This page intentionally left blank Deep Beauty Understanding the Quantum World Through Mathematical Innovation No scientific theory has caused more puzzlement and confusion than quantum theory. Physics is supposed to help us to understand the world, but quantum theory makes it seem a very strange place. This book is about how mathematical innovation can help us gain deeper insight into the structure of the physical world. Chapters by top researchers in the mathematical foundations of physics explore new ideas, especially novel mathematical concepts, at the cutting edge of future physics. These creative developments in mathematics may catalyze the advances that enable us to understand our current physical theories, especially quantum theory. The authors bring diverse perspectives, unified only by the attempt to introduce fresh concepts that will open up new vistas in our understanding of future physics. Hans Halvorson is Professor of Philosophy at Princeton University. He has written ex- tensively on the foundations of quantum physics, with articles appearing in the Journal of Mathematical Physics, Physical Review, Philosophy of Science, and British Journal of Philosophy of Science, among others. He is currently working on applying the tools of category theory to questions in the foundations of mathematics. Halvorson has been the recipient of the Mellon New Directions Fellowhip (2007), the Cushing Memorial Prize in the History and Philosophy of Physics (2004), Best Article of the Year by a Recent Ph.D. (Philosophy of Science Association, 2001), and Ten Best Philosophy Articles of the Year (The Philosopher’s Annual, 2001 and 2002). Deep Beauty Understanding the Quantum World Through Mathematical Innovation Edited by Hans Halvorson Princeton University cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao˜ Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107005709 C Cambridge University Press 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Deep beauty: understanding the quantum world through mathematical innovation / edited by Hans Halvorson. p. cm. Includes bibliographical references and index. ISBN 978-1-107-00570-9 (hardback) 1. Quantum theory – Mathematics. 2. Mathematical physics. I. Halvorson, Hans. QC174.17.M35D44 2011 530.120151–dc22 2010046924 ISBN 978-1-107-00570-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Just as the essence of perception is not sensing objects but apprehending them, even if we can only apprehend them through the mediation of sense, so the paradigm of a real world is not its sensible imaginability but its intelligible apprehensibility. I do not mean by this that anything which can be conceived by the intellect is thereby shown to exist, but I do mean that anything that concretely exists can be grasped by the intellect in its concrete existence. If therefore the universe of modern physics is one in which all attempts to make it intelligible by models of sensory type fail and which requires for its systematisation the kind of concepts that are used by quantum physics, this does not in the least imply that it is unreal or subjective. It simply means that the formulae of quantum physics express the kind of intelligibility that it has. Eric Mascall (1905–1993) Contents Contributors page ix Preface xi Acknowledgments xiii Introduction1 Hans Halvorson I Beyond the Hilbert Space Formalism: Category Theory 1 A Prehistory of n-Categorical Physics 13 John C. Baez and Aaron D. Lauda 2 A Universe of Processes and Some of Its Guises 129 Bob Coecke 3 Topos Methods in the Foundations of Physics 187 Chris J. Isham 4 The Physical Interpretation of Daseinisation 207 Andreas Doring¨ 5 Classical and Quantum Observables 239 Hans F. de Groote 6 Bohrification 271 Chris Heunen, Nicolaas P. Landsman, and Bas Spitters II Beyond the Hilbert Space Formalism: Operator Algebras 7 Yet More Ado about Nothing: The Remarkable Relativistic Vacuum State 317 Stephen J. Summers 8 Einstein Meets von Neumann: Locality and Operational Independence in Algebraic Quantum Field Theory 343 Miklos´ Redei´ vii viii contents III Behind the Hilbert Space Formalism 9 Quantum Theory and Beyond: Is Entanglement Special? 365 Borivoje Dakic´ and Caslavˇ Brukner 10 Is Von Neumann’s “No Hidden Variables” Proof Silly? 