  The time operator canonically conjugated to the Hamiltonian of $N$
interacting particles on the line is constructed using SU(1,1) as a dynamical
symmetry. This hidden conformal symmetry enables us to make a group theoretic
analysis of the time operator in terms of SU(1,1) generators. At distances very
far from the interacting region the time operator is represented as a
generalization of the quantum "time-of-arrival" operator.
