Relativistic Covariance of Scattering

Abstract

We analyze relativistic quantum scattering in the Schr\"odinger picture. The suggestive requirement of translational invariance and conservation of the four-momentum, that the interacting Hamiltonian commute with the four-momentum $P$ of free particles, is shown to imply the absence of interactions. The relaxed requirement, that the interacting Hamiltonian $H'$ commute with the four-velocity $U= P/M$, $M=\sqrt{P^2}$, allows Poincar\'e covariant interactions just as in the nonrelativistic case. If the $S$-matrix is Lorentz invariant, it still commutes with the four-momentum $P$ though $H'$ does not. Shifted observers, whose translations are generated by the four-velocity $U$, just see a shifted superposition of near-mass-degenerate states with unchanged relative phases, while the four-momentum generates oscillated superpositions with changed relative phases.