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In this contribution, we suggest the approach that geometric concepts oughtto be defined in terms of physical operations involving quantum matter. In thisway it is expected that some (presumably nocive) idealizations lying deepwithin the roots of the notion of spacetime might be excluded. In particular,we consider that spacetime can be probed only with physical (and thereforeextended) particles, which can be effectively described by coordinates thatfail to commute by a term proportional to the spin of the particles.

Center of mass in special and general relativity and its role in an effective description of spacetime