A metric on the space of projections admitting nice isometries
Volume 191 / 2009
Studia Mathematica 191 (2009), 271-281
MSC: Primary 47B49; Secondary 54E40.
DOI: 10.4064/sm191-3-8
Abstract
Motivated by the concept of separation between propositions in quantum logic, we introduce the so-called separation metric or Santos metric on the space of all projections in a Hilbert space. We show that the resulting metric space has only “nice” surjective isometries. On the nontrivial projections they are all unitarily or antiunitarily equivalent to the identity or to taking the orthogonal complement. We relate this result to Wigner's classical theorem on the form of quantum mechanical symmetry transformations.
