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We propose a general method for the description of arbitrary single spin-jstates transforming according to (j,0)+(0,j) carrier spaces of the Lorentzalgebra in terms of Lorentz-tensors for bosons, and tensor-spinors forfermions, and by means of second order Lagrangians. The method allows to avoidthe cumbersome matrix calculus and higher \partial^{2j} order wave equationsinherent to the Weinberg-Joos approach. We start with reducible Lorentz-tensor(tensor-spinor) representation spaces hosting one sole (j,0)+(0,j) irreduciblesector and design there a representation reduction algorithm based on one ofthe Casimir invariants of the Lorentz algebra. This algorithm allows us toseparate neatly the pure spin-j sector of interest from the rest, whilepreserving the separate Lorentz- and Dirac indexes. However, the Lorentzinvariants are momentum independent and do not provide wave equations. Genuinewave equations are obtained by conditioning the Lorentz-tensors underconsideration to satisfy the Klein-Gordon equation. In so doing, one alwaysends up with wave equations and associated Lagrangians that are second order inthe 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 thetotally antisymmetric Lorentz tensor-spinor of second rank, \Psi_[ \mu\nu].Moreover, the particle is shown to propagate causally within an electromagneticbackground. In our study of (3/2,0)+(0,3/2) as part of \Psi_[\mu\nu] wereproduce the electromagnetic multipole moments known from the Weinberg-Joostheory. We also find a Compton differential cross section that satisfiesunitarity in forward direction. The suggested tensor calculus presents itselfvery computer friendly with respect to the symbolic software FeynCalc.

Bosonic and fermionic Weinberg-Joos (j,0)+ (0,j) states of arbitrary spins as Lorentz-tensors or tensor-spinors and second order theory