ell ^{\texttt {R}}\)(1,7) representations are determined as algebras over the field of real numbers. The suggested gamma matrix representations of the Lie algebras SO(10), SO(1,9) are constructed on the basis of the Clifford algebras \(\textit{C}\ell ^{\texttt {R}}\)(0,8)\(, \textit{C}\ell ^{\texttt {R}}\)(1,7) representations. Comparison with the corresponded algebras in the space of standard 4-component Dirac spinors is demonstrated. The proposed mathematical objects allow generalization of our results, obtained earlier for the standard Dirac equation, for equations of higher spin and, especially, for equations, describing particles with spin 3/2. The maximal 84-dimensional pure matrix algebra of invariance of the 8-component Dirac equation in the Foldy–Wouthuysen representation is found. The corresponding symmetry of the Dirac equation in ordinary representation is found as well. The possible generalizations of considered Lie algebras to the arbitrary dimensional SO(n) and SO(m,n) are discussed briefly.

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References

  1. M. Benmerrouche, M., Davidson, R.M., Mukhopadhyay, N.C.: Problems of describing spin-3/2 baryon resonances in the effective Lagrangian theory. Phys. Rev. C. 39(6), 2339–2348 (1989). https://doi.org/10.1103/PhysRevC.39.2339

  2. Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. John Wiley and Sons Inc., New York (1980)

    Google Scholar 

  3. Elliott, J,P., Dawber, P.J.: Symmetry in Physics, vol.1. Macmillian Press, London (1979)

  4. Foldy, L.L.: Synthesis of covariant particle equations. Phys. Rev. 102(2), 568–581 (1956). https://doi.org/10.1103/PhysRev.102.568

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Foldy, L.L., Wouthuysen, S.A.: On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78(1), 29–36 (1950). https://doi.org/10.1103/PhysRev.78.29

    Article  ADS  MATH  Google Scholar 

  6. Fushchich, W.I., Krivsky, I.Y., Simulik, V.M.: On vector and pseudovector Lagrangians for electromagnetic field. Nuovo Cim. B. 103(4), 423–429 (1989). https://doi.org/10.1007/BF02874313

    Article  ADS  MathSciNet  Google Scholar 

  7. G\(\ddot{\rm u}\)rsey, F.: Relation of charge independence and baryon conservation to Pauli’s transformation. Nuov. Cim. 7(3), 411–415 (1958). https://doi.org/10.1007/BF02747705

  8. Good, R.H., Jr.: Properties of the Dirac matrices. Rev. Mod. Phys. 27(2), 187–211 (1955). https://doi.org/10.1103/RevModPhys.27.187

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Haouam, I.: On the Fisk–Tait equation for spin 3/2 fermions interacting with an external magnetic field in noncomutative space-time. J. Phys. Stud. 24(1), 1801 (2020). https://doi.org/10.30970/jps.24.1801

  10. Heaviside, O.: On the forces, stresses and fluxes of energy in the electromagnetc field. Phil. Trans. Roy. Soc. Lond. A. 183, 423–480 (1892)

    Article  ADS  MATH  Google Scholar 

  11. Hepner, W.A.: The inhomogeneous Lorentz group and the conformal group, \(j_{z}\)-conserving coupled states approximation. Nuov. Cim. 26(2), 351–368 (1962). https://doi.org/10.1007/BF02787046

    Article  MathSciNet  MATH  Google Scholar 

  12. Ibragimov, N.K.: Invariant variational problems and conservation laws (remarks on Noether’s theorem). Theor. Math. Phys. 1(3), 267–274 (1969). https://doi.org/10.1007/BF01035741

    Article  MathSciNet  Google Scholar 

  13. Johnson, K., Sudarshan, E.C.G.: Inconsistency of the local field theory of charged spin 3/2 particles. Ann. Phys. (N.Y.) 13(1), 126–145 (1961). https://doi.org/10.1016/0003-4916(61)90030-6

  14. Kaloshin, A.E., Lomov, V.P.: Rarita-Schwinger field and multi-component wave equation. Phys. Part. Nucl. Lett. 8(6), 517–520 (2011). https://doi.org/10.1134/S1547477111060100

    Article  Google Scholar 

  15. Khalil, M.A.K., Seetharaman, M.: Fisk–Tait equation for spin-3/2 particles. Phys. Rev. D. 18(6), (1978) 3040–3044 (1978)

