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Library of Congress Cataloging-in-Publication Data

Mehra, Jagdish.

The completion of quantum mechanics, 1926±1941 / Jagdish Mehra, Helmut Rechenberg.

p. cm. Ð (The historical development of quantum theory ; v. 6)

Includes bibliographical references and index.

ISBN 0-387-95086-9 (pt. 1 : alk. paper)

1. Quantum theoryÐHistory. I. Rechenberg, Helmut. II. Title.

QC173.98.M44 vol. 6

530X12H09Ðdc21

00-040039

Printed on acid-free paper.

( 2001 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identi®ed, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

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987654321

ISBN 0-387-95086-9

SPIN 10771857

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gontentsЀ—rt P

Chapter IV The Conceptual Completion and the Extensions of

Quantum Mechanics (1932±1941)

671

Introduction

671

IV.1 The Causality Debate (1929±1935)

678

(a) Introduction: The Principle of Causality in Quantum Theory

678

(b) Heisenberg's Discussions Concerning the Positivism of the `Vienna

Circle' (1929±1932)

683

(c) The Indeterminacy Relations for Relativistic Quantum Fields

(1929±1933)

692

(d) The Continuation of the Debate on Causality with the Berlin

Physicists (1929±1935)

703

IV.2 The Debate on the Completeness of Quantum Mechanics and Its

Description of Reality (1931±1936)

713

(a) Introduction

713

(b) From Inconsistency to Incompleteness of Quantum Mechanics:

The EPR Paradox (1931±1935)

717

(c) The Response of the Quantum Physicists, Notably, Bohr and

Heisenberg to EPR (1935)

725

(d) Erwin SchroÈ dinger Joins Albert Einstein: The Cat Paradox (1935±

1936)

738

(e) Reality and the Quantum-Mechanical Description (1935±1936)

747

IV.3 New Elementary Particles in Nuclear and Cosmic-Ray Physics

(1929±1937)

759

(a) Introduction: `Pure Theory' Versus `Experiment and Theory'

759

(b) The Theoretical Prediction of Dirac's `Holes' and `Monopoles'

(1928±1931)

772

(c) The Discovery of New Elementary Particles of Matter and

Antimatter (1930±1933)

785

(d) Quantum Mechanics of the Atomic Nucleus and Beta-Decay

(1931±1934)

801

(e) Universal Nuclear Forces and Yukawa's New Intermediate Mass

Particle (1933±1937)

822

IV.4 Solid-State, Low-Temperature, and Relativistic High-Density

Physics (1930±1941)

837

(a) Introduction

837

(b) New American and European Schools of Solid-State Physics

(1933±1937)

840

vi

Contents

(c) Low-Temperature Physics and Quantum Degeneracy (1928±1941)

857

(d) Toward Astrophysics: Matter Under High Pressures and High

Temperatures (1926±1939)

877

IV.5 High-Energy Physics: Elementary Particles and Nuclear

Reactions (1932±1942)

898

(a) Introduction

898

(b) Between Hope and Despair: Progress in Quantum

Electrodynamics (1930±1938)

902

(c) New Fields Describing Elementary Particles, Their Properties, and

Interactions (1934±1941)

935

(d) Nuclear Forces and Reactions: Transmutation, Fusion, and

Fission of Nuclei (1934±1942)

964

Epilogue: Aspects of the Further Development of Quantum Theory

(1942±1999)

1015

1. The Elementary Constitution of Matter: Subnuclear Particles and

Fundamental Interactions

1020

1.1 Some Progress in Relativistic Quantum Field Theory and the

Formulation of the Alternative S-Matrix Theory (1941±1947)

1024

(a) E. C. G. Stueckelberg: `New Mechanics (1941)' (b) The Principle of Least Action in Quantum Mechanics (Feynman
and Tomonaga, 1942±1943) (c) Heisenberg's S-Matrix (1942±1947)

1024
1024 1030

1.2 The Renormalized Quantum Electrodynamics (1946±1950)

1033

(a) The Shelter Island Conference (1947) (b) Hans Bethe and the Initial Calculation of the Lamb Shift (1947) (c) The Anomalous Magnetic Moment of the Electron (1947) (d) The Pocono Conference (1948) (e) Vacuum Polarization (1948) (f ) The Michigan Summer School: Freeman Dyson at Julian
Schwinger's Lectures (1948) (g) The Immediate Impact of Schwinger's Lectures (1948) (h) Schwinger's Covariant Approach (1948±1949) (i) Gauge Invariance and Vacuum Polarization (1950) ( j) The Quantum Action Principle (1951) (k) Tomonaga Writes to Oppenheimer (April 1948) (l) Tomonaga's Papers (1946±1948) (m) Feynman's Preparations up to 1947 (n) Richard Feynman after the Shelter Island Conference (1947±
1950) (o) Freeman Dyson and the Equivalence of the Radiation Theories
of Schwinger, Tomonaga, and Feynman (1949±1952) (p) The Impact of Dyson's Work (q) Feynman and Schwinger: Cross Fertilization
1.3 New Elementary Particles and Their Interactions (1947±1964) 1.4 The Problems of Strong-Interaction Theory: Fields, S-Matrix,

1033 1038 1043 1051 1057
1059 1062 1064 1074 1081 1085 1086 1088
1091
1099 1104 1106 1107

Currents, and the Quark Model (1952±1969)

1118

Contents

vii

1.5 The `Standard Model' and Beyond (1964±1999)

1125

(a) The `Electroweak Theory' (1964±1983)

1126

(a1) The `Intermediate Weak Boson'

1126

(a2) Spontaneous Symmetry-Breaking and the Higgs Mechanism 1127

(a3) The Weinberg±Salam Model and Its Renormalization

1127

(a4) Neutral Currents and the Discovery of the Weak Bosons

1128

(b) Quantum Chromodynamics (QCD) (1965±1995)

1130

(b1) The Discovery of Physical Quarks

1130

(b2) Asymptotic Freedom of Strong Interaction Forces

1131

(b3) Quantum Chromodynamics

1132

(b4) The Completion of QCD

1133

(c) Beyond the Standard Model (1970±1999)

1134

2. Quantum E¨ects in the Physical Laboratory and in the Universe

1138

2.1 The Industrial and Celestial Laboratories (1947±1957)

1139

(a) The Transistor in the Industrial Laboratory (1947±1952)

1139

(b) The Celestial Laboratory (1946±1957)

1143

2.2 The Application of Known Quantum E¨ects (1947±1995)

1145

(a) The Casimir E¨ect and Its Applications (1947±1978)

1145

(b) The Maser and the Laser (1955±1961)

1153

(c) The Bose-Einstein Condensation (1980±1995)

1156

2.3 Super¯uidity, Superconductivity, and Further Progress in

Condensed Matter Physics (1947±1974)

1159

(a) Rotons and Other Quasi-Particles (1947±1957)

1159

(b) The Solution of the Riddle of Superconductivity (1950±1959)

1163

(c) Critical Phenomena and the Renormalization Group (1966±1974) 1170

2.4 New Quantum E¨ects in Condensed Matter Physics (1958±1986)

1173

(a) The MoÈ ssbauer E¨ect (1958)

1173

(b) Experimental Proof of Magnetic Flux Quantization (1961)

1175

(c) The Josephson E¨ect (1962)

1176

(d) Super¯uid Helium III: Prediction and Veri®cation (1961±1972)

1177

(e) The Quantum Hall E¨ect and Lower Dimensional Quantization

(1980)

1179

(f ) High-Temperature Superconductors (1986)

1181

2.5 Stellar Evolution, the Neutrino Crisis, and 3 K Radiation (1957±

1999)

1183

(a) Stellar Evolution and New Types of Stars (1957±1971)

1185

(b) The Solar Neutrino Problem and the Neutrino Mass (1964±1999)

1187

(c) 3 K Radiation and the Early Universe (1965±1990)

1190

3. New Aspects of the Interpretation of Quantum Mechanics

1193

3.1 The Copenhagen Interpretation Revisited and Extended (1948±

1966)

1197

3.2 Causality, Hidden Variables, and Locality (1952±1968)

1208

(a) The Hidden Variables and von Neumann's Mathematical

Disproof Revisited (1952±1963)

1212

viii

Contents

(b) The EPR Paradox Revisited, Bell's Inequalities, and Another Return to Hidden Variables (1957±1968)
(c) The Aharonov±Bohm E¨ect (1959±1963)
3.3 Further Interpretations and Experimental Con®rmation of the Standard Quantum Mechanics (1957±1999)
(a) The Many-World Interpretation and Other Proposals (1957±1973) (b) Tests of EPR-Type Gedankenexperiments: Hidden Variables or
Nonlocality (1972±1986) (c) The Process of Disentanglement of States and SchroÈ dinger's Cat:
An Experimental Demonstration (1981±1999)

1216 1222
1224 1224
1229
1235

Conclusion: Four Generations of Quantum Physicists

1244

References

1255

Author Index

1441

Subject Index for Volumes 1 to 6

1469

gh—pter s† „he gon™eptu—l gompletion —nd the ixtensions of
u—ntum we™h—ni™s @IWQP±IWRIA
Introduction
The invention of quantum and wave mechanics and the great, if not complete, progress achieved by these theories in describing atomic, molecular, solid-state andÐto some extentÐnuclear phenomena, established a domain of microphysics in addition to the previously existing macrophysics. To the latter domain of classical theories created since the 17th century appliedÐprincipally, the mechanics of Newton and his successors, and the electrodynamics of Maxwell, Hertz, Lorentz, and Einstein. The statistical mechanics of Maxwell, Boltzmann, Gibbs, Einstein, and others indicated a transition to microphysics; when applied to explain the behaviour of atomic and molecular ensembles, it exhibited serious limitations of the classical approach. Classical theories were closely connected with a continuous description of matter and the local causality of physical processes. The microscopic phenomena exhibited discontinuities, `quantum' features, which demanded changes from the classical description. In the standard scheme of quantum theory that emerged between 1926 and 1928, notably in GoÈ ttingen, Cambridge, and Copenhagen, the following description arose:
(i) Microscopic natural phenomena could be treated on the basis of the theories of matrix and wave mechanics, i.e., formally di¨erent but mathematically equivalent algebraic and operator formulations.
(ii) The quantum-mechanical theories satis®ed the known conservation principles of energy, momentum, angular momentum, electric charge and current, etc.
(iii) The visualizable (anschauliche) particle and wave pictures of the classical theories had to be replaced by `dualistic' or `complementary' aspects of microscopic objects which exhibited simultaneous particle and wave features.
(iv) The causal structure known from the classical lawsÐi.e., the di¨erential equationsÐremained valid for the quantum-mechanical laws, but the behaviour of quantum-mechanical objects deviated from those of classical ones.
(v) Based on Born's statistical interpretation of the wave function and Heisenberg's uncertainty (or indeterminacy) relations, Bohr (on the physical side) and von Neumann (on the mathematical side) proposed a subtle formalism that accounted for the measurement of microscopic properties by macroscopic instruments (and observers), in which the classical subject±object relation introduced 300 years earlier by Rene Descartes was replaced by a di¨erent one.
The completed physical theory of microscopic phenomena that thus arose, and was soon characterized as the `Copenhagen interpretation of quantum mechanics,'

672 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
was by no means accepted by all physicists, not even by all quantum physicists universally. Especially in Middle Europe, a lively debate arose from the late 1920's onward concerning several characteristic aspects of this interpretation. We have mentioned in Chapter II that already since the origin of the complementarity view, Erwin SchroÈ dinger and Albert Einstein vigorously attacked the validity of its very basis, namely, Heisenberg's uncertainty relations. While Bohr and his associates, in particular Heisenberg and Pauli, had emerged victorious in this debate on the uncertainty relationsÐby demonstrating that the quantum-mechanical scheme was fully consistent as a mathematical theory and gave an adequate description of microscopic phenomenaÐa new debate started around 1930 (i.e., after the defeat of Einstein's arguments by Bohr et al. at the sixth Solvay Conference on Physics) about the consequences from the uncertainty relations for the principle of causality in quantum mechanics. Now Planck and SchroÈ dinger argued vigorously against renouncing the (classical) causality concept, which had formed the basis of all previous successful physical theories and beyond. On the other hand, a powerful philosophical movement in Germany and its vicinity, notably positivism and the related views of the `Vienna Circle,' supported, more or less fully, the Copenhagen interpretation. Simultaneously with these epistemological debates, certain theoreticians worked on the problem of whether the uncertainty relations would not break down when one would seek to extend quantum mechanics to relativistic phenomena. These investigations, carried out between 1930 and 1933, ended with the result that uncertainty relations existed also for relativistic ®elds; hence, the Copenhagen interpretation remained valid also in this domain.797 The debate on causality and the extension of the uncertainty relations will be dealt with in Section IV.1.
In spite of his defeat in 1930, Einstein would not yield to the claim of the validity of the quantum-mechanical description of microscopic processes. After several years of preparation, he would publish with two collaborators a new and, as he believed, decisive blow: the so-called `Einstein-Podolsky-Rosen (EPR) paradox' did not argue against the consistency of the modern quantum theory (involving, especially, the validity of the uncertainty relations) but rather attempted to show that the entire, though so successful, scheme violated the very essence of a physical theory, namely, to describe the `reality' of nature completely. Bohr, Heisenberg, and others hurried to reply to Einstein's accusations by demonstrating that the view of physical reality assumed by their distinguished colleague simply did not apply to the microscopic domain. At the same time, Erwin SchroÈ dinger analyzed, partly independently of Einstein, the intuitive (anschauliche) content of quantum mechanics and published his famous `cat paradox.' This nonrelativistic example addressed the same reality problem which had been discussed by Einstein and his quantum-mechanical opponents in the relativistic example of EPR. We shall treat the purely epistemological debate between Einstein and SchroÈ dinger, on
797 We recall from Section II.7 that the most eminent quantum-mechanical experts were ready to accept a breakdown of their theory in the domain of relativistic and nuclear physics.

Introduction

673

the one side, and the partisans of the Copenhagen interpretation, on the other, in Section IV.2.
It has been noted and emphasized by several historians of science that the debates on the philosophical contents of quantum mechanics were carried out mainly in Middle Europe. Paul Forman explained this fact by associating the philosophical ideas leading to the creation of quantum mechanics and its interpretation with the general sociological conditions of the Weimar Republic: the political and economic necessities following World War I in defeated Germany also ultimately nourished the emergence of doubts in the causality of physical phenomena (Forman, 1971). Nancy Cartwright, on the other hand, in an analysis of the response of some American physicists to the Copenhagen interpretation, argued that the limited participation of her fellow countrymen in the philosophical debates rested on `the well-known American doctrines of pragmatism and operationalism':
[This] philosophy stressed two things: (1) hypotheses must be veri®ed by experiment and not accepted merely because of their explanatory power; and (2) the models that physics uses are inevitably incomplete and incompatible, even in studying di¨erent aspects of the same phenomena. (Cartwright, 1987b, p. 417)
The `basic attitude' of the progressive young American quantum physicists (including J. C. Slater, E. U. Condon, J. H. Van Vleck, E. H. Kennard, E. C. Kemble, D. M. Dennison, N. Wiener, H. P. Robertson, and less so J. R. Oppenheimer) around 1930 `was that the task of physics is not to explain but to describe' the natural phenomena (loc. cit.).798 However, by discussing the laws of quantum mechanics in the light of previously recognized principles of physical theory (notably, the relation between cause and e¨ect), Bohr and his associates sought to found the new atomic theory as a generalization of the old dynamics: as in the old theories, they wished to explain rather than just to describe natural processes. And yet, in spite of all their epistemological e¨orts, they could not satisfy the demands of physicists of the older generation, whose ideal seemed to be the status of late nineteenth-century science before the quantum and relativistic phenomena enforced a change of the description. Some of them, like Philipp Lenard and Johannes Stark, were even more extreme and used the new political philosophy of the Nazis, ruling Germany since 1933, to demand a return to what they called the good `German physics (Deutsche Physik)' or `Aryan physics,' i.e., the mathematically less abstract, directly visualizable physics that existed before the advent of the modern quantum and relativity theories. They accused Heisenberg, Sommerfeld, and even the old Max Planck of having createdÐtogether with Einstein and many well-known physicists who were driven out of German universities and

798 In a sense, one might say that the di¨erent attitudes toward quantum mechanics represented by the Central Europeans, on the one hand, and the Americans, on the other, followed the old antipodean schemes of deductively describing the laws of nature from metaphysical principles (e.g., by Leibniz) or inductively deriving otherwise unexplained laws from observations (e.g., by Newton), respectively.

