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Clifford Analysis for Higher Spin Operators

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Abstract

This chapter focuses on the use of Clifford analysis techniques as an encompassing and unifying tool to study higher spin generalizations of the classical Dirac operator. These operators belong to a complete family of conformally invariant first-order differential operators, acting on functions taking their values in an irreducible representation for the spin group (the double cover for the orthogonal group). Their existence follows from a standard classification result due to Fegan (Q. J. Math. 27:513–538, 1976), and a canonical way to construct them is to use the technique of Stein–Weiss gradients. This then gives rise to two kinds of differential operators defined on irreducible tensor fields, the standard language used in, e.g., theoretical physics, where higher spin operators appear in the equations of motion for elementary particles having arbitrary half-integer spin: on the one hand, there are the (elliptic) generalizations of the Dirac operator, acting as endomorphisms on the space of smooth functions with values in a fixed module (i.e., preserving the values), and on the other hand there are the invariant operators acting between functions taking values in different modules for the spin group (the so-called twistor operators and their duals). In this chapter, both types of higher spin operators will be defined on spinor-valued functions of a matrix variable (i.e., in several vector variables): this has the advantage that the resulting equations become more transparent, and it allows using techniques for Clifford analysis in several variables. In particular, it provides an elegant framework to develop a function theory for the aforementioned operators, such as a full description of the (polynomial) null solutions and analogues of the classical Cauchy integral formula.

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References

  1. Ahlfors, L.: Mobius transformations in \(\mathbb{R}^{n}\) expressed through 2 × 2 matrices of Clifford numbers. Complex Var. 5, 215–224 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman, London (1982)

    Google Scholar 

  3. Brackx, F., Eelbode, D., van de Voorde, L., Van Lancker, P.: On the fundamental solution and integral formulae of a higher spin operator in several vector variables. In: AIP Conference Proceedings, vol. 1281, pp.1519–1522 (2010)

    Google Scholar 

  4. Brackx, F., De Schepper, H., Krump, L., Souček, V.: Explicit Penrose transform for massless field equations of general spin in dimension four. In: ICNAAM 2011 AIP Conference Proceedings, vol. 1389, pp. 287–290 (2011)

    Google Scholar 

  5. Brackx, F., Eelbode, D., van de Voorde, L.: Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables. Math. Phys. Anal. Geom. 14(1), 1–20 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brackx, F., Eelbode, D., van de Voorde, L.: The polynomial null solutions of a higher spin Dirac operator in two vector variables. Adv. Appl. Clifford Algebras 21(3), 455–476 (2011)

    Article  MATH  Google Scholar 

  7. Branson, T.: Stein-Weiss operators and ellipticity. J. Funct. Anal. 151(2), 334–383 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Rarita-Schwinger type operators in Clifford analysis. J. Funct. Anal. 185, 425–456 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Symmetric analogues of Rarita-Schwinger equations. Ann. Glob. Anal. Geom. 21(3), 215–240 (2001)

    Google Scholar 

  10. Colombo, F., Sabadini, I., Sommen, F., Struppa, D.: Analysis of Dirac systems and Computational Algebra. Progress in Mathematical Physics, vol. 39. Birkhaüser, Basel (2004)

    Google Scholar 

  11. Constales, D., Sommen. F., Van Lancker, P.: Models for irreducible representations of Spin(m). Adv. Appl. Clifford Algebras 11(S1), 271–289 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Bie, H.: Harmonic and Clifford analysis in superspace. PhD. Dissertation, Ghent University (2008)

    Google Scholar 

  13. Delanghe, R., Sommen, F., Souček, V.: Clifford analysis and spinor valued functions. Kluwer, Dordrecht (1992)

    Book  Google Scholar 

  14. De Schepper, H., Eelbode, D., Raeymaekers, T.: On a special type of solutions of arbitrary higher spin Dirac operators. J. Phys. A 43(32), 1–13 (2010)

    Article  Google Scholar 

  15. De Schepper, H., Eelbode, D., Raeymaekers, T.: Twisted higher spin Dirac operators. Complex. Anal. Oper. Theory (2013). doi:10.1007/s11785-013-0295-5

    Google Scholar 

  16. Eastwood, M.: Higher symmetries of the Laplacian. Ann. Math. 161, 1645–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eelbode, D., Raeymaekers, T., Van Lancker, P.: On the fundamental solution for higher spin Dirac operators. J. Math. Anal. Appl. 405, 555–564 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Eelbode, D., Souček, V.: Conformally invariant powers of the Dirac operator in Clifford analysis. Math. Meth. Appl. Sci. 33(13), 1558–1570 (2010)

