Advertisement

Higher Order Fermionic and Bosonic Operators

  • Chao Ding
  • Raymond Walter
  • John Ryan
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

This paper studies a particular class of higher order conformally invariant differential operators and related integral operators acting on functions taking values in particular finite dimensional irreducible representations of the Spin group. The differential operators can be seen as a generalization to higher spin spaces of kth-powers of the Euclidean Dirac operator. To construct these operators, we use the framework of higher spin theory in Clifford analysis, in which irreducible representations of the Spin group are realized as polynomial spaces satisfying a particular system of differential equations. As a consequence, these operators act on functions taking values in the space of homogeneous harmonic or monogenic polynomials depending on the order. Moreover, we classify these operators in analogy with the quantization of angular momentum in quantum mechanics to unify the terminology used in studying higher order higher spin conformally invariant operators: for integer and half-integer spin, these are respectively bosonic and fermionic operators. Specifically, we generalize arbitrary powers of the Dirac and Laplace operators respectively to spin-\(\frac {3}{2}\) and spin-1.

Keywords

Higher order fermionic and bosonic operators Conformal invariance Fundamental solutions Intertwining operators Ellipticity 

Mathematics Subject Classification (2010)

Primary 53A30; Secondary 20G05 30G35 

Notes

Acknowledgements

The authors wish to thank anonymous referees for helpful suggestions that significantly improved the manuscript. The authors are grateful to Bent Ørsted for communications pointing out that the intertwining operators of our conformally invariant differential operators and our convolution type operators can be recovered as Knapp-Stein intertwining operators and Knapp-Stein operators in higher spin theory. The second author “Raymond Walter” acknowledges this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0957325 and the University of Arkansas Graduate School Distinguished Doctoral Fellowship in Mathematics and Physics.