674 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
research institutesÐa `Jewish,' a `degenerate' physics. The Third Reich, with its racial laws and other anti-democratic, nationalistic measures, certainly damaged seriously the cause of modern atomic theory in GermanyÐmany Jewish and several of the other creators and distinguished representatives of quantum and relativity theories took away with them the fruits of the great tradition established in Middle Europe before and during the Weimar period.
Still, quantum theory continued to ¯ourish during the 1930s, even in impoverished Germany, through many applications and extensions, which we shall sketch in the following sections of this chapter. In particular, the ®elds of nuclear and high-energy physics were incorporated into the descriptions based on quantum mechanics and relativistic quantum ®eld theory. These successes began with the introduction of several new elementary particles, i.e., basic constituents of matter, besides the already known proton and electron, two of them having been predicted and the others detected by surprise. In Section III.7, we have already mentioned the `neutron,' assumed hypothetically by Pauli to rescue energy conservation in the beta-decay (December 1930). Then, in summer 1931, Dirac interpreted the negative-energy states of his relativistic electron equation as a positively charged `anti-electron,' which would be identi®ed in the following year by the American physicist Carl Anderson with a particle detected in cosmic radiation and named the `positron.' The existence of the positron, and in particular, the creation of electron±positron pairs by very energetic gamma-rays, explained many phenomena in high-energy physics, as well as the so-called Meitner±Hupfeld anomaly referred to in Section III.7. On the other hand, a neutral particle, having approximately the mass of the proton, was identi®ed in February 1932 by James Chadwick in certain nuclear reactions. This object, foreseen clearly by Rutherford in 1920, performed a tremendous job in removing most of the previous di½culties encountered by the physicists when they tried to apply the quantum-mechanical scheme to nuclear structure: that is, all of the problems noticed earlier in connection with the existence of electrons in nuclei and the statistics of certain nuclei disappeared all of a sudden. Thus, in 1932, nuclear physics became a wellde®ned branch of standard quantum mechanics. Heisenberg hurried to make use, in the same year, of Chadwick's heavy `neutron' to establish the proton± neutron structure of the atomic nuclei and started to explain their masses, or, more accurately, their binding energies by assuming new exchange forces; two years later, a young Japanese physicistÐHideki YukawaÐassociated these exchange forces with another new elementary particle, which he called the `heavy quantum.' Even earlier, in December 1933, Enrico Fermi developed a consistent quantumtheoretical description of beta-decay by making use of Pauli's light `neutron' of 1930, which he (in 1932) properly renamed the `neutrino.' These wonderful discoveries in nuclear and high-energy physics, which came to a peak in the `annus mirabilis' of 1932, will be treated in Section IV.3.
Also in the low-energy domains of condensed matter physics, namely, solidstate and low-temperature physics, the 1930's proved to be a quite fruitful period for the application of quantum-mechanical methods and principles, as we shall

Introduction

675

summarize in Section IV.4. On the one hand, the theory of metals and solids, established so successfully between 1927 and 1932 (and described in Section III.6), was further developed, especially in the USA (with John Slater and the Hungarian immigrant, Eugene Wigner, and their students playing a leading role) and England (where, for example, Nevill Mott in Bristol formed a new school). On the other hand, the newly investigated anomalous behaviour of helium at temperatures around 2K (notably, the super¯uidity discovered by Peter Kapitza in late 1937) became amenable to treatment by means of quantum theory. Only the old riddle of low-temperature physicsÐsuperconductivityÐstill lacked real theoretical understanding from ®rst, microscopic principles, in spite of the great progress made in the macroscopic description of the phenomenon.
The most exciting results in the second half of the 1930's were again achieved in nuclear and high-energy physics, although the formalism exhibited, at least in the relativistic domain, serious defects as had been noticed already by Heisenberg and Pauli in their pioneering work on quantum ®eld theory of 1929. All of the ®eld theories devised to explain elementary particles and their interactionsÐwhether the original quantum electrodynamics, the so-called `Fermi-®eld' theory (developed from Fermi's beta-decay theory in order to account for binding and scattering processes in nuclear or high-energy physics), or the various forms of Yukawa's heavy quantum (soon to be called `meson') theory of nuclear forcesÐyielded most disturbing in®nite results for fundamental properties, like the masses of particles or cross sections of characteristic processes. These principal in®nities would be handled only later by the procedure of `renormalization,' ®rst in the case of quantum electrodynamics. On the other hand, many experimental ®ndings, in the ®rst place the discovery of the `mesotron' by Anderson and Seth Neddermeyer toward the end of 1936, encouraged the quantum-®eld theorists. The new cosmicray particle not only seemed to have the properties demanded by Yukawa for the `meson'; since it was unstable and decayed in milliseconds when at rest, it also accounted for the existence and the properties of the hitherto unexplained `penetrating component' of cosmic radiation. The special task for the theoreticians remained to select from the available quantum-®eld theoriesÐthe scalar theory of Pauli and Viktor Weisskopf of 1934 or the vector theory proposed in late 1937 independently by Yukawa and his Japanese collaborators, on the one hand, and several theoreticians in Great Britain (Nicholas Kemmer, Homi Bhabha, Herbert FroÈ hlich, and Walter Heitler), on the otherÐthe suitable candidate, which would allow one to calculate both the binding energy of characteristic nuclei (especially of `deuterium,' the nucleus of the heavy water atom) and the high-energy scattering of nuclear particles. While a host of problems remained unanswered in the late 1930's, the general investigation of quantum-®eld theories as the tool for describing elementary particles made great progress; thus, Pauli and Markus Fierz in Zurich proved the general spin-statistics theorem (1938), and Frederick Belinfante in Leyden discovered the symmetry of quantum ®elds under `charge conjugation' (1939a). We shall close Section IV.5 of this chapter with another discovery in the domain of nuclear physics, which would create an even bigger stir among the

676 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
scientists and the public at large: nuclear ®ssion. This new, and completely unexpected, mode of nuclear reactions, observed by the chemists Otto Hahn and Fritz Straûmann (1939a) when scattering slow neutrons by uranium nuclei, could be immediately explained on the basis of the standard quantum-mechanical nuclear theory.
In the early 1930's, the time began when quantum mechanics advanced to the status of an established fundamental theory, on the basis of which the various branches of physics became reorganized: atomic physics, molecular physics, solidstate physics (with its sub®elds of metal, semiconductor physics, etc.) and the physics of condensed matter (especially low-temperature physics), on the one hand, and nuclear and elementary particle physics (emerging, to a large extent, from cosmic-ray physics and still called, until the early 1950's, high-energy nuclear physics), on the other. The community of quantum physicists, whose eminent members earlier covered in their theoretical investigations several, if not all, ®elds, now began to split into well-de®ned parts or groups of specialists dealing with one or at the most two topics. The history of quantum theory consequently branches out into separate histories of all of these ®elds, each of which deserving its own detailed treatment. Such a task would clearly surpass the goal of the present series of volumes on The Historical Development of Quantum Theory. Sections IV.3 to IV.5 therefore address only the essential quantum-theoretical ideas involved in the new ®elds; none of them would discuss any topic in its entirety, as this would require a series of di¨erent historical accountsÐonly a few of which have been attacked so far (e.g., in the book of Hoddeson et al. (1992) on the history of solidstate physics, or in that of Brown and Rechenberg (1996) on the origin of the concept of nuclear forces).
The development of these new topics demanded the work of many scholars; even new schools arose, for example, the Bristol school of Nevill Mott in Great Britain or the MIT school of John Slater in the United States, both devoted to research on solid-state physics. Of course, also in the more specialized physics of the 1930's, the great ®gures, who had ushered in the quantum-mechanical revolution in the 1920's, remained leaders in many of the new developments, especially Bohr, Dirac, Heisenberg, Pauli, and Wigner, supported by their early gifted disciples (from Bloch and Heitler to Peierls and Rosenfeld). On the other hand, the in¯uence of old masters like Born and Sommerfeld became diminished, less by their age than by the formidable di½culties created by the Third Reich in Germany, which forced the former into emigration and denied the latter to choose an appropriate successor to continue the work of his school.
Indeed, the forced emigration of a large part of the best of the older as well as the younger generation from Germany played a decisive role in the contributions from various other countries to quantum physics of the later 1930s. One can say that through the actions of the Nazi Government (which came to power in Germany in early 1933) nearly a whole generation of the most talented quantum theorists got lost to Germany; not all of them were Jews or of Jewish origin (hence, fell under the racial laws of 1933 and 1935), but quite a few also had to

Introduction

677

leave or left voluntarily because of political reasons (because they were associated with liberal to leftist ideas).799 The emigrating quantum theorists then ¯ooded in large numbers into the other countries, preferably to the West (mostly to Great Britain and the USA, less so to France, Spain, and some countries of South America), but some also to the East (from Czechoslovakia to the Soviet Union, Turkey, and even China). Much has been argued about the e¨ect of this emigration, in particular about the role played by the theorists coming from Germany in establishing and strengthening quantum physics in their new home-countries. Paul Hoch, who studied the situation in several cases more closely, arrived at more modi®ed conclusions concerning the immediate in¯uence of the emigrants from Germany, pointing out that their reception by the scienti®c communities abroad was often lukewarm; thus, he warned about overrating the support the emigrants received, especially in the most favoured host countries, Great Britain and the USA, by referring to the historical situation as it existed then:
This is all very well, and was to be very important as a foundation for the growth of theoretical physics in America in the subsequent decades. But in 1933 the number of theoreticians on the physics sta¨ at those institutions considered to be the main centres of this discipline in America wasÐwith the possible exception of the University of MichiganÐtiny compared to those primarily engaged in experimentation, and rarely exceeded one or occasionally two permanent sta¨.
If one is going to write a dispassionate history of the transmission of this new branch of knowledge, one has to bring into the equation not only those factors facilitating it but also those opposing it. The predominant attitude to physics in Britain and America at that time was that it wasÐand should be almost as matter of moralityÐan experimental science (at Oxford it was still known as ``experimental philosophy''). Cambridge had a considerable mathematical aspect to its physics at least since Maxwell, which embraced in part the work of Kelvin, Rayleigh and J. J. Thomson among others. However, by the 1930s it was assumed even within this tradition that Cavendish physicists did their own experiments and that those not doing experiments were not physicists and belonged to the mathematics faculty. This was still the case in the early 1930s, even for the Stokes Lecturer in Mathematical Physics, Ralph Fowler, and for his most promising students Nevill Mott and Alan H. Wilson, all of whom were attached to mathematics. [Moreover, Fowler's] own main interests were within statistical and older mathematical physics, rather than, for example, in the new ``German'' quantum mechanics and its application to solids, in which such visitors to Germany and Denmark as Wilson and Mott were to play a considerable part. In the years after 1933 a great number of refugee theoreticians obtained temporary accommodation at Cambridge including Hans Bethe, Max Born, Rudolf Peierls and P. P. Ewald, among others. But none of these was able to obtain a post on the permanent sta¨ and all went elsewhere. (Hoch, 1990, pp. 24±25)
The situation, as Hoch described further, was similar at Oxford, where actually only the brothers Fritz and Heinz London settled for a longer period.

799 Many others (like Hermann Weyl and Erwin SchroÈ dinger) just resigned from their posts, because they did not wish to live in the atmosphere created by the Third Reich and refused to swear an oath of allegiance to the FuÈhrer Adolf Hitler.

678 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
To these reasons, which arose from the general background of science in Great Britain and USA, often also anti-Semitic sentiments in the faculties (especially in the USA) were added to prevent an early integration of the immigrants. They mostly occupied positions and treated subjects which the local people did not favour, investigating especially theoretical problems of nuclear physics (partially needed by the well-established experimentalists in their host countries).800 Many of them later participated decisively in the nuclear energy projects during World War II in Britain and in the USA and, as a consequence of their meritorious work at that time, established themselves as respected members of the scienti®c community after the war. But this is quite another story which transcends the aim of this chapter and the subjects to be discussed here.
IV.1 The Causality Debate (1929±1935)
(a) Introduction: The Principle of Causality in Quantum Theory
In a dictionary of physics, the concept of causality is de®ned as follows:
The physicist considers causality as identical with determinism, that is, with the unique ®xing of the future events by the present ones according to the laws of nature. (Westphal, ed., 1952, p. 649)
Since in classical physics the fundamental equations of nature were di¨erential equations, the `deterministic hypothesis,' as formulated toward the end of the 19th century, says: If one knows at a given instant the initial values of all parameters describing the system considered, then one can calculate the values of these parameters for all future times. Evidently, this hypothesis worked well in Newtonian mechanics and Maxwell's electrodynamics. It could also be taken over into relativistic dynamics and Einstein's theory of gravitation of 1915, as David Hilbert remarked:
By knowing the [physical quantities and their time derivatives] in the present, once and for all the values of these quantities can be determined in the future, provided they have a physical meaning. (Hilbert, 1917, p. 61)
Clearly, the classical statistical mechanics also did not a¨ect the `deterministic hypothesis' as such: The probabilistic description involved in it was considered only as a device to calculate in a simple and comfortable manner the gross properties of a large assembly of particles, and the clever `Maxwell demon' would then be able to disentangle all individual particle trajectories, which obeyed deterministic classical laws.
800 Particular examples have been discussed with respect to the `British' theoreticians developing the meson theory of nuclear forces in 1937 and 1938 (Brown and Rechenberg, 1996, Section 7.4).

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The situation changed only with the investigation of certain phenomena after 1900, especially the blackbody radiation law of Max Planck (1900f ), the nature of the law of radioactivity by Egon von Schweidler (1905), and the emission and absorption processes of radiation by Albert Einstein (1916d). Hence, the former trained physicist and later philosopher Moritz Schlick attempted to endow the `causal principle' with a more adequate formulation by stating:

The causal principle is not a natural law itself but rather the general expression of the fact, that everything which happens in nature obeys laws which are valid without exception. . . . First, one realized that the events happening at one instant of time are only determined by the events happening at the immediately preceding instant, i.e., the dependence does not extend without intermediate action over distant times. . . . A further extended and increasingly better justi®ed experience has made it very probable that . . . in space there exists as little an action-at-a-distance as in time: the natural processes therefore are completely determined by those in the immediate vicinity and depend only via the intermediate action of the latter on those at a larger distance. The intermediate action could occur also discontinuously, hence ®nite di¨erences would replace the di¨erentials. The experiences of quantum theory warn us not to lose sight of this possibility. (Schlick, 1920, pp. 461±462)801

Around 1920, a new discussion indeed arose, especially in Germany, about the meaning of causality, triggered by the progress of quantum theory.802 Walter Schottky of WuÈ rzburg published in June 1921 a popular article on `Das Kausalproblem der Quantentheorie als eine Grundfrage der modernen Naturforschung

801 We have added emphasis to the last two sentences by italics. Moritz Schlick was born in Berlin on 14 April 1882, the son of an industrialist. He studied at the University of Berlin and received his doctorate in 1904 under Planck's direction with a thesis, `UÈ ber die Re¯exion des Lichtes in einer inhomogenen Schicht (On the Re¯ection of Light in An Inhomogeneous Layer).' Immediately afterward, he turned his attention from the problems of theoretical physics to those of a very general philosophical nature and started a career in the philosophy of science. In 1910 he received his Habilitation at the University of Rostock; in 1917, he was promoted to a professorship there, before moving to Vienna in 1921 as a full professor to occupy the philosophical chair previously held by Ernst Mach and Ludwig Boltzmann. In Vienna, he created a school of the logic of science (Wissenschaftslogik) and the foundations of mathematics, the `Wiener Kreis (Vienna Circle).' Schlick was shot to death on 22 June 1936 by a former student in the University of Vienna. With his publications on the philosophy of modern physical theories, especially relativity theory and quantum theory, and through the `Vienna Circle,' he became one of the most original and in¯uential teachers in the philosophy of science. For more details on the life and work of Moritz Schlick, see his obituary by Zilsel (1937).
802 In an article on the cultural origin of statistical causality, Norton M. Wise traced `the idea of indeterministic statistical causality' back to the period between 1870 and 1920 in Central Europe, referring especially to the physiologist Wilhelm Wundt and his Leipzig colleague, the historian Karl Lamprecht, and later to the Danish philosopher Harald Hù¨ding, who in¯uenced Niels Bohr (Wise, 1987). We might add here the name of Franz Exner, the teacher of Erwin SchroÈ dinger, who pondered at least since his rectorial address of 1908 about the statistical nature of physical laws, and expanded on the subject in his 1919 book on a new concept of causality, arriving at the following conclusion: `There must exist causes which direct the average processes, but only those and not the individual ones, into lawful courses. (Es muÈssen Ursachen vorhanden sein, welche das durchschnittliche Geschehen, aber auch nur dieses, und nicht Einzelheiten bedingen und in gesetzmaÈûige Bahnen leiten.)' (Exner, 1919; quoted from the second edition, 1922, p. 676)

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uÈberhaupt (The Central Problem of Quantum Theory as a Fundamental Problem of Modern Science in General).' (Schottky, 1921b) Schottky argued in particular that the interaction between matter and electromagnetic radiation, as considered by Planck, Einstein, and others, seems to put the strict causal law of classical mechanics into doubt; that is, when considering a quantum jump, one cannot ask the question: `How does such a jump occur from one ``orbit'' to another, under what conditions does it happen, how long does it last, etc.?' (Schottky, loc. cit., p. 507) Rather:
What can be grasped by the concept of causality . . . are the conditions for the frequency of occurrence of elementary events of a de®nite type. However, for this purpose the laws . . . are so strict and general in validity that one never ®nds a deviation, as long as one just takes a su½ciently large number of elementary processes together or adopts a point of view, in which the assumed ``structure'' of processes does not show up anymore. (Schottky, 1921b, p. 511)803
Though based on other results of modern atomic theory, Walther Nernst argued for a similar weakening of the causality principle in his Berlin rectorial address of 15 November 1921 (Nernst, 1922, especially, p. 494).
The turbulent development of quantum theory in the following years led to a series of speculations about the nature of physical laws, of which the radiation theory of Niels Bohr, Hendrik Kramers, and John Slater, proposed in early 1924, suggested the most radical departure from classical causality.804 The quantum and wave mechanics, which emerged in 1925 and 1926, then restricted these speculations again. The conservation laws, violated in the previous Bohr±Kramers± Slater approach, regained full validity in the new atomic theory; however, now Max Born's statistical interpretation of the wave function implied a breakdown of the causality hypothesis for all atomic processes, which Born replaced by the statement: `The motion of particles conforms to the law of probability, but the probability itself is propagated in accordance with the law of causality.' (Born, 1926b, p. 804) This interpretation of the quantum-mechanical formalism initiatedÐas we have shown in Chapter II in this volume (and in Volume 5, Part 2, Section IV.5)Ðthe fundamental debates, especially in Copenhagen, leading to Heisenberg's uncertainty relations and Bohr's principle of complementarity. Then, in March 1927, Heisenberg drew the radical conclusion:
803 Walter Schottky, whom we have mentioned several times in earlier volumes, was born on 23 July 1886, in Zurich and studied at the University of Berlin from 1904 to 1912, obtaining his doctorate under Planck's guidance in 1912 with a dissertation on relativistic dynamics. Then, he became an assistant to Max Wien in Jena and worked on problems of electron tubes and on the thermal emission of electrons in high-voltage electric ®elds (`Schottky e¨ect,' discovered in 1914). In 1915, he accepted a position in the laboratory of Siemens & Halske in Berlin; in 1920, he returned to an academic career at the University of Rostock (extraordinary professor, 1923, full professor, 1926). From 1927 to 1951, he ®nally worked as a leading scientist for the Siemens & Halske and Siemens-Schuckert Companies in Berlin and Pretzfeld, developing in particular the theory of defects in crystals and of semiconductors. Schottky, one of the great pioneers in this ®eld, died on 4 March 1976, at Forchheim.
804 The BKS theory and its implications have been discussed in Volume 1, Part 2, Section V.2.

