Some Properties of the Higher Spin Laplace Operator

TL;DR

This paper analyzes the higher spin Laplace operator, decomposing it into conformally invariant components, and establishes fundamental solutions and integral formulas relevant to higher spin theory.

Contribution

It provides a decomposition of the higher spin Laplace operator into Rarita-Schwinger operators and proves their conformal invariance, linking fundamental solutions and integral formulas.

Findings

Decomposition of the higher spin Laplace operator into Rarita-Schwinger operators

Proof of conformal invariance of the components

Derivation of a Green type integral formula

Abstract

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformally invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.