The Mathematical Apparatus for Quantum-Theories Based on the Theory
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Herausgegeben von
J. L. Doob
. E. Heinz· F. Hirzebruch . E. Hopf H. Hopf . W. Maak . S. MacLane W. Magnus· D. Mumford· F. K. Schmidt· K. Stein
GeJchajtsjiihrende HerauJgeber
B. Eckmann und B. L. van der Waerden
The Mathematical Apparatus for Quantum -Theories Based on the Theory of Boolean Lattices
Otton Martin Nikodym
Springer-Verlag New York Inc. 1966
Geschaftsfilhrende Herausgeber:
Prof. Dr. B. Eckmann Eidgenossische Techuische Hochschule Ziirich
Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitat Ziirich
ISBN-13: 978-3-642-46032-6 e-ISBN-13: 978-3-642-46030-2 DOl: 10.1007/978-3-642-46030-2 All rights reserved, especially that of translation into foreign languages It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard or any other means)
without written permission from the Publishers
© 1966 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1966 Library of Congress Catalog Card Number 66·27977
Tide No.5 I 12
Dedicated to my wife Dr. Stanislawa Nikodym
Preface The purpose of this book is to ,give the theoretical physicist a geometrical, visual and precise mathematical apparatus which would be better adapted to some of their arguments, than the existing and generally applied methods. The theories, presented in this book, are based on the theory of Boolean lattices, whose elements are closed subspaces in the separable and complete Hilbert-Hermite-space. The first paper, in which the outlines of the said mathematical apparatus is sketched, is that of the author: "Un nouvel appareil mathematique pour la theorie des quanta."] The theory exhibited in this paper has been simplified, generalized and applied to several items of the theory of maximal normal operators in Hilbert-space, especially to the theory of multiplicity of the continuous spectrum and to permutable normal operators, based on a special canonical representation of normal operators and on a general system of coordinates in Hilbert-space, which is well adapted not only to the case of discontinuous spectrum, but also to the continuous one. The normal operators, which can be roughly characterized as operators with orthogonal eigen-vectors and complex eigen-values, constitute a generalization of hermitean selfadjoint and of unitary operators. The importance of the methods, sketched in the mentioned paper, has been emphasized in the review in the "Zentralblatt fUr Mathematik", by the physicist G. LUDWIG 2 and later applied by him in his book "Die Grundlagen der Quantenmechanik"3. The mentioned theory has 1 Annales de l'Institut HENRI POINCARE, tome The paper constitutes the content of four lectures by and 13, (1947) at the Institut HENRI POINCARE in 2 Bd.37, 1951, P.278/279. 3 Berlin/Gottingen/Heidelberg: Springer-Verlag. matischen Wissenschaften 52. (1954), XII 460 pp.
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XI, fasc. II, pages 49-112. the author: February 4,6, 11 Paris. Die Grundlehren der Mathe(see the footnote p. 75).
VIII
Preface
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