Time operator for a quantum many-body system with SU(1,1) dynamical symmetry

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.