  We consider the space of probabilities {P(x)}, where the x are coordinates of
a configuration space. Under the action of the translation group there is a
natural metric over the space of parameters of the group given by the
Fisher-Rao metric. This metric induces a metric over the space of
probabilities. Our next step is to set the probabilities in motion. To do this,
we introduce a canonically conjugate field S and a symplectic structure; this
gives us Hamiltonian equations of motion. We show that it is possible to extend
the metric structure to the full space of the {P,S} and this leads in a natural
way to a Kaehler structure; i.e., a geometry that includes compatible
symplectic, metric and complex structures.
  The simplest geometry that describes these spaces of evolving probabilities
has remarkable properties: the natural, canonical variables are precisely the
wave functions of quantum mechanics; the Hamiltonian for the quantum free
particle can be derived from a representation of the Galilean group using
purely geometrical arguments; and it is straightforward to associate with this
geometry a Hilbert space which turns out to be the Hilbert space of quantum
mechanics. We are led in this way to a reconstruction of quantum theory based
solely on the geometry of probabilities in motion.
