From what I read there, it seems that previous assumptions of the relationships of particles (presumed to be on a “flat space”) were giving wonky answers when trying to figure how they moved. By using a different notation describing a curved space then observations “worked out” better.
But I’m probably seriously misguided ;-)
This is my interpretation of it (not understanding Hamiltonians or Hermitian spaces at all):
If a particle is being acted on by some external force that gives it a preference in the direction of travel (for example, gravity making it easier to move down than up, or an electric or magnetic field having the same kind of effect), then we can model that preference as a curved space instead of a flat space. In this case, it is easier for the particle to move ‘down’ the curve instead of ‘up’ the curve.
Therefore, the quantum properties that should be orthogonal but aren’t due to the external force, actually turn out to be orthogonal to the curved surface, just like gravity is always pointing to the center of the earth and the direction changes as you move along the earth’s surface. This happens with gravitational force on a ramp, as well. Gravity makes a ball roll down the ramp because a portion of the gravitational force is down the ramp (and the other part is toward the ramp surface), whereas if the ramp were flat, it all points down toward the center of the earth and there is no motion.
So, the properties are orthogonal to this curved surface and are not orthogonal to the flat surface and this fact can now be used to make better equations that model the behavior. It seems to be just a better way of modeling the behavior of the system.