A geometrical description of the semi-classical limit

We present a general geometrical formulation for the correspondance between quantum dynamics and classical dynamics. In the projective space of quantum states, the classical dynamics is seen as an orthogonal projection of the quantum dynamics on a given family of states, such as the coherent states. This formulation is equivalent to the time-dependant-variational-principle, and thus gives a geometrical interpretation of it. Moreover the angle of this projection informs us on the validity of the approximation. These results are illustrated on a numerical example. P.A.C.S. number : 03.65.Sq 03.65.Ca