# Metric space approach to quantum mechanics

### Seminar

- Date: Jun 23, 2015
- Time: 10:00
- Speaker: Prof. Irene D'Amico
- York University
- Location: Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale)
- Room: Seminarraum A.2.20

Hilbert space combines the properties of
two different types of mathematical spaces: vector space and metric
space. While the vector-space aspects are widely used, the metric-space
aspects are much less exploited. We show that conservation laws in
quantum mechanics naturally lead to metric spaces for the set of related
physical quantities[1]. All such metric spaces have an "onion-shell"
geometry. The related metric stratifies Fock space into concentric
spheres on which maximum and minimum distances between states can be
defined and geometrically interpreted. Unlike the usual Hilbert-space
analysis, our results apply also to the reduced space of only ground
states and to that of particle densities, which are metric, but not
Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground
states, which is highly complex and nonlocal in coordinate description,
is found, for the systems analysed, to be simple in metric space, where
it becomes a monotonic and nearly linear mapping of vicinities onto
vicinities[2]. Similarly, by considering metric spaces associated to
many-body systems immersed in a magnetic field, we consider the mapping
between wave-function and (current and particle) densities at the core
of Current Density functional theory. We find, in the related metric
spaces, regions of allowed and forbidden distances, a "band structure",
directly arising from the conservation of the z component of the angular
momentum[1]. Finally, recent results include appropriate metrics for
the external potential which allow us to directly explore the 'third
leg' of the Hohenberg-Kohn theorem[3].

[1] P. M. Sharp and I. D'Amico, Phys. Rev. B 89, 115137 (2014)

[2] I. D'Amico, J. P. Coe, V. V. Franca and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)

[3] P. M. Sharp and I. D'Amico, preprint (2015).