Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ablamowicz, R., Fauser, B.: Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple. Comput. Phys. Commun. 170, 115–130 (2005)
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)
Bittencourt, A.S.V.V., Bernardini, A.E., Blasone, M.: Global Dirac bispinor entanglement under Lorentz boosts. Phys. Rev. A 97, 032106 (2018)
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)
Bulacu, D.: A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category. J. Algebra 332, 244–284 (2011)
Castro, C.: Extended Lorentz transformations in Clifford space relativity theory. Adv. Appl. Clifford Algebras 25, 553–567 (2015)
Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)
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)
Da Rocha, R., Bernardini, A.E., Vaz Jr., J.: Int. J. Geom. Methods Mod. Phys. 07, 821 (2010)
Doran, C.J.L.: Geometric Algebra and its Application to Mathematical Physics. PhD thesis, Cambridge University (1994)
Faulkner, T., Guica, M., Hartman, T., Myers, R.C., Van Raamsdonk, M.: Gravitation from Entanglement in Holographic CFTs. JHEP 03, 051 (2014)
Fauser, B.: Quantum Clifford Hopf algebra for quantum field theory. Adv. Appl. Clifford Algebras 13, 115–125 (2003)
Fauser, B., Oziewicz, Z.: Clifford Hopf gebra for two-dimensional space. Miscellanea Algebraicae 5(2), 31–42 (2001)
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)
Hestenes, D.: Space Time Algebra. Gordon and Breach, New York (1996)
Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984)
Kassel, C.: Quantum Groups. Springer-Verlag, New York (1995)
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)
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)
Lashkari, N., McDermott, M.B., Van Raamsdonk, M.: Gravitational dynamics from entanglement thermodynamics. JHEP 04, 195 (2014)
Lashkari, N., Rabideau, C., Sabella-Garnier, P., Van Raamsdonk, M.: Inviolable energy conditions from entanglement inequalities. JHEP 06, 067 (2015)
Lopez, E.: Quantum Clifford–Hopf algebras for even dimensions. J. Phys. A 27, 845–854 (1994)
Lu, W.: A Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies. Int. J. Mod. Phys. A 32, 1750159 (2017)
Majid, S.: Hopf algebras for physics at the Planck scale. Class. Quant. Grav. 5(12), 1587–1607 (1988)
Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (2000)
Martel, K., Poisson, E.: Regular coordinate systems for Schwarzschild and other spherical spacetimes. Am. J. Phys. 69, 476 (2001)
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)
Rodrigues, W.A., Oliveira, E.C.: The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach. Springer, New York (2017)
Sakharov, A.D.: Vacuum quantum fluctuations in curved space and the theory of gravitation. Sov. Phys. Dokl. 12, 1040 (1968)
Thiemann, T.: Loop quantum gravity: an inside view. Approaches to fundamental physics. Lect. Notes Phys. 721, 185–263 (2006)
Verlinde, E.P.: On the origin of gravity and the Laws of Newton. JHEP 1104, 029 (2011)
Verlinde, E.P.: Emergent gravity and the dark universe. SciPost Phys. 2, 016 (2017)
Zwiebach, B.: A First Course in String Theory. Cambridge University Press, New York (2004)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is part of the Topical Collection on Homage to Prof. W.A. Rodrigues Jr. edited by Jayme Vaz Jr..
Rights and permissions
About this article
Cite this article
Trindade, M.A.S., Pinto, E. & Floquet, S. Clifford Algebras, Multipartite Systems and Gauge Theory Gravity. Adv. Appl. Clifford Algebras 29, 1 (2019). https://doi.org/10.1007/s00006-018-0917-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-018-0917-0