Observable Causality Implies Lorentz Group: Alexandrov-zeeman-type Theorem for Space-time Regions

The famous Alexandrov-Zeeman theorem proves that causality implies Lorentz group. The physical meaning of this result is that once we observe which event can causally a ect which other events, then, using only this information, we can reconstruct the linear structure of the Minkowski space-time. The original Alexandrov-Zeeman theorem is based on the causality relation between events represented by points in space-time. Knowing such a point means that we know the exact moment of time and the exact location of the corresponding event and that this event actually occurred at a single moment of time and at a single spatial location. In practice, events take some time and occupy some spatial area. Besides, even if we have a point-wise event, we would not be able to know the exact moment of time and exact spatial location since the only way to determine the moment of time and the spatial location is by measurement, and measurements are never absolutely accurate. To come up with a more realistic description of observable causality relation between events, we need to consider events which are not pointwise, but rather represented by bounded regions A in the Minkowski space-time. When we have two events represented by regions A and B, the fact that we have observed that the rst event can causally in uence the second one means that a ≤ b for some points a ∈ A and b ∈ B. In this paper, we show that even if we only know the causal relation between such regions, we can still reconstruct the linear structure on the Minkowski space-time. Thus, already observable causality implies Lorentz group.