Apart from serving as a parameter in describing the evolution of a system, time appears also as an observable property of a system in experiments where one measures ‘the time of occurrence’ of an event associated with the system. However, while the observables normally encountered in quantum theory (and characterized by self-adjoint operators or projection-valued measures) correspond to instantaneous measurements, a time of occurrence measurement involves continuous observations being performed on the system to monitor when the event occurs. It is argued that a time of occurrence observable should be represented by a positive-operator-valued measure on the interval over which the experiment is carried out. It is shown that while the requirement of time-translation invariance and the spectral condition rule out the possibility of a self-adjoint time operator (Pauli’s theorem), they do allow for time of occurrence observables to be represented by suitable positive-operator-valued measures. It is also shown that the uncertainty in the time of occurrence of an event satisfies the time-energy uncertainty relation as a consequence of the time-translation invariance, only if the time of occurrence experiment is performed on the entire time axis.
KeywordsTime operator spectral condition Pauli’s theorem time of occurrence of an event continuous measurements positive-operator-valued measures timeenergy uncertainty relation time of arrival of photons
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