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Clifford Algebra, Lorentz Transformation and Unified Field Theory

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Abstract

According to a framework based on Clifford algebra \(C\ell (1,3)\), this paper gives a classification for elementary fields, and then derives their dynamical equations and transformation laws in detail. These results provide an outline on elementary fields and some new insights into their unusual properties. All elementary fields exist in pairs, and one part of the pair is a complex field. Some intrinsic symmetries and constraints such as Lorentz gauge condition are automatically included in the canonical equation. Clifford algebra \(C\ell (1,3)\) is a natural language to describe the world. In this language, the representation formalism of dynamical equation is symmetrical and elegant with no more or less contents. This paper is also a summary of some previous problem-oriented researches. Solutions to some simple equations are given.

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References

  1. Ablmowicz, R., Sobczyk, G. (eds.): Lectures on Clifford (Geometric) Algebras and Applications. Birkhauser, Basel (2004)

    Google Scholar 

  2. Ablmowicz, R. (ed.): Clifford Algebras Applications to Mathematics, Physics, and Engineering, PIM 34. Birkhauser, Basel (2004)

    Google Scholar 

  3. Ablamowicz, R., Lounesto, P.: On Clifford algebras of a bilinear form with an antisymmetric part. In: Ablamowicz, R., Lounesto, P., Parra, J.M. (eds.) Clifford Algebras with Numeric and Symbolic Computations, pp. 167–188. Birkhauser, Boston (1996)

    Chapter  Google Scholar 

  4. Baake, M., Reinicke, P., Rittenberg, V.: Fierz identities for real Clifford algebras and the number of supercharges. J. Math. Phys. 26(5), 1070–1071 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Balabane, M., et al.: Existence of excited states for a nonlinear Dirac field. Commun. Math. Phys. 119, 153–176 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Balabane, M., et al.: Existence of standing waves for Dirac fields with singular nonlinearities. Commun. Math. Phys. 133, 53–74 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Boudet, R.: Conservation laws in the Dirac theory. J. Math. Phys. 26, 718–724 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  8. Cartan, E.: The Theory of Spinors. The M.T.I. Press, Cambridge (1966)

    MATH  Google Scholar 

  9. Cazenave, T., Vazquez, L.: Existence of localized solutions for a classical nonlinear Dirac field. Commun. Math. Phys. 105, 35–47 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chen, C.-M., Nester, J.M., Tung, R.-S.: Gravitational energy for GR and poincare Gauge theories: a covariant Hamiltonian approach. Int. J. Mod. Phys. D 24, 1530026 (2015). arXiv:1507.07300

  11. Chevalley, C.: The Algebraic Theory of Spinors and Clifford Algebras. Springer, Berlin (1996)

    Book  Google Scholar 

  12. Clifford, W.: Application of Grassmann’s extensive algebra. Am. J. Math. 1, 350–358 (1878)

    Article  MathSciNet  MATH  Google Scholar 

  13. Crawford, J.P.: On the algebra of Dirac bispinor densities: factorization and inversion theorems. J. Math. Phys. 26, 1439–1441 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  14. Delanghe, R., Sommen, F., Soucek, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)

    Book  MATH  Google Scholar 

  15. Dimakis, A., Muller-Hoissen, F.: Clifform calculus with applications to classical field theories. Class. Quantum Grav. 8, 2093 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Dirac, P.: The quantum theory of the electron. Proc. R. Soc. Lond. A 117, 610–624 (1928)

    Article  ADS  MATH  Google Scholar 

  17. Dressel, J., Bliokh, K.Y., Nori, F.: Spacetime algebra as a powerful tool for electromagnetism. Phys. Rep. 589, 1–71 (2015). arXiv:1411.5002

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Esteban, M.J., Sere, E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Fauser, B.: Dirac theory from a field theoretic point of view. Mathematics 94, 89–107 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Finkelsten, R., et al.: Nonlinear spinor fields. Phys. Rev. 83(2), 326–332 (1951)

