Time, classical and quantum

We propose a new point of view regarding the problem of time in quantum mechanics, based on the idea of replacing the usual time operator T with a suitable real-valued function TT on the space of physical states. The proper characterization of the function TT relies on a particular relation with the dynamical evolution of the system rather than with the infinitesimal generator of the dynamics (Hamiltonian). We first consider the case of classical hamiltonian mechanics, where observables are functions on phase space and the tools of differential geometry can be applied. The idea is then extended to the case of the unitary evolution of pure states of finite-level quantum systems by means of the geometric formulation of quantum mechanics. It is found that TT is a function on the space of pure states which is not associated with any self-adjoint operator. The link between TT and the dynamical evolution is interpreted as defining a simultaneity relation for the states of the system with respect to the dynamical evolution itself. It turns out that different dynamical evolutions lead to different notions of simultaneity, i.e., the notion of simultaneity is a dynamical notion.