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Year 2013, Volume 3, Issue 1, 46 - 61, 01.06.2013

Abstract

References

  • [1] Simulik, V.M., Krivsky, I.Yu. and Lamer, I.L., (2012), Generalized Clifford - Dirac algebra and Fermi - Bose duality of the Dirac equation, Proceedings of the 14-th Internat. Conference ”Mathematical Methods in Electromagnetic Theory” - 28-30 August 2012, Institute of Radiophysics and Electronics, Kharkiv, Ukraine. 197-200.
  • [2] Simulik, V.M. and Krivsky, I.Yu., (2009), Extended real Clifford - Dirac algebra and bosonic symmetries of the Dirac equation with nonzero mass, arXiv: math-ph, 0908.3106, 21 Aug 2009, 5 p.
  • [3] Simulik, V.M. and Krivsky, I.Yu., (2010), On the extended real Clifford - Dirac algebra and new physical meaningful symmetries of the Dirac equation with nonzero mass, Reports of the National Academy of Sciences of Ukraine, no 5, 82-88.
  • [4] Krivsky, I.Yu. and Simulik, V.M., (2010), Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra, Cond. Matt. Phys., 13, 43101 (1 -15).
  • [5] Simulik, V.M. and Krivsky, I.Yu., (2011), Bosonic symmetries of the Dirac equation, Phys. Lett. A., 375, 2479-2483.
  • [6] Darwin, C.G., (1928), The wave equation of the electron, Proc. Roy. Soc. London, 118a, no 780, 654-680.
  • [7] Laporte, O. and Uhlenbeck, G.E., (1931), Application of spinor analysis to the Maxwell and Dirac equation, Phys. Rev., 37, 1380-1397.
  • [8] Oppenheimer, J.R., (1931), Note on light quanta and the electromagnetic field, Phys. Rev., 38, 725- 746.
  • [9] Mignani, R., Recami, E. and Baldo, M., (1974), About a Dirac-like equation for the photon according to Ettore Majorana, Lett. Nuovo cim., 11, no 12, 572-586.
  • [10] Archibald, W.J., (1955), Field equations from particle equations, Can. Journ. Phys., 33, 565-572.
  • [11] Good, R.H., (1957), Particle aspect of the electromagnetic field equations, Phys. Rev., 105, 1914-1919.
  • [12] Bludman, S.A., (1957), Some theoretical consequences of a particle having mass zero, Phys. Rev., 107, 1163-1168.
  • [13] Moses, H.E., (1958), A spinor representation of Maxwell’s equations, Nuovo cim. Suppl., 7, no 1, 1-18.
  • [14] Lomont, J.S., (1958), Dirac-like wave equations for particles of zero rest mass and their quantization, Phys. Rev., 111, 1710-1716.
  • [15] Nelson, T.G. and Good, R.H., (1969), Description of massless particles, Phys. Rev., 179, 1445-1449.
  • [16] Sankaranarayanan, A.A., (1970), A Hamilton form of Maxwell’s equations, Prog. Theor. Phys., 43, 1204-1212.
  • [17] Lord, E.A., (1972), Algebraic properties of relativistic equations for zero rest mass, Intern. Journ. Theor. Phys., 5, 349-357.
  • [18] Sallhofer, H., (1978), Elementary derivation of the Dirac equation, Z. Naturforsch. A., 33, 1379-1381.
  • [19] Da Silveira, A., (1979), Dirac-like equation for the photon, Z. Naturforsch. A., 34, 646-647.
  • [20] Hillion, P. and Quinnes, S., (1984), Spinor solutions to Maxwell’s equations in free space, Canad. Journ. Phys., 62, 674-682.
  • [21] Giannetto, E., (1985), A Majorana-Oppenheimer formulation of quantum electrodynamics, Lett. Nuovo Cim. 44, no 3, 140-144.
  • [22] Sallhofer, H., (1986), Maxwell-Dirac - Isomorphism. XI, Z. Naturforsch. A., 41, 1087-1088.
