THE LOGICO-ALGEBRAIC APPROACH TO QUANTUM MECHANICS

VOLUME I

HISTORICAL EVOLUTION

THE UNIVERSITY OF WESTERN ONT ARIO

SERIES IN PHILOSOPHY OF SCIENCE

A SERIES OF BOOKS

ON PHILOSOPHY OF SCIENCE, METHODOLOGY,

AND EPISTEMOLOGY

PUBLISHED IN CONNECTION WITH

THE UNIVERSITY OF WESTERN ONTARIO

PHILOSOPHY OF SCIENCE PROGRAMME

Managing Editor

J. J. LEACH

Editorial Board

J. BUB, R. E. BUTTS, W. HARPER, J. HINTIKKA, D. J. HOCKNEY,

C. A. HOOKER, J. NICHOLAS, G. PEARCE

VOLUME 5

THE LOGICO-ALGEBRAIC

APPROACH TO

QUANTUM MECHANICS

VOLUME I

Historical Evolution

Edited by

C. A. HOOKER University of Western Ontario, Ontario, Canada

D. REIDEL PUBLISHING COMPANY

DORDRECHT-HOLLAND / BOSTON-U.S.A.

Library of Congress Cataloging in Publication Data

Main entry under title:

The Logico-algebraic approach to quantwn mechanies.

(The University of Westem Ontario series in philosophy of science ; v. 5)

Ineludes bibliographical referenees. CONTENTS: v. 1. Historieal evolution. 1. Quantwn theory-Addresses, essays, lectures.

2. Logic, Symbolic and mathematical-Addresses, essays, lectures. I. Hooker, Clitford Alan. II. Series: London, Ont. University of Westem Ontario. Series in philosophy of scienee ; v. 5. QCI74.125.L63 530.1 '2 75-8737

ISBN-13: 978-90-277-0613-3 e-ISBN-13: 978-94-010-1795-4 OOl: 10.1 007/978-94-0 I 0-1795-4

Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland

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by D. Reidel Publishing Company, Inc. 306 Dartmouth Street, Boston,

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All Rights Reserved Copyright © 1975 by D. Reidel Publishing Company, Dordrecht, Holland

No part of the material protected by this copyright notice may be reprodueed or utilized in any form or by any means, electronic or mechanieal,

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T ABLE OF CONTENTS

PREFACE

ACKNOWLEDGEMENTS

G. BIRKHOFF and J. VON NEUMANN / The Logie of Quantum Mechanies (1936)

M. STRAUSS / The Logie of Complementarity and the Foundation

VII

XIII

ofQuantum Theory (1972) 27

M. STRAUSS / Mathematies as Logieal Syntax - A Method to For-malize the Language of a Physieal Theory (1937-38) 45

H. REICHENBACH / Three-Valued Logie and the Interpretation of Quantum Mechanies (1944) 53

H. PUTNAM / Three-Valued Logie (1957) 99

P. FEYERABEND / Reiehenbaeh's Interpretation of Quantum Me-ehanies (1958) 109

A. M. GLEASON / Measures on the ClosOO Subspaees of a Hilbert Space (1957) 123

E. P. SPECKER / The Logie of Propositions Whieh are not Simul-taneously Decidable (1960) 135

D. J. FOULIS / Baer *-Semigroups (1960) 141

N. ZIERLER / Axioms for Non-Relativistie Quantum Mechanies (1961) 149

v. S. VARADARAJAN / Probability in Physies and a Theorem on Simultaneous Observability (1962) 171

J. LOS / Semantie Representation of the Probability of Formulas in Formalized Theories (1963) 205

F. KAMBER / The Structure of the Propositional Caleulus of a Physieal Theory (1964) 221

VI T AB LE OF CONTENTS

N. ZIERLER and M. SCHLESSINGER / Boolean Embeddings of Orthomodular Sets and Quantum Logic (1965) 247

S. KocHENandE. P. SPECKER/ Logical Structures Arisingin Quan-tum Theory (1965) 263

s. KOCHEN and E. P. SPECKER / The Calculus of Partial Proposi-tional Functions (1965) 277

s. KOCHEN and E. P. SPECKER / The Problem of Hidden Variables in Quantum Mechanics (1967) 293

P. SUPPES / Logics Appropriate to Empirical Theories (1965) 329

P. SUPPES / The Probabilistie Argument for a Non-Classical Logic of Quantum Mechanies (1966) 341

M. STRAUSS / Foundations of Quantum Mechanics (1967) 351

J. c. T. POOL / Baer *-Semigroups and the Logic of Quantum Me-chanics (1968) 365

J. c. T. POOL / Semimodularity and the Logic of Quantum Me-chanics (1968) 395

P. D. FINCH / On the Structure of Quantum Logic (1969) 415

J. M. JAUCH and c. PIRON / On the Structure of Quantal Proposi-tion Systems (1969) 427

s. s. HOLLAND, JR. / The Current Interest in Orthomodular Lat-tices (1970) 437

N. s. KRONFLI / Integration Theory of Observables (1970) 497 N. s. KRONFLI / Probabilistie Formulation of Classieal Mechanies

(1970) 503

N. s. KRONFLI / Atomicity and Determinism in Boolean Systems (1971) 509

c. PIRON / Survey of General Quantum Physies (1972) 513

R. J. GREECHIE and s. P. GUDDER / Quantum Logies (1974) 545

B. c. v AN FRAASSEN / The Labyrinth of Quantum Logics (1974) 577

PREFACE

The twentieth century has witnessed a striking transformation in the un­derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi­tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc­ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied.

Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc­tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip­ulation of purely abstract structures. (The earlier history is set out in some detail in e.g. M. Jammer, The Conceptua/ Deve/opment of the Quantum Theory, McGraw-HilI, 1966 and a subsequent companion volume now in press.) A similar story holds for the development of relativistic theories, and of course for the reeent attempts to consistently combine them.

Despite this evolution of pure mathematical sophistication and self­consciousness we are at the present time, so it seerns, neither any closer to an adequate atomic theory nor yet to a satisfactory physical inter­pretation of even elementary quantum theory. At least in regard to the

VIII PREFACE

latter problem, in my opinion, the situation stems directly from the faet that our conceptual understanding of physical theory has been even more slowly evolving than has our understanding of the mathematical struc­ture of those theories themselves. In this respect the philosophy of science has itself undergone its own twentieth century revolution. From the 'everyday' intuitions and debates about detenninism, mechanism and the like there has emerged slowly a tradition of conceptual analysis in which the ideal is to characterize conceptually interesting questions so sharply that they admit of fonnally precise answers. (Thus "Is the theory deter­ministic?" becomes "Does the theory admit such and such mathematical construction?" and an analysis of the predicate '-is detenninistic' be­comes '-has structure S'.) The examination of physical theory from this point of view has led to many rich and fruitful insights.

It is precisely the combination of the emerging mathematical sophis­tication and self-consciousness with the emerging fonnal sophistication and insight into the structure of conceptual schemes that is the founda­tion for the claim that the deep understanding of physical theory lies in the understanding of its abstract structures.

(In respeet of these revolutions we may note: (i) That each had its origins in preceding centuries, though not there the dominant tradition, one thinks of Hamilton and Jacobi or of Boole - even so most of the development lies in the twentieth century, mathematical logic, fonnal syntax and semantics, the use in physics of Hilbert space, group theory, lattice theory and so on all essentially belong to the last 60 years. (ii) In respect of contributions to the various developments the departmental origins of salary were not closely correlated with type of contribution - in­deed the various disciplines involved have never coped with the blurring of their separateness and are stiIl in the early stages of adjusting to the changing intellectual perspective.)

The papers in this volume all belong to one strand of this complex development - the understanding of elementary quantum theory through examination of its fonnal, or abstract, structure. Remarkably, this is the first time (to my knowledge) that anyone has attempted to collect them together, though the body of literature has been weil known to anyone approaching the subject in the last decade. (This is in itselfwitness to the newness of the perspective.) Considering the span of time covered and the diversity of authorship, the collection is satisfyingly complete - a

