Position Operators and Center of Mass: New Perspectives

After reviewing the work of Pryce on Center-of-Mass (CoM) definitions in special relativity, and that of Jordan and Mukunda on position operators for relativistic particles with spin, we propose two new criteria for a CoM candidate: associativity, and compatibility with the Poisson bracket structure. We find that they are not satisfied by all of Pryce's definitions, and they also rule out Dixon's CoM generalization to the curved spacetime case. We also emphasize that the various components of the CoM position do not commute among themselves, in the general case, and thus provide a natural entry point to the arena of noncommutative spacetime, without the ad-hoc assumptions of the standard paradigm.