  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 $\mathbf{T}$
with a suitable real-valued function $T$ on the space of physical states. The
proper characterization of the function $T$ 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 $T$ is a function on the space of pure states which is not
associated to any self-adjoint operator. The link between $T$ 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.
