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 tensorspinors for fermions, and by means of second order Lagrangians. The method allows to avoid the cumbersome matrix calculus and higher ∂ 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 Lorentztensors 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, Ψ [µν] . 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 Ψ [µν] 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 1 itself very computer friendly with respect to the symbolic software FeynCalc.

The gauged Klein–Gordon equation, extended by a gsσμνFμν/4 interaction, the contraction of the electromagnetic field strength tensor, Fμν, with the generators, σμν/2, of the Lorentz group in (1/2, 0) ⊕ (0, 1/2), and gs being the gyromagnetic factor, is examined with the aim to find out as to what extent it qualifies as a wave equation for general relativistic spin-1/2 particles transforming as (1/2, 0) ⊕ (0, 1/2) and possibly distinct from the Dirac fermions. This equation can be viewed as the generalization of the gs = 2 case, known under the name of the Feynman–Gell-Mann equation, the only one which allows for a bilinearization into the gauged Dirac equation and its conjugate. At the same time, it is well-known a fact that a gs = 2 value can also be obtained upon the bilinearization of the nonrelativistic Schrödinger into nonrelativistic Pauli equations. The inevitable conclusion is that it must not be necessarily relativity which fixes the gyromagnetic factor of the electron to g(1/2) = 2, but rather the specific form of the primordial quadratic wave equation obeyed by it, that is amenable to a linearization. The fact is that space-time symmetries alone define solely the kinematic properties of the particles and neither fix the values of their interacting constants, nor do they necessarily prescribe linear Lagrangians. Information on such properties has to be obtained from additional physical inputs involving the dynamics. We here provide an example in support of the latter statement. Our case is that the spin-1/2- fermion residing within the four-vector spinor triad, ψμ ~ (1/2+-1/2--3/2-), whose sectors at the free particle level are interconnected by spin-up and spin-down ladder operators, does not allow for a description within a linear framework at the interacting level. Upon gauging, despite transforming according to the irreducible (1/2, 1) ⊕ (1, 1/2) building block of ψμ, and being described by 16-dimensional four-vector spinors, though of only four independent components each, its Compton scattering cross sections, both differential and total, result equivalent to those for a spin-1/2 particle described by the generalized Feynman–Gell-Mann equation from above (for which we provide an independent algebraic motivation) and with g(1/2-) = -2/3. In effect, the spin-1/2- particle residing within the four-vector spinor effectively behaves as a true relativistic "quadratic" fermion. The g(1/2-) = -2/3 value ensures in addition the desired unitarity in the ultraviolet. In contrast, the spin-1/2+ particle, in transforming irreducibly in the (1/2, 0) ⊕ (0, 1/2) sector of ψμ, is shown to behave as a truly linear Dirac fermion. Within the framework employed, the three spin sectors of ψμ are described on equal footing by representation- and spin-specific wave equations and associated Lagrangians which are of second-order in the momenta.