References

  1. 1.
    L.V. Ahlfors, Möbius transformations in \(\mathbb {R}^n\) expressed through 2 × 2 matrices of Clifford numbers. Complex Var. 5, 215–224 (1986)Google Scholar
  2. 2.
    M.F. Atiyah, R. Bott, A. Shapiro, Clifford modules. Topology 3(Suppl. 1), 3–38 (1964)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, 2nd edn. Graduate Texts in Mathematics (Springer, New York, 2001)Google Scholar
  4. 4.
    F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis (Pitman, London, 1982)zbMATHGoogle Scholar
  5. 5.
    F. Brackx, D. Eelbode, L. Van de Voorde, Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables. Math. Phys. Anal. Geom. 14(1), 1–20 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Branson, Second order conformal covariants. Proc. Am. Math. Soc. 126, 1031–1042 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Bureš, F. Sommen, V. Souček, P. Van Lancker, Rarita-Schwinger type operators in clifford analysis. J. Funct. Anal. 185(2), 425–455 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.L. Clerc, B. Ørsted, Conformal covariance for the powers of the Dirac operator. https://arxiv.org/abs/1409.4983
  9. 9.
    H. De Bie, D. Eelbode, M. Roels, The higher spin Laplace operator. Potential Anal. 47(2), 123–149 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. De Bie, F. Sommen, M. Wutzig, Reproducing kernels for polynomial null-solutions of Dirac operators. Constr. Approx. 44(3), 339–383 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Delanghe, F. Sommen, V. Souček, Clifford Algebra and Spinor-Valued Functions: A Function Theory for the Dirac Operator (Kluwer, Dordrecht, 1992)CrossRefGoogle Scholar
  12. 12.
    H. De Schepper, D. Eelbode, T. Raeymaekers, On a special type of solutions of arbitrary higher spin Dirac operators. J. Phys. A Math. Theor. 43(32) (2010). https://doi.org/10.1088/1751-8113/43/32/325208
  13. 13.
    C. Ding, J. Ryan, On Some Conformally Invariant Operators in Euclidean Space, in Honor of Paul A. M. Dirac, CART 2014, Tallahassee, FL, December 15–17Google Scholar
  14. 14.
    C. Ding, R. Walter, J. Ryan, Construction of arbitrary order conformally invariant operators in higher spin spaces. J. Geom. Anal. 27(3), 2418–2452 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    C.F. Dunkl, J. Li, J. Ryan, P. Van Lancker, Some Rarita-Schwinger type operators. Comput. Methods Funct. Theory 13(3), 397–424 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Eastwood, Higher symmetries of the Laplacian. Ann. Math. 161(3), 1645–1665 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. Eelbode, M. Roels, Generalised Maxwell equations in higher dimensions. Complex Anal. Oper. Theory 10(2), 267–293 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Eelbode, T. Raeymaekers, Construction of conformally invariant higher spin operators using transvector algebras. J. Math. Phys. 55(10) (2014). http://dx.doi.org/10.1063/1.4898772
  19. 19.
    H.D. Fegan, Conformally invariant first order differential operators. Q. J. Math. 27, 513–538 (1976)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory. Graduate Texts in Contemporary Physics (Springer, New York, 1997)Google Scholar
  21. 21.
    T. Fulton, F. Rohrlich, L. Witten, Conformal invariance in physics. Rev. Mod. Phys. 34(3), 442–456 (1962)MathSciNetCrossRefGoogle Scholar
  22. 22.
    W. Fulton, J. Harris, Representation Theory. A First Course. Graduate Texts in Mathematics, Readings in Mathematics, vol. 129 (Springer, New York, 1991)Google Scholar
  23. 23.
    J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  24. 24.
    A.W. Knapp, E.M. Stein, Intertwining operators for semisimple groups. Ann. Math. 93(3), 489–578 (1971)MathSciNetCrossRefGoogle Scholar
  25. 25.
    B. Lawson, M.L. Michelson, Spin Geometry (Princeton University Press, Princeton, 1989)Google Scholar
  26. 26.
    J. Li, J. Ryan, Some operators associated to Rarita-Schwinger type operators. Complex Var. Elliptic Equ. Int. J. 57(7–8), 885–902 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    W. Miller, Symmetry and Separation of Variables (Addison-Wesley, Providence, 1977)Google Scholar
  28. 28.
    J. Peetre, T. Qian, Möbius covariance of iterated Dirac operators. J. Austral. Math. Soc. Ser. A 56, 403–414 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    I. Porteous, Clifford Algebra and the Classical Groups (Cambridge University Press, Cambridge, 1995)CrossRefGoogle Scholar
  30. 30.
    W. Rarita, J. Schwinger, On a theory of particles with half-integral spin. Phys. Rev. 60(1), 60–61 (1941)CrossRefGoogle Scholar
  31. 31.
    M. Roels, A Clifford analysis approach to higher spin fields. Master’s Thesis, University of Antwerp (2013)Google Scholar
  32. 32.
    J. Ryan, Conformally covariant operators in Clifford analysis. Z. Anal. Anwendungen 14, 677–704 (1995)MathSciNetCrossRefGoogle Scholar
  33. 33.
    J. Ryan, Iterated Dirac operators and conformal transformations in \(\mathbb {R}^m\), in Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics (World Scientific, Singapore, 1987), pp. 390–399Google Scholar
  34. 34.
    J.J. Sakurai, J. Napolitano, Modern Quantum Mechanics, 2nd edn. (Addison-Wesley, San Francisco, 2011)zbMATHGoogle Scholar
  35. 35.
    J. Slovák, Natural Operators on Conformal Manifolds. Habilitation thesis, Masaryk University, Brno (1993)zbMATHGoogle Scholar
  36. 36.
    J. Slovák, V. Souček, Invariant operators of the first order on manifolds with a given parabolic structure. Sémin. Congr. 4, 251–276 (2000)MathSciNetzbMATHGoogle Scholar
  37. 37.
    V. Souček, Higher spins and conformal invariance in Clifford analysis, in Proc. Conf. Seiffen, Lecture in Seiffen (1996)Google Scholar
  38. 38.
    P. Van Lancker, F. Sommen, D. Constales, Models for irreducible representations of Spin(m). Adv. Appl. Clifford Algebras 11(1 supplement), 271–289 (2001)MathSciNetCrossRefGoogle Scholar
  39. 39.
    G. Velo, D. Zwanzinger, Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential. Phys. Rev. 186(5), 1337–1341 (1969)CrossRefGoogle Scholar
  40. 40.
    G. Velo, D. Zwanzinger, Noncausality and other defects of interaction Lagrangians for particles with spin one and higher. Phys. Rev. 188(5), 2218–2222 (1969)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Chao Ding
    • 1
  • Raymond Walter
    • 1
    • 2
  • John Ryan
    • 1
  1. 1.Department of Mathematical SciencesUniversity of ArkansasFayettevilleUSA
  2. 2.Department of PhysicsUniversity of ArkansasFayettevilleUSA

Personalised recommendations