Abstract
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.
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