Victor L. Mironov

Hypercomplex numbers

 

 

 

 

 

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Application of hypercomplex numbers
and Clifford algebras

 

 

 

 

 

William Rowan Hamilton
(1805 - 1865) 
Invention of quaternions 1843

 

 

 Hermann Grassmann

 (1809 - 1877)

Invention of noncommutative algebra 1844

 

 

William Clifford
(1845-1879)

Invention of Clifford algebra 1876

 

 

Alexander Macfarlane
(1851 - 1913)

Invention of Hyperbolic quaternions 1900

 

 

John Graves
(1806 - 1870)
Invention of octonions
1844

 

 

Arthur Cayley
(1821 - 1895)
Invention of octonions
(1845)

 

 

James Clerk Maxwell

(1831-1879)

Discovery of equations

 for electromagnetic field

1862

 

 

Oliver Heaviside
(1850 - 1925)

Vector algebra and   gravitoelectromagnetism

 

 

Wolfgang Pauli 
(1900 - 1958)
Invention of spinors 
1927

 

 

Paul Dirac 

(1902 - 1984)

Invention of 4-matrix algebra 1928

Quaternions

  1. W.R.Hamilton, "Lectures on Quaternions". Royal Irish Academy, (1853).
  2. W.R.Hamilton, "Elements of Quaternions". University of Dublin Press. (1866). 
  3. A.Macfarlane "Hyperbolic Quaternions" Proceedings of the Royal Society at Edinburgh, 1899-1900 session, pp. 169-181 (1900).
  4. I.L.Kantor, A.S.Solodovnikov - Hypercomplex numbers: an elementary introduction to algebras, Springer-Verlag, 1989.
  5. A.V.Berezin, Yu.A.Kurochkin and E.A.Tolkachev  -  Quaternions in relativistic physics. Minsk: Nauka y Tekhnika, 1989 (in Russian).
  6. F.Gürsey and H.C.Tze  - On the role of division, Jordan and related algebras in particle physics. Singapore: World Scientific, 1996.
  7. K. Gürlebeck, W. Sprössig - Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, 1997.
  8. G.M.Dixon - Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics (Mathematics and its applications), Springer, 2006.
  9. D.Smith, J.H.Conway - On quaternions and octonions, Pub. AK Peters, 2003.
  10. S.L.Adler - Quaternionic quantum mechanics and quantum fields, New York: Oxford University Press, 1995.
  11. S.L.Adler - Time-dependent perturbation theory for quaternionic quantum mechanics, with application to CP nonconservation in K-meson decays, Phys. Rev. D, 34(6), 1871-1877, (1986).
  12. S.L.Adler - Scattering and decay theory for quaternionic quantum mechanics, and the structure of induced T nonconservation, Phys. Rev. D, 37(12), 3654-3662, (1988).
  13. A.J.Davies, B.H.J.McKellar - Nonrelativistic quaternionic quantum mechanics in one dimension, Phys. Rev. A, 40(8), 4209-4214, (1989).
  14. A.J.Davies - Quaternionic Dirac equation, Phys. Rev. D, 41(8), 2628-2630, (1990).
  15. S.De Leo, P.Rotelli - Quaternion scalar field, Phys. Rev. D, 45(2), 575-579, (1992).
  16. S.De Leo - Quaternionic electroweak theory, Journal of Physics G: Nuclear and Particle Physics, 22(8), 1137, (1996).
  17. A.Gsponer, J.P.Hurni - Quaternions in mathematical and physics (1), e-print arXiv:math-ph/0510059v3 (2006).
  18. A.Gsponer, J.P.Hurni - Quaternions in mathematical and physics (2), e-print arXiv:math-ph/0511092v3 (2006)
  19. V.Majernik - Quaternionic formulation of the classical fields, Advances in Applied Clifford Algebras, 9(1), 119-130 (1999).
  20. K.Imaeda - A new formulation of classical electrodynamics, Nuovo cimento, v. 32, no 1, 138-162 (1976).
  21. M.Tanışlı - Gauge transformation and electromagnetism with biquaternions, Europhysics Letters, 74(4), 569-573 (2006).
  22. S.Demir and K.Özdas - Dual quaternionic reformulation of classical electromagnetism, Acta Physica Slovaca, 53(6), 429-436 (2003).
  23. S.Demir - Matrix realization of dual quaternionic electromagnetism, Central European Journal of Physics, 5(4), 487-506 (2007).
  24. S.Demir, M.Tanişli , N.Candemir - Hyperbolic quaternion formulation of electromagnetism, Advances in Applied Clifford Algebras, 20(3-4), 547-563 (2010).
  25. S.Demir, M.Tanışlı - A compact biquaternionic formulation of massive field equations in gravi-electromagnetism, European Physical Journal-Plus, 126(11), 115 (2011).
  26. M.Tanişli , M.Emre Kansu, S.Demir - Supersymmetric quantum mechanics and euclidean dirac operator with complexified quaternions, Modern Physics Letters A, 28(8), 1350026 (2013).
  27. S.Ulrych - The Poincaré mass operator in terms of a hyperbolicalgebra, Physics Letters B, 612(1-2), 89 (2005) ArXiv.
  28. S.Ulrych - Considerations on the hyperbolic complexKlein-Gordon equation, Journal of Mathematical Physics, 51(6), 063510 (2010) ArXiv.
  29. S.Ulrych - Higher spin quaternion waves in the Klein-Gordon theory, International Journal of Theoretical Physics, 52(1), 279 (2013) ArXiv.
  30. V.V.Kravchenko - Quaternionic reformulation of Maxwell's equations for inhomogeneous media and new solutions, ArXiv: http://arxiv.org/abs/math-ph/0104008
  31. V. G. Kravchenko, V. V. Kravchenko - Quaternionic factorization of theSchrödinger operator and its applications to some first order systems ofmathematical physics. Journal of Physics A, 2003, v. 36, No. 44, 11285-11297. (available from arxiv.org)
  32. S. M. Grudsky, K. V. Khmelnytskaya, V. V. Kravchenko - On a quaternionic Maxwell equation for the time-dependentelectromagnetic field in a chiral medium. Journal of Physics A, 2004, v. 37, 16, 4641-4647 (available from arxiv.org).