PREFACE IX

tribute to the eagemess of all concemed to see their work set in the wider perspective of the emerging field of study. Even casual examination of the dates of the papers - they are arranged nearly chronologically, not alphabetically or by theme - reveals an explosion of interest and fruitful work in the late 1960's. (Occasionally I have foregone strict chronological order so that related articles might be grouped together. This is so in the case of Reichenbach's 3-valued approach and responses, and the work of Kochen and Specker, to which should be attached the earlier paper by Specker.) /

The present vohJme bare1y reaches the beginnings of the recent ex­plosion of interest and productivity, some papers with a broader perspec­tive are included for the reader's benefit (principally those by Holland, Piron, Gudder and Greechie and van Fraassen), but by and large this volume concentrates on the foundational work laid down in the 'long period of initial struggle', 1935-1965. It is my intention to devote a second volume (to appear shortly) to recent research.

These articles do not belong to a single tradition conceming quantum theory, nothing like that could have emerged until the 1970's and the major points of view are only just now emerging clearly. Nor are they written from the same perspective - some are written by mathematicians for mathematicians, others by logicians for philosophers. These articles represent the 'raw material' for study. This volume is designed to be a basic reference text, not the presentation of a particular doctrine. I have my own understanding of the significance of these papers, but that will appear as part of Volume II which will be more devoted to works that analyse and interpret the mathematical material than is this volume.

In keeping with this referential role for the volume I shall refrain here from taking issue with the various interpretive battles now raging and from any attempt to evaluate the relative significances of the various mathematical contributions. Where this text is used - senior under­graduate or graduate course - one assumes a competent leader who will set the material in some preferred order. Some passing remarks may be in order for the benefit of the disciplinarily one-sided reader. The approach of Birkhoff and von Neumann is connected to the structure of logical systems clearly for the first time in the work of Kochen and Specker, who also make clear the significance of Gleason's theorem in

x PREFACE

this context. The connection between this approach and lattice and POset t_heory is discussed at length in Holland and reviewed in Piron and Gudder and Greechie, while its corresponding and dose association with group theory is developed by Poole upon the basis of the funda­mental paper by Foulis. The important paper by Varadarajan ought to be read in conjunction with the discussion by Kochen and Specker con­cerning the probabilistic constraints on acceptable boolean embeddings (ef. Zieler and Schlessinger) and compared with the discussions by Suppes. Finally, it is weIl known to those in philosophy that these researches have generated a heated discussion concerning the nature of logic, a debate of great profundity but one barely hinted at here (ef. Kochen and Specker, Suppes, van Fraassen); rather these papers serve as the background to the debate, the debate itself being taken up in Volume II.

As I remarked earlier, the material spreads across the boundaries between mathematics, physics and philosophy. From a mathematician's point of view this volume is designed to offer some of the basic souree material for a study of the kind ofaxiomatic approach to quantum theory followed by George Mackey, Josef Jauch, and others, to connect it to the physical and conceptual (chiefly logical here) issues and to introduce several areas of mathematical enquiry delightful in their own rights. From the philosopher's point of view it is designed as a basic referenee text to educate in the formalism and results he (she!) must know in order to competently follow the current debates and to contribute thereto. For the physicist the book offers an introduction to that com­plex of mathematical and philosophical argument which constitutes a first example in the new way of studying physical theories.

From the point of view of an adequately complete referenee source it must be admitted that several other entire volumes ought also to be induded! Of course this is not possible. I shaIl mention now several other volumes which, if the reader will employ them to complement the present selection of journal artides, will provide him/her with a weIl rounded referenee library. Among the more important volumes are:

Bub, J., The Interpretation of Quantum Mechanies, D. Reidel Publishing Co., Dordrecht, 1974.

Hooker, C. A., Contemporary Research in the Foundations and Philosophy of Quantum Theory, D. Reide! Publishing Co., Dordrecht, 1974.

PREFACE XI

Jauch, J., The Foundations of Quantum Mechanics, Addison-Wesley, New York, 1968. Mackey, G. W., The MathematicaI Foundations of Quantum Mechanics, W. A. Benjamin,

New York, 1963. Varadarajan, V. S., The Geometry ofQuantum Mechanics, 2 vols., Van Nostrand, Princeton,

N.J.,1968.

London. Ontario. 1974.

