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Two-Component Spinorial Formalism Using Quaternions for Six-Dimensional Spacetimes

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Abstract

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with \(SL(2,\mathbb {H})\), which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of \(SL(2,\mathbb {H})\) do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of SO(5, 1) is presented, providing a physical interpretation for the elements of \(SL(2,\mathbb {H})\). Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by SU(4).

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Notes

  1. Here we have multiplied the bivector representations \(B^A_{\;\;B}\) adopted in [6, 9] by a global factor of \(-4\). This global factor was arbitrarily chosen in these references and here it is conveniently redefined.

  2. Again, the definitions of \(T^{+\,AB}\) and \(T^-_{AB}\) differ from the ones adopted in [9] by a global multiplicative factor of \(-3\), which is of no fundamental relevance.

References

  1. Adler, S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 104(8), 611–656 (1986)

    MathSciNet  MATH  ADS  Google Scholar 

  2. Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields, International Series of Monographs on Physics. Oxford University Press, Oxford (1995)

  3. Arkani-Hamed, N., Dimopoulos, S., Dvali, G.: The hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429(3), 263–272 (1998)

    Article  MATH  ADS  Google Scholar 

  4. Arkani-Hamed, N., Dimopoulos, S., Dvali, G.: Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity. Phys. Rev. D 59(8), 086004 (1999)

    Article  ADS  Google Scholar 

  5. Baez, J.C., Huerta, J.: Division algebras and supersymmetry I, part of superstrings, geometry, topology, and \(C^{*}\)-algebras. Proc. Symp. Pure Math. 81, 65–80 (2010)

    Article  MATH  Google Scholar 

  6. Batista, C., Cunha, B.C.: Spinors and the Weyl tensor classification in six dimensions. J. Math. Phys. 54(5), 052502 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Batista, C.: Killing spinors and related symmetries in six dimensions. Phys. Rev. D 93(6), 065002 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  8. Batista, C.: Conformally invariant spinorial equations in six dimensions. Class. Quantum Gravity 33(1), 015002 (2016)

    Article  MATH  ADS  Google Scholar 

  9. Batista, C.: Generalizing the Petrov Classification. Lambert Academic Publishing, Germany (2014)

    Google Scholar 

  10. Bengtsson, I.: Particles, twistors and the division algebras. Nucl. Phys. B 302(1), 81–103 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  11. Benn, I., Tucker, R.: An Introduction to Spinors and Geometry with Applications in Physics. Adam Hilger, Bristol (1987)

    MATH  Google Scholar 

  12. Britto, R., Cachazo, F., Feng, B., Witten, E.: Direct proof of tree-level scattering amplitude recursion relation in Yang–Mills theory. Phys. Rev. Lett. 94(18), 181602 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  13. Carrion, H.L., Rojas, M., Toppan, F.: Quaternionic and octonionic spinors. A classification. JHEP 2003(04), 040 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  14. Casalderrey-Solana, J., Liu, H., Mateos, D., Rajagopal, K., Wiedemann, U.: Gauge/String Duality, Hot QCD and Heavy Ion Collisions. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

  15. Cavaglia, M.: Black hole and brane production in TeV gravity: a review. Int. J. Mod. Phys. A 18(11), 1843–1882 (2003)

    Article  MATH  ADS  Google Scholar 

  16. Cederwall, M.: Introduction to division algebras, sphere algebras and twistors (1993). arXiv:hep-th/9310115

  17. Csáki, C.: TASI lectures on extra dimensions and branes. From fields to strings 2, 967–1060 (2005). arXiv: hep-ph/0404096

    MathSciNet  MATH  Google Scholar 

  18. Cunha, B.C.: On the six-dimensional Kerr theorem and twistor equation. Eur. Phys. J. C 74, 2854 (2014)

    Article  ADS  Google Scholar 

  19. Dieudonné, J.: Les déterminants sur un corps non commutatif. Bulletin de la S. M. F. tome 71, 27–45 (1943)

  20. Dirac, P.A.M.: Application of quaternions to Lorentz transformations. Proc. R. Irish Soc. A 50, 261–270 (1945)

    MathSciNet  MATH  Google Scholar 

  21. Dobrev, V.K., Petkova, V.B.: Elementary representations and intertwining operators for the group \(SU^{*}(4)\). Rep. Math. Phys. 13(2), 233–277 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Emparan, R., Reall, H.S.: Black holes in higher dimensions. Living Rev. Relativ. 11(1), 6 (2008)

    Article  MATH  ADS  Google Scholar 

  23. Emparan, R., Suzuki, R., Tanabe, K.: The large \(D\) limit of general relativity. JHEP 06, 009 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Erlich, J., Katz, E., Son, D.T., Stephanov, M.A.: QCD and a holographic model of hadrons. Phys. Rev. Lett. 95(26), 261602 (2005)

    Article  ADS  Google Scholar 

  25. Finazzo, S.I., Rougemont, R., Marrochio, H., Noronha, J.: Hydrodynamic transport coefficients for the non-conformal quark-gluon plasma from holography. JHEP 1502, 051 (2015)

    ADS  Google Scholar 

  26. Fiorenza, D., Sati, H., Schreiber, U.: Super-exceptional embedding construction of the heterotic M5: emergence of SU(2)-flavor sector. J. Geom. Phys. 170, 104349 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Giardino, S.: Four-dimensional conformal field theory using quaternions. Adv. Appl. Clifford Algebras 27(3), 2457–2471 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gursey, F.: Applications of quaternions to field equations. Ph.D. Thesis, Imperial College, London (1950)