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Since all experiments obey the equation

‰ p1q1 d hŠ

‰…631†Š

[with p1 and q1 denoting the minimum inaccuracies of momentum and position of quantum-theoretical particles], the incorrectness of the law of causality is a de®nitely established consequence of quantum mechanics itself. (Heisenberg, 1927b, p. 197)

He found, especially, that in the old statement of causalityÐ`The exact knowledge of the present allows the future to be calculated'Ð`not the conclusion but the [initial] hypothesis is false' (Heisenberg, loc. cit.).
About a year later, he had accepted the wider conclusions from his own ®ndings, as formulated by Bohr in the principle of complementarity, and stated in a public address the de®ciency of the causality principle as follows:

The [classical] formulations of the causal law have shown themselves to be untenable in modern physics. . . . To obtain a statement about an object to be agreed upon, one must observe it. This observation implies an interaction between the observer [i.e., subject] and the object, which changes the object. For the smallest particles the interaction becomes so strong that observation often means destruction. Bohr has coined the concept of ``complementarity'' to describe the situation more appropriately. An accurate knowledge of the velocity [of the particle] excludes an accurate knowledge of [its] position; the former is complementary to the latter. Or, the causal description of a system is complementary to the space-time description of the same system. Because, in order to obtain a space-time description, one must observe, and this observation disturbs the system. If the system is disturbed, we cannot follow anymore its causal connection in a pure manner. . . .
[Consequently], the simple deterministic concept of nature that existed in the previous [classical] physics cannot be carried out anymore. The interaction between observer and object renders a clear causal connection impossible. Of course, one can again think of formulations of the causal principle, which are compatible with modern physics. The most trivial example would be, say, ``Everything that happens, also must happen.'' This statement is, however, meaningless, it does not tell us anything. Or, also, ``If one knows the parameters of a system accurately, one can describe the future.'' This statement is equally meaningless. (Heisenberg, 1984d, pp. 26±27)805

When Heisenberg published his paper containing the relation [(631)]Ðin the above quotationÐhe created a considerable echo that reached beyond Europe. For example, E. H. Kennard of Cornell University, whoÐduring his stay in Copenhagen in summer 1927Ðhad already analyzed the derivation of Heisenberg's uncertainty relations and de®ned the inaccuracies as `mean square deviations' (Kennard, 1927), later argued that they referred less to `the errors of a simultane-

805 The quotation is from the address, entitled `Erkenntnistheoretische Probleme in der modernen Physik (Epistemological Problems of Modern Physics)' post-humanly published (in Heisenberg, 1984d, pp. 22±28). This address may have been Heisenberg's inaugural lecture upon his appointment as Professor of Theoretical Physics at the University of Leipzig; lecture delivered in 1928.

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ous observation of both [canonically conjugate] quantities' but rather `primarily to the ``statistical situation'' determined by the experimental conditions and our knowledge of them' (Kennard, 1928, pp. 345±346). Arthur E. Ruark of the University of Pittsburgh, on the other hand, proposed `an arrangement of the apparatus which seemed to make possible the simultaneous determinations of the coordinate q and the momentum p of a free particle, so accurately that Heisenberg's relation hq Á h p d h is violated'; however, he noticed immediately that `the precision of the measurement of both p and q is limited by statistical ¯uctuations in the measuring apparatus,' especially: `The true reason for the validity of the principle is that slight velocity changes occur when the particle passes through a variable slit.' (Ruark, 1928, p. 709) Finally, Howard Percy Robertson of Princeton provided a mathematical proof of the relation for generalized coordinates in a letter of 18 June 1929, published in July of that year (Robertson, 1929). Altogether, the Americans did accept Heisenberg's result readily, but showed little interest in the complementarity philosophy in which the Copenhagen protagonists had embedded the relation [(631)].806 They rather asked practical questions connected with it, for example, about the `the length of light-quanta' (Breit, 1927). Such a question seemed to his philosophically ambitious European colleagues quite irrelevant: Philipp Frank of Prague discussed in a paper, entitled `UÈ ber die ``Anschaulichkeit'' physikalischer Theorien (On the ``Visualizability'' of Physical Theories)' and published in a February 1928 issue of Naturwissenschaften, the consequences derived from Heisenberg's work on the concept of Anschaulichkeit (visualizability, perceptualness, intuitiveness), which was also at the basis of Bohr's investigations leading to the principle of complementarity.807
After sketching the contents of Heisenberg's pioneering paper (Heisenberg, 1927b), Frank wrote the following comments:
If one then thinks that nothing has been stated [in Heisenberg's g-ray experiment] about the position and the velocities of electrons themselves, but only something about the possibilities of obtaining an accurate measurement, we have to reply: One must distinguish between the mathematical concepts of position coordinates and the velocity coordinates as physical events. As to the latter, they are grasped just through the properties of the scattered light; in the former sense, however, quantum mechanics demonstrates that the components of the coordinates of points do not constitute the most suitable quantities to represent radiation phenomena. But there is nothing ``visualizable'' in these material or electrical points [i.e., in the mathematical position coordinates of particles or radiation]. (Frank, 1928, p. 124)
Frank further claimed that `the requirement for a representation [of atomic objects] by moving points or aether vibrations has nothing to do at all with the
806 For details, see the review by Cartwright (1987b). 807 When Frank wrote his paper, he had not yet seen Bohr's papers on the principle of complementarity (which he therefore did not refer to).

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requirement of Anschaulichkeit but is just connected with a certain Weltanschauung (world view) which is composed of two [quite di¨erent] aspects,' namely, the `materialistic view of nature' and the `idealistic view': The former assumes the existence of completely impenetrable small particles in vacuum, while the latter is based on the (Kantian) trinity of space, time and causality. Only a third point of view, the `positivistic view' (represented by Ernst Mach) would allow oneÐin Frank's opinionÐto resolve the contradictions between the ®rst two views and to describe the situation in quantum mechanics on the basis of Heisenberg's results. Clearly, he claimed that the problem of visualizability was very much connected with the problem of causality; Bohr and Heisenberg would agree, in principle, though Frank's answer, given within the framework of Mach's positivistic view, would be somewhat di¨erent (see also Frank, 1929, and the discussion below).

(b) Heisenberg's Discussions Concerning the Positivism of the `Vienna Circle' (1929±1932)
At the opening session of the ®fth Deutsche Physiker und Mathematikertag in Prague, on 16 September 1928, two speakers addressed the philosophical consequences from the new physical theories, the local physicist Philipp Frank and the Berlin mathematician Richard von Mises. Frank (1929) embedded the results of relativity and quantum theories into the more recent epistemological thoughts of Henri Bergson, William James, and Ernst Mach, as well as those of Rudolf Carnap, Moritz Schlick, and Hans Reichenbach. From the quantum-mechanical situation, he concluded: `The question can never be asked, therefore, as the physicist of the [old]-school philosophy puts it: ``Does strict causality govern nature?'' but rather: ``What are the properties of the correlation between the events and the state variables connected by strict [mathematical] laws?'' ' (Frank, 1929, p. 993) He continued:
When looked at from the point of view of the old philosophy [Schulphilosophie], the [physical theories of the 20th century] imply an undermining (Zersetzung) of rational thinking; they simply are prescriptions for representing experimental results but do not yield any recognition of reality which is left to other methods. However, for those who do not accept the non-scienti®c argumentations, the present theories enforce the conviction that even in such questions, as the ones about space, time and continuity, still there exists a scienti®c progress which proceeds with the progress in our experiences; that it is, therefore, not necessary to assume besides the green and growing tree of science a grey region occupied by the problems that never will be solved. . . . , but rather that there are no limitations where physics passes over into philosophy, if one just formulates the task of physics in the sense of Ernst MachÐas formulated by CarnapÐ``to organize the perceptions systematically, and to derive from the existing perceptions conclusion for future perceptions.'' (Frank, loc. cit., pp. 993±994)

684 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Thus, Frank argued vigorously that the theories of the 20th century should deal with the extended Machian positivism of the `Vienna Circle.'808
While Frank discussed the philosophical interpretation of the causal and the statistical features of the new theories, von Mises spoke about the mathematical formulation of these aspects (von Mises, 1930a). First, he pointed out that the causal principle had received many di¨erent expressions in the past and summarized:
If physics, or science in general, based on progressing information, has ®nally adopted fully the methods of reasoning (Schluûweisen) and the ideas of [mathematical] statistics and accepted them as indispensable tools, then after some time nobody will think that thereby any philosophical demand will remain unsatis®ed. In a word: The causal principle will be changed and will be subdued to what physics requires. (von Mises, loc. cit., pp. 145±146)
The previous deterministic AnsaÈtze of classical physics, so Richard von Mises argued, were connected with certain macroscopic concepts, such as density, dielectric constants and with `directly observable' motions, say, of celestial bodies; as soon as one proceeds to the situation involving many bodies (especially atoms), however, the statistical description must be used that naturally implies a certain irregularity, e.g., a molecular disorder. This description then requires, in the ®rst place, the theorem `that every physical statement must represent a fact which can be checked by observation, i.e., with the help of a real experiment,' and `the observations possess as a decisive property that are repeatable arbitrarily often, be it at di¨erent times, at di¨erent places, or by di¨erent means' (von Mises, loc. cit., pp. 150±151). Of course, due to errors in measurement, the individual observations will exhibit a ¯uctuation; hence, one must take as the `true value of a measurement the expectation value of the ensemble under consideration,' or: `A [physical] theory will be veri®ed by experiment, if the value calculated agrees with the ``true value'' of the observation, i.e., the expectation value as determined through the measured object and the measuring device which, strictly speaking, can be obtained only after having performed in®nitely many measurements.' (von Mises, loc. cit., p. 151)
Now the partisans of causal, deterministic theories insisted on a further `arbitrary' assumption, namely: `To each theoretical result there exists an in®nite sequence of di¨erent experimental arrangements having an increasing accuracy
808 The `Wiener Kreis' (`Vienna Circle') was founded, as mentioned earlier, by Moritz Schlick, and was described as follows: `Numerous pupils, eager to learn and devoted [to learning], assembled around the new Ordinarius [in Vienna, 1922] and the more advanced students together with some teachersÐ among them the philosopher Rudolf Carnap, the mathematician Hans Hahn and other mathematical colleaguesÐto form a circle, which discussed under the guidance of Schlick problems of the logic of science and of fundamental mathematical research and investigated together the development of the philosophical results obtained. Through several publications within a common framework and the organization of several philosophical congresses, this working community became knownÐalso outsideÐas the `Wiener Kreis' (Zilsel, 1937, p. 161).

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such that, if measured in constant units, the size of ¯uctuations of the distribution obtained decreases continuously and ®nally approaches zero.' However, the mathematician Richard von Mises continued, the very existences of ®nite atoms rendered this extrapolation impossible, since:

One would have to imagine that there exist measuring devices whose precision supersedes atomic dimensions, which would obviously imply giving up any physical content. Recently, especially Heisenberg has pointed out the necessity of describing the atomic experiments in detail, and by this he has thrown new light on the discussion of causality and statistics. (von Mises, loc. cit., p. 152)

Indeed, the new quantum mechanics allowed one to calculate quantities in agreement with a statistical evaluation, and Heisenberg's considerations led to the conclusion: `The actual measuring process, also in microphysics, does not represent an elementary but rather a statistical situation. (Der konkrete Meûvorgang ist auch in der Mikrophysik kein Elementarvorgang, sondern ein statistisches Geschehen).' Hence, von Mises declared:
Strict determinism, as is ascribed usually to classical physics of di¨erential equations, is only an apparent (scheinbare) property; it cannot be upheld, if one considers a theory in principle only as valid in connection with experiments that allow one to test it, i.e., one restricts oneself to what is perceptible by senses (sinnlich Wahrnehmbare) or observable ``in principle.'' In macroscopic physics, the indeterministic elements are contained in the objects of observation and partly in the measurement processes; every microscopical phenomenon, however, contains intrinsically the statistical element, because this alone permits the transition to a mass phenomenon (Massenerscheinung) [as opposed to an individual one] and each measurement already represents such a thing [i.e., a mass phenomenon]. (von Mises, loc. cit., p. 153)
These quite lively presentations show that the latest results of the quantum theorists were soon understood, accepted, and properly incorporated into the philosophy of science and the mathematical description in Germany.809 The high point of this very positive reception of the recent results of the quantum physicists occurred at the 91.Naturforscherversammlung, held from 6 to 11 September 1930, in KoÈ nigsberg. A special `Tagung fuÈr Erkenntnislehre der exakten Wissenschaften (Conference on the Epistemology of Exact Sciences)' during this large assembly of scientists and physicians dealt with two topics, namely, the epistemology of science and the foundations of mathematics. On the latter topic, distinguished speakers discussed the main lines of these hot developments at that time: for example, R. Carnap treated the mathematical `logic (Logizismus)' (of Bertrand Russell and others), A. Heyting the `institutionalism' (aÁ la L. Brouwer), J. von Neumann the `formalism' (of Hilbert), and F. Waismann a `critique of language' (aÁ la Wittgen-

809 Richard von Mises repeated his positive remarks on the quantum-theoretical results in developing a `world view (Weltbild ) of science' in a public lecture delivered on 27 July 1930 at the University of Berlin (von Mises, 1930b).

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stein); ®nally, D. Hilbert gave his famous talk on the mathematical problems, ending with the enthusiastic statement, `Instead of a stupid Ignorabimus our parole should be: ``We must know, we shall know (Wir muÈssen wissen, wir werden wissen)'' ' (Hilbert, 1930, p. 963). Hilbert's optimistic remark also applied to the discussions of the second theme, dealing with `the philosophical questions arising from quantum mechanics.' In a brief summary, Hans Reichenbach reported about the talks:
W. Heisenberg (Leipzig) delivered a lecture on causality and quantum mechanics, preceded by one by H. Reichenbach (Berlin) on the concept of truth in physics ( physikalischer Wahrheitsbegri¨ ). The latter talk [Reichenbach's] started from a philosophical critique of the previous physics and explained how, already since some time, the emergence of the probability concept has led in physics to a revision of the physical concept of truth via replacing the alternate logic (Alternativlogik)Ðhitherto the only one knownÐby a logic of probability, for which a given theorem may have any degree of probability, chosen from the continuous values between 0 and 1. These ideas connected smoothly with Heisenberg's lecture which argued that [absolutely] rigorous statements about natural processes in microphysics cannot be made anymore, hence they are meaningless. Following these two talks, in which a remarkable agreement was expressed between the results of research in the philosophy of science (Naturphilosophie) and physics, a stimulating discussion took place that further clari®ed many details. (Reichenbach, 1930, p. 1094)
In a paper, entitled `Die KausalitaÈt in der gegenwaÈrtigen Physik (Causality in Present Physics)' and published in February 1931, Moritz Schlick wrote in his introductory remarks:
The turn taken by physics in recent years on the question of causality could not have been foreseen in any case. As much has been philosophized about determinism and indeterminism, about constants, validity and checks of the causality principleÐno one has as yet hit upon the possibility o¨ered by quantum physics as the key to recognize that a kind of causal order exists in reality. Only a posteriori do we recognize where the new ideas branch o¨ from the old ones, and we are a little amazed that we have previously always passed carelessly through the intersection. (Schlick, 1931, p. 145)
What the Viennese philosopher wished to emphasize was thatÐperhaps di¨erent from the impression given earlier by Frank, von Mises, and ReichenbachÐthe crucial new philosophical idea did not arise so much from the previous lines of argument, but was triggered by Heisenberg's `uncertainty relations (Ungenauigkeitsrelationen).' `The new thing which physics has contributed to the problem of causality does not consist in the fact that the validity of the causal law has been challenged at all, nor [in the claim] that the microstructure of nature would have to be described by statistical instead of causal regularities. All these ideas have been expressed earlier, in part a long time ago,' he declared, and further:

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Rather, the new thing consisted in the up to then never anticipated discovery that by natural laws themselves a limitationÐin principleÐin the accuracy of predictions has been ®xed. This is sometimes totally di¨erent from the obvious idea that factually and practically there exists a limit in the accuracy of observations, and that the assumption of absolutely exact natural laws can be dispensed with in any case if one wants to account for the empirical facts. (Schlick, loc. cit., p. 153)

How did Heisenberg, the man responsible for this drastic change in natural philosophy, see the situation?
In his lecture in KoÈ nigsberg on `Kausalgesetz und Quantenmechanik (Causal Law and Quantum Mechanics),' delivered on 6 September 1930, Heisenberg presented in detail what he considered to be the causal law in the old physics and the extent to which it was violated by the new quantum mechanics (Heisenberg, 1931a). His earlier formulation (Heisenberg, 1927b) had been attacked recently in a book by Hugo Bergmann of the Hebrew University in Jerusalem, who had argued in particular that `one cannot talk about a de®nite statement of the causal law being not valid in quantum mechanics, but at the most about its inapplicability' (Bergmann, 1929, p. 39), i.e., Heisenberg's statement `if-then' would not be su½cient to prove the principle as being invalid.810 Heisenberg answered by taking a longer excursion into the concepts which, he said, were empty and uninteresting if they could not be refuted formally. Thus he stated the simplest form of the causal law as: `Everything that happens, also must happen.' (Heisenberg, 1931a, p. 174) On the other hand, the more serious formulation that `if the present state of an isolated system is known through all the parameters, the future state can be calculated,' still remained valid, provided the interaction between the observing subject and the object could be made arbitrarily small. `In the new quantum theory . . . it is impossible, in principle, to determine all the parameters of an isolated system,' Heisenberg emphasized and continued:

Therefore, the just mentioned formulation of the causal law is not proven to be false but just empty; it does not possess any domain of validity or application any more, hence it does not interest the physicist. (Heisenberg, loc. cit., p. 175)
Clearly, Heisenberg wished to say the following. Even the well-known formulation of the causality principle by Immanuel KantÐi.e., `If we ®nd that something happens, we always assume that something precedes it, upon which it follows according to a [well-de®ned] rule'Ðhad not been proven wrong by quantum mechanics, because the great philosopher had assumed it to be `an a priori synthetic judgment' which could not be checked by experience. Now in quantum physics, the statement simply turned out to be `impractical (unpraktisch).'

