Narrow Results

Some years ago it was shown that, in some cases, a notion of locality can arise from the symmetry group of the theory [1–3], i.e. in an intrinsic way. In particular, when the Möbius covariance is present, it is possible to associate some particular transformations with the Tomita–Takesaki modular operator and conjugation of a specific interval of an abstract circle. In this context we propose a way to define an operator representing the coordinate conjugated to the modular transformations. Remarkably this coordinate turns out to be compatible with the abstract notion of locality. Finally a concrete example concerning a quantum particle on a line is given.

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 space–time 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 space–time, without the ad-hoc assumptions of the standard paradigm.