Octonions

  1. S.Okubo - Introduction to octonion and other non-associative algebras in physics (Montroll memorial lecture series in mathematical physics, 2) Cambridge University Press, 1995.
  2. M.Gogberashvili - Octonionic electrodynamics, J. Phys. A.: Math. Gen., 39, 7099-7104, (2006).
  3. A.Gamba - Maxwell equations in octonion form, Nuovo Cimento A, 111(3), 293-302, (1998).
  4. T.Tolan, K.Özdas, M.Tanişli- Reformulation of electromagnetism with octonions, Nuovo Cimento B, 121(1), 43-55, (2006).
  5. R.Penney - Octonions and Dirac equation, Amer. J. Phys., 36, 871-873 (1968).
  6. A.A.Bogush, Yu.A.Kurochkin - Cayley-Dickson procedure, relativistic wave equations and supersymmetric oscillators, Acta Applicandae Mathematicae, 50, 121-129 (1998)
  7. S.De Leo, K.Abdel-Khalek - Octonionic Dirac equation, Progress of Theoretical  Physics, 96, 833-846 (1996). (arXiv:hep-th/9609033v1 (1996)), 
  8. M.Tanişli, M.Emre Kansu - Octonionic Maxwell equations for bi-İsotropic media , Journal of Mathematical Physics, 52(5), 053511 (2011).
  9. M.Tanişli, M. Emre Kansu, S.Demir - A new approach to Lorentz invariance on electromagnetism with hyperbolic octonions, European Physical Journal-Plus, 127(6), 69, (2012).
  10. S.Demir - Hyperbolic octonion formulation of gravitational field equationsInternational Journal of Theoretical Physics, 52(1), 105-116 (2013).
  11. J.Köplinger - Nonassociative quantum theory on octooctonion algebra, Journal of physical Mathematics, 1, S090501 (2009).
  12. B.C. Chanyal, P.S. Bisht, O.P.S. Negi - Generalized octonion electrodynamics, International Journal of Theoretical Physics,  49(6), 1333 (2010).
  13. V. Dzhunushaliev, Nonassociativity, supersymmetry, and hidden variables, Journal of Mathematical Physics 49, 042108 (2008).
  14. V. Dzhunushaliev - Hidden structures in quantum mechanics, Journal of Generalized Lie Theory and Applications, 3(1),  33-38 (2009).
  15. B.C. Chanyal, P.S. Bisht, O.P.S. Negi - Octonion and conservation laws for dyons, International Journal of Modern Physics A, 28(26), 1350125 1-17 (2013).
  16. M.E. Kansu - An analogy between macroscopic and microscopic systems for Maxwell equations in higher dimensions. European Physical Journal - Plus, 128, 149 (2013).

