Fundamental Theories of Physics 188 Klaas Landsman Foundations of Quantum Theory From Classical Concepts to Operator Algebras Fundamental Theories of Physics Volume 188 Series editors Henk van Beijeren, Utrecht, The Netherlands Philippe Blanchard, Bielefeld, Germany Paul Busch, York, UK Bob Coecke, Oxford, UK Dennis Dieks, Utrecht, The Netherlands Bianca Dittrich, Waterloo, Canada Detlef D ü rr, M ü nchen, Germany Ruth Durrer, Gen è ve, Switzerland Roman Frigg, London, UK Christopher Fuchs, Boston, USA Giancarlo Ghirardi, Trieste, Italy Domenico J.W. Giulini, Bremen, Germany Gregg Jaeger, Boston, USA Claus Kiefer, K ö ln, Germany Nicolaas P. Landsman, Nijmegen, The Netherlands Christian Maes, Leuven, Belgium Mio Murao, Bunkyo-ku, Japan Hermann Nicolai, Potsdam, Germany Vesselin Petkov, Montreal, Canada Laura Ruetsche, Ann Arbor, USA Mairi Sakellariadou, London, UK Alwyn van der Merwe, Denver, USA Rainer Verch, Leipzig, Germany Reinhard Werner, Hannover, Germany Christian W ü thrich, Geneva, Switzerland Lai-Sang Young, New York City, USA The international monograph series “ Fundamental Theories of Physics ” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scienti fi c fi elds. Original contributions in well-established fi elds such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fi elds. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high pro fi le and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scienti fi c standard. More information about this series at http://www.springer.com/series/6001 Klaas Landsman Foundations of Quantum Theory From Classical Concepts to Operator Algebras ISSN 0168-1222 ISSN 2365-6425 (electronic) Fundamental Theories of Physics ISBN 978-3-319-51776-6 ISBN 978-3-319-51777-3 (eBook) DOI 10.1007/978-3-319-51777-3 Library of Congress Control Number: 2017933673 © The Author(s) 2017. This book is an open access publication. 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Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Klaas Landsman IMAPP Radboud University Nijmegen The Netherlands To Jeremy Butterfield Preface ‘Der Kopf, so gesehen, hat mit dem Kopf, so gesehen, auch nicht die leiseste ̈ Ahnlichkeit (. . . ) Der Aspektwechsel. “Du w ̈ urdest doch sagen, dass sich das Bild jetzt g ̈ anzlich ge ̈ andert hat!” Aber was ist anders: mein Eindruck? meine Stellungnahme? (. . . ) Ich beschreibe die ̈ Anderung wie eine Wahrnehmung, ganz, als h ̈ atte sich der Gegenstand vor meinen Augen ge ̈ andert.’ (Wittgenstein, Philosophische Untersuchungen II , §§ 127, 129). 1 As the well-known picture above is meant to allegorize, some physical systems admit a dual description in either classical or quantum-mechanical terms. According to Bohr’s “doctrine of classical concepts”, measurement apparatuses are examples of such systems. More generally—as hammered down by decoherence theorists— the classical world around us is a case in point. As will be argued in this book, the measurement problem of quantum mechanics (highlighted by Schr ̈ odinger’s Cat) is caused by this duality (rather than resolved by it, as Bohr is said to have thought). 1 ‘The head seen in this way hasn’t even the slightest similarity to the head seen in that way (. . . ) The change of aspect. “But surely you’d say that the picture has changed altogether now! But what is different: my impression? my attitude? (. . . ) I describe the change like a perception; just as if the object has changed before my eyes.’ Translation: G.E.M. Anscombe, P.M.S. Hacker, & J. Schulte (Wittgenstein, 2009/1953, pp. 205–206). vii viii The aim of this book is to analyze the foundations of quantum theory from the point of view of classical-quantum duality, using the mathematical formalism of operator algebras on Hilbert space (and, more generally, C*-algebras) that was orig- inally created by von Neumann (followed by Gelfand and Naimark). In support of this analysis, but also as a matter of independent interest, the book covers many of the traditional topics one might expect to find in a treatise on the foundations of quantum mechanics, like pure and mixed states, observables, the Born rule and its relation to both single-case probabilities and long-run frequencies, Gleason’s Theo- rem, the theory of symmetry (including Wigner’s Theorem and its relatives, culmi- nating in a recent theorem of Hamhalter’s), Bell’s Theorem(s) and the like, quantiza- tion theory, indistinguishable particle, large systems, spontaneous symmetry break- ing, the measurement problem, and (intuitionistic) quantum logic. One also finds a few idiosyncratic themes, such as the Kadison–Singer Conjecture, topos theory (which naturally injects