Supersymmetry generators of arbitrary spin

Abstract

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 $[{S}_{i},{S}_{j}]=c{({\ensuremath{\alpha}}^{\ensuremath{\mu}}C)}_{\mathrm{ij}}{P}_{\ensuremath{\mu}}$, where a graded commutator is used. Supposing that ${S}_{i}$ belongs to some representation of the Lorentz group we study the conditions on ${\ensuremath{\alpha}}^{\ensuremath{\mu}}$ and $C$ which result from Jacobi identities and Hermitian conjugation. For the three-dimensional case the conditions are satisfied if ${({\ensuremath{\alpha}}^{k}C)}_{\mathrm{ij}}$ is chosen to be a Clebsch-Gordan coefficient. This allows $S$ to have any spin \ensuremath{\ne} 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 ${L}_{k}=i\overline{\ensuremath{\psi}}{\ensuremath{\alpha}}^{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\psi}$ and ${L}_{m}=\overline{\ensuremath{\psi}}\ensuremath{\psi}$, 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.