Relativistic hamiltonian equations for any spin

We discuss 2(2 J + 1)-component Poincaré-invariant Hamiltonian theories that describe free particles of definite mass and spin and that are subject to the conditions (a) every observable O is either Hermitian or pseudo-Hermitian (i.e., O = ϱ₃O⁺ϱ₃) and (b) the theory is invariant under the discrete symmetries. Our treatment is based on the Heisenberg equations of motion and on the Lie algebra of the Poincaré group. Explicit formulas are found for the generators of this algebra, including the Hamiltonian H, and all relations between the operators Γ and H that are both necessary and sufficient for K = 1/2[x , H] ₊ + Γ to generate Lorentz boosts are found. To illustrate the utility of our results, we apply them to obtain explicit generalizations of the Dirac equation to any spin, by requiring that Γ = 0, and of the Sakata-Taketani spin-0 and spin-1 equations to any spin, by requiring that Γ = -ϱ ₃( 1/2m) S × p.