Bhabha first-order wave equations. III. Poincaré generators

Abstract

We construct the Poincaré generators for arbitrary-spin Bhabha fields. After showing that for high-spin fields Hermiticity does not mean that the operators are self-adjoint, but rather satisfy Eqs. (2.16) and (2.17) below, we observe that these generators are Hermitian in this generalized sense. We explicitly demonstrate that the generators algebraically satisfy the associated Lie algebra for arbitrary half-integer-spin representations, but only as an operator algebra on the fields themselves for integer-spin representations. Specifically, of the six independent commutation relations $[K_i,K_j]=-i{\epsilon }_{\mathrm{ijk}}{J}_k$ is not satisfied algebraically, and of the three dependent commutation relations $[K_i,H]=i{P}_i$ is not satisfied algebraically. By looking at the Sakata-Taketani decomposition of the Duffin-Kemmer-Petiau case, we find that it is only the built-in subsidiary components, not the particle components, which need an operator equation on the fields to satisfy the above two commutation relations. We generalize this result and show that the particle components for arbitrary-integer-spin fields satisfy the commutation relations algebraically. Finally, we comment on the interacting field case and problems associated with high-spin interacting field theories.