Covariant field with unique mass and spin 3/2

Abstract

We present the explicit theory of eight-dimen\-sional massive covariant fields with single spin $\frac{3}{2}$ transforming according to the representation $(\frac{3}{2},0)\oplus(0, \frac{3}{2})$ of the group $SL(2,\mathbb{C})$. This is done starting with the reducible representation $(1,0)\otimes(\frac{1}{2},0)$ instead of the irreducible one $(1,\frac{1}{2})=(1,0)\otimes(0,\frac{1}{2})$ we meet in Rarita-Schwinger or Joss-Weinberg frameworks. The resulting $12$-component covariant field transforming according to the representation $[(1,0)\otimes(\frac{1}{2},0)]\oplus [(0,1)\otimes(0, \frac{1}{2})]$ is maximally reducible, up to subspaces of irreducible representations of the $SU(2)$ group. Consequently, after building the theory in direct product basis of the representation $(1,0)\otimes(\frac{1}{2},0)$, the sector of spin half can be separated revealing thus the genuine $(\frac{3}{2},0)\oplus(0, \frac{3}{2})$ field. In this manner the theory of massive field of single spin $\frac{3}{2}$ can be developed naturally from the field equation and associated matrices, Lagrangian formalism and inner product up to closed expressions of orthonormal mode spinors.