    Article  ADS  Google Scholar 

  21. Grassmann, H.: Die Lineale Ausdehnungslehre, ein neuer Zweig derMathematik [The Theory of Linear Extension, a New Branch of Mathematics], (1844)

  22. Greiner, W., Reinhdardt, J.: Field Quantization. Springer, Berlin (1996)

    Book  Google Scholar 

  23. Gu, Y.Q.: A Note on the Representation of Clifford Algebra, to appear

  24. Gu, Y.Q.: Dynamical Constraints on the Cosmological Parameters. arXiv:0709.2414

  25. Gu, Y.Q.: Functions and Relations for an Evolving Star with Spherical Symmetry. arXiv:0801.0294

  26. Gu, Y.Q.: Integrable conditions for Dirac equation and Schrödinger equation. arXiv:0802.1958

  27. Gu, Y.Q.: Light-Cone Coordinate System in General Relativity. arXiv:0710.5792

  28. Gu, Y.Q.: Mass Spectrum of Dirac Equation with Local Parabolic Potential. arXiv:hep-th/0612214

  29. Gu, Y.Q.: Natural Coordinate System in Curved Space-Time. arXiv:gr-qc/0612176, to appear in JGSP

  30. Gu, Y.Q.: Rigorous Solutions to the Gravitational collapse in Comoving Coordinate System. arXiv:0705.2133

  31. Gu, Y.Q.: Space-Time Geometry and Some Applications of Clifford Algebra in Physics, to appear soon

  32. Gu, Y.Q.: Stationary Spiral Structure and Collective Motion of the Stars in a Spiral Galaxy. arXiv:0805.2828

  33. Gu, Y.Q.: Structure of the Star with Ideal Gases. arXiv:0712.0219

  34. Gu, Y.Q.: The Simplification of Spinor Connection and Classical Approximation. arXiv:gr-qc/0610001

  35. Gu, Y.Q.: The Vierbein Formalism and Energy-Momentum Tensor of Spinors. arXiv:gr-qc/0612106

  36. Gu, Y.Q.: A canonical form for relativistic dynamic equation. Adv. Appl. Clifford Algebras 7(1), 13–24 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Gu, Y.Q.: Some properties of the spinor soliton. Adv. Appl. Clifford Algebras 8(1), 17–29 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gu, Y.Q.: Spinor soliton with electromanetic field. Adv. Appl. Clifford Algebras 8(2), 271–282 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gu, Y.Q.: The electromagnetic potential among nonrelativistic electrons. Adv. Appl. Clifford Algebras 9(1), 61–79 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gu, Y.Q.: New approach to N-body relativistic quantum mechanics. Int. J. Mod. Phys. A 22, 2007–2020 (2007). arXiv:hep-th/0610153

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Gu, Y.Q.: A cosmological model with dark spinor source. Int. J. Mod. Phys. A 22, 4667–4678 (2007). arxiv:gr-qc/0610147

    Article  ADS  MATH  Google Scholar 

  42. Gu, Y.Q.: Exact vacuum solutions to the Einstein equation. Chin. Ann. Math. Ser. B 28(5), 499–506 (2007). arXiv:0706.0318

    Article  MathSciNet  MATH  Google Scholar 

  43. Gu, Y.Q.: The series solution to the metric of stationary vacuum with axisymmetry. Chin. Phys. B 19, 030402 (2010). arXiv:0811.0449

    Article  ADS  Google Scholar 

  44. Gu, Y.Q.: Some paradoxes in special relativity and the resolutions. Adv. Appl. Clifford Algebras 21(1), 103–119 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gu, Y.Q.: The quaternion structure of space-time and arrow of time. J. Mod. Phys. 3, 570–580 (2012)

    Article  Google Scholar 

  46. Gu, Y.Q.: Some subtle concepts in fundamental physics. Phys. Essays 30, 356–363 (2017). arXiv:0901.0309

    Article  ADS  Google Scholar 

  47. Gu, Y.Q.: Some characteristic functions for the Eigen solution of nonlinear spinor. Quantum Phys. Lett. 6(2), 123–129 (2017). arXiv:hep-th/0611210

