The present paper deals with the quantum coordinates of an event in space-time, individuated by a quantum object. It is known that these observables cannot be described by self-adjoint operators or by the corresponding spectral projection-valued measure. We describe them by means of a positive-operator-valued (POV) measure in the Minkowski space-time, satisfying a suitable covariance condition with respect to the Poincaré group. This POV measure determines the probability that a measurement of the coordinates of the event gives results belonging to a given set in space-time. We show that this measure must vanish on the vacuum and the one-particle states, which cannot define any event. We give a general expression for the Poincaré covariant POV measures. We define the baricentric events, which lie on the world-line of the centre-of-mass, and we find a simple expression for the average values of their coordinates. Finally, we discuss the conditions which permit the determination of the coordinates with an arbitrary accuracy. PACS: 03.65.Bz -quantum theory; 02.20.+b -group theory.

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Explicit saddle surfaces and return jump rates are calculated for model crystals; the results establish the validity of rate calculations for atomic jumps in solids. A nonphysical effect, in which isotopic substitution causes a "return jump catastrophe," is resolved by use of an exact momentum-dependent criterion to distinguish successful jumps.

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