intuitionism into quantum logic), and an unusual emphasis on both conceptual and mathematical aspects of limits in physical theories. All of this is held together by what we call Bohrification , i.e., the mathematical interpretation of Bohr’s classical concepts by commutative C*-algebras, which in turn are studied in their quantum habitat of noncommutative C*-algebras. Thus the book is mostly written in mathematical physics style, but its real subject is natural philosophy . Hence its intended readership consists not only of mathemati- cal physicists, but also of philosophers of physics, as well as of theoretical physicists who wish to do more than ‘shut up and calculate’, and finally of mathematicians who are interested in the mathematical and conceptual structure of quantum theory. To serve all these groups, the native mathematical language (i.e. of C*-algebras) is introduced slowly, starting with finite sets (as classical phase spaces) and finite- dimensional Hilbert spaces. In addition, all advanced mathematical background that is necessary but may distract from the main development is laid out in extensive appendices on Hilbert spaces, functional analysis, operator algebras, lattices and logic, and category theory and topos theory, so that the prerequisites for this book are limited to basic analysis and linear algebra (as well as some physics). These appendices not only provide a direct route to material that otherwise most readers would have needed to extract from thousands of pages of diverse textbooks, but they also contain some original material, and may be of interest even to mathematicians. In summary, the aims of this book are similar to those of its peerless paradigm: ‘Der Gegenstand dieses Buches ist die einheitliche, und, soweit als m ̈ oglich und angebracht, mathematisch einwandfreie Darstellung der neuen Quantenmechanik (. . . ). Dabei soll das Hauptgewicht auf die allgemeinen und prinzipiellen Fragen, die im Zusammenhange mit dieser Theorie entstanden sind, gelegt werden. Insbesondere sollen die schwierigen und vielfach noch immer nicht restlos gekl ̈ arten Interpretationsfragen n ̈ aher untersucht werden.’ (von Neumann, Mathematische Grundlagen der Quantenmechanik , 1932, p. 1). 2 2 ‘The object of this book is to present the new quantum mechanics in a unified presentation which, so far as it is possible and useful, is mathematically rigorous. (. . . ) Therefore the principal emphasis shall be placed on the general and fundamental questions which have arisen in connection with this theory. In particular, the difficult problems with interpretation, many of which are even now not fully resolved, will be investigated in detail.’ Translation: R.T. Beyer (von Neumann, 1955, p. vii). Preface ix Two other quotations the author often had in mind while writing this book are: ‘And although the whole of philosophy is not immediately evident, still it is better to add something to our knowledge day by day than to fill up men’s minds in advance with the preconceptions of hypotheses.’ (Newton, draft preface to Principia , 1686). 3 ‘Juist het feit dat een genie als D ESCARTES volkomen naast de lijn van ontwikkeling is bli- jven staan, die van G ALILEI naar N EWTON voert (. . . ) [is] een phase van den in de historie zoo vaak herhaalden strijd tusschen de bescheidenheid der mathematisch-physische meth- ode, die na nauwkeurig onderzoek de verschijnselen der natuur in steeds meer omvattende schemata met behulp van de exacte taal der mathesis wil beschrijven en den hoogmoed van het philosophische denken, dat in ́ e ́ en genialen greep de heele wereld wil omvatten (. . . ).’ (Dijksterhuis, Val en Worp , 1924, p. 343). 4 Acknowledgements 1. Research underlying this book has been generously supported by: • Radboud University Nijmegen, partly through a sabbatical in 2014. • The Netherlands Organization for Scientific Research (NWO), initially by funding various projects eventually contributing to this book, and most re- cently by paying the Open Access fee, making the book widely available. • The Templeton World Charity Foundation (TWCF), by funding the Oxford– Princeton–Nijmegen collaboration Experimental Tests of Quantum Reality • Trinity College (Cambridge), by appointing the author as a Visiting Fellow Commoner during the Easter Term 2016, when the book was largely finished. 