ACKNOWLEDGEMENTS

I should like to aeknowledge the generous eo-operation of all of those journal editors and publishing houses whieh granted copyright permis­sion so readily and so generously, thereby making this volume possible. I should like partieularly to aeknowledge the work of Mrs A. Smith, without whose indefatigable efforts and intelligence I should not have been able to assemble the volume in anything but a semi-infinite time.

'The Logie of Quantum Mechanies' by G. Birkhoff and J. von Neumann in Annals of Mathematies 37 (1936), 823--43. Reprinted by permission of the surviving author (Birkhoft) and the Estate of von Neumann and Annals of Mathematies.

'The Logie of Complementarity and the Foundation of Quantum Theory' by M. Strauss in Modem Physies and Its Philosophy (ed. by M. Strauss), D. Reidel Publishing Co., Dordreeht-Holland, 1972. (Note: This is a translation of an article appearing in 1936 together wish a postseript added in 1971.) Reprinted by permission of the author and D. Reidel Publishing Co.

'Mathematies as Logieal Syntax - A Method to FormaHze the Language of a Physieal Theory' by M. Strauss in Erkenntnis 7 (1937-38), 147-153. Reprinted by permission of the author and D. Reidel Publ. Co.

'Three-Valued Logie and the Interpretation of Quantum Mechanies' by H. Reiehenbaeh. An edited portion of seetions 29-37 of Philosophie Fotmdations of Quantwn Meehanies by H. Reiehenbaeh, University of California Press, Los Angeles, 1944. Reprinted by permission of Maria Reiehenbaeh. Originally published by the University of California Press; reprinted by permission of the Regents of the University of California.

'Three-Valued Logie', by H. Putnam in Philosophical Studies VIII (1957), 73-80. Reprinted by permission of the author and Philosophiea/ Studies.

'Reiehenbach's Interpretation of Quantum Mechanies' by P. Feyerabend in Philosophieal Studies IX (1958), 49-59. Reprinted by permission of the author and Philosophical Studies.

'Measures on the Closed Subspaces of a Hilbert Space' by A. M. Gleason in Journal of Mathematics and Meehanics 6 (1957), 885-93. Reprinted by permission of the author and Indiana University Mathematics Journal. (Originally Journal of Mathematics and Mechanies.)

'The Logie of Propositions whieh are not Simultaneously Decidable' by E. P. Specker in Dialeetiea 14 (1960),239--46. (A translation by Mr. A. Stairs.) Reprinted by permission of the author and Dialeetica.

'Baer *-Semigroups' by D. J. Foulis in Proceedings of the American Mathematical Society 11 (1960), 648-54. Reprinted by permission of the author and American Mathe­matieal Society.

'Axioms for Non-Relativistic Quantum Mechanies' by N. Zierler in Paeific Journal of

XIV ACKNOWLEDGEMENTS

Mathematics 11 (1961), 1151-69. Reproduction is by pennission of the author and the copyright owners, the Pacific Journal of Mathematics.

'Probability in Physics and a Theorem on Simultaneous Observability' by V. S. Varadarajan in Communications in Pure and Applied Mathematics 15 (1962), 189-217. (As corrected Loe. eit. 18 (1965).) Reprinted by pennission of John Wiley & Sons, Inc.

'Semantic Representation of the Probability of Fonnulas in Fonnalized Theories' by J. Los in Studia Logico 14 (1963), 183-96. Reprinted by pennission of the author and Studia Logica.

'The Structure of the Propositional Calculus of a Physical Theory' by F. Kamber in Nach­richten der Akademie der Wissenschaften Mathematisch-Physikalische Klasse 10 (1964) 103-124. Reprinted by pennission of the author and Nachrichten der Akademie der Wissenschaften M athematisch-Physikalische Klasse.

'Boolean Embeddings of Orthomodular Sets and Quantum Logic' by N. Zierler and M. Schlessinger in Duke Mathematical Journal 32 (1965), 251-62. Reprinted by per­mission of the author and publisher. © Copyright 1961, Duke University Press, Durham, North Carolina.

'Logical Structures Arising in Quantum Theory' by S. Kochen and E. P. Specker in The Theory of Models (ed. by J. Addison, L. Henkin, and A. Tarski), North-Holland Pub­lishing Co., Arnsterdam, 1965. Reprinted by pennission of the authors and North­Holland Publishing Co.

