Quantum Stochastic Calculus and Representations of Lie Superalgebras
This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to
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Timothy M. W Eyre
Quantum Stochastic Calculus and Representations of Lie Superalgebras
Springer
Author Timothy M. W. Eyre Data Connection Ltd 100 Church Street Enfield, EN2 6BQ, UK e-mail: [email protected] Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Eyre, Timothy M. W.: Quantum stochastic calculus and representations of Lie superalgebras / TImothy M. W. Eyre. - Berlin; Heidelberg; New York ; Barcelona ; Budapest ; Hong Kong; London ; Milan ; Paris ; Santa Clara; Singapore; Tokyo: Springer, 1998 (Lecture notes in mathematics; 1692) ISBN 3-540-64897-6
Mathematics Subject Classification (1991): 81S25 ISSN 0075-8434 ISBN 3-540-64897-6 Springer-Verlag Berlin Heidelberg New York 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Printed in Germany 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. Typesetting: Camera-ready TEX output by the author SPIN: 10649929 41/3143-543210 - Printed on acid-free paper
Preface
These notes are intended to give a self-contained exposition of Z2-graded quantum stochastic calculus and the resulting representations of Lie superalgebras. No pre-knowledge of quantum stochastic calculus is assumed; Chapter 2 presents the Hudson-Parthasarathy formulation of the theory in a straightforward manner. I should like to thank Professor Robin Hudson for his collaboration during the years 1994-1997 which I spent at The University of Nottingham where this research was carried out. I should also like to thank EPSRC for funding this research. Special thanks must go to the Indian Statistical Institute New Delhi Centre, India (especially K. R. Parthasarathy), the Wydzial Matematyki, University of Lods, Poland (especially Stanislaw Goldstein) and the Institute Fourier, Grenoble, France (especially Stephane Attal) for their kind hospitality. These academic visits were all of great value to the research.
Table of Contents
1.
Introduction 1.1 Discussion 1.2 Overview 1.3 Notational Conventions
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2 5
2.
Quantum Stochastic Calculus 2.1 The Kernel Construction and Fock Space 2.2 The Processes of Quantum Stochastic Calculus 2.3 Construction of Simple
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