  There exist instances of dynamical systems possessing symmetry
transformations of which the conserved Noether charges generating these
symmetries feature an explicit time dependence in their functional
representation over phase space. The generators of such symmetries certainly do
not commute with the Hamiltonian, and yet these charges are conserved
observables for the classical and quantised dynamics. Furthermore within the
Hamiltonian formalism and in the case of global symmetries such charges may be
gauged to allow for arbitrary time dependent symmetry transformations, simply
by extending the Hamiltonian to include the Noether charges as first-class
constraints. An explicit illustration of these issues is presented in a simple
and most familiar model that applies also to the constant gravitational force.
This note draws its primary motivation from the quest towards a theory for
quantum gravity, in wanting to understand better the tension existing between
the local Equivalence Principle of the gravitational interaction and the
fundamental principles of Quantum Mechanics by considering the formulation of
quantum systems relative to reference frames that are inertial or noninertial,
and thus accelerated relative to one another through arbitrary time dependent
spatial translations.