  16. Kristiano, J., Clymton, S., Mart, T.: Pure spin-3/2 propagator for use in particle and nuclear physics. Phys. Rev. C 96(5), 052201R (2017). https://doi.org/10.1103/PhysRevC.96.052201

    Article  ADS  Google Scholar 

  17. Krivsky, I.Yu., Simulik, V.M.: Lagrangian for the electromagnetic field in the terms of field strengths and the conservation laws. Ukr. J. Phys. 30(10), 1457–1459 (1985) (in Russian)

  18. Krivsky, I.Y., Simulik, V.M.: The Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations. Theor. Math. Phys. 90(3), 265–276 (1992). https://doi.org/10.1007/BF01036532

    Article  Google Scholar 

  19. Krivsky, I.Y., Lompay, R.R., Simulik, V.M.: Symmetries of the complex Dirac-K\(\ddot{\rm a }\)hler equation. Theor. Math. Phys. 143(1), 541–558 (2005). https://doi.org/10.1007/s11232-005-0089-7

    Article  MATH  Google Scholar 

  20. Larmor, I.: Collected papers. Clarendon Press, London (1928)

    Google Scholar 

  21. Lounesto, P.: Clifford Algebras and Spinors, 2-nd edition. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  22. Napsuciale, M., Kirchbach, M., Rodriguez, S.: Spin 3/2 beyond the Rarita-Schwinger frame-work. Eur. Phys. J. A 29(3), 289–306 (2006). https://doi.org/10.1140/epja/i2005-10315-8

    Article  ADS  Google Scholar 

  23. Okubo, S.: Real representations of finite Clifford algebras. I. Classification. J. Math. Phys. 32(7), 1657–1668 (1991). https://doi.org/10.1063/1.529277

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Pascalutsa, V.: Correspondence of consistent and inconsistent spin-3/2 couplings via the equivalence theorem. Phys. Lett. B 503(1–2), 85–90 (2001). https://doi.org/10.1016/S0370-2693(01)00140-X

    Article  ADS  Google Scholar 

  25. Pauli, W.: On the conservation of the lepton charge. Nuov. Cim. 6(1), 204–215 (1957). https://doi.org/10.1007/BF02827771

    Article  MathSciNet  MATH  Google Scholar 

  26. Penrose, R., Rindler, W.: Spinors and Space-time, vol. 2. Cambridge University Press, New York (1986)

    Book  MATH  Google Scholar 

  27. Penrose, R., Rindler, W.: Spinors and Space-time, vol. 1. Cambridge University Press, New York (1986)

    Book  MATH  Google Scholar 

  28. Petras, M.: The SO(3,3) group as a common basis for Dirac’s and Proca’s equations. Czech J. Phys. 45(6), 455–464 (1995). https://doi.org/10.1007/BF01691683

    Article  ADS  MathSciNet  Google Scholar 

  29. Rainich, G.Y.: Electrodynamics in the general relativity theory. Trans. Amer. Math. Soc. 27, 106–136 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  30. Silenko, A.J.: Exact form of the exponential Foldy-Wouthuysen transformation operator for an arbitrary-spin particle. Phys. Rev. A 94(3), 032104 (2016). https://doi.org/10.1103/PhysRevA.94.032104

    Article  ADS  MathSciNet  Google Scholar 

  31. Simulik, V.M., Krivsky, I.Y., Lamer, I.L.: Bosonic symmetries, solutions and conservation laws for the Dirac equation with nonzero mass. Ukr. J. Phys. 58(6), 523–533 (2013). https://doi.org/10.15407/ujpe58.06.0523

  32. Simulik, V.M., Krivsky, I,Yu.: Bosonic symmetries of the massless Dirac equation. Adv. Appl. Clifford Algebras 8(1), 69–82 (1998). https://doi.org/10.1007/BF03041926

  33. Simulik, V.: (Edit.) What is the electron? Apeiron, Montreal (2005)

  34. Simulik, V.M.: Derivation of the Dirac and Dirac-like equations of arbitrary spin from the corresponding relativistic canonical quantum mechanics. Ukr. J. Phys. 60(10), 985–1006 (2015). https://doi.org/10.15407/ujpe60.10.0985

