Relativistic Covariance of Scattering

TL;DR

This paper investigates how relativistic quantum scattering can be consistent with Poincaré covariance, showing that relaxing certain invariance requirements allows for covariant interactions while preserving Lorentz invariance of the S-matrix.

Contribution

It demonstrates that Poincaré covariant interactions are possible if the interacting Hamiltonian commutes with the four-velocity instead of the four-momentum, clarifying conditions for relativistic invariance.

Findings

Relativistic scattering can be covariant with modified Hamiltonian commutation relations.

Lorentz invariance of the S-matrix does not require the interacting Hamiltonian to commute with four-momentum.

Observers moving with different four-velocities perceive shifted superpositions of states without changing relative phases.

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.