We propose a general method for the description of arbitrary single spin-j
states transforming according to (j,0)+(0,j) carrier spaces of the Lorentz
algebra in terms of Lorentz-tensors for bosons, and tensor-spinors for
fermions, and by means of second order Lagrangians. The method allows to avoid
the cumbersome matrix calculus and higher \partial^{2j} order wave equations
inherent to the Weinberg-Joos approach. We start with reducible Lorentz-tensor
(tensor-spinor) representation spaces hosting one sole (j,0)+(0,j) irreducible
sector and design there a representation reduction algorithm based on one of
the Casimir invariants of the Lorentz algebra. This algorithm allows us to
separate neatly the pure spin-j sector of interest from the rest, while
preserving the separate Lorentz- and Dirac indexes. However, the Lorentz
invariants are momentum independent and do not provide wave equations. Genuine
wave equations are obtained by conditioning the Lorentz-tensors under
consideration to satisfy the Klein-Gordon equation. In so doing, one always
ends up with wave equations and associated Lagrangians that are second order in
the momenta. Specifically, a spin-3/2 particle transforming as (3/2,0)+ (0,3/2)
is comfortably described by a second order Lagrangian in the basis of the
totally antisymmetric Lorentz tensor-spinor of second rank, \Psi_[ \mu\nu].
Moreover, the particle is shown to propagate causally within an electromagnetic
background. In our study of (3/2,0)+(0,3/2) as part of \Psi_[\mu\nu] we
reproduce the electromagnetic multipole moments known from the Weinberg-Joos
theory. We also find a Compton differential cross section that satisfies
unitarity in forward direction. The suggested tensor calculus presents itself
very computer friendly with respect to the symbolic software FeynCalc.Comment: LaTex 34 pages, 1 table, 8 figures. arXiv admin note: text overlap
with arXiv:1312.581
