Orthogonality and Quantum Geometry: Towards a Relational Reconstruction of Quantum Theory [PhD Thesis]

266 Pages·2015·1.257 MB·English

by  Shengyang Zhong

Orthogonality and Quantum Geometry
Towards a Relational Reconstruction of Quantum Theory
Shengyang Zhong

Orthogonality and Quantum Geometry
Towards a Relational Reconstruction of Quantum Theory
ILLC Dissertation Series DS-2015-03
For further information about ILLC-publications, please contact
Institute for Logic, Language and Computation
Universiteit van Amsterdam
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1098 XG Amsterdam
phone: [phone]
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The investigations were supported by the China Scholarship Council (CSC).
Copyright (cid:13)c 2015 by Shengyang Zhong
Cover design by Shengyang Zhong.
Printed and bound by GVO drukkers & vormgevers B.V.
ISBN: 978-90-6464-894-6
Orthogonality and Quantum Geometry
Towards a Relational Reconstruction of Quantum Theory
Academisch Proefschrift
ter verkrijging van de graad van doctor aan de
Universiteit van Amsterdam
op gezag van de Rector Magnificus
prof.dr. D.C. van den Boom
ten overstaan van een door het college voor
promoties ingestelde commissie, in het openbaar
te verdedigen in de Aula der Universiteit
op woensdag 9 september 2015, te 11.00 uur
door
Shengyang Zhong
geboren te Guangdong, China.
Promotor: Prof. dr. J.F.A.K. van Benthem
Co-promotor: Dr. A. Baltag
Dr. S.J.L. Smets
Overige leden: Dr. N. Bezhanishvili
Prof. dr. R.I. Goldblatt
Prof. dr. J.R. Harding
Prof. dr. Y. Venema
Prof. dr. R.M. de Wolf
Prof. dr. M. Ying
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Contents
Acknowledgments ix
1 Introduction 1
1.1 Introduction: Logics of Quantum Theory . . . . . . . . . . . . . . 1
1.1.1 A Brief Survey of Quantum Logic . . . . . . . . . . . . . . 2
1.1.2 Motivation of the Thesis . . . . . . . . . . . . . . . . . . . 8
1.1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . 9
1.1.4 Acknowledgement of Intellectual Contributions . . . . . . . 12
1.2 The Hilbert Space Formalism of Quantum Mechanics . . . . . . . 12
1.2.1 Quantum Systems and Hilbert Spaces . . . . . . . . . . . . 13
1.2.2 Observables and Self-Adjoint Operators . . . . . . . . . . . 14
1.2.3 Evolution and Unitary Operators . . . . . . . . . . . . . . 15
1.2.4 Composite Systems and Tensor Products . . . . . . . . . . 16
2 Quantum Kripke Frames 19
2.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 State Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Geometric Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 From Geometric Frames to Projective Geometries . . . . . 29
2.3.2 From Projective Geometries to Geometric Frames . . . . . 32
2.3.3 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Complete Geometric Frames . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Saturated Sets in Geometric Frames . . . . . . . . . . . . 37
2.4.2 Complete Geometric Frames and Hilbertian Geometries . . 41
2.4.3 Finite-Dimensionality . . . . . . . . . . . . . . . . . . . . . 42
2.5 (Quasi-)Quantum Kripke Frames . . . . . . . . . . . . . . . . . . 46
2.6 Subframes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
v
2.6.1 Subframes of a Geometric Frame . . . . . . . . . . . . . . 51
2.6.2 Subframes of a (Quasi-)Quantum Kripke Frame . . . . . . 52
2.7 Quantum Kripke Frames in Quantum Logic . . . . . . . . . . . . 54
2.7.1 Quantum Kripke Frames and Hilbert Spaces . . . . . . . . 54
2.7.2 Piron Lattices and Other Quantum Structures . . . . . . . 67
2.7.3 Classical Frames . . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Maps between Quantum Kripke Frames 79