  29. Horowitz, G.T., Polchinski, J.: Gauge/Gravity Duality, Approaches to Quantum Gravity. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  30. Howe, P.S., Sierra, G., Townsend, P.K.: Supersymmetry in six dimensions. Nucl. Phys. B 221(2), 331–348 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  31. Hubeny, V.E.: The AdS/CFT correspondence. Class. Quantum Gravity 32(12), 124010 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  32. Huerta, J., Sati, H., Schreiber, U.: Real ADE-equivariant (co)homotopy and super M-branes. Commun. Math. Phys. 371, 425–524 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  33. Huerta, J., Schreiber, U.: M-theory from the superpoint. Lett. Math. Phys. 108, 2695–2727 (2018)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  34. Koller, K.: A six-dimensional superspace approach to extended superfields. Nucl. Phys. B 222(2), 319–337 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  35. Kopczyński, W., Trautman, A.: Simple spinors and real structures. J. Math. Phys. 33(2), 550–559 (1992)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  36. Kugo, T., Townsend, P.: Supersymmetry and the division algebras. Nucl. Phys. B 221(2), 357–380 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  37. Lounesto, P.: Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series, vol. 286, 2nd edn. Cambridge University Press, Cambridge (2001)

  38. Lukierski, J.: Quaternionic six-dimensional (super)twistor formalism and composite (super)spaces. Mod. Phys. Lett. A 6(03), 189–197 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  39. Lukierski, J., Nowicki, A.: Euclidean superconformal symmetry and its relation with Minkowski supersymmetries. Phys. Lett. B 127(1), 40–46 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  40. Maldacena, J.M.: The large \(N\) limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mason, L.J., Reid-Edwards, R.A., Taghavi-Chabert, A.: Conformal field theories in six-dimensional twistor space. J. Geom. Phys. 62(12), 2353–2375 (2012)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  42. Mezincescu, L., Routh, A.J., Townsend, P.K.: Supertwistors and massive particles. Ann. Phys. 346, 66–90 (2014)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  43. Morita, K.: Quaternioninc formulation of the Dirac theory in special and general relativity. Prog. Theor. Phys. 70(6), 1648–1665 (1983)

    Article  MATH  ADS  Google Scholar 

  44. Morita, K.: Quaternionic structure of simple \(D=4\) supergravity. Prog. Theor. Phys. 72(5), 1056–1059 (1984)

    Article  MATH  Google Scholar 

  45. Mukhi, S.: String theory: a perspective over the last 25 years. Class. Quantum Gravity 28(15), 153001 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  46. Nieuwenhuizen, P.V., Warne, N.P.: Integrability conditions for Killing spinors. Commun. Math. Phys. 93, 277–284 (1984)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  47. Penrose, R.: A spinor approach to general relativity. Ann. Phys. 10(2), 171–201 (1960)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  48. Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. A 284(1397), 159–203 (1965)

    MathSciNet  MATH  ADS  Google Scholar 

  49. Penrose, R., Rindler, W.: Spinors and space-time, Cambridge Monographs on Mathematical Physics, vol. 1. Cambridge University Press, Cambridge (1984)

  50. Penrose, R., Rindler, W.: Spinors and space-time, Cambridge Monographs on Mathematical Physics, vol. 2. Cambridge University Press, Cambridge (1986)

  51. Randall, L., Sundrum, R.: An alternative to compactification. Phys. Rev. Lett. 83(23), 4690–4693 (1999)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  52. Rocha, R., Vaz, J.: Conformal structures and twistors in the paravector model of spacetime. Int. J. Geom. Methods Mod. Phys. 4(4), 547–576 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sudbery, A.: Division algebras, (pseudo)orthogonal groups and spinors. J. Phys. A 17(5), 939–955 (1984)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  54. Toppan, F.: Hermitian versus holomorphic complex and quaternionic generalized supersymmetries of the M-theory. A classification. JHEP 2004(09), 016 (2004)

    Article  Google Scholar 

  55. Venâncio, J.: The Spinorial Formalism. Lambert Academic Publishing, Germany (2019)

    Google Scholar 

  56. Venâncio, J., Batista, C.: Separability of the Dirac equation on backgrounds that are the direct product of bidimensional spaces. Phys. Rev. D 95(8), 084022 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  57. Weinberg, S.: Six-dimensional methods for four-dimensional conformal field theories. Phys. Rev. D 82(4), 045031 (2010)

    Article  ADS  Google Scholar 

  58. Weinberg, S.: Six-dimensional methods for four-dimensional conformal field theories II: irreducible fields. Phys. Rev. D 86(8), 085013 (2012)

    Article  ADS  Google Scholar 

  59. Wilker, J.B.: The quaternion formalism for Möbius groups in four or fewer dimensions. Linear Algebra Appl. 190, 99–136 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhang, F.: Quaternions and matrices of quaternions. Math. Fac. Articles 251, 21–57 (1997)

    MathSciNet  MATH  ADS  Google Scholar 

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Acknowledgements

C. B. would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the partial financial support through the research productivity fellowship. Likewise, C. B. thanks Universidade Federal de Pernambuco for the funding through Qualis A project. J. V. thanks CNPq for the financial support. We both thank CAPES for the invaluable funding of the graduation program of our department. We are grateful to Urs Schreiber for pointing out the noteworthy reference [5] after we released the preprint. In addition, we thank the kind and enlightening e-mail of Paul Townsend mentioning Dirac’s hidden article [20].

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  1. Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco, 50670-901, Brazil

    Joás Venâncio & Carlos Batista

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  1. Joás Venâncio
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Correspondence to Joás Venâncio.

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Communicated by Roldão da Rocha.

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Venâncio, J., Batista, C. Two-Component Spinorial Formalism Using Quaternions for Six-Dimensional Spacetimes. Adv. Appl. Clifford Algebras 31, 71 (2021). https://doi.org/10.1007/s00006-021-01172-1

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