Symmetries and time operators

All covariant time operators with normalized probability distribution are derived for Hamiltonians with continuous eigenvalues. Symmetry criteria are invoked to arrive at a unique expression for a given Hamiltonian. As an application, a well-known result for the arrival time distribution of a free particle is generalized and extended. Interestingly, the resulting arrival time distribution operator is connected to a particular, positive, quantization of the classical current. For particles in a potential, we also introduce and study the notion of conditional arrival-time distribution.