Information geometry, dynamics and discrete quantum mechanics

TL;DR

This paper reconstructs discrete quantum mechanics using information geometry, symplectic and Kähler structures, showing that wave functions and unitary transformations naturally emerge from geometric principles.

Contribution

It introduces a geometric framework combining information geometry and symplectic structures to derive quantum mechanics from first principles.

Findings

Derives wave functions as canonical coordinates in a Kähler space.

Shows the full unitary group arises naturally from the geometric structure.

Connects the metric to Wootters' statistical distance in quantum systems.

Abstract

We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the picture. We do this in three steps. Our starting point is information geometry, the natural geometry of the space of probability distributions. Dynamics requires additional structure. To evolve the probabilities $P^k$, we introduce coordinates $S^k$ canonically conjugate to the $P^k$ and a symplectic structure. We then seek to extend the metric structure of information geometry, to define a geometry over the full space of the $P^k$ and $S^k$. Consistency between the metric tensor and the symplectic form forces us to introduce a K\"ahler geometry. The construction has notable features. A complex structure is obtained in a natural way. The canonical coordinates of the K\"ahler space are precisely the wave functions of quantum mechanics. The full group of unitary transformations is obtained. Finally, one may associate a Hilbert space with the K\"ahler space, which leads to the standard version of quantum theory. We also show that the metric that we derive here using purely geometrical arguments is precisely the one that leads to Wootters' expression for the statistical distance for quantum systems.