    Article  Google Scholar 

  48. Gu, Y.Q.: Nonlinear spinors as the candidate of dark matter. OALib. J. 4, e3954 (2017). arXiv:0806.4649

    Google Scholar 

  49. Gu, Y.Q.: Functions of state for spinor gas in general relativity. OALib. J. 4, e3953 (2017). arXiv:0711.1243

    Google Scholar 

  50. Gu, Y.Q.: A procedure to solve the Eigen solution to Dirac equation. Quantum Phys. Lett. 6(3), 161–163 (2017). arXiv:0708.2962

    Google Scholar 

  51. Gu, Y.Q.: Local Lorentz transformation and mass-energy relation of spinor. Phys. Essays 31, 1–6 (2018). arXiv:hep-th/0701030

    Google Scholar 

  52. Gu, Y.Q.: Test of Einstein’s mass-energy relation. Appl. Phys. Res. 10(1), 1–4 (2018)

    Article  MathSciNet  Google Scholar 

  53. Gull, S., Lasenby, A., Doran, Ch.: Imaginary numbers are not real—the geometric algebra of spacetime. Found. Phys. 23(9), 1175–1201 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  54. Hamilton, W.: On Quaternions, or on a New System of Imaginaries in Algebra, Philosophical Magazine (1844)

  55. Hestenes, D., Li, H., Rockwood, A.: New algebraic tools for classical geometry. Geo. Comput. Clifford Algebras 3–26 (2001)

  56. Hestenes, D.: Observables, operators, and complex numbers in the Dirac theory. J. Math. Phys. 16, 556–572 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  57. Hestenes, D.: A unified language for mathematics and physics. Adv. Appl. Clifford Algebra 1(1), 1–23 (1986)

    MathSciNet  MATH  Google Scholar 

  58. Hestenes, D., Sobczyk, G.E.: Clifford algebra to geometric calculus: a unified language for mathematics and physics. Am. J. Phys. 53(5), 510–511 (1984)

    Article  ADS  MATH  Google Scholar 

  59. Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 23(2), 377–404 (2013). arXiv:1305.5663

    Article  MathSciNet  MATH  Google Scholar 

  60. Islam, J.N.: Rotating Fields in General Relativity. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  61. Keller, J.: Spinors and multivectors as a unified tool for spacetime geometry and for elementary particle physics. Int. J. Theor. Phys. 30, 137–184 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  62. Keller, J.: A multivectorial Dirac equation. J. Math. Phys. 31(10), 2501–2510 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. Keller, J.: Spinors and multivectors as a unified tool for spacetime geometry and for elementary particle physics. Int. J. Theor. Phys. 30(2), 137–184 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  64. Keller, J.: The geometric content of the electron theory. Adv. Appl. Clifford Algebras 3(2), 47–200 (1993)

    MathSciNet  Google Scholar 

  65. Keller, J.: The geometric content of the electron theory. Adv. Appl. Clifford Algebras 3(2), 147–200 (1993)

    MathSciNet  MATH  Google Scholar 

  66. Keller, J.: The geometric content of the electron theory. (Part II) Theory of the electron from start. Adv. Appl. Clifford Algebras 9(2), 309–395 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  67. Keller, J., Rodriguez-Romo, S.: Multivectorial representation of Lie groups. Int. J. Theor. Phys. 30(2), 185–196 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  68. Lasenby, A., Gull, S., Doran, C.: STA and the Interpretation of Quantum Mechanics. Clifford (Geometric) Algebras, pp. 147–169. Birkhauser, Boston (1996)

    Book  MATH  Google Scholar 

  69. Lasenby, J., Lasenby, A.N., Doran, C.J.L.: A unified mathematical language for physics and engineering in the 21st century. Philos. Trans. R. Soc. 358(1765), 21–39 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  71. Merle, F.: Existence of stationary states for nonlinear Dirac equations. J. Differ. Equ. 74, 50–68 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  72. Moses, H.E.: A spinor representation of Maxwell’s equations. Nuovo Cimento 8(suppl. 1), 18 (1958)

