Reformulation of Relativistic Quantum Mechanics Equations with Non-Commutative Sedeons

Affiliation(s)

Institute for Physics of Microstructures of the Russian Academy of Sciences, Nizhniy Novgorod, Russia.

Institute for Physics of Microstructures of the Russian Academy of Sciences, Nizhniy Novgorod, Russia.

Abstract

We present sixteen-component values “sedeons”, generating associative non-commutative space-time algebra. The generalized relativistic wave equations based on sedeonic wave function and space-time operators are proposed. We demonstrate that sedeonic second-order wave equation for massive field can be reformulated as the quasi-classical equation for the potentials of the field or in equivalent form as the Maxwell-like equations for the field intensities. The sedeonic first-order Dirac-like equations for massive and massless fields are also discussed.

Keywords

Clifford Algebra; Sedeons; Relativistic Quantum Mechanics; Sedeonic Klein-Gordon and Dirac Equations

Clifford Algebra; Sedeons; Relativistic Quantum Mechanics; Sedeonic Klein-Gordon and Dirac Equations

Cite this paper

V. Mironov and S. Mironov, "Reformulation of Relativistic Quantum Mechanics Equations with Non-Commutative Sedeons,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 53-60. doi: 10.4236/am.2013.410A3007.

V. Mironov and S. Mironov, "Reformulation of Relativistic Quantum Mechanics Equations with Non-Commutative Sedeons,"

References

[1] S. L. Adler, “Quaternionic Quantum Mechanics and Quantum Fields,” Oxford University Press, New York, 1995.

[2] V. Majernik, “Quaternionic Formulation of the Classical Fields,” Advances in Applied Clifford Algebras, Vol. 9, No. 1, 1999, pp. 119-130.

http://dx.doi.org/10.1007/BF03041944

[3] K. Imaeda, “A New Formulation of Classical Electrodynamics,” Nuovo Cimento, Vol. 32, No. 1, 1976, pp. 138162. http://dx.doi.org/10.1007/BF02726749

[4] A. J. Davies, “Quaternionic Dirac Equation,” Physical Review D, Vol. 41, No. 8, 1990, pp. 2628-2630.

http://dx.doi.org/10.1103/PhysRevD.41.2628

[5] S. De Leo and P. Rotelli, “Quaternion Scalar Field,” Physical Review D, Vol. 45, No. 2, 1992, pp. 575-579.

http://dx.doi.org/10.1103/PhysRevD.45.575

[6] C. Schwartz, “Relativistic Quaternionic Wave Equation,” Journal of Mathematical Physics, Vol. 47, No. 12, 2006, Article ID: 122301. http://dx.doi.org/10.1063/1.2397555

[7] Y.-F. Liu, “Triality, Biquaternion and Vector Representation of the Dirac Equation,” Advances in Applied Clifford Algebras, Vol. 12, No. 2, 2002, pp. 109-124.

http://dx.doi.org/10.1007/BF03161242

[8] M. Gogberashvili, “Octonionic Electrodynamics,” Journal of Physics A: Mathematics in General, Vol. 39, No. 22, 2006, pp. 7099-7104.

http://dx.doi.org/10.1088/0305-4470/39/22/020

[9] A. Gamba, “Maxwell’s Equations in Octonion Form,” Nuovo Cimento A, Vol. 111, No. 3, 1998, pp. 293-302.

[10] T. Tolan, K. Ozdas and M. Tanisli, “Reformulation of Electromagnetism with Octonions,” II Nuovo Cimento B, Vol. 121, No. 1, 2006, pp. 43-55.

http://dx.doi.org/10.1393/ncb/i2005-10189-9

[11] S. Demir and M. Tanisli, “A Compact Biquaternionic Formulation of Massive Field Equations in Gravi-Electromagnetism,” European Physical Journal Plus, Vol. 126, 2011, p. 115.

http://dx.doi.org/10.1140/epjp/i2011-11115-8

[12] B. C. Chanyal, P. S. Bisht and O. P. S. Negi, “Generalized Octonion Electrodynamics,” International Journal of Theoretical Physics, Vol. 49, No. 6, 2010, pp. 1333-1343.

http://dx.doi.org/10.1007/s10773-010-0314-5

[13] P. S. Bisht, G. Karnatak and O. P. S. Negi, “Generalized Gravi-Electromagnetism,” International Journal of Theoretical Physics, Vol. 49, No. 6, 2010, pp. 1344-1356.

http://dx.doi.org/10.1007/s10773-010-0315-4

[14] V. Dzhunushaliev, “Nonassociativity, Supersymmetry and Hidden Variables,” Journal of Mathematical Physics, Vol. 49, No. 4, 2008, Article ID: 042108.

