Relativistic hamiltonian equations for any spin
Abstract
We discuss 2(2 J + 1)component Poincaréinvariant 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 pseudoHermitian (i.e., O = ϱ_{3}O^{+}ϱ_{3}) 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 SakataTaketani spin0 and spin1 equations to any spin, by requiring that Γ = ϱ _{3}( 1/2m) S × p.
 Publication:

Annals of Physics
 Pub Date:
 December 1974
 DOI:
 10.1016/00034916(74)901808
 Bibcode:
 1974AnPhy..88..504G