393 Jeffrey Bub 11 Foliable Operational Structures for General Probabilistic Theories 409 Lucien Hardy 12 The Strong Free Will Theorem 443 John H. Conway and Simon Kochen Index 455 Contributors John C. Baez Hans F. de Groote Department of Mathematics, University Formerly of the Fachbereich Informatik of California, Riverside, California, und Mathematik, Goethe-Universitat¨ United States Frankfurt a.M., Germany Caslavˇ Brukner Andreas Doring¨ Faculty of Physics, University of Vienna, Computing Laboratory, University of Austria; Institute of Quantum Optics and Oxford, United Kingdom Quantum Information, Austrian Lucien Hardy Academy of Sciences, Vienna, Austria Perimeter Institute, Waterloo, Ontario, Jeffrey Bub Canada Philosophy Department and Institute for Chris Heunen Physical Science and Technology, Computing Laboratory, University of University of Maryland, College Park, Oxford, United Kingdom Maryland, United States Chris J. Isham Bob Coecke The Blackett Laboratory, Imperial Computing Laboratory, University of College, London, United Kingdom Oxford, United Kingdom Simon Kochen John H. Conway Department of Mathematics, Princeton Department of Mathematics, Princeton University, Princeton, New Jersey, University, Princeton, New Jersey, United States United States Nicolaas P. Landsman Borivoje Dakic´ Institute for Mathematics, Astrophysics, Faculty of Physics, University of Vienna, and Particle Physics, Radboud University Austria Nijmegen, Nijmegen, The Netherlands ix x contributors Aaron D. Lauda Bas Spitters Department of Mathematics, Columbia Department of Computer Science, University, New York, New York, Radboud University Nijmegen, United States Nijmegen, The Netherlands Miklos´ Redei´ Stephen J. Summers Department of Philosophy, Logic and Department of Mathematics, University Scientific Method, London School of of Florida, Gainesville, Florida, United Economics and Political Science, States London, United Kingdom Preface In the fall of 2006, I received a phone call from Charles (Chuck) Harper, then the Senior Vice President and Chief Strategist of the John Templeton Foundation (JTF).1 Chuck asked me, What are the topics that most need attention in your field (philosophy of physics) but that are not receiving funding from other sources? I told Chuck that current academic culture does not provide much encouragement for scientists to pursue “foundational” or “philosophical” issues. Whereas in previous centuries many of the great scientific minds also displayed a sharp philosophical acumen—witness Descartes, Leibniz, Bohr, Einstein, von Neumann, and Weyl—our current generation of great scientists seems to lack the time, ability, or interest to expand their scientific research into the more speculative or conceptual realms. So I told Chuck we should provide some encouragement to those scientists who wish to continue the tradition of philosophical reflection on their subjects. This book is the direct result of Chuck’s, and JTF’s, taking this idea seriously— that is, the idea that “philosophical” is not an antonym of “scientific.” Concretely, the support provided by JTF enabled us to bring a group of distinguished philosophers and scientists from around the globe to Princeton, New Jersey, for a two-day conference in October 2007.2 The invited speakers were given free rein to speak on whatever topic they chose. However, it immediately became apparent that there was a common methodological theme: Although the speakers had diverse research goals, they shared in common the vision of achieving a deeper insight into foundational issues by stretching the resources provided by traditional mathematics. (For elaboration on this point, see the introduction.) After the conference, it was agreed that the speakers would expand their ideas into chapters for a book. The book in your hands (or on your screen, or being read to you) 1 Currently Chancellor for International Distance Learning and Senior Vice President, Global Programs, of the American University System, as well as President of Vision-Five.com Consulting. 2 See http://symposia.templeton.org/deep beauty/. xi xii preface is the result. If our original intention for this book was realized—and we think that it was in excelsis—then its twelve essays carry the tradition of natural philosophy into the twenty-first century. Hans Halvorson Princeton University Acknowledgments Many thanks to the John Templeton Foundation (JTF) for its generous financial backing for this project. I hope that this book offers high yields in the growth of