  35. Simulik, V.: Relativistic quantum mechanics and field theory of arbitrary spin. Nova Science, New York (2020). https://doi.org/10.52305/VFKY2861

  36. Simulik, V.M, Gordievich, I.O.: Symmetries of relativistic hydrogen atom. Ukr. J. Phys. 64(12), 1148–1153 (2019). https://doi.org/10.15407/ujpe64.12.1148

  37. Simulik, V.M. Vyikon. I.I.: On the choice of relativistic wave equation for the particle having spin s=3/2. J. Phys. Commun. 6(7), 075008 (2022). https://doi.org/10.1088/2399-6528/ac7eae

  38. Simulik, V.M.: Link between the relativistic canonical quantum mechanics of arbitrary spin and the corresponding field theory. J. Phys: Conf. Ser. 670, 012047 (2016). https://doi.org/10.1088/1742-6596/670/1/012047

    Article  Google Scholar 

  39. Simulik, V.M.: On the gamma matrix representations of SO(8) and Clifford Algebras. Adv. Appl. Clifford Algebras 28(5), 93 (2018). https://doi.org/10.1007/s00006-018-0906-3

    Article  MathSciNet  MATH  Google Scholar 

  40. Simulik, V., Gordievich, I.: Hidden symmetries of relativistic hydrogen atom. J. Phys: Conf. Ser. 1416, 012034 (2019). https://doi.org/10.1088/1742-6596/1416/1/012034

    Article  Google Scholar 

  41. Simulik, V.M., Krivsky, I.Y.: Clifford algebra in classical electrodynamical hydrogen atom model. Adv. Appl. Clifford Algebras 7(1), 25–34 (1997). https://doi.org/10.1007/BF03041213

    Article  MathSciNet  MATH  Google Scholar 

  42. Simulik, V.M., Krivsky, I.Y.: Bosonic symmetries of the Dirac equation. Phys. Lett. A 375(25), 2479–2483 (2011). https://doi.org/10.1016/j.physleta.2011.03.058

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Simulik, V.M., Krivsky, I.Y., Lamer, I.L.: Some statistical aspects of the spinor field Fermi-Bose duality. Cond. Matt. Phys. 15(4), 43101 (2012). https://doi.org/10.5488/CMP.15.43101

    Article  ADS  Google Scholar 

  44. Vaz, J., Jr., da Rocha R., Jr.: An Introduction to Clifford Algebras and Spinors. Oxford University Press, Oxford (2016). https://doi.org/10.1093/acprof:oso/9780198782926.001.0001

  45. Vaz, J., Jr.: The Clifford algebra of physical space and Dirac theory. Eur. J. Phys. 37(5), 055407 (2016). https://doi.org/10.1088/0143-0807/37/5/055407

    Article  MATH  Google Scholar 

  46. Velo, G., Zwanziger, D.: Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential. Phys. Rev. 186(5), 1337–1341 (1969). https://doi.org/10.1103/PhysRev.186.1337

    Article  ADS  Google Scholar 

  47. Williams, H.T.: Misconceptions regarding spin 3/2. Phys. Rev. C 31(6), 2297–2299 (1985). https://doi.org/10.1103/PhysRevC.31.2297

    Article  ADS  Google Scholar 

  48. Wybourne, B.J.: Classical Groups for Physicists. John Wiley and Sons, New York (1974)

    MATH  Google Scholar 

  49. Zou, L., Zhang, P., Silenko, A.J.: Position and spin in relativistic quantum mechanics. Phys. Rev. A 101(3), 032117 (2020). https://doi.org/10.1103/PhysRevA.101.032117

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

Volodimir Simulik is very grateful for the two month Fellowship at the Erwin Schödinger International Institute for Mathematics and Physics of the University of Vienna. The authors are much grateful for the unknown referees for kind and very useful remarks.

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Authors and Affiliations

  1. Institute of Electron Physics, National Academy of Sciences, Universitetska 21, 88017, Uzhgorod, Ukraine

    V. M. Simulik

  2. Erwin Schödinger Institute for Mathematics and Physics, University of Vienna, Vienna, Austria

    V. M. Simulik & I. I. Vyikon

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  1. V. M. Simulik
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Simulik, V.M., Vyikon, I.I. On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation. Adv. Appl. Clifford Algebras 33, 53 (2023). https://doi.org/10.1007/s00006-023-01295-7

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