Abstract
The classical Dirac operator is a conformally invariant first order differential operator mapping spinor-valued functions to the same space, where the spinor space is to be interpreted as an irreducible representation of the spin group. In this article we twist the Dirac operator by replacing the spinor space with an arbitrary irreducible representation of the spin group. In this way, the operator becomes highly reducible, whence we determine its full decomposition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman, London (1982)
Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Rarita-Schwinger type operators in Clifford analysis. J. Funct. Anal. 185, 425–456 (2001)
Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Symmetric analogues of Rarita-Schwinger equations. Ann. Glob. Anal. Geom. 21(3), 215–240 (2001)
Constales, D., Sommen. F., Van Lancker, P.: Models for irreducible representations of Spin\((m)\). Adv. Appl. Clifford Algebras 11(S1), 271–289 (2001)
De Schepper, H., Eelbode, D., Raeymaekers, T.: On a special type of solutions for arbitrary higher spin Dirac Operators. J. Phys. A: Math. Theor. 43(32), 1–13 (2010)
De Schepper, H., Eelbode, D., Raeymaekers, T.: Twisted higher spin Dirac operators. Complex Anal. Oper. Theory 8, 429–447 (2014)
Delanghe, R., Sommen, F., Souček, V.: Clifford Analysis and Spinor Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)
Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. Lond. 117 (1928)
Eelbode, D., Raeymaekers, T.: Construction of higher spin operators using transvector algebras. J. Math. Phys. 55(10), 101703 (2015)
Eelbode, D., Smid, D.: Factorization of Laplace operators on higher spin representations. Complex Anal. Oper. Theory 6, 1011–1023 (2012)
Fegan, H.D.: Conformally invariant first order differential operators. Q. J. Math. 27, 513–538 (1976)
Fulton, W., Harris, J.: Representation Theory: A First Course. Springer, New York (1991)
Gilbert, J., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
Howe, R., Tan, E., Willenbring, J.: Stable branching rules for classical symmetric pairs. Trans. AMS 357(4), 1601–1626 (2004)
Humphreys, J.: Introduction to Lie Algebra and Representation Theory. Springer, New York (1972)
Klimyk, A.U.: Infinitesimal operators for representations of complex Lie groups and Clebsch-Gordan coefficients for compact groups. J. Phys. A: Math. Gen 15, 3009–3023 (1982)
Molev, A.I.: Yangians and Classical Lie Algebras. Mathematical surveys and monographs, vol. 143. AMS Bookstore (2007)
Stein, E.W., Weiss, G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Amer. J. Math. 90, 163–196 (1968)
Tolstoy, V.N.: Extremal projections for reductive classical Lie superalgebras with a non-degenerate generalised Killing form. Russ. Math. Surv. 40, 241–242 (1985)
Zhelobenko, D.P.: Transvector algebras in representation theory and dynamic symmetry, group theoretical methods in physics. In: Proceedings of the Third Yurmala Seminar, vol. 1 (1985)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Raeymaekers, T. (2018). Decomposition of the Twisted Dirac Operator. In: Cerejeiras, P., Nolder, C., Ryan, J., Vanegas Espinoza, C. (eds) Clifford Analysis and Related Topics. CART 2014. Springer Proceedings in Mathematics & Statistics, vol 260. Springer, Cham. https://doi.org/10.1007/978-3-030-00049-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-00049-3_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-00047-9
Online ISBN: 978-3-030-00049-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)