2. The author was fortunate in having been surrounded by outstanding students and postdocs, who made essential contributions to the insights described in this book. In alphabetical order these were Christian Budde, Martijn Caspers, Ronnie Her- mens, Jasper van Heugten, Chris Heunen, Bert Lindenhovius, Robin Reuvers, Bas Spitters, Marco Stevens, and Sander Wolters. Those were the days! 3. The author is indebted to Jeremy Butterfield, Peter Bongaarts, Harvey Brown, Dennis Dieks, Siegfried Echterhoff, Aernout van Enter, Jan Hamhalter, Jaap van Oosten, and Bas Terwijn for comments on the manuscript. In addition, through critical feedback on a Masterclass at Trinity, Owen Maroney and Fred Muller indirectly (but considerably) improved Chapter 11 on the measurement problem. 4. Angela Lahee from Springer thoughtfully guided the publication process of this book from the beginning to the end. Thanks also to her colleague Aldo Rampioni. Finally, it is a pleasure to dedicate this book to Jeremy Butterfield, in recognition of his ideas, as well as of his unrelenting support and friendship over the last 25 years. 3 Newton (1999), p. 61. 4 ‘The very fact that a genius like Descartes was completely sidelined in the development leading from Galilei to Newton (. . . ) represents a phase in the struggle—that has so often been repeated throughout history—between the modesty of the approach of mathematical physics, which af- ter precise investigations attempts to describe natural phenomena in increasingly comprehensive schemes using the exact language of mathematics, and the haughtiness of philosophical thought, which wants to comprehend the entire world in one dazzling grasp.’ Translation by the author. Preface Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I C 0 ( X ) and B ( H ) 1 Classical physics on a finite phase space . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.1 Basic constructions of probability theory . . . . . . . . . . . . . . . . . . . . . . . 24 1.2 Classical observables and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3 Pure states and transition probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4 The logic of classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5 The GNS -construction for C ( X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Quantum mechanics on a finite-dimensional Hilbert space . . . . . . . . . . 39 2.1 Quantum probability theory and the Born rule . . . . . . . . . . . . . . . . . . . 40 2.2 Quantum observables and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3 Pure states in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 The GNS -construction for matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5 The Born rule from Bohrification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6 The Kadison–Singer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.7 Gleason’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.8 Proof of Gleason’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.9 Effects and Busch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.10 The quantum logic of Birkhoff and von Neumann . . . . . . . . . . . . . . . 75 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3 Classical physics on a general phase space . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Vector fields and their flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2 Poisson brackets and Hamiltonian vector fields . . . . . . . . . . . . . . . . . . 88 3.3 Symmetries of Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.4 The momentum map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 xi xii Contents 4 Quantum physics on a general Hilbert space . . . . . . . . . . . . . . . . . . . . . . 103 4.1 The Born rule from Bohrification ( II ) . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2 Density operators and normal states . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 The Kadison–Singer Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Gleason’s Theorem in arbitrary dimension . . . . . . . . . . . . . . . . . . . . . . 119 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5 Symmetry in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1 Six basic mathematical structures of quantum mechanics . . . . . . . . . 126 5.2 The case H = C 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.3 Equivalence between the six symmetry theorems . . . . . . . . . . . . . . . . 137 5.4 Proof of Jordan’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.5 Proof of Wigner’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.6 Some abstract representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.7 Representations of Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . 155 5.8 Irreducible representations of SU ( 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.9 Irreducible representations of compact Lie groups . . . . . . . . . . . . . . . 162 5.10 Symmetry groups and projective representations . . . . . . . . . . . . . . . . 