Second order theory of $(j,0)\oplus (0,j)$ single high spins as Lorentz tensors

TL;DR

This paper presents a covariant, second-order tensor approach to describing pure high-spin states, avoiding complex differential equations and matrix spinor calculus, and successfully reproducing known electromagnetic properties and scattering behaviors.

Contribution

It introduces a novel tensor-based method for high-spin representations that simplifies calculations and extends to any $(j,0)igoplus(0,j)$ state, avoiding matrix couplings.

Findings

Correctly reproduces electromagnetic multipole moments for spin-3/2.

Shows unitarity in Compton scattering at $g=2/3$ for spin-3/2.

Method extends straightforwardly to arbitrary high spins.

Abstract

We show that higher order differential equations and matrix spinor calculus are completely avoidable in the description of pure high spin-$j$ Weinberg-Joos states, $(j,0)\oplus (0,j)$. The case is made on the example of $(3/2,0)\oplus(0,3/2)$, for the sake of concreteness and without loss of generality. Namely, we use as a vehicle for the aforementioned covariant single spin-$3/2$ description the antisymmetric tensor of second rank with Dirac spinor components, $\Psi_{[\mu\nu]}=B_{[\mu\nu]}\otimes\psi$. The $(3/2,0)\oplus(0,3/2)$ sector of interest is tracked down in two steps. First we search for spin-$3/2$ by means of a covariant spin projector constructed from the Casimir invariants of the Poincar\'e algebra, and then we identify the wanted irreducible representation space by means of a momentum independent (static) projector designed on the basis of the Casimir invariants of the Lorentz algebra. The latter projectors unambiguously identify any irreducible $so(1,3)$ subspace of any Lorentz tensor and without rising the order of the differential equation. The method proposed correctly reproduces the electromagnetic multipole moments earlier calculated for single spin-$3/2$ particles in treating it in the standard way as eight dimensional spinor. We furthermore calculate Compton scattering off the pure spin-$3/2$ under discussion, and show that the differential cross section satisfies unitarity in forward direction for a gyromagnetic ratio of $g=2/3$. Suggesting possible validity of Belinfante's conjecture for pure spin-states, while the natural value of $g=2$ seems more likely to characterize the highest spins in the Rarita-Schwinger representation spaces. The scheme straightforwardly extends to any $(j,0)\oplus (0,j)$ Weinberg-Joos state and brings the advantage of avoiding rectangular matrix couplings between states of different spins, replacing them by simple Lorentz contractions.