    MATH  Google Scholar 

  73. Nester, J.M.: Special orthonormal frames. J. Math. Phys. 33, 910 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. Ohmura, T.: A new formulation on the electromagnetic field. Prog. Theor. Phys. 16, 684–685 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. Oliveira, E.C., Rodrigues Jr., W.A.: Dotted and undotted algebraic spinor fields in general relativity. J. Mod. Phys. D 13(8), 1637–1659 (2004). arXiv:math-ph/0407024

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. Randa, A.F., Soler, M.: Perturbation theory for an exactly soluble spinor model in interaction with its electromagnetic field. Phys. Rev. D 8(10), 3430–3433 (1973)

    Article  ADS  Google Scholar 

  77. Riesz, M.: Clifford Numbers and Spinors. Springer, Netherlands (1993)

    Book  MATH  Google Scholar 

  78. Rodrigues Jr., W.A., Oliveira, E.C.: Clifford valued differential forms, and some issues in gravitation, electromagnetism and ’Unified’ theories. Int. J. Mod. Phys. D 13(9), 1879–1915 (2004). arXiv:math-ph/0311001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. Rodrigues Jr., W.A., Wainer, S.A.: Equations of motion and energy-momentum 1-forms for the coupled gravitational, Maxwell and Dirac fields. Adv. Appl. Clifford Algebras 27(1), 1–17 (2016)

    MathSciNet  Google Scholar 

  80. Rylov, Y.A.: Dirac equation interms of hydrodynamical variables. Adv. Appl. Clifford Algebras 5(1), 1–40 (1995)

    MathSciNet  Google Scholar 

  81. Sachs, M.: General Relativity and Matter, (Ch. 3). D. Reidel, Dordrecht (1982)

    Book  Google Scholar 

  82. Shirokov, D.S.: Clifford algebras and their applications to Lie groups and spinors. arXiv:1709.06608

  83. Soler, M.: Classical, stable, nonlinear spinor field with positive rest energy. Phys. Rev. D 1(10), 2766–2767 (1970)

    Article  ADS  Google Scholar 

  84. Soler, M.: Classical electrodynamics for a nonlinear spinor field: perturbative and exact approaches. Phys. Rev. D 8, 3424–3429 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  85. Takahashi, Y.: The Fierz identities—a passage between spinors and tensors. J. Math. Phys. 24, 1783–1790 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  86. Todorov, I.: Clifford algebras and spinors. Bulg. J. Phys. 38, 3–28 (2011). arXiv:1106.3197

    MathSciNet  Google Scholar 

  87. Tucker, R.W.: A Clifford calculus for physical theories. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 177–200. D. Reidel Publ. Co., Dordrecht (1985)

    Google Scholar 

  88. Vargas, J.H.G., Torr, D.G.: A Geometric Language for Dirac Equation. In: Ablamowicz R., and Fauser B. (eds.) Clifford Algebras and their Applications in Mathematical Physics (Ixtapa-Zihuatanejo, Mexico 1999), Algebra and Physics, Progress In: Physics 18, vol. 1, pp. 135–154. Birkh¡auser, Boston (2000)

  89. Varlamov, V.V.: Discrete symmetries and Clifford algebras. Int. J. Theor. Phys. 40, 769–805 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  90. Vaz, J., Rodrigues, W.: Equivalence of Dirac and Maxwell equations and quantum mechanics. Int. J. Theor. Phys. 32, 945–959 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

It is my pleasure to thank my friends Prof. Hao Wang, Mr. Hai-Bin Lei for their encouragement and help. Prof. J. M. Nester reminded me to have a more systematical study on Clifford algebra. Prof. D. S. Shirokov gave many times enlightening discussion on representation of Clifford algebra. Most works were fulfilled under the support of my supervisor Prof. Ta-Tsien Li. The paper has been modified according to the suggestions of a referee.

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    Ying-Qiu Gu

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Communicated by Vladislav Kravchenko.

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Gu, YQ. Clifford Algebra, Lorentz Transformation and Unified Field Theory. Adv. Appl. Clifford Algebras 28, 37 (2018). https://doi.org/10.1007/s00006-018-0852-0

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