Abstract
We consider the characteristic time operator introduced in Galapon (2002) [26] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian with some growth condition, satisfies the canonical relation for in a dense subspace of the Hilbert space. While is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set. In the neighborhood of each time t in, is still canonically conjugate to and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of, and is proportional to the Pauli matrix. Thus, one can construct a quantum clock that tells the time in the neighborhood of by measuring a compatible observable.
https://www.sciencedirect.com/science/article/pii/S0375960124008867?dgcid=coauthor