167 5.11 Position, momentum, and free Hamiltonian . . . . . . . . . . . . . . . . . . . . . 177 5.12 Stone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Part II Between C 0 ( X ) and B ( H ) 6 Classical models of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.1 From von Neumann to Kochen–Specker . . . . . . . . . . . . . . . . . . . . . . . 193 6.2 The Free Will Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.3 Philosophical intermezzo: Free will in the Free Will Theorem . . . . . 205 6.4 Technical intermezzo: The GHZ -Theorem . . . . . . . . . . . . . . . . . . . . . . 210 6.5 Bell’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.6 The Colbeck–Renner Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7 Limits: Small ̄ h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.1 Deformation quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.2 Quantization and internal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.3 Quantization and external symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.4 Intermezzo: The Big Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.5 Induced representations and the imprimitivity theorem . . . . . . . . . . . 262 7.6 Representations of semi-direct products . . . . . . . . . . . . . . . . . . . . . . . . 268 7.7 Quantization and permutation symmetry . . . . . . . . . . . . . . . . . . . . . . . 275 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Contents xiii 8 Limits: large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.1 Large quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.2 Large systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 8.3 Quantum de Finetti Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 8.4 Frequency interpretation of probability and Born rule . . . . . . . . . . . . 310 8.5 Quantum spin systems: Quasi-local C*-algebras . . . . . . . . . . . . . . . . . 318 8.6 Quantum spin systems: Bundles of C*-algebras . . . . . . . . . . . . . . . . . 323 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 9 Symmetry in algebraic quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 9.1 Symmetries of C*-algebras and Hamhalter’s Theorem . . . . . . . . . . . . 334 9.2 Unitary implementability of symmetries . . . . . . . . . . . . . . . . . . . . . . . 344 9.3 Motion in space and in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 9.4 Ground states of quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 9.5 Ground states and equilibrium states of classical spin systems . . . . . 352 9.6 Equilibrium ( KMS ) states of quantum systems . . . . . . . . . . . . . . . . . . . 358 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 10 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 10.1 Spontaneous symmetry breaking: The double well . . . . . . . . . . . . . . . 371 10.2 Spontaneous symmetry breaking: The flea . . . . . . . . . . . . . . . . . . . . . . 375 10.3 Spontaneous symmetry breaking in quantum spin systems . . . . . . . . 379 10.4 Spontaneous symmetry breaking for short-range forces . . . . . . . . . . . 383 10.5 Ground state(s) of the quantum Ising chain . . . . . . . . . . . . . . . . . . . . . 386 10.6 Exact solution of the quantum Ising chain: N < ∞ . . . . . . . . . . . . . . . 390 10.7 Exact solution of the quantum Ising chain: N = ∞ . . . . . . . . . . . . . . . 397 10.8 Spontaneous symmetry breaking in mean-field theories . . . . . . . . . . . 409 10.9 The Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 10.10 The Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 11 The measurement problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 11.1 The rise of orthodoxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 11.2 The rise of modernity: Swiss approach and Decoherence . . . . . . . . . . 440 11.3 Insolubility theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 11.4 The Flea on Schr ̈ odinger’s Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 12 Topos theory and quantum logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 12.1 C*-algebras in a topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 12.2 The Gelfand spectrum in constructive mathematics . . . . . . . . . . . . . . 466 12.3 Internal Gelfand spectrum and intuitionistic quantum logic . . . . . . . . 471 12.4 Internal Gelfand spectrum for arbitrary C*-algebras . . . . . . . . . . . . . . 476 12.5 “Daseinisation” and Kochen–Specker Theorem . . . . . . . . . . . . . . . . . 485 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 xiv Contents A Finite-dimensional Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 A.