    MATH  Google Scholar 

  19. Eelbode, D., Šmíd, D.: Factorization of Laplace operators on higher spin representations. Complex. Anal. Oper. Theory 6(5), 1011–1023 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Eelbode, D., Van Lancker, Total Rarita-Schwinger operators in Clifford analysis, Ann. Glob. Anal. Geom. 42, 473–493 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fefferman, C., Graham, C.R.: Conformal invariants. In: Elie Cartan et les Mathématique d’aujourd’hui, Numéro hors série, pp. 95–116 (1985)

    Google Scholar 

  22. Fegan, H.D.: Conformally invariant first order differential operators. Q. J. Math. 27, 513–538 (1976)

    Article  MathSciNet  Google Scholar 

  23. Fischer, E.: Über die Differentiationsprozesse der Algebra. J. für Math. 148, 1–78 (1917)

    MATH  Google Scholar 

  24. Fulton, W., Harris, J.: Representation Theory: A First Course. Springer, New York (1991)

    MATH  Google Scholar 

  25. Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  26. Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. ISNM, vol. 89. Birkhäuser, Basel (1990)

    Google Scholar 

  27. Holland, J., Sparling, G.: Conformally invariant powers of the ambient Dirac operator (2001). arXiv:math/0112033v2

    Google Scholar 

  28. Humphreys, J.: Introduction to Lie algebras and Representation Theory. Springer, New York (1972)

    Book  MATH  Google Scholar 

  29. Latvamaa, E., Lounesto, P.: Conformal transformations and Clifford algebras. Proc. Am. Math. Soc. 79, 533–538 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, H., Ryan, J.: Clifford analysis techniques for spherical PDE. J. Fourier Anal. Appl. 8(6), 535–563 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Miller, W. Jr.: Symmetry and Separation of Variables. Addison-Wesley, Massachusetts (1977)

    MATH  Google Scholar 

  32. Peetre, J., Qian, T.: Moebius covariance of iterated Dirac operators. J. Aust. Math. Soc. Ser. A 56, 1–12 (1994)

    Article  MathSciNet  Google Scholar 

  33. Penrose, R., Rindler, W.: Spinors and Space-Time, vols. 1 and 2. Cambridge University Press, Cambridge (1986)

    Book  Google Scholar 

  34. Porteous, I.R.: Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics, vol. 50. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  35. Qian, T., Ryan, J.: Conformal transformations and Hardy spaces arising in Clifford analysis. J. Oper. Theory 35, 349–372 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Rarita, W., Schwinger, J.: On a theory of particles with half-integer spin. Phys. Rev. 60, 61 (1941)

    Article  MATH  Google Scholar 

  37. Slovak, J., Natural operators on conformal manifolds, PhD. Dissertation, Masaryk University, Brno (1993)

    Google Scholar 

  38. Šmíd, D.: Conformally invariant higher order higher spin operators on the sphere. AIP Conf. Proc. 1493, 911 (2012). doi:10.1063/1.4765596

    Google Scholar 

  39. Somberg P.: Twistor Operators on Conformally Flat Spaces. Suppl. ai Rend. del Circ. Matematico di Pal., Ser. II, Num. 66, 179–197 (2001)

    Google Scholar 

  40. Sommen, F., Van Acker, N.: Monogenic differential operators. Results Math. 22(3–4), 781–798 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sommen, F., Van Acker, N.: Invariant differential operators on polynomial-valued functions. In: Clifford Algebras and Their Applications in Mathematical Physics. Fundamental Theories and Physics, vol. 55, pp. 203–212. Kluwer, Dordrecht (1993)

    Google Scholar 

  42. Souček, V.: Conformal invariance of higher spin equations. In: Proc. Symp. Analytical and Numerical Methods in Clifford Analysis, Seiffen, pp. 175–186 (1996)

    Google Scholar 

  43. Stein, E.W., Weiss, G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tolstoy, V.N.: Extremal projections for reductive classical Lie superalgebras with a non-degenerate generalised Killing form. Russ. Math. Surv. 40, 241–242 (1985)

    Article  Google Scholar 

  45. Van Lancker, P.: Higher spin fields on smooth domains. In: Clifford Analysis and Its Applications. NATO Science Series, vol. 25, pp. 389–398. Springer, Dordrecht (2001)

    Google Scholar 

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Correspondence to David Eelbode .

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Eelbode, D. (2015). Clifford Analysis for Higher Spin Operators. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_23

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