Abstract
It is argued that the affine structure of spacetime can be operationally defined only in classical mechanics. The behaviour of free quantum particles, or subject only to gravity, depends on their mass and thus cannot be ascribed to some universal geometrical property. Consequently, either quantum theory is not fundamental, or spacetime has no fundamental affine structure. The latter conclusion would not challenge the formal apparatus of general relativity, but would constitute a serious obstruction to its geometrical interpretation.
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