Free-motion time-of-arrival operator and probability distribution

We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. 122, 1649 (1961). We use Naimark’s dilation theorem to reproduce the generalized decomposition of unity (or positive-operator-valued measures) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the same form.