Quantum Mechanics in Metric Space:

"Our findings on the distance between wave functions may complement the use of the fidelity[13] in quantum information theory and in the study of quantum phase transitions. In conclusion, we emphasize that the metric and the vector-space viewpoints are complementary, and that only together do they exhaust the full richness of Hilbert space."

PRL 106, 050401 (2011)
PHYSICAL    REVIEW    LETTERS    week ending 4 FEBRUARY 2011
Quantum Mechanics in Metric Space: Wave Functions and Their Densities
I.D’Amico,1 J.P.Coe,1 V.V.Franc ?a,2,3 andK.Capelle4
1Department of Physics, University of York, York YO10 5DD, United Kingdom 2Physikalisches Institut, Albert-Ludwigs Universitat, Hermann-Herder-Strasse 3, Freiburg, Germany 3Capes Foundation, Ministry of Education of Brazil, Caixa Postal 250, Brasilia, 70040-20, Brazil 4Centro de Cieˆncias Naturais e Humanas, Universidade Federal do ABC, Santo Andre ?, 09210-170 Sa ?o Paulo, Brazil (Received 30 May 2010; published 1 February 2011)

"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. Here we show that a suitable 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 three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities."