  We show that conservation laws in quantum mechanics naturally lead to metric
spaces for the set of related physical quantities. All such metric spaces have
an "onion-shell" geometry. We demonstrate the power of this approach by
considering many-body systems immersed in a magnetic field, with a finite
ground state current. In the associated metric spaces we find regions of
allowed and forbidden distances, a "band structure" in metric space directly
arising from the conservation of the $z$ component of the angular momentum.
