Characteristic time operators as quantum clocks

Abstract

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 $[\mathsf{T},\mathsf{H}]|\ddot{I}ˆ⟩=i\hslash |\ddot{I}ˆ⟩$ for $|\ddot{I}ˆ⟩$ 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 $\mathcal{T}$. In the neighborhood of each time $t$ in $\mathcal{T}$, $\mathsf{T}$ 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 $\mathcal{T}$, and is proportional to the Pauli matrix $\ddot{I}{ƒ}_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $\mathcal{T}$ by measuring a compatible observable.

Abstract revised, references updated