Quantum theory from the geometry of evolving probabilities

Abstract

We consider the space of probabilities {P(x)}, where the x are coordinates of a config- uration space. Under the action of the translation group, P(x)→P(x+θ ), there is a natural metric over the parameters θ 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 canon- ically 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 introducing a Kähler 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.