Advances in Applied Clifford Algebras

, Volume 9, Issue 2, pp 309–395 | Cite as

The geometric content of the electron theory. (Part II) theory of the electron from start

  • Jaime Keller


From the considerations of the previous paper on the geometric content of the electron theory and the basic principles of the space-time-action relativity theory (START) we formulate a comprehensive and complete theory of the electron. Our approach contains, being a deductive theory, the results of density functional theory, wave function quantum mechanics, the classical theory of the electron, the description of the electron as a lepton in elementary particles theory and the fundamentals of both electrodynamics and electroweak interactions. The approach is otherwise selfcontained.


Dirac Equation Lorentz Transformation Clifford Algebra Geometric Algebra Interaction Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Abraham M. (1905), “Theorie der Electrizität”, Teubner, Leipzig; Abraham M. and Lorentz H. A. (1909), “Theorie der Electrizität”, Teubner, Leipzig.Google Scholar
  2. [2]
    Ahlfors L. and Lounesto P. (1989), Some remarks on Clifford algebrasComplex Variables 12 201–209.MATHMathSciNetGoogle Scholar
  3. [3]
    Altmann S. L. (1986), “Rotations, Quaternions and Double Groups”, Oxford, Clarendon Press.MATHGoogle Scholar
  4. [4]
    Artin E. (1957), “Geometric Algebra”, New York Interscience.Google Scholar
  5. [5]
    Atiyah M. F., Bott R. and Shapiro A. (1964), Clifford modulesTopology 3 (Suppl. 1) 3–38.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Baylis W. E. (1996), “Clifford (Geometric) Algebras”, Boston, Birkhäuser.MATHGoogle Scholar
  7. [7]
    Benn I. M. and Tucker R. W. (1987), “An Introduction to Spinors and Geometry in Applications to Physics”, Bristol, Hilger.Google Scholar
  8. [8]
    Boudet R. (1971),C. R. H. S. Acad. Sci. (Paris) Serie A 272 767.Google Scholar
  9. [9]
    Boudet R. (1974),C. R. H. S. Acad. Sci. (Paris) Serie A278 1063.MathSciNetGoogle Scholar
  10. [10]
    Boudet R. (1985),C. R. H. S. Acad. Sci. (Paris) Serie II300 157.MathSciNetGoogle Scholar
  11. [11]
    Brackx F., Delanghe R. and Serras H. (1993), “Clifford Algebras and their Applications in Mathematical Physics”, Dordrecht, Kluwer A. P.MATHGoogle Scholar
  12. [12]
    Burinskii A. (1998), Kerr spinning particle, strings, and superparticle models,Phys. Rev. 57 (4) 2392–2396 and references therein.ADSMathSciNetGoogle Scholar
  13. [13]
    Campbell J. E. (1926), “A course on Differential Geometry”, Oxford, Clarendon.Google Scholar
  14. [14]
    Campolattaro A. A. (1980),Int. J. Theor. Phys. 19 99; (1980),Int. J. Theor. Phys. 19 127.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Campolattaro A. A. (1990),Int. J. Theor. Phys. 29 (2) 141; (1990),Int. J. Theor. Phys. 29 (5) 477.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Cartan E (1981), “The Theory of Spinors”, New York, Dover (corresponding to the French edition, Hermann Ed. 1937).MATHGoogle Scholar
  17. [17]
    Carter B. (1968),Phys. Rev. 174 1579.ADSGoogle Scholar
  18. [18]
    Casanova G. (1970),C. R. H. S. Acad. Sci. (Paris) Serie A270 1202.MathSciNetGoogle Scholar
  19. [19]
    Casanova G. (1976), “L’algebre vectorielle” Paris, Presses Universitaires de France.Google Scholar
