Fundamental Theories of Physics 188 Klaas Landsman Foundations of Quantum Theory From Classical Concepts to Operator Algebras www.com 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 Dürr, München, Germany Ruth Durrer, Genève, Switzerland Roman Frigg, London, UK Christopher Fuchs, Boston, USA Giancarlo Ghirardi, Trieste, Italy Domenico J. Giulini, Bremen, Germany Gregg Jaeger, Boston, USA Claus Kiefer, Köln, 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 Wüthrich, Geneva, Switzerland Lai-Sang Young, New York City, USA www.com 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 scientific fields. Original contributions in well-established fields 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 fields. 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 profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard. More information about this series at http://www.com/series/6001 www.com Klaas Landsman Foundations of Quantum Theory From Classical Concepts to Operator Algebras www.com Klaas Landsman IMAPP Radboud University Nijmegen The Netherlands 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.
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The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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 www.com To Jeremy Butterfield www.com Preface ‘Der Kopf, so gesehen, hat mit dem Kopf, so gesehen, auch nicht die leiseste Ähnlichkeit (. “Du würdest doch sagen, dass sich das Bild jetzt gänzlich geändert hat!” Aber was ist anders: mein Eindruck? meine Stellungnahme? (. ) Ich beschreibe die Änderung wie eine Wahrnehmung, ganz, als hätte sich der Gegenstand vor meinen Augen geändert.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 Schrödinger’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. Schulte (Wittgenstein, 2009/1953, pp.com viii Preface 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., 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. 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 möglich 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 geklärten Interpretationsfragen näher untersucht werden.’ (von Neumann, Mathematische Grundlagen der Quantenmechanik, 1932, p.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. Beyer (von Neumann, 1955, p.com 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.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 één genialen greep de heele wereld wil omvatten (.’ (Dijksterhuis, Val en Worp, 1924, p.
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.
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. 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.
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.com Contents Introduction. 1 Part I C0 (X) and B(H) 1 Classical physics on a finite phase space .1 Basic constructions of probability theory .2 Classical observables and states .3 Pure states and transition probabilities .4 The logic of classical mechanics .5 The GNS-construction for C(X). 38 2 Quantum mechanics on a finite-dimensional Hilbert space .1 Quantum probability theory and the Born rule .2 Quantum observables and states .3 Pure states in quantum mechanics .4 The GNS-construction for matrices .5 The Born rule from Bohrification .6 The Kadison–Singer Problem .8 Proof of Gleason’s Theorem .9 Effects and Busch’s Theorem .10 The quantum logic of Birkhoff and von Neumann .