810 For a brief account of the early discussion of the philosophical consequences of Heisenberg's relations, see Jammer, 1974, pp. 75±78.

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(Heisenberg, loc. cit., p. 176) From the properties of atomic systems, it rather followed that:
The indeterminacy relations show ®rst that an accurate knowledge of the parameters, which is needed in classical physics to ®x the causal connection, cannot be achieved. A further consequence of the indeterminacy is that also the future behaviour of such an inaccurately known system can be predicted only inaccurately, i.e., only statistically. It is evident that through the indeterminacy relations the foundation for the precise causal law of physics gets lost, both whether it applies to the particle or the wave picture. (Heisenberg, loc. cit., p. 177)
By just referring to the SchroÈ dinger equation, which appears to be a causal equation (in the sense of any classical theory), Heisenberg said, one cannot reinstate the classical causal law, because the wave function does not determine the state of the system uniquely in space and time: To reach this situation, one had to observe the system, but then the indeterminacy relations would spoil the case. Even the idea of describing the observer and the system by a single wave function would not solve the problem, as there exists no space-time description in that case either.
In a lecture on `Die Rolle der Unbestimmtheitsrelationen in der neuen Physik (The Role of the Indeterminacy Relations in the Recent Physics,' presented on 9 December 1930, in Vienna, Heisenberg returned in detail to the causality question (Heisenberg, 1931b).811 He now made the following statement about causality:
In classical physics the causal law was formulated as: ``If at a certain time all data are known for a given system, then it is possible to predict unambiguously the physical behaviour of the system also for the future.'' In quantum theory one may consider as data practically the representative [SchroÈ dinger] function. . . . Then the prescription of the classical law is certainly wrong, because the physical behaviour of a system can in general be predicted only statistically from the SchroÈ dinger function. (Heisenberg, loc. cit., p. 370)
That is, the mathematical formalism of the theory `does not realize anything from the indeterminacy relations'; just the transition from the SchroÈ dinger function to the physical behaviour implies the statistical hypothesis; hence, `one can always consider the perturbations created by the measuring apparatus on the system as the cause of the degeneracy' (Heisenberg, loc. cit.). Heisenberg ®nally concluded:
If nature has built the universe from small constituents of ®nite size, namely electrons and protons, then the question: ``What happens in regions smaller than these constituents?'' should not make a reasonable sense. Therefore, these constituents should behave ``unanschaulich,'' i.e., di¨erent from the objects of the daily life, in order that
811 It should be noted that in 1930 Heisenberg always talked about the `indeterminacy relations (Unbestimmtheitsrelationen)' rather than the earlier `uncertainty relations' (which Heisenberg had referred to as `Ungenauigkeit' or `Unsicherheit' in that context, e.g., in Heisenberg, 1927b). But from 1930 onward, he systematically replaced the word `uncertainty' always by `indeterminacy.'

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nature in the small can be considered to be a closed system (abgeschlossen). Modern physics, for the ®rst time has shown how such a closure of the microworld might be conceivable in principle; the epistemological (erkenntnistheoretische) discussions, which have led to this goal, have clari®ed our thinking, made the language precise, and o¨ered us a deep insight into the essence (Wesen) of human knowledge about nature. (Heisenberg, loc. cit., pp. 371±372)
As we have mentioned earlier, the Viennese philosopher Moritz Schlick took up Heisenberg's results on causality and embedded them into a more professional philosophical system. He also rejected the criticisms of his colleagues like Hugo Bergmann's, though with a slightly di¨erent argument:
There do not exist synthetic judgments a priori. If a theorem states anything at all about reality (and only if it does so, it of course contains some knowledge), then by observing reality one must be able to show whether it is right or wrong. If there exists no possibility, in principle, for testing, i.e., the theorem is compatible with any possible experience, then it cannot contain any knowledge about nature. If, by assuming the theorem to be wrong, anything in the world of experience were di¨erent from the situation for which the theorem is right, a test would be possible; hence the impossibility of a test by experience means that: the view of the world does not depend at all on the theorem being right or wrong, hence it says nothing about nature. (Schlick, 1931, pp. 153±154)
In general, Schlick continued, three di¨erent attitudes could be taken toward the principle of causality, namely:
1. The principle of causality is a tautology. In this case it would always be true but without content (nichtssagend ).
2. It is an empirical theorem. Then it is either true or false, either knowledge or error (Erkenntnis oder Irrtum).
3. It constitutes a postulate, a demand to look further for causes. In this case, it can be neither true nor false, but at most useful or not useful. (Schlick, loc. cit., p. 154)
Now, a tautology was certainly not what was needed in science; on the other hand, the causality principle used so far did not seem to have the character of a physical law; hence, only the third interpretation remained. Indeed, from Heisenberg's indeterminacy relation, de®nitely a physical law, there followed `a rejection of determinism.' However, Schlick continued that this `rejection cannot be considered as a proof for a statement to be untrue, but rather as the demonstration that a rule is not suitable,' and `there always remains the hope that the causality principle will again become triumphant as knowledge progresses.' He concluded:
The rejection of determinism by modern physics means neither that a statement is wrong nor that it is empty; but the prescription, which as the ``causal principle'' shows the path to every induction and every natural law, is unsuitable. The unsuitability is claimed only for a well-de®ned, limited domain, but there it is connected with every certainty implied in the physical experience of today's research. (Schlick, loc. cit., p. 156)

690 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Evidently, Schlick wished to retain as much of the causality principle as possible for future physics.812
It seems that Heisenberg either met Schlick in Vienna or got into direct contact with the philosopher about that time. In any case, toward the end of December 1930, he wrote him a letter thanking him for his `interesting essay on the law of causality' and said that he had `learned much from it' and that:
the tendency of it [Schlick's essay] is extraordinarily pleasant (auûerordentlich sympathisch) to me. In particular, the clear distinction among the three possibilities [to interpret the principle of causality] was very instructive for me; I have tried to present something similar in my lecture at KoÈ nigsberg, but I did not succeed in bringing it out clearly. (Heisenberg to Schlick, 27 December 1930)
Still, Heisenberg had a few objections. He did not understand really `the di¨erence between [the terms] order, lawfulness and ``statistical lawfulness'' ' used by Schlick, and he especially criticized the latter's description of Born's interpretation of the wave function as being `split into two parts: in the strictly lawful propagation of the ™-wave and the existence of a particle or a quantum that is absolutely accidental (schlechthin zufaÈllig) within the limits of ``probability,'' as given by the ™value at the position under consideration' (Schlick, 1931, p. 157). After proposing an example in atomic physics to discuss this point, Heisenberg wrote:
Now what does ``absolutely accidental within the limits of probability'' mean? I cannot see any di¨erence between your ``statistical lawfulness'' and that which we know from atomic physics. Further, I do not see which intermediate between full causality and disorder plus probability law can still be found. . . .
I am also a bit unhappy that I am always quoted along with the statement of the ``invalidity of the causal law,'' as if I were in opposition to Born's conceptions. At that time I considered the phrase ``invalidity'' quite carefully, intending to express two things: ®rst, that the principle of causality has lost its applicability in physics . . .Ðwhich is not the same as the assertion that ``it is wrong''; second, that a theorem having no domain of validity can really not be interesting. The word ``invalid'' seemed to me to lie just right in the middle between ``wrong'' and ``inapplicable,'' but unfortunately it has always been identi®ed with the word ``wrong.'' (Heisenberg to Schlick, 27 December 1930)
Needless to say, Heisenberg agreed with Schlick's refutation of Hugo Bergmann's position. He closed the letter to the `highly esteemed colleague (sehr verehrte Herr Kollege)' by correcting a few statements of the latter about Bohr's ideas of complementarity when applied to biological systems, and thanked him again for his `extraordinarily instructive essay.'
In an immediate reply, dated 2 January 1931, Schlick expressed his apprecia-
812 Schlick also contradicted the proposal of Hans Reichenbach that the causal law could only be extended to the future; hence, it ®xed a direction in time for natural phenomena (see Reichenbach, 1925 and 1931).

IV.1 The Causality Debate

691

tion of Heisenberg's quick reading of his manuscript, especially the clari®cation of Bohr's position. However, he still hoped to be able to retain the di¨erentiation between strict lawfulness and pure accident in atomic theory, as stated in his manuscript (and later paper: Schlick, 1931). More than a year later, Schlick sent Heisenberg a new essay with the title `Positivismus und Realismus (Positivism and Realism),' published earlier in the journal Erkenntnis (Schlick, 1932). In it, he tried to summarize the principles of `the philosophical methods (Denkweisen) known under the name positivism' since the invention of this concept by Auguste Comte in the ®rst half of the 19th century, which consisted in the removal of the contrast between the `true' or `transcendental existence' of reality and the `apparent existence,' as noticed by perceptions. In particular, Schlick focused on the di¨erent attitudes assumed by the `realists,' on the one hand, and the `positivists,' on the other, by investigating the two sets of problems: `The meaning of statements' (in Section II) and `What is the meaning of reality? What does the ``external world'' signify?' (in Section III). We shall return especially to the second part of Schlick's arguments in our next Section IV.2, but here we shall refer to Heisenberg's reaction, as it throws light on his position with respect to the positivistic movement which had thus far embraced his physical results.
In a letter dated 21 November 1932, Heisenberg thanked Schlick for a copy of his essay, and began by stating: `Most of your assertions concerning your programme, I consider to be absolutely right, or, as you indicate on p. 8 yourself, completely trivial. To doubt the statement, which you regard as the central theorem of positivism, seems to me completely absurd.'813 But then he immediately pointed out disagreements about their understanding of what is philosophy: In particular, Heisenberg did not appreciate the establishment of systems of `arti®cial de®nitions' which seemed to him to suppress the `important values' also of philosophy (which he felt to be closer to art than to photography). Hence, he wrote especially:
Your de®nition of philosophy on p. 6 seems to meÐplease forgive meÐcompletely o¨ the track (abwegig). The question whether a certain philosophical statement is true or false, in most cases is completely uninteresting and irrelevant for the value of philosophy. For many deep statements of truth rather the fact applies [as Bohr once said] that the opposite of [a deeply true statement] is also a deep truth. . . . Of course, one may say that ``these truths therefore contain only statements about experiences of sentiments,'' but this excuse appears to me as very suspicious. If we say, ``Here is a table,'' what else is that but ``the expression of the existence of certain feelings, which induce us to de®nite reactions of speech or other nature'' (p. 28). If you reply, ``I can show the table to everybody else,'' I'll tell you, ``similarly one can create in every person the experiences, which are meant by the statements of philosophy.'' Perhaps, you will object, that philosophy is partly art and therefore valuable, but therefore no ``science.'' I would, at this point, at the most admit that philosophy is a type of ``chemical compound'' of science and art (not just a mixture!) . . . , at any rate, a compound transferring knowledge. (Heisenberg to Schlick, 21 December 1932, p. 2)

813 In the published paper, there is just one statement marked as the `central theorem (Hauptsatz),' namely: `Only the given things are real (Nur das Gegebene ist real ).' (Schlick, 1932, p. 4)

692 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Even in science, Heisenberg added, the nonanalytically discovered `suddenly sparkling recognitions,' such as Newton's discovery that the gravity of all bodies causes the planetary motion, constitute `valuable knowledge.'
Heisenberg concluded by emphasizing the importance of the logical calculus for philosophyÐ`though this instrument [Heisenberg called it a ``brilliant (herrliches)'' system] is not yet philosophy.' While he did not believe at all in the possibility of a `really ``clear'' language,' he believed `that the best [thing] to achieve is to create clarity at the one little place, where a contradiction directs our attention to an obscurity.' As an example, he cited simultaneity in Einstein's investigation leading to special relativity theory. `Please forgive me that I have used the words lightheartedly, . . . I hoped you would prefer to hear a natural opinion rather than read a learned (ausgetuÈftelte) essay,' Heisenberg concluded his letter on philosophy to Schlick, and signed it with `herzlicher Hochachtung (cordial respects).' He had indeed written quite clearly about what he considered to be the central message of quantum mechanics for philosophy, when he remarked: `It seems to me more than an unfortunate accident that [Philipp] Frank and [Hans] Reichenbach in their work hardly mention the real point of quantum theory, namely Bohr's [principle of ] complementarity, and instead of it reproduce the much more super®cial aspects in Born's papers and mine.' (Heisenberg to Schlick, loc. cit.)

(c) The Indeterminacy Relations for Relativistic Quantum Fields (1929±1933)

In spring 1931, the young Carl Friedrich von WeizsaÈcker (not yet quite 19 years of age) submitted his doctoral thesis under the direction of Werner Heisenberg in Leipzig, dealing with the determination of the position of an electron by a microscope (von WeizsaÈcker, 1931). He carried out, in particular, `a rigorous calculation of the problem . . . with the help of the Heisenberg-Pauli formulation of quantum electrodynamics' (von WeizsaÈcker, loc. cit., p. 114). According to the standard discussion of the g-ray Gedankenexperiment of Heisenberg, the uncertainty of the position hq assumed at least the value

hq

d

l sin

e

Y

…632†

with l denoting the wavelength of the light used and 2e the angle of aperture of the bundle of rays used for the imaging procedure. Von WeizsaÈcker then found that the procedures involved in the measurement of position were more complex than Heisenberg and Bohr had assumed in their 1927 discussions of the Gedankenexperiment; especially, they included the illumination of the original electron at a space point P, the emission of radiation by this electron under the angle of aperture …2e†, and the stimulation of a second electron at the point PH of the observation screen. If he treated the problem according to proper quantum electro-

IV.1 The Causality Debate

693

dynamics (with the Dirac equation describing the electron), he indeed obtained for the position probability a wave packet of size lasin e.814
Von WeizsaÈcker's thesis completed the demonstration of the `elementary' relations by using ®eld-theoretical methods. Two years earlier, Heisenberg `contemplated how one could elucidate the uncertainty relations (Unsicherheitsrelationen) for the [electromagnetic] wave amplitudes.' As Heisenberg wrote to Bohr:

As a matter of course, any measurement would yield not [the electric ®eld strength] E and the [magnetic ®eld strength] H at an exact point but average values over perhaps very small spatial regions. Let hV be the volume of this spatial region, then the commutation relations between Ei and pk look like this,

Ei pk

À

pk Ei

ˆ

dik 2hci

1 hV

‰…633†Š

where Ei and pk are now to be interpreted as average values over the spatial volume hV …ˆ …hL†3†. Consequently, one would expect indeterminacy relations of the form

hEi

pi

†

dik

hc hV

or

Ei hHk

†

hc hV hL

‰…634†Š

(Heisenberg to Bohr, 16 June 1929; English translation in Bohr, 1996, pp. 5±6)

Heisenberg then indicated twoÐas he admitted, not `quite solid'Ðmethods to derive Eq. [(634)]. In spite of the shaky derivation, however, he took over the last Eq. [(634)] into his lectures at Chicago [Heisenberg, 1930a, p. 50, Eq. (38)].
Bohr did not respond to the detailed contents of Heisenberg's letter until early in 1931 when the situation had changed drastically, as LeÂon Rosenfeld recalled nearly two decades later:

When I arrived at the [Copenhagen] Institute on the last day of February 1931, for my annual stay, the ®rst person I saw was Gamow. As I asked him about the news, he replied in his own picturesque way by showing me a neat pen drawing he had just made. It represented Landau tightly bound to a chair and gagged, while Bohr standing before him with upraised fore®nger, was saying: ``Bitte, bitte, Landau muss ich nur ein Wort sagen!'' [``Please, please, Landau, may I just say one word!''] I learned that Landau and Peierls had just come for a few days before with some new paper of theirs which they wanted to show Bohr, ``but'' (Gamow added airily) ``he does not seem to agreeÐand this is the kind of discussion which has been going on all the time.'' Peierls had left the day before, ``in a state of complete exhaustion,'' Gamow said. Landau stayed for a few weeks longer, and I had the opportunity of ascertaining that Gamow's representation of the situation was only exaggerated to the extent usually conceded to artistic fantasy. (Rosenfeld, 1955, p. 70)

814 Von WeizsaÈcker added the remark: `Our result that the uncertainty of imaging quantum-

theoretically

will

not

be

larger

than

l sin

e,

cannot

be

guaranteed

for

l

ˆ

l0

d

h mc

because

of

the

size

of

the momentum transfer in the Compton e¨ect.' (von WeizsaÈcker, 1931, p. 130)

694 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

Rosenfeld's reference was to the investigation, entitled `Erweiterung des Unbestimmtheitsprinzips fuÈr die relativistische Quantentheorie (Extension of the Indeterminacy Principle to Relativistic Quantum Theory),' which Lev Landau and Rudolf Peierls had completed in late January and would eventually submit in early March 1931 to Zeitschrift fuÈ r Physik (Landau and Peierls, 1931). They started with the observation that the application of wave-mechanical methods to relativistic problems led to several `senseless' results: ®rst, the negative energy states of Dirac's electron equation; second, to `hopelessly in®nite divergence' of the interaction of a charged particle with itself; and third, to `in®nite matrix elements of the energy density.' Hence, they concluded:

It is shown that by considering possible methods of measurement that all the physical quantities occurring in wave mechanics can in general no longer be de®ned in the relativistic range. This is related to the well-known failure of the methods of wave mechanics in that range. (Landau and Peierls, loc. cit., p. 56; English translation, p. 152)

In order to support their special claim `that in the range considered the physical requirements of the applicability of the methods of wave mechanics are no longer satis®ed,' Landau and Peierls turned to what they considered a generalization of Bohr's ideas on the concept of measurement (as presented by Bohr in his lectures at Como and Brussels, 1928a, e, respectively). In particular, they argued that according to Bohr, for every quantum-mechanical system there should exist predictable measurements; i.e., `measurements such that for every result there is a state of the system in which the measurement certainly gives the result' (Landau and Peierls, loc. cit., p. 57); hence, they also concluded: `If the wave function of the system cannot be determined by the measurement, it can have no meaning,' or `the existence of predictable measurements is an absolutely necessary condition for the validity of wave mechanics' (Landau and Peierls, loc. cit., p. 58). Now, in Bohr's scheme, every momentum measurement in time ht is connected with a de®nite change hP (in addition to the unknown change which restricts the accuracy of the measurement due to the indeterminacy relation), given by the relation

…v

À

v H †hP

b

h ht

Y

…635†

where v and vH denote the velocities of the particle before and after the change. In the relativistic case, v À vH assumes at most the value c; hence, they found

hP

Á

ht

b

h c

Y

…635 H †

or `the concept of momentum has a sharp meaning only for long times.' (Landau and Peierls, loc. cit., p. 61) This applies, in particular, for free particles, while for

IV.1 The Causality Debate

695

charged particles emitting radiation another additional momentum uncertainty, hp, would result; i.e.,

h

p

Á

ht

b

e2 c3

…v

À

v H †X

…636†

Now, for electrons (where vH À v is of the order c), the uncertainty is smaller than the uncertainty (635H), because then h pht b e2 (and the small ®ne-structure
c constant); but for macroscopic bodies Eq. (636) becomes important; hence, both uncertainties have to be combined to give the ®nal relation

r

h

pht

b

h c

he c2 X

…637†

Landau and Peierls thus derived, e.g., in the case of the Compton e¨ect, an additional scattering e¨ect, consisting of `a further, uncontrollable radiation . . . obtained when higher approximations are taken into account in the perturbationtheoretical calculation for the interaction between radiation and particle' (Landau and Peierls, loc. cit., p. 62).815
With these preparations, Landau and Peierls proceeded to consider the measurement of electric and magnetic ®eld strengths. For the observation of the electric ®eld E, they employed a body of very large mass (hence small velocity, to keep the magnetic disturbance small), whose momentum accuracy, h p, after the measurement processes was given by Eq. (637). Then, the accuracy hE of the measured ®eld strength was given by

p

hE

b

1 eht

h

p

ˆ

hc …cht†2

X

…638†

Similarly, for the accuracy of the magnetic ®eld strength H followed in the case of a separate measurement

p

hH

b

hc …cht†2

X

…639†

In the case of simultaneous measurements of both electric and magnetic ®eld strengths, the magnetic ®eld of the charged test body had to be considered as well,

815 In the case of the Compton e¨ect, though, this extra radiation became quite negligible due to the

smallness p. 62).

of

e2 pc

(the

®ne-structure

constant),

as

Landau

and

Peierls

noted

(Landau

and

Peierls,

1931,

696 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

yielding an additional inaccuracy; thus, ®nally, there followed the relation

hE

Á

hH

b

hc …cht†2

1 …hl†2

Y

…640†

where hl was the distance between the test body and the magnetic needle (measuring the magnetic ®eld strength H). From Eqs. (638) to (640), Landau and Peierls concluded `that for ht ˆ y, the measurement can be made arbitrarily accurate for both E and H'; hence:

Thus static ®elds can be completely de®ned in the classical sense. . . . In the quantum range, on the other hand, the ®eld strengths are not measurable quantities. (Landau and Peierls, loc. cit., p. 63)

That is, neither light-quanta nor material particles (such as electrons) could be measuredÐthis impossibility then might explain also the well-known di½culties with energy conservation in beta-decay, Landau and Peierls argued at the end of their paper.
The investigation by Landau and Peierls caused considerable stir, not only in Copenhagen.816 After some time, Pauli raised objections; in a letter to Peierls, he wrote:

Obviously,

it

is

wrong

that

the

radiation

energy

e2 c3

…v À vH†2 ht

represents

an

uncertain

energy change. . . . It may be that the radiation energy also contains some uncertainty

in time development, but in the ®rst approximation the radiation energy certainly

represents a de®nite change. Hence the equation [(636)] is certainly wrong as an

uncertainty relation. This is already clear from the fact that it does not contain h

[Planck's constant] and, if correct, would postulate a fundamental uncertainty of

charged particles in the classical theory. (Pauli to Peierls, 3 July 1931, in Pauli, 1985,

p. 91; English translation in Bohr, 1996, p. 10)

However, in January 1933, when he read the proofs of his Handbuch article on wave mechanics, Pauli admitted the validity of the indeterminacy relations (638) and (639), while still denying Eq. (636). (See Pauli's letter to Heisenberg, dated 18 January 1933, in Pauli, 1985, especially, p. 150.) In his published Handbuch treatise, Pauli wrote:

At this point, however, the argument of Landau and Peierls contains an essential gap, since the emitted-radiation momentum and the emitted-radiation energy can be measured accurately. The change of energy and momentum of the charged [test] body caused by them therefore cannot be regarded just as an indeterminate change. Because of this the further consequences are connected with an essential uncertainty, and the question of the ®eld-strength measurement must be considered to be one that has not yet been clari®ed. (Pauli, 1933c, p. 257)

816 See, for instance, the letters of Heisenberg to Peierls and Landau, dated 26 January 1931, and Heisenberg to Pauli, dated 12 March 1931, in Pauli, 1985, pp. 53±54, 66±67.

IV.1 The Causality Debate

697

Pauli, in his letter to Heisenberg mentioned above, claimed that Bohr also considered Eqs. (638) and (639) to be correct. At that time, however, the Copenhagen teamÐnow consisting of Bohr and RosenfeldÐhad nearly completed their own investigation on the subject, leading to quite di¨erent conclusions. We do not know exactlyÐnot even from Rosenfeld's recollections which we have quoted earlierÐ when they really began their work actively. It might have been already rather early, i.e., soon after Rosenfeld's arrival in March 1931, because the latter also recalled: `My ®rst task was to lecture Bohr on the fundamentals of ®eld quantization; the mathematical structure of the commutation relations and the underlying physical assumptions of the theory were subjected to unrelenting scrutiny.' (Rosenfeld, 1955, p. 71) But it is clear that the main results were in hand on 2 December 1932, when Bohr presented them to the Danish Academy, although the ®nally published paper was signed only in April 1933.817 In any case, Rosenfeld reported that Bohr took over the lead `after a short time' and then `he was pointing out to me essential features to which nobody had yet paid su½cient attention,' especially:
His ®rst remark, which threw decisive light on the problem, was that ®eld components taken at de®nite space-time points are used in the formalism as idealizations without immediate physical meaning; the only meaningful statements of the theory concern averages of such ®eld components over ®nite space-time regions. This meant that in studying the measurability of ®eld components we must use as test bodies ®nite distributions of charge and current, and not point charges as has been loosely de®ned so far. The consideration of ®nite test bodies immediately disposed of Landau and Peierls' argument concerning the perturbation of the momentum measurements by the radiation reaction; it is easily seen that this reaction is so much reduced for ®nite test bodies, as to be always negligible. (Rosenfeld, 1955, p. 71)
However, the problem of constructing and using test bodies proved to be a long story which began with a quick result, namely, the case given by Heisenberg's Eq. (634)Ðin fact, the only case written by anybodyÐ`was one in which unlimited accuracy had to be expected from the correctly integrated commutation law.' On the other hand, the correct relativistic treatment of extended bodies presented many di½cult situations, especially when they were investigating whether relativity implied further restrictions to the measurability of momentum. `This necessitated a much more detailed analysis of the measuring process than one was wont [to carry out] in an ordinary quantum mechanics,' recalled Rosenfeld, and: `Bohr succeeded in showing that the measurement of the total momentum can even be performed in such a way that the displacements of the elements, though uncontrollable within a

817 See the report in Overs. Dan. Vidensk. Selsk. Virks Juni 1932±Maj 1933, p. 35: `Niels Bohr gav en Meddelelse: Om den begroensede Maarlelighed af elektromagnetiske Kraftfelter'; or the announcement in Nature 132, 75 (1933): `Dec. 2, Niels Bohr: The limited measurability of electromagnetic ®elds of force. An investigation in collaboration with L. Rosenfeld proves the existence of a limitation of the measurability of electromagnetic ®eld components, conforming with the tentative rational formulation of quantum electrodynamics, and analogous to the characteristic complementary limitations of the mechanical quantities, which secures the consistency of quantum mechanics.' (Reprinted in Bohr, 1996, p. 54)

698 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
®nite latitude hx, are equal, and that the determination of the total momentum is only limited by the uncertainty of the common displacement hx to the extent pahx, indicated by the indeterminacy relation.' (Rosenfeld, loc. cit., p. 75) `The reading of the fourteen or so successive proofs only took about one more year,' in which a ®nal great trouble had to be resolved, namely, the role played by the ®eld ¯uctuation in the logical structure of the theory. (Rosenfeld, loc. cit., p. 77)
Bohr and Rosenfeld embarked upon their fundamental paper, `Zur Frage der Meûbarkeit der elekromagnetischen FeldgroÈûen (On the Question of the Measurability of the Electromagnetic Field Quantities),' with the ®rm conviction that `the quantum theory of ®elds should be viewed as a consequent, correspondence-like reformulation of the classical electrodynamic theory, just as quantum mechanics constitutes a reshaping of classical mechanics corresponding to the existence of the quantum of action' (Bohr and Rosenfeld, 1933, pp. 3±4). In dealing with the topic properly, the fact had to be considered that the quantum-electrodynamical formalism did not depend per se on the atomic constitution of matter; hence, the e¨ects of retardationÐwhich played an essential role in the earlier investigationsÐ could be neglected by choosing suitably extended test bodies (i.e., large compared to atomic dimensions) having an approximately constant charge distribution. Further, `the ®eld quantities are not represented by genuine point-functions but by functions of space-time regions, which correspond formally to the average values of the idealized ®eld components over the regions under investigation' (Bohr and Rosenfeld, loc. cit., p. 5). In relativistic ®eld theory, an essential complication of measurement arose, because `when comparing ®eld averages over di¨erent spacetime regions, we cannot speak generally in a unique manner about a time sequence of measuring processes, but already the interpretation of single results of a ®eld measurement requires a still greater caution than in the case of usual [i.e., nonrelativistic] quantum-mechanical measurement problems,' Bohr and Rosenfeld emphasized, and then sketched the main aspects of their treatment as follows:
For measurements of ®eld quantities, each result measured is well de®ned on the basis of the classical ®eld concept; the limited application of the classical ®eld theory for describing the unavoidable electromagnetic ®eld actions of the test bodies in the measurements leads, as we shall see, to the consequence that those ®eld actions in¯uence to a certain extent the very result of the measurement in an uncontrollable manner. A closer study of the principally statistical character of the consequences from the quantum-electrodynamical formalism, however, demonstrates that this in¯uence of the measuring process on the measured object does not restrict the possibilities to check such consequences in any way; it must rather be considered to constitute an essential feature of the intimate ®t (innige Anpassung) of the quantum theory of ®elds to the problem of measurability. (Bohr and Rosenfeld, loc. cit., pp. 6±7)
With these ideas, Bohr and Rosenfeld attacked their problem, emphasizing at once, however, that they would leave out completely the discussion of the wellknown di½culties of quantum electrodynamics, primarily the in®nite self-energy. This meant that they were able to deal in their programme entirely with the charge-free theory.

IV.1 The Causality Debate

699

In that approach, which had been prepared several years earlier by Pascual Jordan and Wolfgang Pauli (1928), the commutation relations (see Section II.7) between the electromagnetic ®eld components at the space-time points 1 and 2 assumed the form,

‰Ex…1†Y Ex…2†Š ˆ ‰Ex…1†Y Ey…2†Š ˆ

‰Hx…1†Y Hx…2†Š ‰Hx…1†Y Hy…2†Š

ˆ ˆ

i

h 2p

…Ax…1x2†

i

h 2p

…Ax…1y2†

À Ax…2x1††Y À Ax…2y1††Y

W bbbbbbbbbbbbbba

‰Ex…1†Y Hx…2†Š ‰Ex…1†Y Hy…2†Š

ˆ ˆ

0Y À‰Hx…1†Y Ey…2†Š

ˆ

i

h 2p

…Bx…1y2†

À

Bx…2y1††Y

bbbbbbbbbbbbbbY

…641†

with the help of the relativistic generalizations of the Dirac d-function,

Ax…1x2† Ay…1y2† Bx…1y2†

ˆ ˆ ˆ

2

À

q2 qx1qx2

À

1 c2

q2 qt1qt2

3&1 r

 d t2

À

q2 qx1qx2

&1 r

 d t2

À

t1

À

cr'Y

À

1 c

q qt1qz2

& 1 r

 d t2

À

t1

À

cr'X

À

t1

À

cr'Y

W bbbbbbbbbbbba bbbbbbbbbbbbY

…641a†

Since Bohr and Rosenfeld considered the averages of the ®eld quantities (denoted

by bars) over a space-time region having the volume V and the time duration T,

i.e., Ax…IxY

Ex…I † , II† ˆ

eVtIcV., IIo1TnIlyTItIhÁe…

average…d (an…d,

dt1 dt2 dv1

vI

vII

therefore, dv2 Ax…1x2† ,

regular) relativistic d-functions, etc., entered into their quantum-

electrodynamical commutation relations, which they wrote explicitly,

hEx…I † hEx…II † hEx…I † hEy…II †

d d

h 2p

jAx…IxY

II

†

h 2p

jAx…IyY

II

†

À À

W

Ax…IxI Ax…IyI

Y Y

I I

†jY †jY

bbbbbbbbbbbbbba

hEx…I † hHx…II † hEx…I † hHy…II †

d d

0Y
h 2p

jBx…IyY II†

À

Bx…IyIY I†jX

bbbbbbbbbbbbbbY

…642†

700 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

From the relations (642), they derived immediately `that the averages of all ®eld components over the same space-time region commute, and therefore can be measured independently of each other,' and further `that the averages of two di¨erent-types of components, like Ex and Hy, over arbitrary time intervals commute if the respective space regions coincide' (Bohr and Rosenfeld, 1933, p. 12). The di¨erent result, concluded in the latter cases by Heisenberg earlier, depended on his peculiar limiting procedure: He ®rst took equal times t1 ˆ t2, and then equal space regions, which actually led to an ambiguous result. Such an ambiguity could be avoided if one took, as Bohr and Rosenfeld insisted upon, extended test bodies.
For spatial dimensions, i.e., L b cT, the above results corresponded to those of the classical theory; for L … ct, peculiar ¯uctuations arose in quantum ®eld theory, `which are most intimately connected with the impossibility to visualize the lightquantum picture characterizing quantum ®eld theory in terms of classical concepts' (Bohr and Rosenfeld, loc. cit., p. 15). Bohr and Rosenfeld calculated these ¯uctuations explicitly and found that for ®eld averages surpassing a critical value S (which was the square root of the vacuum ¯uctuations), the ¯uctuations might be neglected. From the commutation relations (642), there resulted also another critical value U (being about the square root of the right-hand side of Eqs. (642) for regions shifted by distances L and T ); for ®eld strengths larger than U, all quantum-theoretical features would disappear. These critical values were given, respectively, by

r

UdSd

h 2p

ca…L

Á

cT

†

for L ` cT

…643a†

and

r

r

Ud

ha2p L3T

c

and

Sd

…ha2p† Á c L2

for L b cTX

…643b†

Hence, in the latter case, for L g cT, where U becomes much larger than S, no ®eld ¯uctuations occur in the formalism.
With these background preparations, Bohr and Rosenfeld turned to their main problem, the physical measurement of the ®eld quantities, which is based on the process of transporting momentum onto electrical and magnetic test bodies brought into the ®elds. Thus, for instance, to determine Ex by a test body of volume V …ˆ L3†, having a homogeneous electric density r, they used the relation

pxHH À pxH ˆ rExVT Y

…644†

if pxH and pxHH denoted the momentum of the test body at initial and ®nal times, tH and tHH, respectively …tHH À tH ˆ T†. Upon inserting the fundamental indeterminacy

IV.1 The Causality Debate

701

relation … px hx d ha2p†, they obtained for the uncertainty of Ex,

hEx

d

ha2p rhx Á V Á

T

Y

…645†

which could be made arbitrarily small by choosing the electric density r large enough. By selecting the particularly suitable situation L b ct, hEx could be written as

r

hEx d lQY with Q ˆ hVa2TpY

r

…645 H †

where

l

denoted

a

small

dimensionless

factor

(namely,

1 rhx

hVa2Tp).818 Now, by

taking into account the acceleration of the test body, the measured ®eld received a

slight change; indeed, an elementary charge as a test body would then give rise to a

minimum uncertainty of the electric ®eld strength Ex; i.e.,

p

hmEx d

…ha2p†c c 2 T ht

X

…646†

Upon this result, Bohr and Rosenfeld commented as follows:

If one further, like Landau and Peierls, does not distinguish between T and ht, this expression agrees with the absolute limit of measurability of a ®eld component, on which they based their criticism of the foundations of the quantum-electrodynamical formalism. (Bohr and Rosenfeld, loc. cit., pp. 24±25)
However, in the case of an extended test chargeÐas considered by Bohr and Rosenfeld, in contrast to Landau and Peierls and Heisenberg beforeÐthe retardation e¨ect became much smaller, namely of the order of l2. Hence, the authors concluded:
For the discussion of the measurability of [electromagnetic] ®eld quantities, it is of fundamental importance to assume that the test bodies used [behave] like a uniformly charged rigid body, whose momentum can be determined within any given, arbitrarily small time interval with an accuracy derived from ‰h phx d ha2pŠ, complementary to the accompanying, uncontrollable shift in position. (Bohr and Rosenfeld, loc. cit., p. 27)
A detailed evaluation of the test body (if split into many parts) con®rmed that conclusion.