  20. [20]
    Chisholm J. S. R. and Common A. K. (1986), “Clifford Algebras and their Applications in Mathematical Physics”, Dordrecht, Kluwer APMATHGoogle Scholar
  21. [21]
    Clifford W. K. (1876), Preliminary sketch of biquaternions,Proc. London Math. Soc. 4 381–395CrossRefGoogle Scholar
  22. [22]
    Clifford W. K. (1876), On the Classification of Geometric Algebras, published as Paper XLIII in “Mathematical Papers”, (Tucker R. ed.), London, Macmillan (1882).Google Scholar
  23. [23]
    Cohen J. M. and Mustafa E. (1986),Int. J. Theor. Phys. 25 717–726.CrossRefGoogle Scholar
  24. [24]
    Crawford J. P. (1985), On the algebra of Dirac bispinor densities: factorization and inversion theorems,J. Math. Phys. 26 1439.ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    Crowe F. (1992), “A History of Vector Algebra”. New York Dover.Google Scholar
  26. [26]
    Davian C. (1989),Ann. Fondation. Louis de Broglie 14 273; (1989),Ann. Fondation Louis de Broglie 14 373; (1993),Ann. Fondation Louis de Broglie 23 1431–1443.Google Scholar
  27. [27]
    Davian C. and Lochak G. (1991),Ann. Fondation Louis de Broghie 16 43.Google Scholar
  28. [28]
    Dirac Paul A. M. (1928),Proceedings of the Royal Society A 117 610; (1928),Proceedings of the Royal Society A 118 35.CrossRefGoogle Scholar
  29. [29]
    Fermi E. (1927), Applications of statistical gas methods to electronic systemsAtti. Acad. Naz. Lincei (ser. 6) 6 602–606; (1928), Statistical deduction of atomic properties,ibid., Atti. Acad. Naz. Lincei (ser. 6) 7 342–346; (1928), Statistical methods of investigating electrons in atoms,Z. Phys. 48 73–79.Google Scholar
  30. [30]
    Flores J. A. and Keller J. (1992), Differential equations for the square root of the electronic density in symmetry-constrained density-functional theory,Phys. Rev. A 45 (9) 6259–6262; Keller J., Keller A. and Flores J. A. (1990), Una ecuación para la raiz cuadrada de la densidad,Acta Chimica Teoretica Latina XVIII (4) 175–186.ADSCrossRefGoogle Scholar
  31. [31]
    Frankei T. (1997), “The Geometry of Physics: An Introduction”, Cambridge, Cambridge U.P.Google Scholar
  32. [32]
    Fock V. (1929), Geometrisierung der Diracshen Theorie del Electrons,Zeitschrift für Physik 55, 261.ADSCrossRefGoogle Scholar
  33. [33]
    Fock V. (1929), Sur les équation de Dirac dans la Théorie de Relativité GénéraleC. R. Acad. Sciences (Paris)189, 25.MATHGoogle Scholar
  34. [34]
    Fock V. and Iwanenko D. (1929), Über eine Mögliche Geometrische Deutung der Relativistischen QuantentheorieZeitschrift für physik 54, 798.ADSCrossRefGoogle Scholar
  35. [35]
    Fock V. and Iwanenko D. (1929), Géométrie Quantique Linéaire et Deplacement ParaliléleAcad. Sciences (Paris)188, 1470.MATHGoogle Scholar
  36. [36]
    Grassmuan H. (1844), “Die Wissenschaft der extension Grösse oder die Ausdehnunglehre, eine neue mathematischen Disciplin”, Leipzig.Google Scholar
  37. [37]
    Greider T. K. (1980),Phys. Rev. Lett. 44 1718ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    Gueret Ph. and Vigier P. J. (1982),Found. Phys. 12, 1057; (1982),Lett. Nuovo Cimento 35, 256;35, 260 (1982);38, 125 (1982).ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    Habetha K., Dietrich V. and Jank G. (1998), “Clifford Algebras and their Applications in Mathematical Physics”, Dordrecht, Kluwer Academic Publishers.Google Scholar
