First-order differential equations for a particle with spin $S = 1$

TL;DR

This paper develops a new set of first-order differential equations for a spin-1 particle with nonzero mass, featuring a Hamiltonian linear in momentum and a positive-definite probability density, similar to Dirac and Maxwell equations.

Contribution

The paper introduces a novel first-order differential equation system for spin-1 particles that differs from the Proca-Duffin-Kemmer equations, with a Hamiltonian linear in momentum and positive-definite probability density.

Findings

System has a Hamiltonian linear in momentum.

Wave function yields positive-definite probability density.

Shares features with Dirac and Maxwell equations.

Abstract

A system of first-order differential equations for a particle with nonzero mass and spin $S = 1$ is constructed. As distinct from the Proca-Duffin-Kemmer (PDK) equations, the system has the form of the dynamical equation $i\hbar\partial_t\hat{\Psi}=\hat{\mathrm{H}}\hat{\Psi}$ (with constraints) with a Hamiltonian linear in momentum. The six-component wave function yields the positive-definite probability density $\hat{\Psi}^{\dagger}\hat{\Psi}\geq 0$. The system of equations has much in common with the Dirac and Maxwell equations.