818 Evidently Q ˆ U, due to Eq. (643b). For hx f L, the small factor lrmeansthat the test body

carries

a

large

number

N

of

elementary

charges

e,

namely

N

ˆ

rV ae

ˆ

1 l

L cT

2hpcae2.

702 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

Before proceeding to their ®nal goalÐi.e., to calculate the accuracy of ®eld measurementsÐBohr and Rosenfeld evaluated the e¨ect of the ®elds on the test bodies: They found the classical result, with a ¯uctuation determined by S, Eq. (643b); hence, it decreased quickly with increasing size L of the region of measurement. Now, every ®eld component observed, say, Ex, constituted a superposition of the corresponding ®eld components arising from all sources (including the test bodies); hence, Eq. (644) had to be written explicitly as

px…I†HH À px…I†H ˆ rIVITI…Ex…I† ‡ Ex…IY I††Y

…644 H †

where Ex…I† momentum

denoted the measurement

average value would be made

of at

Ex in time t

the observed region I on the test body, and Ex…I

Y

if
I†

no the

contribution of the latter obtained from the measurement. Thus, a minimum value

followed for the uncertainty Ex…I†, given by the relation

q hmEx…I† d ha2pjAx…IxY I†j Y

…647†

which for LI b cTI became identical with the critical quantity QI . This limit could be reduced still further by an additional mechanism (involving a spring), even to zero, apart from the inevitable ®eld ¯uctuations. Hence, the accuracy of a single ®eld measurement in quantum electrodynamics was `restricted only by the limit of the classical description of the test body's ®eld action' (Bohr and Rosenfeld, loc. cit., p. 46), a result which appeared to be justi®ed by the fact `that one must deal in all measurements of physical quantities, by de®nition, with the application of classical conceptions, and that for ®eld measurement any reference to a limitation of the strict applicability of classical electrodynamics would contradict the concept of measurement itself ' (Bohr and Rosenfeld, loc. cit., p. 47). On the other hand, this conclusion must be compensated, Bohr and Rosenfeld continued, in the complementary view, namely by `the fact that the knowledge of the light-quantum composition of the ®elds [i.e., the quantum-mechanical constitution] gets lost by the ®eld actions of the test bodies, . . . the more the greater is the accuracy demanded from the measurement' (Bohr and Rosenfeld, loc. cit., p. 48). That complementary feature of the theory also ensured `that every attempt to restore the knowledge of the light-quantum composition of the ®eld by a later measurement with any suitable apparatus would simultaneously prevent any further use of the ®eld measurement in question' (Bohr and Rosenfeld, loc. cit.).
The measurement of two average values of a given ®eld component could be carried out just as well along these lines, yielding eventually the result,

hEx…I †

Á

hEx…II †

d

h 2p

jAx…IxY II†

À

Ax…IxIY I†j

…648†

in agreement with the ®rst Eq. (642). Herewith, one had to consider a special feature of the relativistic ®eld theory, notably: `When measuring two ®eld averages, one can only speak about a sequence of measurements if the corresponding time

IV.1 The Causality Debate

703

intervals T1 and T2 are completely separated.' (Bohr and Rosenfeld, loc. cit., pp. 57±58) Finally, in the case of measuring two average values of di¨erent ®eld components, Bohr and Rosenfeld calculated the indeterminacy relation

hEx…I †

Á

hRy…II †

d

h 2p

jC x…IyY

II

†

À

Cx…IyIY I†j Y

…649†

where

Ry…II† ˆ Ey…II †

or

Hy…II †

and

C

…I Y xy

II

†

ˆ

Ax…IyY II†

or

Bx…IyY II†,

respectively.

`We

therefore arrive at the conclusion mentioned already in the beginning that the

quantum theory of ®elds represents, as far as the problem of measurability is

concerned, an idealization which is free from contradictions insofar as we can

forget about all restrictions created by the atomistic structure of ®eld sources and

of the measurement apparatus,' Bohr and Rosenfeld ®nished their long memoir

(Bohr and Rosenfeld, loc. cit., p. 64), for whose extensive details they excused

themselves on account of the complicated character of the mathematical formal-

ism of quantum electrodynamics which required, in addition the use of certain

features not known in the nonrelativistic measurement problem.819 LeÂon Rosen-

feld, with whom Bohr had worked out the ®eld-theoretical measurement prob-

lems, would become one of his favourite helpers and a long-term associate in

Copenhagen.820

(d) The Continuation of the Debate on Causality with the Berlin Physicists (1929±1935)
In the early discussions of the causality problem immediately following Heisenberg's derivation of the uncertainty relations, we have thus far missed certain voices that one would have expected to hear from the conservative side, notably,

819 Bohr summarized this work in the general discussion at the seventh Solvay Conference on Physics in Brussels (in Institut International de Physique Solvay, ed, 1934). He also returned to the problem in an unpublished manuscript, entitled `Field and Charge Measurements in Quantum Theory' of 1937 (reproduced in Bohr, 1996, pp. 195±209), and after many further years he wrote a ®nal paper on the topic, again with Rosenfeld, which was published in the Physical Review after World War II (Bohr and Rosenfeld, 1950).
820 LeÂon Rosenfeld was born on 14 August 1904 at Charleroi, Belgium, and studied physics and mathematics at the University of LieÁge, obtaining his doctorate in 1926. He then went to the EÂ cole Normale SupeÂrieure and ColleÁge de France (to work with Louis de Broglie), and in spring 1927 to Brussels (to work with TheÂophile de Donder), before he joined Max Born in GoÈ ttingen as an assistant (1927±1929). During 1929±1930, Rosenfeld worked with Pauli in Zurich, and from 1930 to 1940 he occupied positions at the University of LieÁge (1930±1935 as Reader, 1935±1940 as Professor), spending simultaneously longer periods at Copenhagen, assisting Bohr. From 1940 to 1947, he held a professorship in Utrecht, and from 1947 to 1958 one at the University of Manchester; then he moved to Copenhagen as professor at the newly established Nordic Institute for Theoretical Physics (NORDITA). He died on 23 March 1974 at Copenhagen.
Rosenfeld worked especially on nuclear physics and quantum ®eld theory, principally quantum electrodynamics, and in the 1940s he became an expert on the problem of nuclear forces (on which topic he published a book in 1948). He also investigated basic problems of statistical mechanics and quantum theory, but was always attracted to work on epistemological questions; thus, in later years, he was considered one of the principal advocates and defenders of the `true' Copenhagen interpretation of quantum mechanics and a great admirer of Niels Bohr.

704 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
those of Einstein, Planck, and SchroÈ dinger. Of course, Einstein, between the ®fth and sixth Solvay Conferences in 1927 and 1930, respectively, had tried to undermine the very cornerstone of the acausal interpretation of quantum mechanics, namely, Heisenberg's uncertainty relations, by considering clever Gedankenexperiments in the atomic domain; this was his contribution to the discussion.821 Since all his e¨orts had failed in this direction, he would retire for some years, especially from the public debate, and rather work very eagerly on what he considered to be the big question in physics: the extension of general relativity to obtain a uni®ed ®eld theory of matter which would even incorporate such features as revealed by Dirac's electron theory.822 Still, there existed a further reason for Einstein's temporary absence from the causality debate: In the years after 1928, he spent much time away from home, especially in the United States, where he ®nally established a new home ready to receive him after the political change in Germany drove him away from Europe. However, Berlin did not only have Einstein as a representative of the anti-Copenhagen view, but Planck and SchroÈ dinger also belonged to the same group of critics, and they expressed themselves several times in the 1930's, though expounding di¨erent reasons individually.
On 4 July 1929, Erwin SchroÈ dingerÐwho had been appointed as Max Planck's successor in the chair of theoretical physics at the University of Berlin in fall 1927Ðdelivered his inaugural lecture as a member of the Prussian Academy of Sciences. After sketching the scienti®c development within the Viennese scienti®c community (due to Ludwig Boltzmann, Franz Exner, and Fritz HasenoÈ hrl) and indicating his own ®eld of interest in theoretical physics, he turned to `the most burning questions' of the theory in those days, namely, `whether along with classical mechanics its method had to be given up as well, i.e., the fundamental theorem (Grundsatz) that de®nite laws together with accidental initial conditions determined the natural processes in each single case: it is the question of the usefulness (ZweckmaÈûigkeit) of the infallible postulate of causality' (SchroÈ dinger, 1929d, p. CI). SchroÈ dinger recalled how he had learned already in Vienna (through Exner and HasenoÈ hrl) that the strictly deterministic view of nature might not be upheld because of the practical impossibility to ®x the state of a body consisting of millions of atoms, and he pointed out that the recent development of quantum theory seemed to demand even more, namely, the abandonment altogether of the possibility of determining the initial state of an atomic system. However, he continued:
I do not believe that [the causality problem] will ever be answered in this way. In my opinion, this question does not decide about the real property of nature (wirkliche Bescha¨enheit der Natur) as we are confronted with, but about the suitability and
821 For details, see Section II.6. 822 He worked on this topic especially with the Austrian mathematician Walther Mayer, focusing on a ®ve-dimensional theory of what they called `semi-vectors' (Einstein and Mayer, 1931; 1932a, 1932b). These e¨orts, had they succeeded, would have opened vistas beyond the limitations of the existing quantum mechanics (removing also, in particular, the unwanted statistical foundation).

IV.1 The Causality Debate

705

convenience of one or the other view in our thinking about nature. Henri Poincare has stated that we may be allowed to apply to real space the Euclidean as well as nonEuclidean geometry without fearing to be contradicted by facts. The physical laws which we discover, however, are functions of the geometry applied, and it may happen that one geometry leads to complicated and the other to simpler physical laws. Then one geometry turns out to be convenient, the other inconvenient, and the words ``right'' or ``wrong'' should not be used. The situation may be similar with the postulate of strict causality. There may hardly be [any] imaginable facts of experience which will ®nally decide whether a process of nature is absolutely determined or partially determined in reality, but at the most they will decide whether one or the other view allows a simpler survey of the facts observed. Even to reach this decision a long time will pass. Because also with respect to the geometry of the world we have become less sure, since we grasped with Poincare our freedom of choice. (SchroÈ dinger, loc. cit., pp. CI±CII)
SchroÈ dinger had expressed a similar view already several years earlier in a letter to Hans Reichenbach, dated 25 January 1924 (but published only in 1932). After calling the causality conclusions `nothing but a tautology,' he had added:
However, it perhaps still appears that our idea of causality has something to do with realism. Just because we consider our surrounding as something real which persists for a certain while, we can go as far as giving this reality the property of being causally connected. Of course, behind this concept of a ``relatively continuous reality (relativ bestaÈndigen Realen)'' is hidden only what has been asked originally: why can past experience state something about future experience? Namely, [we say] now: just because of this the organizing property of reality, which has to be imagined as being eternally durable. (SchroÈ dinger, 1932a, p. 66)
Then, he further emphasized that he did not, `in fact, believe this organizational property [of reality],' as was evident already from his inaugural lecture at the University of Zurich in 1922 (SchroÈ dinger, 1929a).823 Evidently, also in 1929, SchroÈ dinger had not moved away much farther from his earlier, uncommitted point of view with respect to causality, as was felt clearly by Max Planck, who responded to SchroÈ dinger as follows:
I cannot resist the temptation to express here some words in favor of strictly causal physics, even with the danger of appearing to you to be a narrow-minded reactionary. . . . The question whether the lawful connections (GesetzmaÈûigkeiten) which we encounter in nature all possess basically only an accidental character, i.e., are of a statistical type, can also be formulated thus: should we search for an explanation of the actually ever present uncertainty and accuracy, connected with every single observation, always only in the peculiar properties of the case under investigation, say, in the complex structure of the observed object or the incompleteness of the measuring apparatus including our senses; or should we trace back the uncertainty still further back into the formulation of the fundamental laws of physics? (Planck, 1929b, p. C II)

823 Indeed, SchroÈ dinger enclosed in the letter to Reichenbach of 1924 a copy of the earlier Zurich lecture, which was eventually published only in 1929.

706 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Certainly, Planck admitted, the problem constituted to some extent one of the usefulness (ZweckmaÈûigkeit); but he also emphasized that `the scheme [of physical theories], in any case, needs a solid basis, . . . and if the postulate of strict causality fails to serve anymore such a basis, then the question arises about the basis of ``acausal physics.'' ' The older Planck did not consider the situation in quantum mechanicsÐnamely, the fact that `the conditions which determine a process causally cannot always be experimentally realized up to a, in principle unrestricted, degree of accuracy'Ðto present a new experience in the history of science. But science must be taken as a whole enterprise based on the causal lawÐe.g., `in biology the real science starts only once the causal law has been introduced' (Planck, loc. cit., p. C III)Ðand he (Planck) rather hoped that SchroÈ dinger's own work on wave mechanicsÐ`which has ®rst demonstrated how the space-time processes in an atomic system can indeed be formulated as strictly [causally] determined' (loc. cit., p. CIV) would make it possible to restore strict causality again in atomic theory.824
Both Planck and SchroÈ dinger participated also in the causality debate of the early 1930's with their younger colleagues in Germany by developing and expounding partly on the viewpoints mentioned so far. Thus, Planck delivered on 17 June 1932, the Seventh Guthrie Lecture on `The Concept of Causality' (Planck, 1932a); later, he elaborated on the topic in a brochure entitled `Der Kausalbegri¨ in der Physik (The Concept of Causality in Physics)' (Planck, 1932b). There Planck admitted that the strict causality entering into the world view of the classical theories (including the one describing Brownian motion) failed vis a vis quantum mechanics, in particular, Heisenberg's uncertainty relations, but he also claimed that a `®nal refutation of the causal law . . . rested on a confusion of the world view (Weltbild ) with the world of senses (Sinnenwelt),' which he called a `premature step' because:
A di¨erent, more obvious way out of the di½culties exists, which has often served in similar situations rather well: it consists in the assumption that the question asking for simultaneous values of the coordinates and momenta of a material point makes no physical sense at all. The impossibility to answer a meaningless question, however, should not be held against the causal law per se but rather against the assumptions leading to ask the question, hence in the present about the assumed structure of the physical world view (Weltbild ). Now, since the classical world view has failed, it must be replaced by another one. (Planck, 1932b, pp. 13±14)
The concept of matter waves, which described atomic particles by a wave packet, in Planck's opinion admittedÐthough it satis®ed Heisenberg's relationÐas considering the same determinism to be at work as in classical point mechanics. Of
824 In a lecture on `Zwanzig Jahre Arbeit am physikalischen Weltbild (Twenty Years of Work on the Physical World View),' which Planck gave at Leyden on 18 February 1929, he had addressed the problem of causality in modern physics in some detail and argued that the wave-mechanical description provided a `di¨erent determinism' from the one existing in classical physics: It now determined just the matter waves (Planck, 1929a, especially, p. 220).

