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4Et moi, ..., si j'avait su comment en revenir, One service
mathematics has rendered the je n'y serais point alle.' human race.
It has put common sense back Jules Verne where it belongs, on the
topmost shelf next to the dusty canister labelled 'discarded non*
The series is divergent; therefore we may be sense'. able to do
something withit. Eric T. Bell O. Heaviside Mathematicsis a tool
for thought. A highly necessary tool in a world whereboth feedback
and non- linearities abound. Similarly, all kinds of parts of
mathematics serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above
one finds such statements as: 'One service topology has rendered
mathematical physics ...'; 'One service logic has rendered com-
puter science ...'; 'One service category theory has rendered
mathematics ...'. All arguably true. And all statements obtainable
this way form part of the raison d' tre ofthis series.
TO SUPERANAL YSIS Edited by A. A. KIRILLOV Translated from the
Russian by J. Niederle and R. Kotecky English translation edited
and revised by Dimitri Leites SPRINGER-SCIENCE+BUSINESS MEDIA, B.
V. Library of Congress Cataloging-in-Publication Data Berezin, F.
A. (Feliks Aleksandrovich) Introduction to superanalysis.
(Mathematical physics and applied mathematics; v. 9) Part I is
translation of: Vvedenie v algebru i analiz s
antikommutirurushchimi peremennymi. Bibliography: p. Includes
index. 1. Mathetical analysis. I. Title. II. Title: Superanalysis.
III. Series. QA300. B459 1987 530. 15'5 87-16293 ISBN
978-90-481-8392-0 ISBN 978-94-017-1963-6 (eBook) DOI 10.
1007/978-94-017-1963-6 All Rights Reserved (c) 1987 by Springer
Science+Business Media Dordrecht Originally published by D. Reidel
Publishing Company, Dordrecht, Holland in 1987 No part of the
material protected by this copyright notice may be reproduced in
whole or in part or utilized in any form or by any means electronic
or mechanical including photocopying recording or storing in any
electronic information system without first obtaining the written
permission of the copyright owner. CONTENTS EDITOR'S FOREWORD ix
INTRODUCTION 1 1. The Sources 1 2. Supermanifolds 3 3. Additional
Structures on Supermanifolds 11 4. Representations of Lie
Superalgebras and Supergroups 21 5. Conclusion 23 References 24
PART I CHAPTER 1. GRASSMANN ALGEBRA 29 1. Basic Facts on
Associative Algebras 29 2. Grassmann Algebras 35 3. Algebras A(U)
55 CHAPTER 2. SUPERANAL YSIS 74 1. Derivatives 74 2. Integral 76
CHAPTER 3. LINEAR ALGEBRA IN Zz-GRADED SPACES 90 1.
This volume deals with those topics of mathematical physics,
associated with the study of the SchrAdinger equation, which are
considered to be the most important. Chapter 1 presents the basic
concepts of quantum mechanics. Chapter 2 provides an introduction
to the spectral theory of the one-dimensional SchrAdinger equation.
Chapter 3 opens with a discussion of the spectral theory of the
multi-dimensional SchrAdinger equation, which is a far more complex
case and requires careful consideration of aspects which are
trivial in the one-dimensional case. Chapter 4 presents the
scattering theory for the multi-dimensional non-relativistic
SchrAdinger equation, and the final chapter is devoted to
quantization and Feynman path integrals. These five main chapters
are followed by three supplements, which present material drawn on
in the various chapters. The first two supplements deal with
general questions concerning the spectral theory of operators in
Hilbert space, and necessary information relating to Sobolev spaces
and elliptic equations. Supplement 3, which essentially stands
alone, introduces the concept of the supermanifold which leads to a
more natural treatment of quantization. Although written primarily
for mathematicians who wish to gain a better awareness of the
physical aspects of quantum mechanics and related topics, it will
also be useful for mathematical physicists who wish to become
better acquainted with the mathematical formalism of quantum
mechanics. Much of the material included here has been based on
lectures given by the authors at Moscow State University, and this
volume can also be recommended as a supplementary graduate level
introduction to the spectral theory ofdifferential operators with
both discrete and continuous spectra. This English edition is a
revised, expanded version of the original Soviet publication.
TO SUPERANAL YSIS Edited by A. A. KIRILLOV Translated from the
Russian by J. Niederle and R. Kotecky English translation edited
and revised by Dimitri Leites SPRINGER-SCIENCE+BUSINESS MEDIA, B.
V. Library of Congress Cataloging-in-Publication Data Berezin, F.
A. (Feliks Aleksandrovich) Introduction to superanalysis.
(Mathematical physics and applied mathematics; v. 9) Part I is
translation of: Vvedenie v algebru i analiz s
antikommutirurushchimi peremennymi. Bibliography: p. Includes
index. 1. Mathetical analysis. I. Title. II. Title: Superanalysis.
III. Series. QA300. B459 1987 530. 15'5 87-16293 ISBN
978-90-481-8392-0 ISBN 978-94-017-1963-6 (eBook) DOI 10.
1007/978-94-017-1963-6 All Rights Reserved (c) 1987 by Springer
Science+Business Media Dordrecht Originally published by D. Reidel
Publishing Company, Dordrecht, Holland in 1987 No part of the
material protected by this copyright notice may be reproduced in
whole or in part or utilized in any form or by any means electronic
or mechanical including photocopying recording or storing in any
electronic information system without first obtaining the written
permission of the copyright owner. CONTENTS EDITOR'S FOREWORD ix
INTRODUCTION 1 1. The Sources 1 2. Supermanifolds 3 3. Additional
Structures on Supermanifolds 11 4. Representations of Lie
Superalgebras and Supergroups 21 5. Conclusion 23 References 24
PART I CHAPTER 1. GRASSMANN ALGEBRA 29 1. Basic Facts on
Associative Algebras 29 2. Grassmann Algebras 35 3. Algebras A(U)
55 CHAPTER 2. SUPERANAL YSIS 74 1. Derivatives 74 2. Integral 76
CHAPTER 3. LINEAR ALGEBRA IN Zz-GRADED SPACES 90 1.
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