  40. [40]
    Hamilton, W. B. (1844), On quaternions or on a new system of imaginaries in algebraPhil. Mag. 25, 489–495.Google Scholar
  41. [41]
    Hamilton W. R. (1853), “Lectures on Quaternions”, Dublin, Hodges & Smith.Google Scholar
  42. [42]
    Hecht L. (1996), The significance of the 1845 Gauss-Weber correspondance,21st Century Science & Technology 9 (3) 22–34.ADSGoogle Scholar
  43. [43]
    Hestenes D. (1966), “Spacetime Algebra”, New York, Gordon and Breach.Google Scholar
  44. [44]
    Hestenes D. (1975).J. Math. Phys. 16 556 and references therein.ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    Hestenes D. (1979),American Journal of Physics 47 5.MathSciNetCrossRefGoogle Scholar
  46. [46]
    Hestenes D. and Sobczyk G. (1984), “Clifford Algebra to Geometric Calculus”, Dordrecht, Reidel.MATHGoogle Scholar
  47. [47]
    Hestenes D. (1991),Advances in Applied Clifford Algebras 1 (1) 5–29; (1991),Foundation of Physics 20 1231; (1991), Zitterbewegung in radiative processes, in “The Electron, New Theory and Experiment”, (edited by Hestenes D., and Weingartsofer A.), Kluwer Academic Publishers Dordrecht, pp. 21–36.MATHGoogle Scholar
  48. [48]
    Hohenberg P. and Kohn W. (1964), Inhomogeneous electron gasPhys. Rev. B 136 864–867.ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    Juvet G. (1930),Commentarii Mathematici Helvetici 2 225.MATHMathSciNetCrossRefGoogle Scholar
  50. [50]
    Juvet G. (1932),Bulletin de la Societé Neuchateloise des Sciences Naturelles 57, 127.Google Scholar
  51. [51]
    Kaluza T. (1921),Sitz. Preuss. Akad. Wiss. 33 966.Google Scholar
  52. [52]
    Keiler J. (1981), Masa y Carga como un problema de eigenvalores de variables estructuralesRev. Soc. Quim. Mex. 25 28–31.Google Scholar
  53. [53]
    Keller J. (1982), Spacetime symmetries corresponding to elementary particle symmetries, in “Mathematics of the Physical Spacetime”, México, UNAM, pp. 117–32.Google Scholar
  54. [54]
    Keller J. (1982), Wave equation of symmetry constrained Dirac particlesInt. J. of Theor. Phys. 21 (10/11) 829–836.CrossRefGoogle Scholar
  55. [55]
    Keller J. (1984), Spacetime dual geometry theory of elementary particles and their interaction fieldsInt. J. of Theor. Phys. 23 (9) 817–37.CrossRefGoogle Scholar
  56. [56]
    Keller J. and Megy F. (1984), Geometría y teoría del spin en mecánica cuánticaContactos 1 (1) 51–54.Google Scholar
  57. [57]
    Keller J. (1986), A system of vectors and spinors in complex spacetime and their application to elementary particle physics, inProceedings of the NATO. & S.E.R.C. Workshop on Clifford Algebras and Their Application in Mathematical Physics (Kent 1985), (Chisholm R. and Common, A. K. eds.).Google Scholar
  58. [58]
    Keller J. (1986), A generalization of the Dirac equation admitting isospin and color symmetriesInt. J. of Theor. Phys. 25 (8) 779–806.MATHCrossRefGoogle Scholar
  59. [59]
    Keller J. (1986),Int. J. Quantum Chem. Symp. 20, 767; Keller J. and Ludeña E. (1987).Int. J. Quantum Chem. Symp. 21 171.CrossRefGoogle Scholar
  60. [60]
