Summary
A connection between the group/spin(8) and octonions is investigated and on this basis of octonions the algebra which faithfully represents spin(8) is constructed. This octonion algebra is used to describe the superstring theory. Furthermore, using these spin(8)-covariant octonions, we construct the ten-dimensional space-time octonionic spinors together with finite spinor transformations. In the ten-dimensional spacetime, the octonionic forms of Dirac equations, the action of a supersymmetric Yang-Mills field and the super symmetry transformations are introduced.
Riassunto
Si studia una connessione tra il gruppo spin(8) e gli ottonioni e su questa base degli ottonioni si costruisce l–algebra che rappresenta fedelmente lo spin(8). L–algebra ottonionica è usata per descrivere la teoria di superstring. Inoltre, usando questi ottonioni a spin(8) covariante, si costruiscono gli spinori ottonionici dello spazio-tempo 10-dimensionali insieme con trasformazioni spinoriali finite. Nello spazio-tempo 10-dimensionale, si introducono le forme ottoniane delle equazioni di Dirac, l–azione di un campo supersimmetrico di YangMills e le trasformazioni di supersimmetria.
Резюме
Исследуется связь между группой Spin (8) и октонионами. На основе октонионов конструируется алгебра, которая верно представляет spin (8). Эта алгебра октонионов используется для описания суперструнной теории. Кроме того, используя эти spin (8)-ковариантные октонионы, мы конструируем десятимерные пространственно-вре менные октонионные операторы вместе с конечными спинорными преобразованиями. В десятимерном пространствевремен и вводятся октонионные формы уравнений Дирака, действие суперсимметричного поля Янга-Миллса и преобразования суперсимметрии.
Similar content being viewed by others
References
K. Imaeda, H. Tachibana, M. Imaeda andS. Ohta:Nuovo Cimento B,100, 53 (1987);K. Imaeda, M. Imaeda, S. Ohta andH. Tachibana:Bull. Okayama Univ. Sci. A,22, 155 (1987).
K. Imaeda:Nuovo Cimento B,32, 138 (1976);K. Imaeda andM. Imaeda:Bull. Okayama Univ. Sci. A,19, 93 (1984);M. Imaeda: inClifford Algebras and Their Applications in Mathematical Physics, edited byJ. S. R. Chisholm andA. K. Common (Reidel Pub. Co., Dordrecht, 1986), p. 565.
M. B. Green andJ. H. Schwarz:Phys. Lett. B,136, 367 (1984);Nucl. Phys. B,243, 285 (1984).
J. M. Evans:Nucl. Phys. B,298, 92 (1988).
T. Kugo andP. Townsend:Nucl. Phys. B,221, 357 (1983).
A. J. Davies andG. C. Joshi:J. Math. Phys. (N.Y.),27, 3036 (1986).
G. C. Joshi:Lett. Nuovo Cimento,44, 449 (1985).
K. W. Chung andA. Sudbery:Phys. Lett. B,198, 161 (1987).
I. Oda, T. Kimura andA. Nakamura:Prog. Them. Phys.,80, 367 (1988).
For example, seeJ. F. Adams: inSuperspace and Supergravity, edited byS. W. Hawking andM. Roček (Cambridge University Press, Cambridge, 1981), p. 435.
A. Gamba:J. Math. Phys. (N.Y.),8, 775 (1966).
R. Foot andG. C. Joshi:Phys. Lett. B,199, 203 (1987);Phys. Rev. D,36, 1169 (1987).
P. Truini andL. C. Biedenharn:J. Math. Phys. (N.Y.),23, 1327 (1982).
É. Cartan:The Theory of Spinors (Hermann, Paris, 1966).
See ref. (10, 11, 14), and also seeM. Günaydin andF. Gürsey:J. Math. Phys. (N.Y.),14, 1651 (1973);J. F. Moffat:J. Math. Phys. (N.Y.),25, 347 (1984).
See ref. (3), and also seeM. B. Green, J. H. Schwarz andE. Witten:Superstring Theory (Cambridge University Press, Cambridge, 1986).
H. Tachibana:Bull. Okayama Univ. Sci. A,24 (1989).
H A. Sudbery:J. Phys. A,17, 939 (1984).
L. Brink, J. H. Schwarz andJ. H. Scherk:Nucl. Phys. B,121, 77 (1977).
Author information
Authors and Affiliations
Additional information
To seed up publication, the authors of this paper have agreed not to receive the proofs for correction.
Rights and permissions
About this article
Cite this article
Tachibana, H., Imaeda, K. Octonions, superstrings and ten-dimensional spinors. Nuov Cim B 104, 91–106 (1989). https://doi.org/10.1007/BF02742828
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02742828