### 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.

Original language | English |
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Pages (from-to) | 271-281 |

Number of pages | 11 |

Journal | Studia Mathematica |

Volume | 191 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 |

### Fingerprint

### Keywords

- Gap metric
- Hilbert space
- Projections
- Santos metric
- Surjective isometries

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Studia Mathematica*,

*191*(3), 271-281. https://doi.org/10.4064/sm191-3-8

**A metric on the space of projections admitting nice isometries.** / Molnár, L.; Timmermann, Werner.

Research output: Contribution to journal › Article

*Studia Mathematica*, vol. 191, no. 3, pp. 271-281. https://doi.org/10.4064/sm191-3-8

}

TY - JOUR

T1 - A metric on the space of projections admitting nice isometries

AU - Molnár, L.

AU - Timmermann, Werner

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Gap metric

KW - Hilbert space

KW - Projections

KW - Santos metric

KW - Surjective isometries

UR - http://www.scopus.com/inward/record.url?scp=67649961337&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649961337&partnerID=8YFLogxK

U2 - 10.4064/sm191-3-8

DO - 10.4064/sm191-3-8

M3 - Article

AN - SCOPUS:67649961337

VL - 191

SP - 271

EP - 281

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 3

ER -