Time Interval Operators

In this work we will advance farther along a line previously developed concerning our proposal of a time interval operator, on finite dimensional spaces. The time interval operator is Hermitian, and its eigenvalues are time values with a precise and interesting role on the dynamics. With the help of the Discrete Phase Space Formalism (DPSF) previously developed, we show that the time interval operator is the complementary pair of the Hamiltonian. From that, a simple system is proposed as a quantum clock. The only restriction is that our results do not apply to all possible Hamiltonians.