quantum mechanics - Is there an observable of time?

In Quantum Mechanics, position is an observable, but time may be not. I think that time is simply a classical parameter associated with the act of measurement, but is there an observable of time? And if the observable will exist, what is an operator of time?

Answer

The problem of extending Hamiltonian mechanics to include a time operator, and to interpret a time-energy uncertainty relation, first posited (without clear formal discussion) in the early days of quantum mechanics, has a large associated literature; the survey article

P. Busch. The time-energy uncertainty relation, in Time in quantum mechanics (J. Muga et al., eds.), Lecture Notes in Physics vol. 734. Springer, Berlin, 2007. pp 73-105. doi:10.1007/978-3-540-73473-4_3, arXiv:quant-ph/0105049.

carefully reviews the literature up to the year 2000. (The book in which Busch's survey appears discusses related topics.) There is no natural operator solution in a Hilbert space setting, as Pauli showed in 1958,

W. Pauli. Die allgemeinen Prinzipien der Wellenmechanik, in Handbuch der Physik, Vol V/1, p. 60. Springer, Berlin, 1958. Engl. translation: The general principles of quantum mechanics, p. 63. Springer, Berlin 1980.

by a simple argument that a self-adjoint time operator densely defined in a Hilbert space cannot satisfy a CCR with the Hamiltonian, as the CCR would imply that $H$ has as spectrum the whole real line, which is unphysical.

Time measurements do not need a time operator, but are captured well by a positive operator-valued measure (POVM) for the time observable modeling properties of the measuring clock.