www.EngineeringBooksPDF.com The Geometry of the Octonions www.EngineeringBooksPDF.com 8456_9789814401814_tp.indd 18/3/15 3:58 pm May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank www.EngineeringBooksPDF.com PST˙ws The Geometry of the Octonions Tevian Dray Department of Mathematics Oregon State University, USA Corinne A Manogue Department of Physics Oregon State University, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI www.EngineeringBooksPDF.com 8456_9789814401814_tp.indd 18/3/15 3:58 pm Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Dray, Tevian The geometry of the octonions / Tevian Dray (Oregon State University, USA) & Corinne A Manogue (Oregon State University, USA) pages cm Includes bibliographical references and index ISBN 978-9814401814 Cayley numbers (Algebra) Cayley algebras Nonassociative algebras Geometry, Algebraic I Manogue, Corinne A II Title QA252.5.D73 2015 512'.5 dc23 2015003048 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright © 2015 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher In-house Editors: Lai Fun Kwong/V Vishnu Mohan Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore www.EngineeringBooksPDF.com Vishnu - The geometry of the octonions.indd 16/2/2015 2:49:10 PM March 19, 2015 10:44 The Geometry of the Octonions - 9in x 6in b2077-OctoGeom To David and Tony, for getting us started www.EngineeringBooksPDF.com v page v May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank www.EngineeringBooksPDF.com PST˙ws March 19, 2015 10:44 The Geometry of the Octonions - 9in x 6in b2077-OctoGeom Preface This is a book about the octonions, a bigger and better version of the complex numbers, albeit with some subtle properties Bigger, because there are more square roots of −1 Better, because an octonionic formalism provides natural explanations for several intriguing results in both mathematics and physics Subtle, because the rules are more complicated; order matters Some readers may be familiar with the quaternions, which lie halfway between the complex numbers and the octonions Originally developed more than 100 years ago to be the language of electromagnetism, an effort that lost out in the end to the use of vector analysis, the quaternions have been reborn as a useful tool for applications as diverse as aeronautical engineering, computer graphics, and robotics What will the octonions be good for? This authors believe that the octonions will ultimately be seen as the key to a unified field theory in physics But that is a topic for another day, although hints of this vision can be found here This book is intended as an introduction to the octonions It is not a mathematics text; theorems and proofs (and references!) are few and far between Nonetheless, the presentation is reasonably complete, with most results supported by at least the outline of the underlying computations The only true prerequisite for reading this book is the ability to multiply matrices, and a willingness to follow computational arguments Familiarity with linear algebra is a plus, up to the level of finding eigenvalues and eigenvectors And of course comfort with the complex numbers is a must, or rather a willingness to become comfortable with them The book is divided into three parts Part I discusses several different number systems, emphasizing the octonions Part II is the heart of the book, taking a detailed look at a particular collection of symmetry groups, including orthogonal, unitary, symplectic, and Lorentz groups, all vii www.EngineeringBooksPDF.com page vii March 19, 2015 viii 10:44 The Geometry of the Octonions - 9in x 6in b2077-OctoGeom The Geometry of the Octonions expressed in terms of division algebras, up to and including the octonions As we demonstrate, octonions provide the language to describe the so-called exceptional Lie groups Finally, Part III contains a rather eclectic collection of applications of the octonions, in both mathematics and physics A companion website for the book is available at http://octonions.geometryof.org which is (partially) mirrored at http://physics.oregonstate.edu/coursewikis/GO/bookinfo Tevian Dray Corinne Manogue Corvallis, OR October 2014 www.EngineeringBooksPDF.com page viii March 19, 2015 10:44 The Geometry of the Octonions - 9in x 6in b2077-OctoGeom Acknowledgments This book has its origins in the research conducted by one of us (Corinne) nearly 30 years ago She in turn introduced her husband (Tevian) to the octonions more than 20 years ago; we have collaborated on further research in this area ever since This book reflects Tevian’s efforts to understand what Corinne has taught him; although most of the actual writing was done by Tevian, the final product has been shaped every bit as much by Corinne, however indirectly Rob Wilson has been a part of this collaboration for the last five years, and has shown us that the mathematical structure of the octonions is even richer than we had dreamed, going far beyond the topics presented here Special mention and thanks are due our students, Jă org Schray, Jason Janesky, Aaron Wangberg, Joshua Kincaid, and Henry Gillow-Wiles, whose published work has been incorporated where appropriate, and to our most recent collaborator, John Huerta This book is however dedicated to David Fairlie and Tony Sudbery, Corinne’s collaborators and mentors for her early work relating the octonions to string theory Our ongoing work with the octonions, including this book, is a testament to their vision; they got us started This book would not have been possible without the financial support and encouragement of the John Templeton Foundation, for which we are very grateful An early draft of the material in Chapter 14 formed the basis for formal and informal seminars at several institutions, including Mount Holyoke and Grinnell Colleges in 2002, and Oregon State University in 2004 Sections 11.5, 12.4, and much of Chapters 13 and 15 are based on the authors’ published research, as cited there Finally, all figures in this book appeared initially on the authors’ website at http://physics.oregonstate.edu/coursewikis/GO with a Creative Commons by-nc-nd license, and are used by permission ix www.EngineeringBooksPDF.com page ix ... The Geometry of SO(2) The Geometry of SO(3) The Geometry of SO(4) Lorentz Transformations The Geometry of SO(3, 1) The Geometry of SO(4, 2) Unitary Transformations The Geometry of U(1)... 11 The Exceptional Groups 11.1 11.2 11.3 11.4 11.5 11.6 The The The The The The Geometry of G2 Albert Algebra Geometry of F4 Geometry of E6 Geometry of E7 Geometry of E8 ... www.EngineeringBooksPDF.com 10 11 13 March 19, 2015 10:44 The Geometry of the Octonions - 9in x 6in The Geometry of the Octonions 4.1 4.2 4.3 4.4 page xii The Geometry of the Octonions