  • [23] Ljolje, K., (1988), Some remarks on variational formulations of physical fields, Fortschr. Phys., 36, 9-32.
  • [24] Fushchich, W.I., Shtelen, W.M. and Spichak, S.V., (1991), On the connection between solutions of Dirac and Maxwell equations, dual Poincare invariance and superalgebras of invariance and solutions of nonlinear Dirac equation // J. Phys. A., 24, 1683-1698.
  • [25] Foldy, L., (1956), Synthesis of covariant particle equations, Phys. Rev., 102, 568-581.
  • [26] Garbaczewski, P., (1986), Boson - Fermion duality in four dimensions: comments on the paper of Luther and Schotte, Internat. Journ. Theor. Phys., 25, 1193-1208.
  • [27] Simulik, V.M., (1994), Some algebraic properties of Maxwell-Dirac isomorphism, Z. Naturforsch. A., 49, 1074-1076.
  • [28] Simulik, V.M. and Krivsky, I.Yu., (1997), Clifford algebra in classical electrodynamical hydrogen atom model, Adv. Appl. Cliff. Algebras, 7, 25-34.
  • [29] Simulik, V.M. and Krivsky, I.Yu., (1998), Bosonic symmetries of the massless Dirac equations, Adv. Appl. Cliff. Algebras, 8, 69-82.
  • [30] Simulik, V.M. and Krivsky, I.Yu., (2002), Slightly generalized Maxwell classical electrodynamics can be applied to inneratomic phenomena // Annales de la Fondation Louis de Broglie (Special issue: Contemporary Electrodynamics), 27, 303-329.
  • [31] Simulik, V.M. and Krivsky, I.Yu., (2002), Relationship between the Maxwell and Dirac equations: symmetries, quantization, models of atom, Reports on Mathematical Physics, 50, P. 315-328.
  • [32] Simulik, V.M. and Krivsky, I.Yu., (2003), Classical electrodynamical aspect of the Dirac equation, Electromagnetic phenomena, 3, no 1(9), 103-114.
  • [33] Simulik, V.M., (2005), The electron as a system of classical electromagnetic and scalar fields, In book: What is the electron? Edited by V.M. Simulik, Montreal, Apeiron, 109-134.
  • [34] Sallhofer, H., (1990), Hydrogen in electrodynamics. VI. The general solution, Z. Naturforsch. A., 45, 1361-1366.
  • [35] Sallhofer, H., (1991), Hydrogen in electrodynamics. VII. The Pauli principle, Z. Naturforsch. A., 46, 869-872.
  • [36] Campolattaro, A.A., (1980), New spinor representation of Maxwell’s equations. I. Generalities, Intern. Journ. Theor. Phys. 19, 99-126.
  • [37] Campolattaro, A.A., (1990), Generalized Maxwell equations and quantum mechanics. I. Dirac equation for the free electron, Intern. Journ. Theor. Phys.,29, 141-155.
  • [38] Daviau, C., (1989), Electromagnetisme, monopoles, magnetiques et ondes de matiere dans l’algebre d’espace-temps (lere partie), Ann. Fond. L. de Broglie, 14, 273-300.
  • [39] Rodrigues, W.A. Jr., Vaz, J. Jr. and Recami, E., (1993), Free Maxwell equations, Dirac equation and Non-Dispersive de Broglie wave packets, In: Directions in Microphysics, Foundation Louis de Broglie, Paris, 379-392.
  • [40] Keller, J., (1999), The geometric content of the electron theory, Adv. Appl. Cliff. Algebras, 9, 309-395.
  • [41] Keller, J., (1997) On the electron theory, Proceedings of the International Conference ”The theory of electron”, Mexico (Mexico): 24-27 September 1995, Adv. Appl. Cliff. Algebras 7(Special), .3-26.
  • [42] Kruglov, S.I., (2001), Generalized Maxwell equations and their solutions, Ann. Fond. L. de Broglie, 26, 725-734.
