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El. knyga: Wigner-Type Theorems for Hilbert Grassmannians

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"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)"--

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An accessible introduction to the geometric approach to Wigner's theorem and its role in quantum mechanics.
Preface vii
Introduction 1(3)
1 Two Lattices
4(18)
1.1 Lattices
4(4)
1.2 The Lattice of Subspaces of a Vector Space
8(3)
1.3 The Orthomodular Lattice of Closed Subspaces of a Hilbert Space
11(11)
2 Geometric Transformations of Grassmannians
22(31)
2.1 Semilinear Maps
23(5)
2.2 Proof of Theorem 2.6
28(3)
2.3 Grassmann Graphs
31(4)
2.4 Automorphisms of Grassmann Graphs
35(6)
2.5 Complementary Preserving Transformations
41(3)
2.6 Apartments Preserving Transformations
44(9)
3 Lattices of Closed Subspaces
53(15)
3.1 Linear and Conjugate-Linear Operators
53(4)
3.2 Lattice Isomorphisms
57(4)
3.3 Kakutani-Mackey Theorem
61(3)
3.4 Extensions of Isomorphisms
64(4)
4 Wigner's Theorem and Its Generalizations
68(34)
4.1 Linear and Conjugate-Linear Isometries
69(1)
4.2 Orthogonality Preserving Transformations
70(5)
4.3 Non-bijective Version of Wigner's Theorem
75(5)
4.4 The Principal Angles between Subspaces
80(5)
4.5 Transformations of Grassmannians Preserving the Principal Angles
85(2)
4.6 Proofs of Theorem 4.26 and Proposition 4.28
87(5)
4.7 Proof of Theorem 4.29
92(1)
4.8 Proof of Theorem 4.25
93(7)
4.9 Transformations of Grassmannians Preserving the Gap Metric
100(2)
5 Compatibility Relation
102(23)
5.1 Compatibility Preserving Transformations
102(4)
5.2 Proofs of Theorems 5.2 and 5.3
106(5)
5.3 Proof of Theorem 5.5
111(2)
5.4 Characterizing Lemmas
113(3)
5.5 Proofs of Theorems 5.6 and 5.7
116(3)
5.6 Proof of Theorem 5.9
119(3)
5.7 A Metric Connected to the Compatibility Relation
122(3)
6 Applications
125(16)
6.1 Automorphisms of the Convex Set of Quantum States
125(3)
6.2 Linear Transformations which Preserve Projections
128(4)
6.3 Proofs of Theorems 6.7 and 6.11
132(9)
References 141(4)
Index 145
Mark Pankov is Professor of Mathematics at Uniwersytet Warmisko-Mazurski, Poland. His research interests include preserver problems in operator theory related to quantum mechanics, geometry of linear codes, and zig-zags in discrete objects. He is the author of Grassmannians of Classical Buildings (2010) and Geometry of Semilinear Embeddings: Relations to Graphs and Codes (2015).
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