Higher Order Fermionic and Bosonic Operators

TL;DR

This paper introduces a class of higher order conformally invariant differential and integral operators acting on functions valued in specific Spin group representations, generalizing higher spin powers of the Dirac operator.

Contribution

It constructs and classifies higher order conformally invariant operators for higher spin spaces, unifying bosonic and fermionic cases within a Clifford analysis framework.

Findings

Operators act on functions in homogeneous harmonic or monogenic polynomial spaces.

Fundamental solutions exhibit conformal invariance.

Classification aligns with quantum angular momentum quantization.

Abstract

This paper studies a particular class of higher order conformally invariant dif- ferential 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 represen- tations 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 de- pending 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. Fundamental solutions and their conformal invariance are presented here.