'The Calculus ofPartial Propositional Funetions' by S. Kochen and E. P. Specker in Logic, Methodology and the Philosophy of Seience (ed. by U. Bar-Hillel), North-Holland Publishing Co., Amsterdam, 1965. Reprinted by pennission of the authors and North­Holland Publishing Co.

'The Problem ofHidden Variables in Quantum Mechanies' by S. Kochen and E. P. Specker in Journal of Mathematics and Mechanics 17 (1967), 59-67. Reprinted by pennission of the authors and Indiana Universily Mathematics Journal. (OriginaUy Journal of Mathe­matics and Mechanies.)

'Logics Appropriate to Empirical Theories' by P. Suppes in The Theory of Models (ed. by J. W. Addison, L. Henkin, and A. Tarski), North-HoUand Publishing Co., Arnsterdam, 1965. Reprinted by pennission of the authors and North-Holland Publishing Co.

'The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanies' by P. Suppes in Philosophy of Seience 33 (1966), 14--21. Reprinted by pennission of the author and Philosophy of Science.

'Foundations of Quantum Mechanies' by M. Strauss in Modem Physics and Its Philosophy, D. Reidel Publishing Co., Dordrecht-Holland, 1972. (Note: This is a translation of an artide appearing in Gennan in Mikrokosmos-makrokosmos, vol. II (ed. by H. Leyand R. Lother), Berlin, 1967.) Reprinted by pennission of the author and D. Reidel Pub­lishing Co.

'Baer *-Semigroups and the Logic of Quantum Mechanics' by J. C. T. Pool in Communica­tions on Mathematical Physics 9 (1948), 118--41. Reprinted by pennission of the author and Communications on Mathematical Physics.

'Semimodularity and the Logic of Quantum Mechanies' by J. C. T. Pool in Communications on Mathematical Physics 9 (1968), 212-28. Reprinted by pennission of the author and Communications on Mathematical Physics.

'On the Structure of Quantum Logic' by P.D. Finch in The Journal of Symbolic Logic 34 (1969),275-82. Reprinted by pennission of the author and the publisher, The Associa­tion for Symbolic Logic, from The Journal of Symbolic Logic.

'On the Structure of Quantal Proposition Systems' by J. M. Jauch and C. Piron in Helvetica

ACKNOWLEDGEMENTS xv

Physica Acta 43 (1969), 842-8. Reprinted by permission of the author and Helvetica Physica Acta.

'The Current Interest in Orthomodular Lattices' by S. S. Holland Jr. in Trends in Lattice Theory, Van Nostrand, New York, 1970. Reprinted by permission of the author and the publisher. © Copyright 1970 by Litton Educational Publishing, Inc.

'Integration Theory ofObservables' by N. S. Kronfli in International Journal of Theoretical Physics 3 (1970), 199-204. Reprinted by permission of the Estate of the author and International Journal of Theoretical Physics.

'Probabilistic Formulation of Classical Mechanics' by N. S. Kronfli in International Journal of Theoretical Physics 3 (1970), 395-400. Reprinted by permission of the author and International Journal of Theoretical Physics.

'Atomicity and Determinism in Boolean Systems' by N. S. Kronfli in International Journal of Theoretical Physics 4 (1971), 141-3. Reprinted by permission of the Estate of the author and International Journal of Theoretical Physics.

'Survey of General Quantum Physics' by C. Piron in Foundations of Physics 2 (1972), 287-314. Reprinted with permission of the author and Foundations of Physics. Copyright held by the publisher, Plenum Publishing Corp.

'Quantum Logics' by S. P. Gudder and R. J. Greechie in Contemporary Research in the Foundations and Phi/osophy of Quantum Theory (ed. by C. A. Hooker), D. Reidel Pub­lishing Co., Dordrecht-Holland, 1974. Reprinted by permission of the authors and D. Reidel Publishing Co.

'The Labyrinth of Quantum Logics' by B. C. van Fraassen in Boston Studies in the Phi­losophy of Science, vol. XIII, D. Reidel Publishing Co., Dordrecht-Holland, 1974. Re­printed by permission of the author and D. Reide! Publishing Co.