Fischer Decomposition for Massless Fields of Spin 1 in Dimension 4

The massless field equations for lower integer and half-integer values of spin in Minkowski space are fundamental equations in mathematical physics. Their counterpart in Euclidean spacetime is a system of elliptic equations, which was already studied from the viewpoint of function theory in the framework of so-called Hodge systems for differential forms of various degrees. In dimension 4 it is possible to substitute spinor calculus for the usual tensor notation. In the present paper we concentrate on the case of the massless field equation for spin 1 in dimension 4, and we treat, in a spinor formalism, a fundamental concept of its function theory: the Fischer decomposition of polynomial spinor fields, for which we give simple and independent proofs.