How to generate spinor representations in any dimension in terms of projection operators

Abstract

We present a method to find solutions of the Weyl or the Dirac equation without specifying a representation choice for $\gamma^a$'s. By taking $\gamma^a$'s formally as independent variables, we construct solutions out of $2^d$ orthonormal polynomials of $\gamma^a$'s (in $d$-dimensional space) operating on a ``vacuum state''. Polynomials reduce into $2^{d/2}$ or $2^{(d+1)/2}$ repetitions of the Dirac spinors for $d$ even or odd, respectively. We further propose the corresponding graphic presentation of basic states, which offers an easy way to see all the quantum numbers of states with respect to the generators of the Lorentz group, as well as transformation properties of the states under any operator.