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The higher spin $Π$-operator in Clifford analysis

Wanqing Cheng, Chao Ding

Abstract

Rarita-Schwinger fields are solutions to the relativistic field equation of spin-$3/2$ fermions in four dimensional flat spacetime, which are important in supergravity and superstring theories. Bure\v s et al. generalized it to arbitrary spin $k/2$ in 2002 in the context of Clifford algebras. In this article, we introduce the higher spin $Π$-operator related to the Rarita-Schwinger operator. Further, we investigate norm estimates, mapping properties and the adjoint operator of the higher spin $Π$-operator. As an application, a higher spin Beltrami equation is introduced, and existence and uniqueness of solutions to this higher spin Beltrami equation is established by the norm estimate of the higher spin $Π$-operator.

The higher spin $Π$-operator in Clifford analysis

Abstract

Rarita-Schwinger fields are solutions to the relativistic field equation of spin- fermions in four dimensional flat spacetime, which are important in supergravity and superstring theories. Bure\v s et al. generalized it to arbitrary spin in 2002 in the context of Clifford algebras. In this article, we introduce the higher spin -operator related to the Rarita-Schwinger operator. Further, we investigate norm estimates, mapping properties and the adjoint operator of the higher spin -operator. As an application, a higher spin Beltrami equation is introduced, and existence and uniqueness of solutions to this higher spin Beltrami equation is established by the norm estimate of the higher spin -operator.

Paper Structure

This paper contains 8 sections, 18 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Clifford algebras
  4. Rarita-Schwinger operators
  5. Function spaces in higher spin Clifford analysis
  6. Integral formulas for Rarita-Schwinger operators
  7. Definitions and properties for the higher spin $\Pi$-operator
  8. Existence of solutions to the higher spin Beltrami equation

Key Result

Proposition 2.2

The space of $\mathcal{C}l_{m-1}$-valued $k-$homog-eneous harmonic polynomials $\mathcal{H}_k(\boldsymbol{u})$ can be decomposed as

Theorems & Definitions (37)

  • Definition 2.1
  • Proposition 2.2: Fischer decomposition
  • proof
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Theorem 3.2: Stokes' Theorem for the Dirac operator
  • Theorem 3.3: Rarita-Schwinger Stokes' Theorem
  • Theorem 3.4: Borel-Pompeiu formula
  • Proposition 3.5
  • ...and 27 more