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 A.2 Functionals and the adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 A.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 A.4 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 A.5 Positive operators and the trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 B Basic functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 B.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 B.2 p spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 B.3 Banach spaces of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 522 B.4 Basic measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 B.5 Measure theory on locally compact Hausdorff spaces . . . . . . . . . . . . . 526 B.6 L p spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 B.7 Morphisms and isomorphisms of Banach spaces . . . . . . . . . . . . . . . . . 538 B.8 The Hahn–Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 B.9 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 B.10 The Krein–Milman Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 B.11 Choquet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 B.12 A pr ́ ecis of infinite-dimensional Hilbert space . . . . . . . . . . . . . . . . . . . 562 B.13 Operators on infinite-dimensional Hilbert space . . . . . . . . . . . . . . . . . 568 B.14 Basic spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 B.15 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 B.16 Abelian ∗ -algebras in B ( H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 B.17 Classification of maximal abelian ∗ -algebras in B ( H ) . . . . . . . . . . . . . 601 B.18 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 B.19 Spectral theory for self-adjoint compact operators . . . . . . . . . . . . . . . 611 B.20 The trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 B.21 Spectral theory for unbounded self-adjoint operators . . . . . . . . . . . . . 625 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 C Operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 C.1 Basic definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 C.2 Gelfand isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 C.3 Gelfand duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 C.4 Gelfand isomorphism and spectral theory . . . . . . . . . . . . . . . . . . . . . . 657 C.5 C*-algebras without unit: general theory . . . . . . . . . . . . . . . . . . . . . . . 660 C.6 C*-algebras without unit: commutative case . . . . . . . . . . . . . . . . . . . . 664 C.7 Positivity in C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 C.8 Ideals in Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 C.9 Ideals in C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 C.10 Hilbert C*-modules and multiplier algebras . . . . . . . . . . . . . . . . . . . . . 677 C.11 Gelfand topology as a frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 C.12 The structure of C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 Contents xv C.13 Tensor products of Hilbert spaces and C*-algebras . . . . . . . . . . . . . . . 697 C.14 Inductive limits and infinite tensor products of C*-algebras . . . . . . . . 707 C.15 Gelfand isomorphism and Fourier theory . . . . . . . . . . . . . . . . . . . . . . . 714 C.16 Intermezzo: Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725 C.17 C*-algebras associated to Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . 730 C.18 Group C*-algebras and crossed product algebras . . . . . . . . . . . . . . . . 734 C.19 Continuous bundles of C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 C.20 von Neumann algebras and the σ -weak topology . . . . . . . . . . . . . . . . 742 C.21 Projections in von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 746 C.22 The Murray–von Neumann classification of factors . . . . . . . . . . . . . . 750 C.23 Classification of hyperfinite factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 C.24 Other special classes of C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 758 C.25 Jordan algebras and (pure) state spaces of C*-algebras . . . . . . . . . . . 763 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 D Lattices and logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 D.1 Order theory and lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 D.2 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784 D.3 Intuitionistic propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 D.4 First-order (predicate) logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 D.5 Arithmetic and set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 E Category theory and topos theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 E.