Clifford Algebras, Multipartite Systems and Gauge Theory Gravity

  • Marco A. S. Trindade
  • Eric PintoEmail author
  • Sergio Floquet
Part of the following topical collections:
  1. Homage to Prof. W.A. Rodrigues Jr


In this paper we present a multipartite formulation of gauge theory gravity based on the formalism of space–time algebra for gravitation developed by Lasenby and Doran (Philos Trans R Soc Lond A 582:356–487, 1998). We associate the gauge fields with a description of fermionic and bosonic states using the generalized graded tensor product. Einstein’s equations are deduced from the graded projections and an algebraic Hopf-like structure naturally emerges from formalism. A connection with quantum information theory is performed through the minimal left ideals and entangled qubits are derived. In addition, applications to black holes physics and standard model are outlined.


Clifford algebra Hopf algebra Gravity Qubits Standard model 

Mathematics Subject Classification

15A66 16T05 



Eric Pinto thanks to FAPESB and CNPq for partial financial support. We would also like to thank the anonymous referees for their detailed comments and suggestions.


  1. 1.
    Ablamowicz, R., Fauser, B.: Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple. Comput. Phys. Commun. 170, 115–130 (2005)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Baylis, W.E.: Quantum/classical interface: a geometric from the classical side. In: Byrnes, J. (ed.) Proceedings of the NATO Advanced Study Institute. Dordrecht Kluwer Academic, Dordrecht (2004)Google Scholar
  3. 3.
    Bittencourt, A.S.V.V., Bernardini, A.E., Blasone, M.: Global Dirac bispinor entanglement under Lorentz boosts. Phys. Rev. A 97, 032106 (2018)ADSCrossRefGoogle Scholar
  4. 4.
    Bittencourt, A.S.V.V., Bernardini, A.E., Blasone, M.: Bilayer graphene lattice-layer entanglement in the presence of non-Markovian phase noise. Phys. Rev. B 97, 125435 (2018)ADSCrossRefGoogle Scholar
  5. 5.
    Bulacu, D.: A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category. J. Algebra 332, 244–284 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Castro, C.: Extended Lorentz transformations in Clifford space relativity theory. Adv. Appl. Clifford Algebras 25, 553–567 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  8. 8.
    Chiang, Hsu-Wen, Yoo-Chieh, Hu, Chen, Pisin: Quantization of space-time baased on a space time interval operator. Phys. Rev. D 93, 084043 (2016)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Da Rocha, R., Bernardini, A.E., Vaz Jr., J.: Int. J. Geom. Methods Mod. Phys. 07, 821 (2010)CrossRefGoogle Scholar
  10. 10.
    Doran, C.J.L.: Geometric Algebra and its Application to Mathematical Physics. PhD thesis, Cambridge University (1994)Google Scholar
  11. 11.
    Faulkner, T., Guica, M., Hartman, T., Myers, R.C., Van Raamsdonk, M.: Gravitation from Entanglement in Holographic CFTs. JHEP 03, 051 (2014)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fauser, B.: Quantum Clifford Hopf algebra for quantum field theory. Adv. Appl. Clifford Algebras 13, 115–125 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fauser, B., Oziewicz, Z.: Clifford Hopf gebra for two-dimensional space. Miscellanea Algebraicae 5(2), 31–42 (2001)Google Scholar
  14. 14.
    Havel, T.F., Doran, C.J.: Interaction and entangled in the multiparticle spacetime algebra. In: Dorst, L., Doran, C.J., Lasenby, J. (eds.) Applications of Geometric Algebra in Computer Science and Engineering. Birkhauser, Basel (2002)Google Scholar
  15. 15.
    Hestenes, D.: Space Time Algebra. Gordon and Breach, New York (1996)zbMATHGoogle Scholar
  16. 16.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984)CrossRefGoogle Scholar
  17. 17.
    Kassel, C.: Quantum Groups. Springer-Verlag, New York (1995)CrossRefGoogle Scholar
  18. 18.
    Lasenby, A.N., Doran, C.J.L.: Geometric algebra, dirac wavefunctions and black holes. In: Bergmann P.G., de Sabbata V. (eds) Advances in the Interplay Between Quantum and Gravity Physics. NATO Science Series (Series II: Mathematics, Physics and Chemistry), vol 60. Springer, Dordrecht (2002)Google Scholar
  19. 19.
    Lasenby, A.N., Doran, C.J.L., Gull, S.F.: Gravity, gauge theories and geometric algebra. Philos. Trans. R. Soc. Lond. A 582, 356–487 (1998)zbMATHGoogle Scholar
  20. 20.
    Lashkari, N., McDermott, M.B., Van Raamsdonk, M.: Gravitational dynamics from entanglement thermodynamics. JHEP 04, 195 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    Lashkari, N., Rabideau, C., Sabella-Garnier, P., Van Raamsdonk, M.: Inviolable energy conditions from entanglement inequalities. JHEP 06, 067 (2015)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lopez, E.: Quantum Clifford–Hopf algebras for even dimensions. J. Phys. A 27, 845–854 (1994)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lu, W.: A Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies. Int. J. Mod. Phys. A 32, 1750159 (2017)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Majid, S.: Hopf algebras for physics at the Planck scale. Class. Quant. Grav. 5(12), 1587–1607 (1988)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (2000)Google Scholar
  26. 26.
    Martel, K., Poisson, E.: Regular coordinate systems for Schwarzschild and other spherical spacetimes. Am. J. Phys. 69, 476 (2001)ADSCrossRefGoogle Scholar
  27. 27.
    Rodrigues Jr., W.A., Maiorino, J.E.: A unified theory for construction of arbitrary speeds \(0<v<\infty \) solutions of the relativistic wave equations. Random Oper. Stoch. Equ. 4, 355–400 (1996)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rodrigues, W.A., Oliveira, E.C.: The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach. Springer, New York (2017)zbMATHGoogle Scholar
  29. 29.
    Sakharov, A.D.: Vacuum quantum fluctuations in curved space and the theory of gravitation. Sov. Phys. Dokl. 12, 1040 (1968)ADSGoogle Scholar
  30. 30.
    Thiemann, T.: Loop quantum gravity: an inside view. Approaches to fundamental physics. Lect. Notes Phys. 721, 185–263 (2006)ADSCrossRefGoogle Scholar
  31. 31.
    Verlinde, E.P.: On the origin of gravity and the Laws of Newton. JHEP 1104, 029 (2011)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Verlinde, E.P.: Emergent gravity and the dark universe. SciPost Phys. 2, 016 (2017)ADSCrossRefGoogle Scholar
  33. 33.
    Zwiebach, B.: A First Course in String Theory. Cambridge University Press, New York (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Colegiado de Física, Departamento de Ciências Exatas e da TerraUniversidade do Estado da BahiaSalvadorBrazil
  2. 2.Instituto de FísicaUniversidade Federal da BahiaSalvadorBrazil
  3. 3.Colegiado de Engenharia CivilUniversidade Federal do Vale do São FranciscoJuazeiroBrazil

Personalised recommendations