Wigner-type theorems for Hilbert Grassmannians / Mark Pankov.

Format:

Book

Author/Creator:

Pankov, Mark, author.

Series:

London Mathematical Society lecture note series ; 460.

Language:

English

Subjects (All):

Hilbert space.

Grassmann manifolds.

Geometry, Projective.

Quantum theory--Mathematics.

Quantum theory.

Physical Description:

vii, 145 pages ; 23 cm.

Place of Publication:

Cambridge, United Kingdom ; New York, NY, USA : Cambridge University Press, 2020.

Summary:

"Wigner's theorem (67) provides a geometric characterization of unitary and anti-unitary operators as transformations of the set of rays of a complex Hilbert space, or equivalently, rank one projections. This statement plays an important role in mathematical foundations of quantum mechanics (11; 50; 63), since rays (rank one projections) can be identified with pure states of quantum mechanical systems. We present various types of extensions of Wigner's theorem onto Hilbert Grassmannians and their applications. Most of these results were obtained after 2000, but to completeness of the exposition we include some classical theorems closely connected to the main topic (for example, Kakutani-Mackey's result on the lattice of closed subspaces of a complex Banach space (31), Kadison's theorem on transformations preserving the convex structure of the set of states of quantum mechanical systems (30)). We use geometric methods related to the Fundamental Theorem of Projective Geometry and results in spirit of Chow's theorem (13)"-- Provided by publisher.

Contents:

Two lattices

Geometric transformations of Grassmannians

Lattices of closed subspaces

Wigner's theorem and its generalizations

Compatibility relation

Applications.

Notes:

Includes bibliographical references and index.

Other Format:

ebook version :

ISBN:

9781108790918

1108790917

OCLC:

1114348842