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 E.2 Toposes and functor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 E.3 Subobjects and Heyting algebras in a topos . . . . . . . . . . . . . . . . . . . . . 820 E.4 Internal frames and locales in sheaf toposes . . . . . . . . . . . . . . . . . . . . . 826 E.5 Internal language of a topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 Introduction After 25 years of confusion and even occasional despair, in March 1926 physicists suddenly had two theories of the microscopic world (Heisenberg, 1925; Schr ̈ odinger, 1926ab), which hardly could have looked more differently. Heisenberg’s matrix me- chanics (as it came to be called a bit later) described experimentally measurable quantities (i.e., “observables”) in terms of discrete quantum numbers, and appar- ently lacked a state concept. Schr ̈ odinger’s wave mechanics focused on unobserv- able continuous matter waves apparently playing the role of quantum states; at the time the only observable within reach of his theory was the energy. Einstein is even reported to have remarked in public that the two theories excluded each other. Nonetheless, Pauli (in a letter to Jordan dated 12 April 1926), Schr ̈ odinger (1926c) himself, Eckart (1926), and Dirac (1927) argued—it is hard to speak of a complete argument even at a heuristic level, let alone of a mathematical proof (Muller, 1997ab)— that in fact the two theories were equivalent! A rigorous equiv- alence proof was given by von Neumann (1927ab), who (at the age of 23) was the first to unearth the mathematical structure of quantum mechanics as we still under- stand it today. His effort, culminating in his monograph Mathematische Grundlagen der Quantenmechanik (von Neumann, 1932), was based on the abstract concept of a Hilbert space , which previously had only appeared in examples (i.e. specific real- izations) going back to the work of Hilbert and his school on integral equations. The novelty of von Neumann’s abstract approach may be illustrated by the advice Hilbert’s former student Schmidt gave to von Neumann even at the end of the 1920s: ‘Nein! Nein! Sagen Sie nicht Operator, sagen Sie Matrix!” (Bernkopf, 1967, p. 346). 5 Von Neumann proposed that observables quantities be interpreted as (possibly un- bounded) self-adjoint operators on some Hilbert space, whilst pure states are real- ized as rays (i.e. unit vectors up to a phase) in the same space; finally, the inner prod- uct provides the probabilities introduced by Born (1926ab). In particular, Heisen- berg’s observables were operators on 2 ( N ) , whereas Schr ̈ odinger’s wave-functions were unit vectors in L 2 ( R 3 ) . A unitary transformation between these Hilbert spaces then provided the mathematical equivalence between their competing theories. 5 ‘No! No! You shouldn’t say operator, you should say matrix!’ 1 2 Introduction This story is well known, but it is worth emphasizing (cf. Zalamea, 2016, § I.1) that the most significant difference between von Neumann’s mathematical axiom- atization of quantum mechanics and Dirac’s heuristic but beautiful and systematic treatment of the same theory (Dirac, 1930) was not so much the lack of mathemat- ical rigour in the latter—although this point was stressed by von Neumann (1932, p. 2) himself, who was particularly annoyed with Dirac’s δ -function and his closely related assumption that every self-adjoint operator can be diagonalized in the naive way of having a basis of eigenvectors—but the fact that Dirac’s approach was rela- tive to the choice of a (generalized) basis of a Hilbert space, whereas von Neumann’s was absolute . In this sense, as a special case of his (and Jordan’s) general transfor- mation theory, Dirac showed that Heisenberg’s matrix mechanics and Schr ̈ odinger’s wave mechanics were related by a (unitary) transformation, whereas for von Neu- mann they were two different realizations of his abstract (separable) Hilbert space. In particular, von Neumann’s approach a priori dispenses with a basis choice alto- gether; this is precisely the difference between an operator and a matrix Schmidt al- luded to in the above quotation. Indeed, von Neumann’s abstract approach (which as a co-founder of functional analysis he shared with Banach, but not with his mentor Hilbert) was remarkable even in mathematics; in physics it must have been dazzling. It is instructive to compare this situation with special relativity, where, so to speak, Dirac would write down the theory in terms of inertial frames of reference, so as to subsequently argue that due to Poincar ́ e-invariance