http://dx.doi.org/10.1063/1.2907868

[15] M. Gogberashvili, “Octonionic Version of Dirac Equations,” International Journal of Modern Physics A, Vol. 21, No. 17, 2006, pp. 3513-3523.

http://dx.doi.org/10.1142/S0217751X06028436

[16] S. De Leo and K. Abdel-Khalek, “Octonionic Dirac Equation,” Progress in Theoretical Physics, Vol. 96, No. 4, 1996, pp. 833-846.

http://dx.doi.org/10.1143/PTP.96.833

[17] D. Hestenes, “Observables, Operators, and Complex Numbers in the Dirac Theory,” Journal of Mathematical Physics, Vol. 16, No. 3, 1975, pp. 556-572.

http://dx.doi.org/10.1063/1.522554

[18] K. Imaeda and M. Imaeda, “Sedenions: Algebra and Analysis,” Applied Mathematics and Computations, Vol. 115, No. 2-3, 2000, pp. 77-88.

http://dx.doi.org/10.1016/S0096-3003(99)00140-X

[19] K. Carmody, “Circular and Hyperbolic Quaternions, Octonions, and Sedenions,” Applied Mathematics and Computations, Vol. 28, No. 1, 1988, pp. 47-72.

http://dx.doi.org/10.1016/0096-3003(88)90133-6

[20] K. Carmody, “Circular and Hyperbolic Quaternions, Octonions, and Sedenions—Further Results,” Applied Mathematics and Computations, Vol. 84, No. 1, 1997, pp. 27-47. http://dx.doi.org/10.1016/S0096-3003(96)00051-3

[21] J. Koplinger, “Dirac Equation on Hyperbolic Octonions,” Applied Mathematics and Computations, Vol. 182, No. 1, 2006, pp. 443-446.

http://dx.doi.org/10.1016/j.amc.2006.04.005

[22] S. Demir and M. Tanisli, “Sedenionic Formulation for Generalized Fields of Dyons,” International Journal of Theoretical Physics, Vol. 51, No. 4, 2012, pp. 1239-1252.

http://dx.doi.org/10.1007/s10773-011-0999-0

[23] W. P. Joyce, “Dirac Theory in Spacetime Algebra: I. The Generalized Bivector Dirac Equation,” Journal of Physics A: Mathematics in General, Vol. 34, No. 10, 2001, pp. 1991-2005.

http://dx.doi.org/10.1088/0305-4470/34/10/304

[24] C. Cafaro and S. A. Ali, “The Spacetime Algebra Approach to Massive Classical Electrodynamics with Magnetic Monopoles,” Advances in Applied Clifford Algebras, Vol. 17, No. 1, 2006, pp. 23-36.

http://dx.doi.org/10.1007/s00006-006-0014-7

[25] V. L. Mironov and S. V. Mironov, “Octonic Representation of Electromagnetic Field Equations,” Journal of Mathematical Physics, Vol. 50, No. 1, 2009, Article ID: 012901. http://dx.doi.org/10.1063/1.3041499

[26] V. L. Mironov and S. V. Mironov, “Octonic SecondOrder Equations of Relativistic Quantum Mechanics,” Journal of Mathematical Physics, Vol. 50, No. 1, 2009, Article ID: 012302.

http://dx.doi.org/10.1063/1.3058644

[27] V. L. Mironov and S. V. Mironov, “Octonic First-Order Equations of Relativistic Quantum Mechanics,” International Journal of Modern Physics A, Vol. 24, No. 22, 2009, pp. 4157-4167.

http://dx.doi.org/10.1142/S0217751X09045480

[28] V. L. Mironov and S. V. Mironov, “Sedeonic Generalization of Relativistic Quantum Mechanics,” International Journal of Modern Physics A, Vol. 24, No. 32, 2009, pp. 6237-6254.

http://dx.doi.org/10.1142/S0217751X09047739

[29] L. D. Landau and E. M. Lifshits, “Classical Theory of Fields,” 4th Edition, Pergamon Press, New York, 1975.

[30] P. A. M. Dirac, “The Principles of Quantum Mechanics,” Clarendon Press, Oxford, 1958.

[31] A. Macfarlane, “Hyperbolic Quaternions,” Proceedings of the Royal Society at Edinburgh, 1899-1900 Sessions, pp. 169-181.

[32] W. Pauli, “Zur Quantenmechanik des Magnetischen Elektrons,” Zeitschrift für Physik, Vol. 43, No. 9-10, 1927, pp. 601-623. http://dx.doi.org/10.1007/BF01397326

[33] P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of Royal Society at London. Series A, Vol. 117, No. 778, 1928, pp. 610-624.

http://dx.doi.org/10.1098/rspa.1928.0023