    Keller J. and Rodríguez-Romo S. (1990), A multivectorial Dirac equationJ. Math. Phys. 31 (10) 2501–2510.MATHADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    Keller J. and Rodríguez-Romo S. (1991), Multivectorial generalization of the Cartan map,J. Math. Phys. 32 (6) 1591–1598.MATHADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    Keller J. and Rodríguez-Romo S. (1991), Multivectorial representation of lie groupsInt. J. of Theor. Phys. 30 (2) 185–196.MATHCrossRefGoogle Scholar
  63. [63]
    Keller J. and Rodríguez A. (1992), Geometric superalgebra and the Dirac equation,J. Math. Phys. 33 (1) 161–170.MATHADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    Keller J. and Keller J. (1991), Spinors and multivectors as a unified tool for spacetime geometry and for elementary particle physicsInt. J. of Theor. Phys. 30 (2) 137–184.MATHCrossRefGoogle Scholar
  65. [65]
    Keller J. and Viniegra F. (1992), The multivector structure of the matter and interaction field theories, inClifford Algebras and Their Applications in Mathematical Physics (Helmsteller J., Micali A. and Boudet R. (eds.), Dordrecht, Kluwer Academic Pub., pp. 437–445.Google Scholar
  66. [66]
    Keller J. (1991), Spinors as a basis of a geometric superalgebraAdvances in Applied Clifford Algebras 1 (1), 31–50.MATHGoogle Scholar
  67. [67]
    Keller J., (1992), Dirac equations with electroweak and color symmetry, in: “Differential Geometric Methods in Theoretical Physics”, (Catto S. and Rocha A. eds.), 1 Singapore, World Scientific, pp. 355–361.Google Scholar
  68. [68]
    Keller J. (1992), The geometric content of the electron theory, Fock and Iwanenko 1929.Advances in Applied Clifford Algebras 2 (2), 195–196.MATHGoogle Scholar
  69. [69]
    Keller, J. (1993), Tautology of quantum mechanics and spacetime, inVistas in Astronomy 37 ofProceedings of the Symposium on Quantum Physics and the Universe Tokyo, Japan, August 19–22, 1992, pp. 283–286.Google Scholar
  70. [70]
    Keller J. (1993), Unified mathematical approach to spinors and multivectors, geometric superalgebra.Int. J. of Modern Phys. A. 3A 511–514.Google Scholar
  71. [71]
    Keller J. (1993), Dual space and Clifford algebras.Advances in Applied Clifford Algebras 3 (1), 1–6.MATHGoogle Scholar
  72. [72]
    Keller J. (1993), The geometric content of the electron theory (part I).Advances in Applied Clifford Algebras 3 (2), 147–200.MATHMathSciNetGoogle Scholar
  73. [73]
    Keller J. (1994), Clifford algebra and the construction of a theory of elementary particle fields.Advances in Applied Clifford Algebras 4 (S1) 379–393.MathSciNetGoogle Scholar
  74. [74]
    Keller J. (1994), Twistors as geometric objects in spacetime, in “Clifford Algebras and Spinor Structures” (Lounesto P. and Ablamowicz R., eds.) Dordrecht, Kluwer Academic Publishers, pp. 133–135.Google Scholar
  75. [75]
    Keller J. (1994), Factorization of the Laplacian and families of elementary particles, in “Symmetry Methods in Physics”, (Sissakian A. N., Pogosyan G. S. and Vinitsky S. I. eds.), I, Dubna, JIRN, pp 236–241.Google Scholar
  76. [76]
    Keller J. (1997), Spinors, Twistors, Screws, Mexors and the massive spinning electron “The Theory of the Electron”,Advances in Applied Clifford Algebras 7 (S) 439–455.MathSciNetGoogle Scholar
  77. [77]
