Characteristic time operators as quantum clocks

We consider the characteristic time operator $T$ introduced in Galapon (2002) [26] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian $H$ with some growth condition, $T$ satisfies the canonical relation $[T,H]|\psi 〉=iħ|\psi 〉$ for $|\psi 〉$ in a dense subspace of the Hilbert space. While $T$ 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 $T$. In the neighborhood of each time t in $T$, $T$ is still canonically conjugate to $H$ and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of $T$, and is proportional to the Pauli matrix ${\sigma }_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $T$ by measuring a compatible observable.