Sedenions

  1. K.Imaeda, M.Imaeda - Sedenions: algebra and analysis, Appl. Math. Comp., 115, 77-88 (2000).
  2. K.Carmody - Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput., 28, 47-72 (1988).
  3. K.Carmody - Circular and hyperbolic quaternions, octonions, and sedenions - further results, Appl. Math. Comput., 84, 27-47 (1997).
  4. J.Köplinger - Dirac equation on hyperbolic octonions, Appl. Math. Comput., 182, 443-446 (2006).
  5. S.Demir, M.Tanişli - Sedenionic formulation for generalized fields of dyons, International Journal of Theoretical Physics, 51(4), 1239-1252 (2012).
  6. V. Dzhunushaliev - Toy Models of a Nonassociative Quantum Mechanics, Advances in High Energy Physics, Volume 2007, Article ID 12387, 10 pages.
  7. V.L. Mironov, S.V. Mironov - Associative space-time sedenions,  http://vixra.org/abs/1401.0162 (2014).
  8. V.L. Mironov, S.V. Mironov - Reformulation of relativistic quantum mechanics and field theory equations with space - time sedenions, http://vixra.org/abs/1402.0157  

Clifford algebras

  1. B.Jancewicz - Multivectors and Clifford Algebra in Electrodynamics, World Scientific, 1988. 
  2. W.M.Pezzaglia - Clifford algebra derivation of the characteristic hypersurfaces of Maxwell equations, e-print arXiv:hep-th/9211062v1 (1992).
  3. W.E.Baylis, G.Jones - The Pauli algebra approach to special relativity, J. Phys. A: Math. Gen., 22, 1-15, (1989).
  4. A.M.Shaarawi - Clifford algebra formulation of an electromagnetic charge-current wave theory, Foundations of physics, 30(11), 1911-1941, (2000).
  5. D.Hestenes - Observables, operators, and complex numbers in the Dirac theory, Journal of Mathematical Physics, 16, 556-572 (1975).
  6. D.Hestenes - Clifford algebra and the interpretation of quantum mechanics. In: Clifford algebra and their applications in mathematical physics. (Eds. J.S.R.Chisholm, A.K.Commons) Reidel, Dordrecht / Boston, 321-346 (1986).
  7. W.P.Joyce - Dirac theory in spacetime algebra: I. The generalized bivector Dirac equation, J.Phys. A: Math. Gen., 34, 1991-2005 (2001).
  8. C.Cafaro, S.A. Ali - The spacetime algebra approach to massive classical electrodynamics with magnetic monopoles, Advances in Applied Clifford Algebras, 17, 23-36 (2006).
  9. V.L.Mironov, S.V.Mironov -Octonic representation of electromagnetic field equations, Journal of Mathematical Physics, 50, 012901 1-10 (2009).
  10. V.L.Mironov, S.V.Mironov - Octonic second-order equations of relativistic quantum mechanics, Journal of Mathematical Physics, 50, 012302 1-13 (2009).
  11. V.L.Mironov, S.V.Mironov - Octonic first-order equations of relativistic quantum mechanics, International Journal of Modern Physics A, 24(22), 4157-4167 (2009).
  12. V.L.Mironov, S.V.Mironov - Noncommutative sedeons and their application in field theory, ArXiv: http://arxiv.org/abs/1111.4035
  13. T.Tolan, M.Tanisli, S. Demir - Octonic form of Proca-Maxwell equations and relativistic derivation of electromagnetism, International Journal of Theoretical Physics, 52(12), 4488-4506 (2013).
  14. S. Demir, M.Tanisli, T.Tolan - Octonic gravitational field equations,International Journal of Modern Physics A, 28(21), 1350112 (2013).
  15. S.V.Mironov, V.L.Mironov - Sedeonic theory of massive fields, ViXra: http://vixra.org/abs/1311.0005 
  16. V.L.Mironov, S.V.Mironov - Reformulation of relativistic quantum mechanics equations with non-commutative sedeons // Applied Mathematics, 4(10C), 53-60 (2013).
  17. V.L. Mironov, S.V. Mironov - Associative space-time sedenions, http://vixra.org/abs/1401.0162 (2014).

Gravitoelectromagnetism

  1. J.C.Maxwell - A dynamical theory of the electromagnetic field, Philosophical Transactions of the Royal Society of London, 155, 459-512 (1865.)
  2. O.Heaviside - A gravitational and electromagnetic analogy, The Electrician, 31, 281-282 (1893).
  3. V.Majernik - Quaternionic formulation of the classical fields, Advances in Applied Clifford Algebras, 9(1), 119-130 (1999).
  4. S.Ulrych - Gravitoelectromagnetism in a complex Clifford algebra, Physics Letters B, 633, 631-635 (2006).
  5. S.Demir, M.Tanışlı - Sedenionic formulation for generalized fields of dyons, International Journal of Theoretical Physics, 51(4), 1239-1253 (2012).
  6. P.S. Bisht, G. Karnatak, O.P.S. Negi - Generalized gravi-electromagnetism, International Journal of Theoretical Physics,  49(6), 1344 (2010).
  7. V.L.Mironov, S.V.Mironov, S.A.Korolev - Sedeonic theory of massless fields, ArXiv: http://arxiv.org/abs/1206.5969
  8. S.V.Mironov, V.L.Mironov - Sedeonic theory of massive fields, ViXra: http://vixra.org/abs/1311.0005  < PDF >

 

 

 

 

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