IV.1 The Causality Debate

707

course, now the conventional world of senses (Sinnenwelt) deviated from the world view (Weltbild ) of the quantum physicist, about which Planck did not worry but preferred to insist upon `retaining determinism ®rst of all in the world view (Weltbild )' (Planck, loc. cit., p. 15). Even the fact that the wave function did not yield the values of the coordinates as functions of time but only the probabilities that the coordinates possess at a given time `somehow given values' would not disturb him (Planck). There still existed `the saving way out,' namely, the assumption that the question about the meaning of a given symbol of the causal quantum-physical Weltbild, say, of a matter wave, makes `no de®nite sense as long as one does not simultaneously say in which state the peculiar measuring apparatus is used to translate the symbol into the Sinnenwelt' (Planck, loc. cit., p. 17). The latter argument raised then (by Bohr, Heisenberg and others) might be refuted perhaps by referring to `indirect test methods which have yielded good results in many cases, where the direct ones have failed' (Planck, loc. cit., pp. 16±17).
In a word, PlanckÐwho initiated quantum theory in the ®rst placeÐwas not prepared to succumb to the central argument of the `indeterminists' stating: Since the wave function in quantum physics is a probabilistic quantity, also strict causality must be necessarily abandoned; all that remains to understand is how strict laws, such as Coulomb's law for electric forces, can arise. Planck rather expounded his credo as follows:
The determinist thinks quite the opposite about all these points. He declares the Coulomb law to have the satisfactory character of a completely ®nal law: on the other hand, he recognizes the wave function as a probabilistic quantity only as long as one can forget about the measuring apparatus by which the wave is analyzed; and he searches for strict theoretical relations between the properties of the wave function and the processes in the measuring apparatus. To achieve this purpose, he must ®rst turn the measuring apparatus, like the wave function, into an object of research: he must not only translate the total experimental setup creating matter wavesÐsay, the high-voltage battery, the heated wire, the radioactive probeÐbut also the registering apparatusÐsay, the photographic plate, the ionization chamber or the Geiger counterÐplus the processes occurring in them into his physical Weltbild, and must deal with all these objects together as a single object, as a closed unit. But the problem would not be ®nished even then, as it has rather become more complex, because: since the total object must neither be cut into parts nor be subject to external actions for otherwise it would lose its characteristics, hence a direct test cannot be made at all. However, now it would be possible to establish new hypotheses concerning the internal processes [within the total object] and then to test their consequences. (Planck, loc. cit., p. 20)
After all of these complications, Planck frankly admitted that `only future will tell us' whether one might really be able to proceed successfully on the path indicated (Planck, loc. cit., pp. 20±21). But with respect to the causality problem, Planck remained optimistic, provided one would assume the following interpretation:
The causal law is neither right nor wrong; it is rather a heuristic principle, a pathindicator (Wegweiser)Ðin my opinion the most valuable indicator we have at

708 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
handÐfor us to ®nd our way in the colourful jumble of events and to indicate the direction in which physical research must go on to reach ®nal results. Just as it occupies from the very beginning the awakening spirit of a child and puts into its mouth the never-fatiguing question ``why?'' it also guides the scholar throughout his life presenting to him unceasingly new problems. Indeed, science does not mean resting leisurely in the possession of cognition already obtained, but it means restless work and steadily progressive development. (Planck, loc. cit.. p. 26)
In presenting this `deterministic' world view (Weltbild ) of quantum theory, Max Planck certainly followed his previous line of arguments, especially the stand he had taken since 1908 against the philosophical attitude of Ernst Mach.825 In Planck's opinion, physical theories should not be restricted to represent an economical connection of sensations or observational data, but had to follow ideal guidelinesÐin the ®rst place, the causal law. To support this view, Planck referred to that form of modern atomic theory which he favoured, namely, SchroÈ dinger's wave mechanics. The wave-mechanical scheme indeed seemed to provide the best chance of retaining the causal principle formulation, which was similar to that of the classical theories. The opponents of the causal interpretation of quantum mechanics, on the other hand, stuck (in Planck's view) too much to the ancient concept of a mass point. Max von Laue, Planck's former student (and later, his colleague and friend in Berlin), agreed in this opinion when he published a note `Zu den EroÈrterungen uÈber KausalitaÈt (About the Discussions on Causality)' in the Naturwissenschaften (von Laue, 1932b). In it, he wrote:
The present forms of quantum mechanics attempt to rescue the life of the ``mass point'' [of the old Newtonian theory]. Then they immediately arrive, because of those wave motions [as found in wave mechanics], necessarily at the uncertainty relations; from the latter, they conclude further that physics must renounce the causal interpretation of the individual [atomic] process and restrict itself to state [only] statistical laws. We do not wish to reproach this procedure; for the moment it may represent the best way out. (von Laue, loc. cit., p. 916)
However, von Laue continued, history may decide for a di¨erent method and eventually return to the older conceptions. `Hence in the case of the quantum riddle it is possible that time is not yet ripe for a [de®nitive] solution,' he claimed. In any case, he concluded: `These di½culties cannot force anybody to change his epistemological point of view, whatever it may be.' (von Laue, loc. cit.) That is, like Planck, von Laue favoured the causal point of view.
The third senior Berlin theoretician, Erwin SchroÈ dinger, also pondered in those years about the consequences arising from quantum mechanics. Having studied in some detail the derivation of the uncertainty relations, especially for relativistic mechanics (SchroÈ dinger, 1930), he declared in a popular talk on `Indeterminismus
825 In a way, Planck's lecture at Leyden, referred to in footnote 824, constituted a modernized version of his previous talk at Leyden in 1908.

IV.1 The Causality Debate

709

in der Physik (Indeterminism in Physics)' two years later that the (uncertainty) relations themselves contained an internal conceptual contradiction if applied to a mass point (SchroÈ dinger, 1932b, ®rst essay). Since a mass point in mechanics has to be de®ned by position, velocity and mass, he now argued, the statement that position and velocity cannot be determined simultaneously with arbitrary accuracy would dissolve the very concept. Evidently, he agreed with Planck and von Laue in hoping for a satisfactory solution of the quantum riddle by applying the purely wave-mechanical description.
As Planck noted, in the beginning of the 1930's, the majority of the quantum physicists believed in the violation of the causality principle, while only a small minority protested. Was this perhaps the matter of the generational di¨erence, since even a scientist like Paul Ehrenfest, friendly to the young revolutionaries, became worried that he might not understand the unanschauliche (non-intuitive) trends taken by the later developments?826 However, Planck, von Laue, and SchroÈ dinger certainly did not adhere to old classical theories; they did not wish to renounce any of the achievements of the modern relativity and quantum theories, but only complained about the Copenhagen interpretation of quantum mechanics and proposed to retain more `Objektivierkeit (objecti®ability)' in the sense accepted since centuries by scientists in many di¨erent ®elds. Bohr and Heisenberg, the spokesmen of the Copenhagen Weltbild, saw the situation quite di¨erently and they criticized the Berlin `conservatives.' Especially, Heisenberg argued that the causal principle did not belong to the old traditions of science: The physicists had accepted it only since about 150 years as an `important consequence of the postulate of Objektivierbarkeit of the observed facts,' he said in a lecture on `Atomtheorie und Naturerkenntnis (Atomic Theory and Understanding of Nature)' presented on 22 November 1933, at Munich (Heisenberg, 1934b). Immanuel Kant had initially expressed this consequence in his Kritik der reinen Vernunft (Critique of Pure Reason) of 1781, and strict determinism had sneaked into the classical theories since the early 19th century; the development of quantum mechanics and its interpretation in the mid-1930s had then shown `that the requirements of perception to be objectivierbar (objecti®able) and of connections being describable by mathematical equations do not depend on each other,' but:

Rather the requirement of clarityÐand more is not attempted by the application of mathematicsÐcan be retained absolutely, even in a ®eld of science, in which Objektivierbarkeit (objecti®ability) of perceptions ceases to be possible. (Heisenberg, loc. cit., p. 13)

In his talk, Heisenberg stated a little later: `For the indivisible constituents of matter, i.e., for the lightest bodies, every irradiation, or every act of observation at all, constitutes a remarkable perturbation (Eingri¨ ) which changes the behaviour

826 See Paul Ehrenfest's `Erkundigungsfragen (scienti®c queries) (1932),' which we shall discuss in the next section.

710 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
of the observed body decisively.' (Heisenberg, loc. cit., p. 14). These and similar arguments were reproached by Max von Laue in a further note, `UÈ ber Heisenbergs Ungenauigkeitsbeziehungen und ihre erkenntnistheoretische Bedeutung (On Heisenberg's Uncertainty Relations and Their Epistemological Meaning,' published in June of the following year (von Laue, 1934). He wrote:
It seems to me altogether doubtful to derive from the present status of physical knowledge too far-reaching conclusions concerning the theory of cognition. Quite apart from the fundamental doubt to abandon the principle that nature can be experienced (Prinzip der Erforschbarkeit der Natur), because one is not able to apply it so far completely, one must at least start from a foundation which is logically ®rm and does not contain contradictions. This cannot be said of the present physics. (von Laue, loc. cit., p. 440)
Here, von Laue pointed to the fact that the concept of smallest particles followed only from the most recent experiments if interpreted according to the old corpuscular view of matter, while wave mechanics and its relativistic extensions rather spoke of extended electrons and the like. Again, he repeated: `The uncertainty relations limit in my opinion every corpuscular mechanics but not every physical cognition.' (von Laue, loc. cit., p. 441) Since he considered causality as the key to any physical cognition, von Laue hoped that `the uncritical pessimism'Ðwhich seemed to him `in spite of all the given physically spurious arguments (Scheinargumente), to be a result of that deep general cultural pessimism forming the fundamental tendency of our times'Ðmight soon be overcome (von Laue, loc. cit.).827
`Cultural pessimism' or `positivism'Ðthese were the accusations directed against the Bohr-Heisenberg interpretation of quantum mechanics, though the originators did not really feel to be victims of such verdicts.828 No, the successful Heisenberg of those daysÐwho had recently explained the structure of atomic nuclei (see Section IV.3 below) and was about to deal with cosmic-ray phenomena
827 In contrast to the other critics of the Copenhagen interpretation, von Laue did not worry about the `Unanschaulichkeit' of quantum phenomena, arguing (as Heisenberg also did): `What one calls nonvisualizable, depends on time. A theory which forces us to give up the usual conceptions to describe the external world, seems to the witnesses of its origin always necessarily unanschaulich, mostly even to the originators themselves.' (von Laue, 1934, pp. 440±441)
828 Heisenberg was even less worried about an argument raised by the Viennese Karl Popper against the validity of the indeterminacy relations (Popper, 1934). Popper claimed that `for `non-prognostic' measurements, e.g., to determine the momentum of a particle when arriving at an exactly given space point,' the relations would not apply; he proposed to demonstrate this point by a Gedankenexperiment involving the crossing of an electron-ray A and an X-ray B, with both rays representing `pure cases' (i.e., a monochromatic parallel beam of electrons interacting with a monochromatic spherical X-ray). Heisenberg let his student Carl Friedrich von WeizsaÈcker analyze the experiment and demonstrate that nothing was wrong with the relations. Von WeizsaÈcker rather concluded:
The uncertainty relations cannot be applied to ``non-prognostic measurements'' because of the only reason: the theorems stating their results do not contain statements about physically possible measurements; on the other hand, conclusions about the past obey the same accuracy as those about the future, due to the symmetry of quantum-mechanical laws with respect to the time direction. (von WeizsaÈcker in Popper, 1934, p. 808)

IV.1 The Causality Debate

711

(see Section IV.5)Ðcould hardly be accused of being in¯uenced by any feeling of cultural pessimism.
Moreover, Heisenberg's Weltbild also did not follow any philosophical doctrine, such as positivism, as we have mentioned earlier in this section. He rather developed his own epistemological conclusions from the quantum-mechanical revolution, which he embedded into the grand historical schemes of physical descriptions in the talk entitled `Wandlungen der Grundlagen der exakten Naturwissenschaft in juÈngster Zeit (Recent Changes in the Foundations of Exact Science)' and delivered on 17 September 1934, at the Hanover Naturforscherversammlung (Heisenberg, 1934f ). Heisenberg spoke in this programmatic lecture about the alterations in the physical concepts achieved by the modern relativity and quantum theories, which showed the limitations of the previous theories, and then stated:

Modern physics has rather purged classical physics from some obscurities connected with the assumption of their unlimited applicability and shown that the single parts of our scienceÐmechanics, electricity, quantum theoryÐconstitute schemes, closed in themselves and being rationally penetrable to their limits, which probably represent the corresponding laws of nature for all future times. (Heisenberg, loc. cit., p. 701)

Such `closed systems' then do not contradict but rather complement each other, as Heisenberg explained in more detail in the talk on `Prinzipiellen Fragen der modernen Physik (The Fundamental Questions of Modern Physics),' given on 27 November 1935, at the University of Vienna (where Moritz Schlick taught). Classical physics, he said there, is built `on a system of sharply formulated axioms whose physical content is determined by the fact that through the choice of words appearing in the axioms their application to nature is uniquely prescribed,' he began his remarks (Heisenberg, 1936a, p. 91). That means, classical physics rested on the range of its concepts, like mass, velocity, and force. The modern theories, ®rst relativity and then quantum mechanics, had restricted the range of the systems of classical concepts. The di½culty in understanding the results of modern theories arose from the necessity to leave `the domain of the daily human experience,' while one had simultaneously to continue using the concepts of those classical theories which can be regarded as the limiting cases of the modern theories. That is, `the classical concepts remain still an indispensable part of the scienti®c language, without which one cannot speak at all about scienti®c results,' Heisenberg concluded the introductory part of his lecture. (Heisenberg, loc. cit., p. 95)
The necessity to go beyond the classical theories had grown out of the experimental observations of new phenomena; e.g., the new experience that `no signals can be transmitted with velocities faster than light,' led to new systems of axioms and concepts which allowed one to formulate new laws describing new experiences. For the physicist, `even the mathematically formulated statements of physics are so-to-speak only ``pictures in words (WortgemaÈlde)'' through which we try to interpret our experiences about nature for us and other people in a de®nite and

712 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
understandable way' (Heisenberg, loc. cit., pp. 97±98), but one always had to transcend the conventional concepts in essential aspects, for example:
Thus relativity theory and classical theory constitute the ®rst decisive steps from the region of visualizable concepts into a more abstract land, and the character of the here discovered connections leaves no doubt that these steps can never be taken back. . . . Actually, the discovery of a new system of concepts means nothing else but the discovery of a new way of thinking which as such can never be taken back. (Heisenberg, loc. cit., p. 98)
That is, the hope expressed by some people that one might return ®nally to the classical concepts must be given up. Especially, unless the results of quantum mechanics were proven to be wrong, the statistical character of the theory would remain ®nal. Further, in treating arbitrary experiments in quantum mechanics, Heisenberg continued, a `cut (Schnitt)' must be introduced between the measuring apparatus and the physical system observed; while this cut can be chosen largely at an arbitrary point, it is responsible for the statistical behaviour of the quantummechanical laws. That is, any possible deterministic reformulation of quantum mechanics would have ®rst to remove the cut, which appears to be quite an impossible task; hence, any revision of the present atomic theory must move away further from the classical theory. Perhaps, the `hole theory' of Paul Dirac might open the way to understand the properties of electrons and even the strength of the electromagnetic coupling constant, Heisenberg argued at the end of his paper, and further continued:
Quite generally, one may say in conclusion: the assumption that even the concepts of modern physics will have to be revised should not be taken as skepticism [or even ``cultural pessimism'']; quite the contrary, it is just another expression for the conviction that the extension of our range of experience will bring to light new harmonies of nature. (Heisenberg, loc. cit., p. 102)
Returning to the topic of causality discussed in this section, we should ®nally mention the attempt of a young student of philosophy, Grete Hermann from éstrupgaard (Denmark), who discussed in her doctoral dissertation of 1935 the `natural-philosophical foundations of quantum mechanics' (Hermann, 1935a, b). The contents of her work, which she carried out in Leipzig and Copenhagen (staying in close contact with Heisenberg and Bohr), may be derived from a review written by Carl Friedrich von WeizsaÈcker:
The present memoir is perhaps the ®rst work from the philosophical side, which provides a positive and incontestable contribution to derive the epistemological consequences of quantum mechanics. She [Grete Hermann] achieves her goal by pursuing a single problem to its depth. [On the one hand,] quantum mechanics claims the impossibility of [arriving at] certain results. On the other hand, because our experiences are not closed, it is always possible to search for the causes of an observed phenomenon as long as they are not yet known. Hence, does not quantum mechanics, when stating the impossibility of a causal description of nature determining all events,

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 713
exceed its competence? The author [Grete Hermann] provides an answer, which on ®rst inspection sounds paradoxical but hits the point exactly: ®nding real, still unknown causes is impossible because quantum mechanics already provides the causes for a given event in any case completely. The impossibility of [making] certain predictions is not based on the fact that a causal chain investigated turns out to be interrupted somewhere, but rather on the fact that the di¨erent causal chains cannot be organized to form a uni®ed picture embracing all aspects of the process, thus it rather remains to the whim (WillkuÈr) of the observer which of the di¨erent ``virtual causal chains'' has been realized. (von WeizsaÈcker, 1936c, p. 527)
The physicists might not be tempted to embed their results too strictly into any philosophical schoolÐas Grete Hermann did by appealing strongly to the traditions of Immanuel Kant, Herbert Fries and Leonard NelsonÐvon WeizsaÈcker noted, and concluded that `a fruitful discussion on the topic could not be opened, at any rate, in a clearer and more pertinent manner.' (von WeizsaÈcker, loc. cit., p. 528)
To complete the story in the words of Grete Hermann herself, a few sentences from the summary of her work might be quoted. In particular, she wrote:
The di½culties, in which the partisans of causality are placed by the discoveries of quantum mechanics, seem in proper light not to arise from the causality principle itself. They rather emerge from the tacit assumption connected with it that the physical cognition grasps natural phenomena adequately and independently of the observational connection (Beobachtungszusammenhang). This assumption is expressed in the prerequisite that every causal connection between processes yields a calculable action due to the cause, even more, that the causal connection is identical with the possibility of such a calculation.
Quantum mechanics forces us to dissolve this mixing of di¨erent principles of natural philosophy, to drop the assumption of the absolute character of the cognition of nature, and to use the causal principle independently of the latter. By no means has it disproved the causal law, but it has clari®ed its status and freed it from other principles which must not be combined with it necessarily. (Hermann, 1935b, p. 721)
When Grete Hermann wrote her dissertation, the debate among the quantum physicists on causality and the prerequisites for cognition of nature had reached a new climax in Albert Einstein's new attack on the question of the completeness of quantum mechanics.
IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality (1931±1936)
(a) Introduction
Expressed in whichever formulation, quantum mechanics o¨ered even to experienced experts puzzling features to ponder about. Thus, Paul Ehrenfest, since 1906 an active contributor to the theory of quanta, wrote in summer 1932 `Einige

