Supersymmetry generators of arbitrary spin

The infinitesimal generators of supersymmetry and translation form a solvable invariant subalgebra of the full graded Lie algebra. In O'Raifeartaigh's classification scheme this belongs to Case (iii). In general we may define the degree-$n$ supersymmetry generators by requiring their nth derived algebra to be equal to translations. In this paper we study degree-1 supersymmetries for which $[S_i,S_j]=c{\left({\alpha }^{\mu }C\right)}_{\mathrm{ij}}{P}_{\mu }$, where a graded commutator is used. Supposing that $S_i$ belongs to some representation of the Lorentz group we study the conditions on ${\alpha }^{\mu }$ and $C$ which result from Jacobi identities and Hermitian conjugation. For the three-dimensional case the conditions are satisfied if ${\left({\alpha }^{k}C\right)}_{\mathrm{ij}}$ is chosen to be a Clebsch-Gordan coefficient. This allows $S$ to have any spin ≠ 0 and also gives the correct spin-statistics connection (grading). In the four-dimensional case we show how the problem is related to that of finding Lagrangian densities $L_k=i\overline{\psi }{\alpha }^{\mu }{\partial }_{\mu }\psi$ and $L_m=\overline{\psi }\psi$, which are Hermitian scalars. There are an infinite number of possible representations to which $S_i$ can belong, including those of Bhabha type, for which the spin-statistics connection comes naturally from the representation. At the same time there can be supersymmetry generators of several different spins. The Volkov-Akulov nonlinear realization works in all cases and a supersymmetry-invariant Lagrangian can be constructed. Anticommutators seem to be important only in the sense that then we can have finite-dimensional linear realizations.