Higher Order Fermionic and Bosonic Operators

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


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.


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

Mathematics Subject Classification (2010)

Primary 53A30; Secondary 20G05 30G35 



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.


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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

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