Is there an observable of time?

Question

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 1

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.

Answer 2

In QM, the temporal variable $t$ is not an observable in the technical sense (i.e., in the same sense that position and momentum are). In order to be an observable, it should have to exist a linear self-adjoint operator $\hat T$ whose eigenvalues $t$ were the outcomes of measurements. But then (at least in the most naive way and according with the Schr. equation) the Hamiltonian and the temporal operator should be non-compatible observables with canonical commutation relations like position and momentum. And this is not possible because in a quantum theory the Hamiltonian must be bounded from bellow and this would imply that its conjugate (time operator) were no self-adjoint.

However, there exist mean lifetimes which are quantum-mechanically computable (they are inverses of probabilities per unit of time) and have units of time. In some sense (arguably vague sense), this is a quantum notion of time.