the physical content of the theory does not depend on such a choice. Von Neumann, on the other hand (had he ever written a treatise on relativity), would immediately present Minkowski’s space-time picture of the theory and develop it in a coordinate-free fashion. However, this analogy is also misleading. In special relativity, all choices of iner- tial frames are genuinely equivalent, but in quantum mechanics one often does have preferred observables: as Bohr would argue from his Como Lecture in 1927 onwards (Bohr, 1928), these observables are singled out by the choice of some experimental context, and they are jointly measurable iff they commute (see also below). Though not necessarily developed with Bohr’s doctrine in mind, Dirac’s approach seems tailor-made for this situation, since his basis choice is equivalent to a choice of “preferred” physical observables, namely those that are diagonal in the given basis (for Heisenberg this was energy, while for Schr ̈ odinger it was position). Von Neumann’s abstract approach can deal with preferred observables and ex- perimental contexts, too, though the formalism for doing so is more demanding. Namely, for reasons ranging from quantum theory to ergodic theory via unitary group representations on Hilbert space, from 1930 onwards von Neumann devel- oped his theory of “rings of operators” (nowadays called von Neumann algebras ), partly in collaboration with his assistant Murray (von Neumann, 1930, 1931, 1938, 1940, 1949; Murray & von Neumann, 1936, 1937, 1943). For us, at least at the moment the point is that Dirac’s diagonal observables are formalized by maximal commutative von Neumann algebras A on some Hilbert space. These often come naturally with some specific realization of a Hilbert space; for example, on Heisen- berg’s Hilbert space 2 ( N ) on has A d = ∞ ( N ) , while Schr ̈ odinger’s L 2 ( R 3 ) is host to A c = L ∞ ( R 3 ) , both realized as multiplication operators (cf. Proposition B.73). Introduction 3 Although the second (1931) paper in the above list shows that von Neumann was well aware of the importance of the commutative case of his theory of operator al- gebras, he—perhaps deliberately—missed the link with Bohr’s ideas. As explained in the remainder of this Introduction, providing this link is one of the main themes of this book, but we will do so using the more powerful formalism of C*-algebras Introduced by Gelfand & Naimark (1943), these are abstractions and generaliza- tions of von Neumann algebras, so abstract indeed that Hilbert spaces are not even mentioned in their definition. Nonetheless, C*-algebras remain very closely tied to Hilbert spaces through the GNS -construction originating with Gelfand & Naimark (1943) and Segal (1947b), which implies that any C*-algebra is isomorphic to a well-behaved algebra of bounded operators on some Hilbert space (see § C.12). Starting with Segal (1947a), C*-algebras have become an important tool in math- ematical physics, where traditionally most applications have been to quantum sys- tems with infinitely many degrees of freedom, such as quantum statistical mechan- ics in infinite volume (Ruelle, 1969; Israel, 1979; Bratteli & Robinson, 1981; Haag, 1992; Simon, 1993) and quantum field theory (Haag, 1992; Araki, 1999). Although we delve from the first body of literature, and were at least influenced by the second, the present book employs C*-algebras in a rather different fashion, in that we exploit the unification they provide of the commutative and the noncom- mutative “worlds” into a single mathematical framework (where one should note that as far as physics is concerned, the commutative or classical case is not purely C*-algebraic in character, because one also needs a Poisson structure, see Chapter 3). This unified language (supplemented by some category theory, group(oid) the- ory, and differential geometry) gives a mathematical handle on Wittgenstein’s As- pektwechsel between classical and quantum-mechanical modes of description (see Preface), which in our view lies at the heart of the foundations of quantum physics. This “change of perspective”, which roughly speaking amounts to switching (and interpolating) between commutative and noncommutative C*-algebras, is added to Dirac’s transformation theory (which comes down to switching between generalized bases, or, equivalently, between maximal commutative von Neumann algebras). The central conceptual importance of the Aspektwechsel for this book in turn derives from our adherence to Bohr’s doctrine of classical concepts , which forms part of the Copenhagen Interpretation of quantum mechanics (here defined strictly as a bod