Linear relativistic hamiltonians

A 2(2J + 1)-component relativistic Hamiltonian H that describes free particles of mass m and spin J is said to be linear if it has the form $\text{H = h}\text{x}\text{?}\text{·}\text{p}\text{+ gm}$, where $\text{x}\text{\_.}\text{= i[H,}\text{x}\text{]}{}_{\mathrm{?}}$, h is a numerical factor, and g commutes with x and p. All such Hamiltonians are found, provided that the metric is either the unit matrix or ?₃ and provided that the theory is invariant under the discrete symmetries. If the operator Γ in the generator $\text{K}\text{=}\text{1}\text{2}\text{[}\text{x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ of Lorentz boosts is required to be local, there are only two possibilities; either Γ = 0, which generalizes the Dirac spin-$\text{1}\text{2}$ theory, or $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$, which generalizes the Sakata-Taketani spin-0 and spin-1 theories. The relationship to linear manifestly covariant equations and its significance is discussed.