"Square Root" of the Proca Equation: Spin-3/2 Field Equation

TL;DR

This paper derives new relativistic equations for spin-3/2 particles, establishing a 'square root' relation to the Proca equation, and provides detailed solutions, projection operators, and interaction properties.

Contribution

It introduces a novel first-order 20-dimensional wave equation for spin-3/2 particles, including multi-spin states, projection operators, and a consistent Lagrangian formulation.

Findings

Derived a non-local 'square root' of the Proca equation for spin-3/2.

Constructed projection operators for energy, spin, and spin projections.

Confirmed causal wave propagation in the proposed framework.

Abstract

New equations describing particles with spin 3/2 are derived. The non-local equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein-Gordon-Fock equation. The local equation describes spin 3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multi-spin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing non-minimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.