  • [43] Xuegang, Y., Shuma, Z. and Quinan, H., (2001), Clifford algebraic spinor and the Dirac wave equation, Adv. Appl. Cliff. Algebras, 11, 27-38.
  • [44] Grudsky, S.M., Khmelnytskaya, K.V. and Kravchenko, V.V., (2004), On a quaternionic Maxwell equation for the time-dependent electromagnetic field in a chiral medium, J. Phys. A., 37, 4641-4647.
  • [45] Armour, R.S. Jr., (2004), Spin 1/2 Maxwell field, Found. Phys., 34, 815-842.
  • [46] Varlamov, V.V., (2005), Maxwell field on the Poincare group, Intern. J. Mod. Phys. A., 20, 4095-4112.
  • [47] Rozzi, T., Mencarelli, D. and Pierantoni, L.,(2009), Deriving electromagnetic fields from the spinor solution of the massless Dirac equation, IEEE: Trans. Microw. Theor. Techn., 57, 2907-2913.
  • [48] Okninski, A., (2012), Duffin-Kemmer-Petiau and Dirac equations – a supersymmetric content, Symmetry, 4, 427-440.
  • [49] Foldy, L. and Wouthuysen, S., (1950), On the Dirac theory of spin 1/2 particles and its non- relativistic limit, Phys. Rev. 78. 29-36.
  • [50] Krivsky, I.Yu.,Zajac, T.M., Simulik, V.M. and Lamer, I.L., (2012), On the bosonic solutions for the Dirac equation for the free field, Uzhgorod Univ. Scientific Herald, Ser. Physics, 31, 163-173.
  • [51] Elliott, J.P. and Dawber, P.J., (1979), Symmetry in Physics, Vol.1, Macmillian Press, London.
  • [52] Wybourne, B.G., (1974), Classical groups for Physicists, John Wiley and sons, New York.
  • [53] Gursey, F., (1958), Relation of charge independence and baryon conservation to Pauli’s transformation, Nuovo Cim., 7, 411-415.
  • [54] Ibragimov, N.H., (1969), Invariant variation problems and the conservation laws (remarks on the Noether theorem), Theor. Math. Phys., 1, 267-274.
  • [55] Hepner, W.A., (1962), The inhomogeneous Lorentz group and the conformal group, Nuovo Cim., 26, 351368.
  • [56] Petras, M., (1995), The SO(3,3) group as a common basis for Diracs and Procas equations, Czech. J. Phys., 45, 455464.
  • [57] Bogoliubov, N.N. and Shirkov, D.V., (1980), Introduction to the theory of quantized fields, John Wiley and Sons, New York.
  • [58] Krech, W., (1969), Einige Bemerkungen zur Klassischen Theorie des Anschaulichen Wellenbildes fur Kraftefreie Materie mit Spin, Wissenschaftliche Zeitschrift der Friedrich-Schiller Universitat Jena, Mathematisch-Naturwissenschaftliche Reine, 18, no 1, 159-163.
  • [59] Krech, W., (1972), Erhaltungssatze des quantisierten Foldy - Wouthuysen Felde, Wissenschaftliche Zeitschrift der Friedrich-Schiller Universitat Jena, Mathematisch-Naturwissenschaftliche Reine, 21, no 1, 51-54.
  • [60] Neznamov, V.P., (2006), On the theory of interacting fields in the FW representation, Phys. Part. Nucl., 37, 86-115.
  • [61] Krivsky, I., Simulik, V., Zajac, T. and Lamer, I., Derivation of the Dirac and Maxwell equations from the first principles of relativistic canonical quantum mechanics // Proceedings of the 14-th Internat. Conference ”Mathematical Methods in Electromagnetic Theory” - 28-30 August 2012, Institute of Radiophysics and Electronics, Kharkiv, Ukraine. 201-204.
  • [62] Thaller B, (1992), The Dirac equation, Springer, Berlin.
  • [63] Neznamov, V.P. and Silenko, A.J., (2009), Foldy-Wouthuysen wave functions and conditions of transformation between Dirac and Foldy-Wouthuysen representations, J. Math. Phys., 50, 122302 (1-15).

