This work has grown out of the lecture notes that were prepared for a series of seminars on some selected topics in quantum logic. The seminars were delivered during the first semester of the 1993/1994 academic year in the Unit for Foundations of Science of the Department of History and Foundations of Mathematics and Science, Faculty of Physics, Utrecht University, The Netherlands, while I was staying in that Unit on a European Community Research Grant, and in the Center for Philosophy of Science, University of Pittsburgh, U.S.A., where I was staying during the 1994/1995 academic year as a Visiting Fellow on a Fulbright Research Grant, and where I also was supported by the Istvan Szechenyi Scholarship Foundation. The financial support provided by these foundations, by the Center for Philosophy of Science and by the European Community is
greatly acknowledged, and I wish to thank D. Dieks, the professor of the Foundations Group in Utrecht and G. Massey, the director ofthe Center for Philosophy of Science in Pittsburgh for making my stay at the respective institutions possible.
I also wish to thank both the members of the Fouridations Group in Utrecht, especially D. Dieks, C. Lutz, F. Muller, J. Uffink and P. Vermaas and the participants in the seminars at the Center for Philosophy of Science in Pittsburgh, especially N. Belnap, J. Earman, A. Janis, J. Norton, and J. Forge not only for their interest in the seminars and in the subsequent stimulating discussions but also for their hospitality in Utrecht and Pittsburgh, which made my stay in Utrecht and Pittsburgh a most pleasant experience. Special thanks go to my friend and colleague D. Petz, professor of mathematics in the Mathematics Institute of the Technical University in Budapest, Hungary, who encouraged me to complete the lecture notes and who corrected a number of errors in the manuscript. Needless to say, neither he, nor any of those mentioned here bear any responsibility whatsoever for any errors that might remain in the work.
Author(s): Miklós Rédei
Series: Fundamental Theories of Physics, 91
Edition: 1998
Publisher: Kluwer Academic Publishers
Year: 1998
Language: English
Pages: 253
Tags: Quantum Logic; Quantum Mechanics
Preface ix
Introduction 1
1.1 Bibliographic notes . 8
2 Observables and states in the Hilbert space formalism of quantum mechanics 11
2.1 Observables . . . . . 11
2.2 States ........ 20
2.3 Bibliographic Notes. 27
3 Lattice theoretic notions 29
3.1 Basic notions in lattice theory . 29
3.2 Bibliographic notes . . . . . . . 43
4 Hilbert lattice 45
4.1 Hilbert space and the lattice of subspaces 45
4.2 Subspaces and projections 54
4.3 Bibliographic notes........... 60
5 Physical theory in semantic approach 61
5.1 Physical theory as semi-interpreted language 61
5.2 The logic of classical mechanics 64
5.3 Hilbert lattice as logic 68
5.4 Bibliographic notes . . . . . . . 74
6 Von Neumann lattices 77
6.1 Von Neumann algebras .................... 77
6.2 Von Neumann lattices ..................... 82
6.3 Appendix: proofs of propositions related to the classification theory of von Neumann algebras 90
6.4 Bibliographic notes....................... 100
7 The Birkhoff-von Neumann concept of quantum logic 103
7.1 Quantum logic as event structure of non-commutative probability ........................... 103
7.1.1 Digression: von Neumann's concept of probability in quantum mechanics in the years 1926-1932 105
7.1.2 Back to the type II_1 factor 112
7.2 Probability is logical 113
7.3 Bibliographic notes......... 116
8 Quantum conditional and quantum conditional probability 119
8.1 Minimal implicative criteria and quantum conditional 119
8.2 Conditional probability and statistical inference . . 127
8.3 Breakdown of Stalnaker's Thesis in quantum logic 134
8.4 Bibliographic notes.................. 137
9 The problem of hidden variables 139
9.1 Historical remarks ...................... 139
9.2 Notion of and no-go results on dispersive hidden theories. 144
9.2.1 Definition of dispersive hidden theory . . . . 144
9.2.2 Negative results on dispersive hidden theories 149
9.3 No-go results on entropic hidden theories . 156
9.4 The problem of local hidden variables . . . . . . . . 160
9.4.1 Bell's question and Bell's inequality . . . . . 160
9.4.2 No-go proposition on dispersive local hidden theories 164
9.5 Bibliographic notes . . . . . . . . . . . . . . . . . . . . . 169
10 Violation of Bell's inequality in quantum field theory 171
10.1 Basic notions of algebraic quantum field theory . . . . 171
10.2 Bell correlation and Bell's inequality . . . . . . . . . . 180
10.3 Violation of Bell's inequality in quantum field theory . 184
10.4 Superluminal correlations in quantum field theory 188
10.5 Bibliographic notes . . . . . . . . . . . . . . . . . . 190
11 Independence in quantum logic approach 191
11.1 Logical independence in quantum logic 193
11.1.1 Logical notions of independence 193
11.1.2 Logical and statistical independence 197
11.2 Counterfactual probabilistic independence 204
11.2.1 Concept of counterfactual probabilistic independence 205
11.2.2 Counterfactual probabilistic independence in quantum field theory . 207
11.3 Bibliographic notes....................... 213
12 Reichenbach's common cause principle and quantum field theory 215
12.1 Reichenbach's common cause principle . . . . . . . . . . . . 216
12.2 Do superluminal correlations have a probabilistic common cause? ........ 219
12.3 Bibliographic notes....................... 224
References 227
Index 235
