Fischer Decomposition of Massless Fields for Spin 3/2 in Dimension 4

Abstract

As an analogue of the massless field equations in Euclidean space, we consider the so-called generalized Cauchy–Riemann equations introduced by E. Stein and G. Weiss. In the spin 1/2 case these equations reduce to the Dirac equation for spin 1/2 fields, which was thoroughly and intensively studied in Clifford analysis. For general spin it was recently shown that, in dimension 4, homogenous solutions form irreducible Spin modules. The next step then is to describe the corresponding Fischer decomposition, i.e. an irreducible decomposition of the space of spinor fields, which is well-known for spin 1/2 and for spin 1. The main aim of the present paper is to describe, still in dimension 4, the Fischer decomposition for spin 3/2.