APPLICATION OF THE GENERALIZED CLIFFORD-DIRAC ALGEBRA TO THE PROOF OF THE DIRAC EQUATION FERMI-BOSE DUALITY

Year 2013, Volume 3, Issue 1, 46 - 61, 01.06.2013

Abstract

The consideration of the bosonic properties of the Dirac equation with arbitrary mass has been continued. As the necessary mathematical tool the structure and different representations of the 29-dimensional extended real Clifford-Dirac algebra Phys. Lett. A., 2011, v.375, p.2479 are considered briefly. As a next step we use the start from the Foldy-Wouthuysen representation. On the basis of these two ideas the property of Fermi-Bose duality of the Dirac equation with nonzero mass is proved. The proof is given on the three maim examples: bosonic symmetries, bosonic solutions and bosonic conservation laws. It means that Dirac equation can describe not only the fermionic but also the bosonic states.

References

  • [1] Simulik, V.M., Krivsky, I.Yu. and Lamer, I.L., (2012), Generalized Clifford - Dirac algebra and Fermi - Bose duality of the Dirac equation, Proceedings of the 14-th Internat. Conference ”Mathematical Methods in Electromagnetic Theory” - 28-30 August 2012, Institute of Radiophysics and Electronics, Kharkiv, Ukraine. 197-200.
  • [2] Simulik, V.M. and Krivsky, I.Yu., (2009), Extended real Clifford - Dirac algebra and bosonic symmetries of the Dirac equation with nonzero mass, arXiv: math-ph, 0908.3106, 21 Aug 2009, 5 p.
  • [3] Simulik, V.M. and Krivsky, I.Yu., (2010), On the extended real Clifford - Dirac algebra and new physical meaningful symmetries of the Dirac equation with nonzero mass, Reports of the National Academy of Sciences of Ukraine, no 5, 82-88.
  • [4] Krivsky, I.Yu. and Simulik, V.M., (2010), Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra, Cond. Matt. Phys., 13, 43101 (1 -15).
  • [5] Simulik, V.M. and Krivsky, I.Yu., (2011), Bosonic symmetries of the Dirac equation, Phys. Lett. A., 375, 2479-2483.
  • [6] Darwin, C.G., (1928), The wave equation of the electron, Proc. Roy. Soc. London, 118a, no 780, 654-680.
  • [7] Laporte, O. and Uhlenbeck, G.E., (1931), Application of spinor analysis to the Maxwell and Dirac equation, Phys. Rev., 37, 1380-1397.
  • [8] Oppenheimer, J.R., (1931), Note on light quanta and the electromagnetic field, Phys. Rev., 38, 725- 746.
  • [9] Mignani, R., Recami, E. and Baldo, M., (1974), About a Dirac-like equation for the photon according to Ettore Majorana, Lett. Nuovo cim., 11, no 12, 572-586.
  • [10] Archibald, W.J., (1955), Field equations from particle equations, Can. Journ. Phys., 33, 565-572.
  • [11] Good, R.H., (1957), Particle aspect of the electromagnetic field equations, Phys. Rev., 105, 1914-1919.
  • [12] Bludman, S.A., (1957), Some theoretical consequences of a particle having mass zero, Phys. Rev., 107, 1163-1168.
  • [13] Moses, H.E., (1958), A spinor representation of Maxwell’s equations, Nuovo cim. Suppl., 7, no 1, 1-18.
  • [14] Lomont, J.S., (1958), Dirac-like wave equations for particles of zero rest mass and their quantization, Phys. Rev., 111, 1710-1716.
  • [15] Nelson, T.G. and Good, R.H., (1969), Description of massless particles, Phys. Rev., 179, 1445-1449.
  • [16] Sankaranarayanan, A.A., (1970), A Hamilton form of Maxwell’s equations, Prog. Theor. Phys., 43, 1204-1212.
  • [17] Lord, E.A., (1972), Algebraic properties of relativistic equations for zero rest mass, Intern. Journ. Theor. Phys., 5, 349-357.
  • [18] Sallhofer, H., (1978), Elementary derivation of the Dirac equation, Z. Naturforsch. A., 33, 1379-1381.
  • [19] Da Silveira, A., (1979), Dirac-like equation for the photon, Z. Naturforsch. A., 34, 646-647.
  • [20] Hillion, P. and Quinnes, S., (1984), Spinor solutions to Maxwell’s equations in free space, Canad. Journ. Phys., 62, 674-682.
  • [21] Giannetto, E., (1985), A Majorana-Oppenheimer formulation of quantum electrodynamics, Lett. Nuovo Cim. 44, no 3, 140-144.
