Lecture Notes in Physics

Transcription

1 Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, M(Jnchen, K. Hepp, Z~rich R. Kippenhahn, MLinchen, H. A. Weidenmeller, Heidelberg and J. Zittartz, K61n 190 Karl Kraus States, Effects, and Operations Fundamental Notions of Quantum Theory Lectures in Mathematical Physics at the University of Texas at Austin Edited by A. BShm, J. D. Dollard and W. H. Wootters Springer-Verlag Berlin Heidelberg New York Tokyo 1983

2 Author Karl Kraus Physikalisches Institut der Universit~t WGrzburg Am Hubland, D-8700 WGrzburg Editors A. B0hm Physics Department, University of Texas Austin, TX 78712, USA J.D. Dollard Mathematics Department, University of Texas Austin, TX 78712, USA W. H. Wootters Department of Physics and Astronomy, Williams College Williamstown, Mass , USA ISBN Springer-Verlag Berlin Heidelberg New York Tokyo ISBN Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Kraus, Karl, States effects, and operations. (Lecture notes in physics; 190) 1. Quantum theory. I. Title. II. Series. QC K ' ISBN (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/

3 Dem Andenken meiner Mutter Marie Kraus geb. Paus 4. Juli Oktober 1980

4 Preface The lecture notes of this volume are based on a series of lectures given in the mathematical physics program at the University of Texas at Austin. These lecture series were started in 1976 with the purpose of establishing communication between mathematicians and physicists and of informing graduate students of both departments about recent developments in mathematics and physics. The lectures are directed at "non-speclallsts", and this volume also should be suitable for a general audience. The subject of this volume is the interpretation and the foundations of quantum theory. There exist diverging opinions about this subject. Some people believe that quantum physics is so remote from the usual way of thinking and so foreign to everyday language that its meaning cannot be communicated directly. According to them, the only way to an understanding of quantum mechanics is to study many separate problems and thereby acquire an intuitive feeling for the foundations of quantum mechanics. The premise for the work discussed here is that the foundations of quantum mechanics can be verbalized, and the present lectures are an attempt to communicate them. But the champions of this field also have different points of view concerning the interpretation of quantum mechanics. The purpose of these lecture notes is not to give a comparative review of the different approaches - this can be found in "Interpretations and Foundations of Quantum Theory", H. Neumann (Ed.), BI-Wissenschaftsverlag (1981). Here just one approach is presented, the one developed by Ludwig and his school, which appears to be sufficiently general to encompass most of the other orthodox (Copenhagen) interpretations of quantum mechanics.

5 Vl The usual custom in this field is to start from physically meaningful postulates in a very general mathematical language, and derive from them the usual formalism of quantum mechanics in terms of linear operators in linear scalar product spaces. Even restricting oneself to only one approach, this is still a very tedious procedure. A more accessible road to the foundations of quantum mechanics is to start with the mathematically more familiar objects of operators and vectors in a Hilbert space, rather than deriving them, and to formulate the fundamental statements in terms of these mathematical quantities. This is the road that has been followed in these lectures in order to make this important and fundamental subject accessible to a larger audience. Arno B~hm J.D. Dollard W.H. Wootters (Editors)

6 Author's Preface The subject of these notes is ordinary quantum mechanics. More specifically, they deal with some particular aspects of the conceptual structure of that theory, which - at least in this form and to this extent - are not treated in the usual textbooks. The reader I had in mind when writing these notes, therefore, should already be familiar with the standard formulation of quantum mechanics, and should be interested not only in applications to specific physical problems but also in the general structure and the physical interpretation of the theory. As quantum mechanics is usually formulated in terms of vectors and linear operators in Hilbert space, some basic knowledge of this mathematical framework has been assumed. Whenever more advanced mathematical tools are required, I have tried to explain them - including suitable references, if necessary - in order to facilitate the reading. For the same reason, most mathematical deductions are presented in full detail. I hope that readers who are not too much interested in such technicalities will just glance over them, and concentrate their attention on the more essential statements. These notes are based on a series of lectures given at the University of Texas at Austin in September An earlier account of the topics discussed in these lectures has been published in the proceedings of the conference on "Foundations of Quantum Mechanics and Ordered Linear Spaces" at Marburg in 1973 [I]. The invitation to lecture on these matters at the University of Texas at Austin provided me with an occasion to rethink the whole subject, to extend my results in several aspects, and to reformulate everything in a more selfcontalned way.

7 VIII For this invitation, and for their kind hospitality during my pleasant stay at Austin, I want to thank Profs. A. Bohm, J. Dollard, T.A. Griffy, and J.A. Wheeler. I am also very grateful to Dr. W.H. Wootters (Austin), who read a preliminary version of my notes, corrected my English, and suggested several other improvements. Last but not least, I want to express my hearty thanks to Mrs. Ch. Steinbauer and Dr. W. Petzold (WUrzburg), who have been burdened with the tedious task of typewrinting and proofreading, and to Prof. W. Beiglb~ck from Springer Verlag for his great patience and constant encouragement during the preparation of the final typescript.

8 Contents I States and Effects Operations The First Representation Theorem Composite Systems The Second Representation Theorem Coexistent Effects and Observables References