    Keller J. (1998), Twistors and clifford algebras, in “Clifford Algebras and their Applications in Mathematical Physics”, (Habetha K., Dietrich V. and Jank G. eds.) Dordrecht, Kluwer Academic Publishers, pp. 161–173.Google Scholar
  78. [78]
    Keller J. (1999), Complex Spacetime Formulation of a Theory of Elementary Particles.Physics of Atomic Nuclei 61 (12).Google Scholar
  79. [79]
    Keller, J. (2000),Turkish Journal of Physics to be published.Google Scholar
  80. [80]
    Keller J. and Weinberger P. (2000),Phil. May to be published.Google Scholar
  81. [81]
    Klein O. (1926).Z. Phys. 37 895.ADSCrossRefGoogle Scholar
  82. [82]
    Kròlikowski W. (1990),Acta Physica Polonica B21 871–879.Google Scholar
  83. [83]
    Kròlikowski W. (1992),Phys. Rev. D45 3222–3227.ADSGoogle Scholar
  84. [84]
    Kohn W. and Sham L. J. (1965), Self-consistent equations including exchange and correlation effects,Phys. Rev. A. 140 1133–1138.ADSMathSciNetCrossRefGoogle Scholar
  85. [85]
    Liu Y. F. and Keller J. (1996), A symmetry of massless fields,J. Math. Phys. 37 (9) 4320–4332; (2000) to be published.MATHADSMathSciNetCrossRefGoogle Scholar
  86. [86]
    Lozada A. (1989),J. Math. Phys. 30 (8) 1713–1720.MATHADSMathSciNetCrossRefGoogle Scholar
  87. [87]
    Mackinuon L. (1981)Lett. Nuovo Cimento 32 311.CrossRefGoogle Scholar
  88. [88]
    Mercier A. (1934),Actes de la Societé Helvetique des Sciences Naturelles Zürich.278.Google Scholar
  89. [89]
    Mercier A. (1935), These, Université de Génève,Archives des Sciences Physiques et Naturelles (Suisse) 17 278.Google Scholar
  90. [90]
    Micall A., Boudet R. and Helmstetter J. (1991), “Clifford Algebras and their Applications in Mathematical Physics,” Dordrecht, Kluwer AP.Google Scholar
  91. [91]
    Nakahara M. (1990), “Geometry, Topology and Physics”, Bristol, IOP.MATHCrossRefGoogle Scholar
  92. [92]
    Ohanian H. C. (1986),American Journal of Physics 54 504.ADSGoogle Scholar
  93. [93]
    Pauli Wolfgang (1921),Zeitschrift für Physik 31 765.Google Scholar
  94. [94]
    Poincaré Henrí (1906),Rend. Circ. Mat. Palermo 21 129–176.CrossRefGoogle Scholar
  95. [95]
    Porteous I. R. (1981), “Topological Geometry”, Cambridge, Cambridge U.P.MATHGoogle Scholar
  96. [96]
    Porteous I. R. (1994), “Geometric Differentiation,” Cambridge, Cambridge U.P.MATHGoogle Scholar
  97. [97]
    Porteous I. R. (1995), “Clifford Algebras and the Classical Groups”, Cambridge, Cambridge U.P.MATHGoogle Scholar
  98. [98]
    Proca A. (1930),C. R. Acad. Sci. Paris190 1377; (1930),C. R. Acad. Sci. Paris191 26; (1930),Journal of Physics VII 1 236.MATHGoogle Scholar
  99. [99]
    Quilichini P. (1971)C. R. H. S. Acad. Sci. (Paris) Serie B273 829.MathSciNetGoogle Scholar
  100. [100]
    Raghavacharyulu I. V. V. and Menon N. B. (1970),J. Math. Phys. 11 3055.MATHADSMathSciNetCrossRefGoogle Scholar
  101. [101]
    Ravsevskii P. K. (1957),Transactions of the American Mathematical Society.6 1.Google Scholar
  102. [102]
    Riesz M. (1946),Comptes Rendus du Dixieme Congres des Mathematiques des Pays Scandinaves 123, Copenhagen; (1953),Comptes Rendus du Douzieme Congres des Mathematiques des Pays Scandinaves 241, Copenhagen; (1958) Clifford Numbers and Spinors,Lecture Series 38, University of Maryland, The Institute for Fluid Dynamics and Applied Mathematics Reprinted as facsimile, (Bolinder E. F. and Lounesto P. eds.), Kluwer 1993.Google Scholar