3.1 Continuous Homomorphisms . . . . . . . . . . . . . . . . . . . . . 80
3.1.1 Definition and Basic Properties . . . . . . . . . . . . . . . 80
3.1.2 Special Continuous Homomorphisms . . . . . . . . . . . . 85
3.2 Parties of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . 91
3.2.1 Rulers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2.2 The Comrades of a Homomorphism . . . . . . . . . . . . . 97
3.2.3 Parties of Homomorphisms . . . . . . . . . . . . . . . . . . 102
3.3 Orthogonal Continuous Homomorphisms . . . . . . . . . . . . . . 106
3.3.1 Uniformly Scaled Rulers . . . . . . . . . . . . . . . . . . . 106
3.3.2 Trace of a Continuous Homomorphism . . . . . . . . . . . 113
3.3.3 Three Assumptions . . . . . . . . . . . . . . . . . . . . . . 116
3.3.4 (Non-)Orthogonality between Continuous Homomorphisms 118
4 Logics of Quantum Kripke Frames 123
4.1 First-Order Definability in Quantum Kripke Frames . . . . . . . . 124
4.1.1 Infinite-Dimensional or -Codimensional Sets . . . . . . . . 125
4.1.2 Automorphisms of a Quasi-Quantum Kripke Frame . . . . 127
4.1.3 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . 131
4.1.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.2 Undecidability in Quantum Kripke Frames . . . . . . . . . . . . . 135
4.2.1 The Decision Problem for Fields . . . . . . . . . . . . . . . 135
4.2.2 The Decision Problem for Quantum Kripke Frames . . . . 136
4.2.3 The Decision Problem for Piron Lattices . . . . . . . . . . 144
4.3 Modal Logics of State Spaces . . . . . . . . . . . . . . . . . . . . 145
4.3.1 Axiomatization of State Spaces . . . . . . . . . . . . . . . 145
4.3.2 Axiomatization of State Spaces with Superposition . . . . 148
4.3.3 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5 Probabilistic Quantum Kripke Frames 155
5.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . 156
5.2 Probabilistic Quantum Kripke Frames and Hilbert Spaces . . . . . 157
5.3 Connection with Quantum Probability Measures . . . . . . . . . . 160
5.4 Quantum Kripke Frames That Can Be Probabilistic . . . . . . . . 166
5.5 Quantum Transition Probability Spaces . . . . . . . . . . . . . . . 169
vi
6 Conclusion 173
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A Quantum Structures 185
B Geometry and Algebra 189
B.1 Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 189
B.1.1 Basic Notions in Projective Geometry . . . . . . . . . . . . 189
B.1.2 Dimensions in Projective Geometry . . . . . . . . . . . . . 191
B.1.3 Projective Geometries with Additional Structures . . . . . 193
B.1.4 Maps between Projective Geometries . . . . . . . . . . . . 196
B.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
B.2.1 Division Rings and Vector Spaces . . . . . . . . . . . . . . 198
B.2.2 Hermitian Forms on Vector Spaces . . . . . . . . . . . . . 201
B.2.3 Maps between Vector Spaces . . . . . . . . . . . . . . . . . 203
B.3 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
B.3.1 Projective Geometries and Vector Spaces . . . . . . . . . . 205
B.3.2 Orthogonality and Hermitian Forms . . . . . . . . . . . . . 208
B.3.3 Homomorphisms and Linear Maps . . . . . . . . . . . . . 211
B.4 Harmonic Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . 213
C Linear Maps 217
C.1 Adjoint of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . 217
C.2 Trace of a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . 218
C.3 The Structure of Hom(V ,V ) . . . . . . . . . . . . . . . . . . . . 221
1 2
C.4 Orthogonal Basis without Eigenvectors . . . . . . . . . . . . . . . 223
Bibliography 227
Index 235
Samenvatting 241
Abstract 245
vii