A metric on the space of projections admitting nice isometries

Lajos Molnár ¹   ; Werner Timmermann ²

¹ Institute of Mathematics University of Debrecen P.O. Box 12 H-4010 Debrecen, Hungary
² Institut für Analysis Technische Universität Dresden D-01062 Dresden, Germany

Studia Mathematica, Tome 191 (2009) no. 3, pp. 271-281

Résumé

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.

Détail

DOI : 10.4064/sm191-3-8

Keywords: motivated concept separation between propositions quantum logic introduce so called separation metric santos metric space projections hilbert space resulting metric space has only nice surjective isometries nontrivial projections unitarily antiunitarily equivalent identity taking orthogonal complement relate result wigners classical theorem form quantum mechanical symmetry transformations