In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom.
References
[1]
Jackiw, R.; Nair, V.P. Relativistic wave equation for anyons. Phys. Rev. D 1991, 43, 1933–1942, doi:10.1103/PhysRevD.43.1933.
[2]
Plyushchay, M. Fractional spin: Majorana-Dirac field. Phys. Lett. B 1991, 273, 250–254, doi:10.1016/0370-2693(91)91679-P.
[3]
Horváthy, P.; Plyushchay, M.; Valenzuela, M. Bosons, fermions and anyons in the plane, and supersymmetry. Ann. Phys. 2010, 325, 1931–1975, doi:10.1016/j.aop.2010.02.007.
[4]
Horváthy, P.A.; Plyushchay, M.S.; Valenzuela, M. Supersymmetry between Jackiw–Nair and Dirac–Majorana anyons. Phys. Rev. D 2010, 81, doi:10.1103/PhysRevD.81.127701.
[5]
Horváthy, P.; Plyushchay, M.; Valenzuela, M. Supersymmetry of the planar Dirac–Deser–Jackiw–Templeton system and of its nonrelativistic limit. J. Math. Phys. 2010, 51, doi:10.1063/1.3478558.
[6]
Horváthy, P.; Plyushchay, M.; Valenzuela, M. Bosonized supersymmetry from the Majorana–Dirac–Staunton theory and massive higher-spin fields. Phys. Rev. D 2008, 77, doi:10.1103/PhysRevD.77.025017.
[7]
Simulik, V.; Krivsky, I. Bosonic symmetries of the massless Dirac equation. Adv. Appl. Clifford Algebr. 1998, 8, 69–82, doi:10.1007/BF03041926.
[8]
Simulik, V.; Krivsky, I. Bosonic symmetries of the Dirac equation. Phys. Lett. A 2011, 375, 2479–2483, doi:10.1016/j.physleta.2011.03.058.
[9]
Okninski, A. Supersymmetric content of the Dirac and Duffin–Kemmer–Petiau equations. Int. J. Theor. Phys. 2011, 50, 729–736, doi:10.1007/s10773-010-0608-7.
[10]
Bagrov, V.; Gitman, D. Exact Solutions of Relativistic Wave Equations; Springer: Berlin, Germany, 1990; Volume 39.
[11]
Berestetskii, V.; Lifshitz, E.; Pitaevskii, V. Relativistic Quanturn Theory, Part 1; McGraw-Hill Science: New York, NY, USA, 1971.
[12]
Misner, C.; Thorne, K.; Wheeler, J. Gravitation; WH Freeman Co.: New York, NY, USA, 1973.
[13]
Corson, E. Theory of Tensors, Spinors, and Wave Equations; Blackie: London, UK, 1953.
[14]
Dirac, P. The quantum theory of the electron. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Character 1928, 117, 610–624, doi:10.1098/rspa.1928.0023.
[15]
Dirac, P. The quantum theory of the electron. Part II. Proc. R. Soc. Lond. Ser. A 1928, 118, 351–361, doi:10.1098/rspa.1928.0056.
[16]
Bjorken, J.; Drell, S. Relativistic Quantum Mechanics; McGraw-Hill: New York, NY, USA, 1964; Volume 3.
[17]
Thaller, B. The Dirac Equation; Springer-Verlag: New York, NY, USA, 1992.
[18]
Zralek, M. On the possibilities of distinguishing dirac from majorana neutrinos. Acta Phys. Pol. B 1997, 28, 2225–2257.
[19]
Perkins, D. Introduction to High Energy Physics; Cambridge University Press: Cambridge, UK, 2000.
[20]
Fukugita, M.; Yanagida, T. Physics of Neutrinos and Applications to Astrophysics; Springer Verlag: Berlin, Germany, 2003.
[21]
Szafron, R.; Zralek, M. Can we distinguish dirac and majorana neutrinos produced in muon decay? Acta Phys. Pol. B 2009, 40, 3041–3047.
[22]
Majorana, E. Teoria simmetrica dell’elettrone e del positrone. Il Nuovo Cimento (1924–1942) 1937, 14, 171–184, doi:10.1007/BF02961314.
[23]
Feynman, R.; Gell-Mann, M. Theory of the Fermi interaction. Phys. Rev. 1958, 109, 193–198, doi:10.1103/PhysRev.109.193.
[24]
Horvathy, P.; Feher, L.; O’Raifeartaigh, L. Applications of chiral supersymmetry for spin fields in selfdual backgrounds. Int. J. Mod. Phys. 1989, A4, 5277–5285.
[25]
Comtet, A.; Horvathy, P. The Dirac equation in Taub-NUT space. Phys. Lett. B 1995, 349, 49–56, doi:10.1016/0370-2693(95)00219-B.
[26]
Duffin, R. On the characteristic matrices of covariant systems. Phys. Rev. 1938, 54, doi:10.1103/PhysRev.54.1114.
[27]
Kemmer, N. The particle aspect of meson theory. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 1939, 173, 91–116, doi:10.1098/rspa.1939.0131.
[28]
Kemmer, N. The Algebra of Meson Matrices. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1943; 39, pp. 189–196.
[29]
Petiau, G. Contribution a la théorie des équations d’ondes corpusculaires. Acad. R. Belg. Classe Sci. Mem. 1936, 16, 1–136.
[30]
Proca, A. Wave theory of positive and negative electrons. J. Phys. Radium 1936, 7, 347–353, doi:10.1051/jphysrad:0193600708034700.
[31]
Proca, A. Sur les equations fondamentales des particules elementaires. Comp. Ren. Acad. Sci. Paris 1936, 202, 1366–1368.
[32]
Lanczos, C. Die erhaltungss？tze in der feldm？？igen darstellung der diracschen theorie. Zeitschrift Phys. Hadron. Nuclei 1929, 57, 484–493.
[33]
Bogush, A.; Kisel, V.; Tokarevskaya, N.; Red’kov, V. Duffin–Kemmer–Petiau formalism reexamined: Non-relativistic approximation for spin 0 and spin 1 particles in a Riemannian space-time. Ann. Fond. Louis Broglie 2007, 32, 355–381.
[34]
Okninski, A. Effective quark equations. Acta Phys. Pol. B 1981, 12, 87–94.
[35]
Okninski, A. Dynamic theory of quark and meson fields. Phys. Rev. D 1982, 25, 3402–3407, doi:10.1103/PhysRevD.25.3402.
[36]
Okninski, A. Splitting the Kemmer–Duffin–Petiau equations. Available online: http://arxiv.org/pdf/math-ph/0309013v1.pdf (accessed on 30 July 2012).
[37]
Colladay, D.; Kostelecky, V.A. CPT violation and the standard model. Phys. Rev. D 1997, 55, 6760–6774.
[38]
Visser, M. Lorentz symmetry breaking as a quantum field theory regulator. Phys. Rev. D 2009, 80, doi:10.1103/PhysRevD.80.025011.
[39]
Kostelecky, V.A.; Russell, N. Data tables for Lorentz and CPT violation. Rev. Mod. Phys. 2011, 83, 11–31, doi:10.1103/RevModPhys.83.11.
