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

TL;DR

This paper introduces a second order Lorentz-tensor and tensor-spinor framework for describing arbitrary spin-j states, simplifying the Weinberg-Joos approach and ensuring causal propagation and unitarity.

Contribution

It develops a novel method to describe single spin-j states using Lorentz-tensors and tensor-spinors with second order Lagrangians, avoiding complex matrix calculus and higher derivatives.

Findings

Successfully describes spin-3/2 as a second order tensor-spinor.

Reproduces known electromagnetic multipole moments.

Ensures causality and unitarity in scattering processes.

Abstract

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.