  103. [103]
    Rodrigues Waldyr A. Jr., Vaz Jr. Jayme and Recami Erasmo (1993), Free Maxwell equations, Dirac equation and non-dispensive de Broglie wave packets, in “Courants, Amers, Écueils en Microphysique”, (edited by Lochak G. and Lochak P.),Ann. Fondation Louis de Broglie Paris, pp. 380–392.Google Scholar
  104. [104]
    Rodrigues Waldyr A. Jr. and Vaz Jr. Jayme (1994),Int. Journal of Modern Physics A 7.Google Scholar
  105. [105]
    Rodríguez-Romo S., Viniegra F. and Keller J. (1992), Geometrical content of the Fierz identities, inClifford Algebras and Their Applications in Mathematical Physics’ Vol.47 of “Fundamental Theories of Physics”, (Helmsteller J., Micall A. and Boudet R., eds.). Dordrecht, Kluwer Academic Pub., pp. 479–497.Google Scholar
  106. [106]
    Sauter F. (1930),Z. Phys. 63 803;64 295.ADSCrossRefGoogle Scholar
  107. [107]
    Salingaros N. and Dresden M. (1979),Phys. Rev. Lett. 43.Google Scholar
  108. [108]
    Sommerfeld Arnold (1939), “Atombau und Spektrallinien II”, 217, Braunschwieg.Google Scholar
  109. [109]
    Sutherhind R. I. (1989),J. Math. Phys. 30 (8) 1721–1726.ADSMathSciNetCrossRefGoogle Scholar
  110. [110]
    Teitler S. (1965a),Nuovo Cimento Supplemento 3 1.MathSciNetGoogle Scholar
  111. [111]
    Teitler S. (1965b),Nuovo Cimento Supplemento 3 15.MathSciNetGoogle Scholar
  112. [112]
    Teitler S. (1965c),J. Math. Phys. 6, 1976.ADSMathSciNetCrossRefGoogle Scholar
  113. [113]
    Teitler S. (1966a).J. Math. Phys. 7 1730.ADSMathSciNetCrossRefGoogle Scholar
  114. [114]
    Teitler S. (1966b),J. Math. Phys. 7 1739.ADSMathSciNetCrossRefGoogle Scholar
  115. [115]
    Thomas L. H. (1927), Calculation of atomic fields.Proc. Cambridge Philos. Soc. 33 542–548.CrossRefGoogle Scholar
  116. [116]
    Uhlenbeck G. E. and Goudsmit S. (1925),Naturwiss,19 953; (1926),Nature 117 261.Google Scholar
  117. [117]
    Vaz Jayme Jr. and Rodrigues Waldyr A. Jr. (1993), Equivalence of the Dirac and Maxwell equations and quantum mechanics,Int. J. Theor. Phys. 32 945–958.MATHMathSciNetCrossRefGoogle Scholar
  118. [118]
    Weber W. E. (1871), Electrodynamische Massbestimmungen, insbesondere über das Princip der Erhaltung der Energie,Abhandlungen der Mathematische Physische Klasse der Koenigles Sachsischen Geselschaft der Wissenschaften,X January; (1872),Phil. Mag. 43 (283) 1–20, 119–149.Google Scholar
  119. [119]
    Weinberger P. (1989),Lectures Notes in Quantum Mechanics Vienna, TU-Wien.Google Scholar
  120. [120]
    Wesson P. S. (1999), “Space-Time-Matter: Modern Kaluza-Klein Theory”, World Scientific Publishing Co London.MATHGoogle Scholar
  121. [121]
    Weyl H. (1952) “Space-Time-Matter” (translated from the 4th german edition by Brose H.L), New York, Dover.Google Scholar

Copyright information

© Birkhäuser-Verlag AG 1999

Authors and Affiliations

  1. 1.División de Estudios de Posgrado, Facultad de Química, and Facultad de Estudios Superiores-CuautitlánUniversidad Nacional Autónoma de MéxicoMéxicoMexico
  2. 2.Federal Institute of Technology (ETH), FestkoerperphysikETH-ZuerichSwitzerland

Personalised recommendations