Quantum Potential Theory

This volume contains the revised and completed notes of lectures given at the school "Quantum Potential Theory: Structure and Applications to Physics," held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald from February 26 to March 10, 2007. Quantum

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1954

Philippe Biane · Luc Bouten · Fabio Cipriani Norio Konno · Nicolas Privault · Quanhua Xu

Quantum Potential Theory Editors: Michael Schürmann Uwe Franz

ABC

Editors Uwe Franz

Michael Schürmann

Département de mathématiques de Besancon Université de Franche-Comté 16, route de Gray 25030 Besancon cedex France [email protected] and Graduate School of Information Sciences Tohoku University 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai 980-8579 Japan

Institut für Mathematik und Informatik Ernst-Moritz-Arndt-Universität Greifswald Jahnstrasse 15a 17487 Greifswald Germany [email protected]

ISBN: 978-3-540-69364-2 e-ISBN: 978-3-540-69365-9 DOI: 10.1007/978-3-540-69365-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008932185 Mathematics Subject Classification (2000): 58B34, 81R60, 31C12, 53C21 c 2008 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

This volume contains the notes of lectures given at the School “Quantum Potential Theory: Structure and Applications to Physics”. This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald from February 26 to March 9, 2007. We thank the lecturers for the hard work they accomplished in preparing and giving these lectures and in writing these notes. Their lectures give an introduction to current research in their domains, which is essentially self-contained and should be accessible to Ph.D. students. We hope that this volume will help to bring together researchers from the areas of classical and quantum probability, functional analysis and operator algebras, and theoretical and mathematical physics, and contribute in this way to developing further the subject of quantum potential theory. We are greatly indebted to the Alfried Krupp von Bohlen und HalbachStiftung for the financial support, without which the school would not have been possible. We are also very thankful for the support by the University of Greifswald and the University of Franche-Comt´e. One of the organisers (UF) was supported by a Marie Curie Outgoing International Fell