Reflections on Quanta, Symmetries, and Supersymmetries
Unitary representation theory has great intrinsic beauty which enters other parts of mathematics at a very deep level. In quantum physics it is the preferred language for describing symmetries and supersymmetries. Two of the greatest figures in its histor
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V. S. Varadarajan
Reflections on Quanta, Symmetries, and Supersymmetries
1C
V. S. Varadarajan University of California Department of Mathematics Los Angeles, CA 90095-1555 USA
ISBN 978-1-4419-0666-3 e-ISBN 978-1-4419-0667-0 DOI 10.1007/978-1-4419-0667-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928909 Mathematics Subject Classification (2010): 22D10, 22E46, 34N99, 81P99 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
In memory of Mackey and Harish-Chandra
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Reality and its description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A quantum education and evolution . . . . . . . . . . . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2
Quantum Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The quantum algebra of Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The von Neumann perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The measurement algebra of Schwinger . . . . . . . . . . . . . . . . . . . 2.4 Weyl–Moyal algebra and the Moyal bracket . . . . . . . . . . . . . . . 2.5 Quantum algebras over phase space . . . . . . . . . . . . . . . . . . . . . . 2.6 Moshe Flato remembered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 15 19 23 35 49 51 55
3
Probability in the quantum world . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The statistical interpretation of quantum theory . . . . . . . . . . . . 3.2 The uncertainty principle of Heisenberg . . . . . . . . . . . . . . . . . . 3.3 Hidden variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Transition probabilities in
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