  • [22] Sallhofer, H., (1986), Maxwell-Dirac - Isomorphism. XI, Z. Naturforsch. A., 41, 1087-1088.
  • [23] Ljolje, K., (1988), Some remarks on variational formulations of physical fields, Fortschr. Phys., 36, 9-32.
  • [24] Fushchich, W.I., Shtelen, W.M. and Spichak, S.V., (1991), On the connection between solutions of Dirac and Maxwell equations, dual Poincare invariance and superalgebras of invariance and solutions of nonlinear Dirac equation // J. Phys. A., 24, 1683-1698.
  • [25] Foldy, L., (1956), Synthesis of covariant particle equations, Phys. Rev., 102, 568-581.
  • [26] Garbaczewski, P., (1986), Boson - Fermion duality in four dimensions: comments on the paper of Luther and Schotte, Internat. Journ. Theor. Phys., 25, 1193-1208.
  • [27] Simulik, V.M., (1994), Some algebraic properties of Maxwell-Dirac isomorphism, Z. Naturforsch. A., 49, 1074-1076.
  • [28] Simulik, V.M. and Krivsky, I.Yu., (1997), Clifford algebra in classical electrodynamical hydrogen atom model, Adv. Appl. Cliff. Algebras, 7, 25-34.
  • [29] Simulik, V.M. and Krivsky, I.Yu., (1998), Bosonic symmetries of the massless Dirac equations, Adv. Appl. Cliff. Algebras, 8, 69-82.
  • [30] Simulik, V.M. and Krivsky, I.Yu., (2002), Slightly generalized Maxwell classical electrodynamics can be applied to inneratomic phenomena // Annales de la Fondation Louis de Broglie (Special issue: Contemporary Electrodynamics), 27, 303-329.
  • [31] Simulik, V.M. and Krivsky, I.Yu., (2002), Relationship between the Maxwell and Dirac equations: symmetries, quantization, models of atom, Reports on Mathematical Physics, 50, P. 315-328.
  • [32] Simulik, V.M. and Krivsky, I.Yu., (2003), Classical electrodynamical aspect of the Dirac equation, Electromagnetic phenomena, 3, no 1(9), 103-114.
  • [33] Simulik, V.M., (2005), The electron as a system of classical electromagnetic and scalar fields, In book: What is the electron? Edited by V.M. Simulik, Montreal, Apeiron, 109-134.
  • [34] Sallhofer, H., (1990), Hydrogen in electrodynamics. VI. The general solution, Z. Naturforsch. A., 45, 1361-1366.
  • [35] Sallhofer, H., (1991), Hydrogen in electrodynamics. VII. The Pauli principle, Z. Naturforsch. A., 46, 869-872.
  • [36] Campolattaro, A.A., (1980), New spinor representation of Maxwell’s equations. I. Generalities, Intern. Journ. Theor. Phys. 19, 99-126.
  • [37] Campolattaro, A.A., (1990), Generalized Maxwell equations and quantum mechanics. I. Dirac equation for the free electron, Intern. Journ. Theor. Phys.,29, 141-155.
  • [38] Daviau, C., (1989), Electromagnetisme, monopoles, magnetiques et ondes de matiere dans l’algebre d’espace-temps (lere partie), Ann. Fond. L. de Broglie, 14, 273-300.
  • [39] Rodrigues, W.A. Jr., Vaz, J. Jr. and Recami, E., (1993), Free Maxwell equations, Dirac equation and Non-Dispersive de Broglie wave packets, In: Directions in Microphysics, Foundation Louis de Broglie, Paris, 379-392.
  • [40] Keller, J., (1999), The geometric content of the electron theory, Adv. Appl. Cliff. Algebras, 9, 309-395.
  • [41] Keller, J., (1997) On the electron theory, Proceedings of the International Conference ”The theory of electron”, Mexico (Mexico): 24-27 September 1995, Adv. Appl. Cliff. Algebras 7(Special), .3-26.
  • [42] Kruglov, S.I., (2001), Generalized Maxwell equations and their solutions, Ann. Fond. L. de Broglie, 26, 725-734.
  • [43] Xuegang, Y., Shuma, Z. and Quinan, H., (2001), Clifford algebraic spinor and the Dirac wave equation, Adv. Appl. Cliff. Algebras, 11, 27-38.
  • [44] Grudsky, S.M., Khmelnytskaya, K.V. and Kravchenko, V.V., (2004), On a quaternionic Maxwell equation for the time-dependent electromagnetic field in a chiral medium, J. Phys. A., 37, 4641-4647.
  • [45] Armour, R.S. Jr., (2004), Spin 1/2 Maxwell field, Found. Phys., 34, 815-842.
