51. Clifford Analysis for Higher Spin Operators

This chapter focuses on the use of Clifford analysis techniques as an encompassing and unifying tool to study higher spin generalizations of the classical Dirac operator. These operators belong to a complete family of conformally invariant first-order differential operators, acting on functions taking their values in an irreducible representation for the spin group (the double cover for the orthogonal group). Their existence follows from a standard classification result due to Fegan (Q. J. Math. 27:513–538, 1976), and a canonical way to construct them is to use the technique of Stein–Weiss gradients. This then gives rise to two kinds of differential operators defined on irreducible tensor fields, the standard language used in, e.g., theoretical physics, where higher spin operators appear in the equations of motion for elementary particles having arbitrary half-integer spin: on the one hand, there are the (elliptic) generalizations of the Dirac operator, acting as endomorphisms on the space of smooth functions with values in a fixed module (i.e., preserving the values), and on the other hand there are the invariant operators acting between functions taking values in different modules for the spin group (the so-called twistor operators and their duals). In this chapter, both types of higher spin operators will be defined on spinor-valued functions of a matrix variable (i.e., in several vector variables): this has the advantage that the resulting equations become more transparent, and it allows using techniques for Clifford analysis in several variables. In particular, it provides an elegant framework to develop a function theory for the aforementioned operators, such as a full description of the (polynomial) null solutions and analogues of the classical Cauchy integral formula.