Characteristic time operators as quantum clocks

We consider the characteristic time operator $\mathsf{T}$ introduced in [E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian $\mathsf{H}$ with some growth condition, $\mathsf{T}$ satisfies the canonical relation $[\mathsf{T},\mathsf{H}]|\psi\rangle=i\hbar|\psi\rangle$ for $|\psi\rangle$ in a dense subspace of the Hilbert space. While $\mathsf{T}$ 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 $\mathscr{T}$. In the neighborhood of each time $t$ in $\mathscr{T}$, $\mathsf{T}$ is still canonically conjugate to $\mathsf{H}$ and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of $\mathscr{T}$, and is proportional to the Pauli matrix $\sigma_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $\mathscr{T}$ by measuring a compatible observable.