  • [46] Varlamov, V.V., (2005), Maxwell field on the Poincare group, Intern. J. Mod. Phys. A., 20, 4095-4112.
  • [47] Rozzi, T., Mencarelli, D. and Pierantoni, L.,(2009), Deriving electromagnetic fields from the spinor solution of the massless Dirac equation, IEEE: Trans. Microw. Theor. Techn., 57, 2907-2913.
  • [48] Okninski, A., (2012), Duffin-Kemmer-Petiau and Dirac equations – a supersymmetric content, Symmetry, 4, 427-440.
  • [49] Foldy, L. and Wouthuysen, S., (1950), On the Dirac theory of spin 1/2 particles and its non- relativistic limit, Phys. Rev. 78. 29-36.
  • [50] Krivsky, I.Yu.,Zajac, T.M., Simulik, V.M. and Lamer, I.L., (2012), On the bosonic solutions for the Dirac equation for the free field, Uzhgorod Univ. Scientific Herald, Ser. Physics, 31, 163-173.
  • [51] Elliott, J.P. and Dawber, P.J., (1979), Symmetry in Physics, Vol.1, Macmillian Press, London.
  • [52] Wybourne, B.G., (1974), Classical groups for Physicists, John Wiley and sons, New York.
  • [53] Gursey, F., (1958), Relation of charge independence and baryon conservation to Pauli’s transformation, Nuovo Cim., 7, 411-415.
  • [54] Ibragimov, N.H., (1969), Invariant variation problems and the conservation laws (remarks on the Noether theorem), Theor. Math. Phys., 1, 267-274.
  • [55] Hepner, W.A., (1962), The inhomogeneous Lorentz group and the conformal group, Nuovo Cim., 26, 351368.
  • [56] Petras, M., (1995), The SO(3,3) group as a common basis for Diracs and Procas equations, Czech. J. Phys., 45, 455464.
  • [57] Bogoliubov, N.N. and Shirkov, D.V., (1980), Introduction to the theory of quantized fields, John Wiley and Sons, New York.
  • [58] Krech, W., (1969), Einige Bemerkungen zur Klassischen Theorie des Anschaulichen Wellenbildes fur Kraftefreie Materie mit Spin, Wissenschaftliche Zeitschrift der Friedrich-Schiller Universitat Jena, Mathematisch-Naturwissenschaftliche Reine, 18, no 1, 159-163.
  • [59] Krech, W., (1972), Erhaltungssatze des quantisierten Foldy - Wouthuysen Felde, Wissenschaftliche Zeitschrift der Friedrich-Schiller Universitat Jena, Mathematisch-Naturwissenschaftliche Reine, 21, no 1, 51-54.
  • [60] Neznamov, V.P., (2006), On the theory of interacting fields in the FW representation, Phys. Part. Nucl., 37, 86-115.
  • [61] Krivsky, I., Simulik, V., Zajac, T. and Lamer, I., Derivation of the Dirac and Maxwell equations from the first principles of relativistic canonical quantum mechanics // Proceedings of the 14-th Internat. Conference ”Mathematical Methods in Electromagnetic Theory” - 28-30 August 2012, Institute of Radiophysics and Electronics, Kharkiv, Ukraine. 201-204.
  • [62] Thaller B, (1992), The Dirac equation, Springer, Berlin.
  • [63] Neznamov, V.P. and Silenko, A.J., (2009), Foldy-Wouthuysen wave functions and conditions of transformation between Dirac and Foldy-Wouthuysen representations, J. Math. Phys., 50, 122302 (1-15).

Details

Primary Language English
Journal Section Research Article
Authors

Volodimir SİMULİK This is me
Institute of Electron Physics, National Academy of Sciences of Ukraine, 21 Universitetska Str., 88000 Uzhgorod, Ukraine


Ivan KRİVSKY This is me
Institute of Electron Physics, National Academy of Sciences of Ukraine, 21 Universitetska Str., 88000 Uzhgorod, Ukraine


Irina LAMER This is me
Institute of Electron Physics, National Academy of Sciences of Ukraine, 21 Universitetska Str., 88000 Uzhgorod, Ukraine

Publication Date June 1, 2013
Published in Issue Year 2013, Volume 3, Issue 1

Cite

Bibtex @ { twmsjaem761695, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2013}, volume = {3}, number = {1}, pages = {46 - 61}, title = {APPLICATION OF THE GENERALIZED CLIFFORD-DIRAC ALGEBRA TO THE PROOF OF THE DIRAC EQUATION FERMI-BOSE DUALITY}, key = {cite}, author = {Simulik, Volodimir and Krivsky, Ivan and Lamer, Irina} }