714 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
die Quantenmechanik betre¨ende Erkundigungsfragen (Certain Queries Concerning Quantum Mechanics)' and submitted them to Zeitschrift fuÈ r Physik (Ehrenfest, 1932). In particular, he listed the following queries: A. The (role of the) imaginary unit in the SchroÈ dinger equation and the Heisenberg±Born commutation relations. B. The limitation of the analogy between photons and electrons. C. The convenient accessibility of the spinor calculus. He concentrated there on what he thought might be called by most quantum physicists as being `senseless questions (sinnlose Fragen)'; e.g., why did SchroÈ dinger, in formulating wave mechanics, start from a real wave function but soon introduce the complex notation, for it seemed to be more convenient, and never returned later to the real formulation; or, how to express the analogy between photons and electrons in a di¨erential equation formulation, and not in the formulation of a non-local integral equation (as suggested by Lev Landau and Rudolf Peierls, 1930)?
Wolfgang Pauli soon replied to these `senseless questions' of his senior friend in some detail, ®rst by letters exchanged from October to December 1932 (Pauli, 1985), and then openly in a paper published also in Zeitschrift fuÈ r Physik (Pauli, 1933d). Concerning query A, he pointed out that it was the assumption of a positive normalized probability which demanded the imaginary unit, especially: `The imaginary unit enters into the search for an expression for the probability density W, which satis®es the requirements and does not contain the time derivatives of [the wave function] ™.' (Pauli, loc. cit., p. 576) This probability density then depended quadratically on the wave function ™r(x, t0) at a given instant of time t0 and could be expressed both in nonrelativistic and relativistic cases only with complex wave functions. With respect to query B, the photon±electron analogy, Pauli proposed to distinguish between `large ®elds (groûe Felder) gr and E, H ' describing many electrons and photons, on the one hand, and `small ®elds (kleine Felder)' ™r and e, h describing single photons and electrons, on the other hand. In the latter case, the photon would not possess a four-current satisfying a continuity equation and having positive-de®nite density; hence, the electromagnetic ®elds e, h of a photon could not be associated with a local space-time density W(x, t) for a particle. Moreover, in the photon situation, particles with positive energies could always be kept in the processes of interaction, while in the electron situation negative-energy particles might result. The large-®eld case also revealed di¨erences: When many photons were present, the E, H constituted classically measurable ®elds (though the number of quanta N did not commute with E and H ); however, the gr ®eld could not be measured like a classical ®eld.
Ehrenfest had mentioned another problem of the quantum theory that bothered him: `If we recall what an uncanny theory of action-at-a-distance is represented by SchroÈ dinger's wave mechanics, we shall preserve a healthy nostalgia for a fourdimensional theory of action by contact.' (Ehrenfest, 1932, p. 557, footnote 1) To that, Pauli replied in detail in §3 of his paper. He noted that already in classical electrodynamics action-at-a-distance forces formally occurred, but the situation could be easily reformulated in the action-by-contact language when introducing the di¨erential equation (div E ˆ 4pr) which the electrostatic ®eld obeyed. Simi-

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 715
larly, he argued, the Coulomb force in the SchroÈ dinger equation (mentioned by Ehrenfest) might be replaced by an action-by-contact (Pauli, 1933a, pp. 584± 586).
For several years, Ehrenfest had been bothered by his lack of understanding as he re¯ected about the decisive features of the modern development.829 Somehow, he still felt attracted by the nostalgic arguments of his friend Albert Einstein. Already at the ®fth Solvay Conference in Brussels in fall 1927, Einstein had criticized the point of view that `quantum mechanics is considered to be a complete theory of individual [atomic] processes,' and stated: If a particle, somehow described by the absolute square of the SchroÈ dinger function j™j2, `is localized, a peculiar action-at-a-distance must be assumed to occur which prevents the continuously distributed wave in space from producing an e¨ect at two places on a screen' (Einstein, 1928, p. 255). In the early 1930's, Einstein continued to worry about this particular problem, as Ehrenfest (with whom he conferred in those times quite regularly) in his paper on the Erkundigungsfragen (queries) mentioned that `certain thought experiments, designed by Einstein but never published, are particularly suited for [clarifying] that purpose' (Ehrenfest, 1932, p. 557). The answers given by Pauli to Ehrenfest did not satisfy Einstein (see the discussion in Jammer, 1974, pp. 117±119), and the Gedankenexperiments recalled by Ehrenfest in 1932 would ®nally lead to the paper containing the famous `Einstein-PodolskyRosen (EPR) paradox,' as Max Jammer concluded from an examination of Einstein's correspondence between 1927 and 1935 (partly supported by a letter which Einstein wrote to Paul Epstein later, on 10 November 1945). Jammer summarized Einstein's steps on the way to this decisive paper as follows:830
The point of departure is Einstein's well-known photon-box experiment which he presented at the sixth Solvay Conference in October 1930 in Brussels in order to disprove the Heisenberg energy-time uncertainty relation. . . . Although defeated, Einstein continued to ponder about this argument and understood that in order to eliminate the unwanted gravitational e¨ect only horizontal motion should be admitted. As described in his [later] letter to Epstein, he thus designed the following modi®cation. He imagined an ideally re¯ecting box B which contains a clock operating a shutter V and a quantum of radiation of unknown frequency; the box is assumed to be movable in a horizontal direction along a frictionless rail which serves as a reference system K, but can also be rigidly connected with K. At one end of the rail an absorbing screen or re¯ecting mirror can be mounted. An observer sitting on top of the box B and in possession of all measuring devices releases the shutter at a precisely determinable moment to emit a photon in the direction of the screen. Thereupon the observer can either immediately connect B with K, read the position of B and predict the time of arrival of the photon at the screen or he can measure the
829 Paul Ehrenfest occasionally mentioned to his friends and colleagues that he would have to vacate his university chair for another, more capable, person. It is di½cult to say how much such feelings may have contributed to his suicide on 25 September 1933.
830 Besides Max Jammer (1974, 1985), especially, Arthur Fine (1986, 1993) has worked on the historical reconstruction of the EPR paper.

716 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

momentum p of B relative to K by means of the Doppler e¨ect with arbitrarily low

frequency

and

the

recoil

formula

p

ˆ

hn c

and

predict

the

energy

of

the

photon

arriving

at the screen. As stated in the letter, Einstein already at that time conceived the idea

that the light-quantum, after leaving the box, represents a ``real state of a¨airs (einen

realen Sachverhalt)'' which can hardly be thought to depend on what kind of mea-

surement is being performed with B. Hence any property of the light-quantum, found

by a measurement on B, must also exist if the measurement would not have been

performed at all. The light-quantum must consequently possess a de®nite position as

well as a de®nite colour, a situation not describable in terms of a wave function.

Hence a description in terms of wave functions cannot be a complete description of

the physical reality. It is clear that the scenario of this thought-experiment is the same

as that of the Brussels photon-box experiment apart from being, so to say, rotated

into the horizontal direction. But it intends to show not the inconsistency but rather

the incompleteness of the theory. And to this end the additional feature of introduc-

ing the idea of a ``real state of a¨airs'' was imperative. It vaguely foreshadowed what

became later known as the ``Einstein separability principle.'' (Jammer, 1985, pp. 133±

134)

Thus, after a preparation of several years, Albert EinsteinÐwith Boris Podolsky and Nathan RosenЮnally sent a paper to the Physical Review (where it was received on 25 March 1935); it was entitled `Can Quantum-Mechanical Description of Physical Reality be Considered Complete?' This EPR-paper concluded with a bold statement:

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question whether or not such a description exists. We believe, however, that such a theory is possible. (EPR, 1935, p. 780)

The EPR-paper, which appeared in the Physical Review (issue of 15 May 1935), aroused an abundant response, ®rst in America, then in Europe (especially from Niels Bohr in Copenhagen). It initiated an extended debate among the physicists on what `physical reality' was all about. In the fall of 1935 the EPR-arguments were supported by Erwin SchroÈ dinger, who in the context of a review on `Die gegenwaÈrtige Situation in der Quantenmechanik (The Present Situation in Quantum Mechanics),' also developed his famous `cat paradox' (SchroÈ dinger, 1935a). The response of the Copenhagen representatives, especially, Niels Bohr and Werner HeisenbergÐas well as certain philosophical supportersÐmingled with the political situation in Germany. The debate on `What is Real?' in physics and whether quantum mechanics is, or ever could be, able to provide a complete description of nature has been going on till the present day.831

831 We shall later brie¯y indicate the development of this debate during the past several decades in the Epilogue.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 717
(b) From Inconsistency to Incompleteness of Quantum Mechanics: The EPR Paradox (1931±1935)
In the early 1930's Albert Einstein, besides becoming deeply involved in a programme on the development of quantum ®eld theory (Einstein and Mayer, 1931, 1932a, b), addressed the problem of the `Knowledge of Past and Future in Quantum Mechanics' in a note written during his visit to California in the winter semester 1930±1931, together with Richard Chace Tolman (the dean of the graduate school of the California Institute of Technology) and the young Russian-born physicist Boris Podolsky.832 In particular, they discussed `a simple ideal experiment which showed that the possibility of describing the past path of a particle would lead to predictions as to the future behaviour of a second particle of a kind not allowed in quantum mechanics' (Einstein, Tolman, and Podolsky, 1931, p. 780). Contrary to some earlier suppositions, stating `that the quantum mechanics would permit an exact prescription of the past path of a particle,' the authors obtained from their analysis `an uncertainty in the description of past events which is analogous to the uncertainty in the prediction of future events.' (Einstein, Tolman, and Podolsky, loc. cit.)
The Einstein±Tolman±Podolsky (ETP) Gedankenexperimental setup worked with a box B containing a number of identical particles in thermal agitation and provided with two small openings to be closed and opened by a shutter S, which releases for a short time particles in two directions: (i) directly toward an observer O, and (ii) after re¯ection at a wall at the point R to the observer on a second, larger path SRO. An energy measurement (by weighing the box B) and a time determination were to be carried out. Then, `knowing the momentum of the particle in the past, and hence also its past velocity and energy, it would seem possible to calculate the [instant of ] time when the shutter must have been open from the known time of arrival of the ®rst particle [on the direct path SO], and to calculate the energy and momentum of the second particle [on the longer path SRO] from the known loss of the energy content of the box when the shutter opened' (ETP, loc. cit., p. 781). This `paradoxical result' of a prediction of exact energy and time of the arrival of the second particle could only be explained by `the circumstance
832 Boris Podolsky, born in Taganrog, Russia, on 29 June 1896, emigrated to the United States in 1913. After receiving a B.S. degree in electrical engineering from the University of Southern California (USC) in 1918, he served in the U. S. Army and then obtained employment in the Los Angeles Bureau of Power and Light. After further studies in mathematics at USC (M.S. in 1926) and physics at the California Institute of Technology, he received his doctorate at Caltech (under the supervision of Paul Sophus Epstein) in 1928. With a National Research Council Fellowship, he spent a year at the University of California at Berkeley, followed by a year in Leipzig as an International Education Board Fellow. In 1930, Podolsky returned to Caltech for a year and worked with Richard C. Tolman, and then spent two years at the Ukrainian Physico-Technical Institute at Kharkov, collaborating there with Vladimir Fock, Paul Dirac (who was on a visit to the U.S.S.R.), and Lev Landau. He returned to the Institute for Advanced Study in Princeton with a fellowship in 1933; from there, he moved to the University of Cincinnati in 1935 as a professor of mathematical physics, and in 1961, he changed to Xavier University in Cincinnati. He died on 28 November 1966, in Cincinnati.

718 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
that the past momentum of the ®rst particle cannot be accurately determined as was assumed' (ETP, loc. cit.).833 `Finally, it is of special interest to emphasize the remarkable conclusion that the principles of quantum mechanics would actually impose limitations upon the localization in time of a macroscopic phenomenon such as the opening and closing of a shutter,' their letter stated (ETP, loc. cit.).834
The idea of using re¯ected particles entered into the next Gedankenexperiment of Einstein, about which Paul Ehrenfest reported to Niels Bohr in a letter, dated 9 July 1931.835 Ehrenfest wrote, in particular, that Einstein no longer intended to make use of the box experiment as an argument `against the indeterminacy relations' but `for a totally di¨erent purpose'; indeed, Einstein now constructed a `machine' which ejects a projectile and considered the following situation: After the projectile had been ejected, an `interrogator (Frager)' asks the `machinist' to predict, by examining the `machine' alone, either what value a quantity A or what value a conjugate quantity B would have if the projectile were subjected to the respective measurements after a long period of time (when the projectile returns after being re¯ected by a distant re¯ector). As Ehrenfest reported further, Einstein believed that a photon box might represent such a machine and proposed to carry out the following experiment:
1. Set the clock's pointer to time O hour and arrange that at the pointer position 1,000 hours [later] the shutter will be released for a short time interval.
2. Weigh the box during the ®rst 500 hours and screw it ®rmly to the fundamental reference frame.
3. Wait for 1,500 hours to be sure that the quantum has left the box on its way to the ®xed re¯ector (mirror), placed at the distance of 1/2 light-year away.
4. Now let the interrogator choose what prediction he wants: (—) either the exact time of arrival of the re¯ected quantum, or (˜) the colour (energy) of it. In case (—), open the still ®rmly screwed box and compare the clock reading (which during the ®rst 500 hours was a¨ected, due to the gravitational red-shift formula) with the standard time and ®nd out the correct standard time for the pointer position ``1,000 hours''; then the exact time of arrival [of the photons] can be computed. In case (˜), weigh the box again after 500 hours; then the exact energy can be determined. (Ehrenfest to Bohr, 9 July 1931; see Jammer, 1974, pp. 171±172)
`The interesting point is that the projectile, while ¯ying around isolated on its own, must be able to satisfy totally di¨erent non-commutative predictions, without
833 Einstein, Tolman, and Podolsky substantiated the above argument to be correct by referring to the measurement of the particle's momentum by a Doppler e¨ect in re¯ected infrared light, which would lead to an uncertainty in the position of the ®rst particle, and thus also in the exact openinginstant of the shutter.
834 The ETP-paradox received some publicity, because a little later another visitor from Europe to USA, Charles Galton Darwin, concluded di¨erently from a Gedankenexperiment working with two shutters. In particular, he stated: `The uncertainty principle is essentially only concerned with the future; we can install instruments which will tell us as much of the past as we like.' (Darwin, 1931, p. 653) See the discussion of this point in Jammer, 1974, p. 169.
835 For a detailed discussion of the contents of this letter and the further development of the story until 1934, we refer to Jammer, 1974, p. 170±178, and Jammer, 1985, pp. 134±137.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 719
knowing as yet which of the predictions will be made,' Ehrenfest concluded the description of Einstein's new Gedankenexperiment in his letter to Bohr, and proposed that Bohr might visit Leyden in the fall to discuss the situation with Einstein (who was also expected to visit Leyden at that time). However, the meeting of Bohr and Einstein did not materialize; but on 4 November 1931, Einstein presented a talk in the Berlin colloquium entitled `UÈ ber die Unbestimmtheitsrelationen (On the Uncertainty Relations),' dealing with a photon-box experiment (Einstein, 1932). The aim of this talk was to point out that, whatever quantity had to be measured accurately, could be decided well after the photon had left the box.
On 4 April 1932, when Einstein was on his way back to Germany from another visit to the United States, he met Ehrenfest again in Rotterdam (where the ship docked for several days). Evidently, the two friends discussed further the Gedankenexperiment, because the next day Einstein wrote a letter to Ehrenfest, and said:
Yesterday you prodded me to modify the ``box experiment'' in such a way that it employs concepts more familiar to the wave-theoretician. This I do in the following by applying only such idealizations which, as I know, you will accept unhesitatingly. It operates as a schematized Compton e¨ect. (Einstein to Ehrenfest, 5 April 1932)
The new experiment now suggested involved the interaction of a photon and a massive particle, and Einstein showed how either the momentum or the position of the heavy particle might be determined by observing the corresponding quantities of the photon. `This is the reason why I ®nd myself inclined to ascribe objective ``reality'' to both [observables, i.e., momentum and position],' he concluded (Einstein to Ehrenfest, loc. cit.). Apparently, he addressed here for the ®rst time explicitly the question of `reality' in quantum mechanics, and what he meant by it became clearer about one-and-a-half years later. Indeed, shortly before Einstein left Europe for good in fall 1933, he attended a lecture given by LeÂon Rosenfeld (who was then a lecturer at the University of LieÁge) in Brussels on the Bohr± Rosenfeld theory of the measurability of electromagnetic ®eld quantities; he then expressed a certain uneasiness about the results obtained and asked Rosenfeld:
What would you say about the following situation? Suppose two particles are set in motion towards each other with the same, very large momentum and that they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum; then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the ®rst particle, he will be able to tell where the particle is. This is a perfectly correct and straightforward deduction from the principles of quantum mechanics. (Rosenfeld, 1967, pp. 127±128)
However, Einstein considered the situation to be `very paradoxical,' because: `How can the ®nal state of a second particle be in¯uenced by a measurement performed on the ®rst, after all physical interaction has ceased between them?' (Rosenfeld, loc. cit., p. 128)

720 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Thus, between spring and fall 1933, Einstein's Gedankenexperiment ®nally took the direction toward what would be formulated in 1935 as the EPR-argument. It may be that the ®nal write-up was also in¯uenced by a paper of Karl Popper criticizing the uncertainty relations.836 Popper had sent a copy of his note (Popper, 1934)Ðaccording to which the path of one particle determined via the conservation laws the path of its partner with which it had collidedÐto Einstein; and a similar situation was considered in the EPR-paper.837 Still missing was only the `completeness' argument, which could perhaps be obtained from the mathematical literature or conversations with John von Neumann (who was also at the Institute for Advanced Study in Princeton).838 In any case, in spring 1935, the Princeton team of Einstein, Podolsky and Rosen connected the hitherto mathematical concept of completeness with the metaphysical concept of `physical reality,' when they stated in the preamble of their paper:
In a complete theory there is an element corresponding to each element of reality. A su½cient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. (EPR, 1935, p. 777)
`Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false,' EPR continued, and then sharply concluded that `the physical description of reality as given by the wave function is not complete.' (EPR, loc. cit.)
Max Jammer, in his classic book on The Philosophy of Quantum Mechanics, organized the analysis of EPR's four-page note (containing two sections) as follows:
The paper contains four parts: (A) an epistemological-metaphysical preamble; (B) a general characterization of quantum-mechanical description; (C) the applciation of this description to a speci®c example; and (D) a conclusion drawn from parts (A) and (C). (Jammer, 1974, p. 181)
836 We have mentioned it above in Footnote 828. 837 For a detailed analysis of Popper's paper, see Jammer, 1974, pp. 174±178. In his reply to Popper, Einstein criticized the conclusion because it contradicted the indeterminacy relations. 838 Max Jammer, in his detailed analysis, referred to remarks on the `completeness of quantum mechanics' by Bohr and other physicists, and to the studies of the Polish logician Alfred Tarski (Jammer, 1985, pp. 137±139). As we have discussed in previous volumes, especially Volume 3, the concept of `completeness' entered into the quantum-mechanical literature (Born, Heisenberg, and Jordan, 1926) quite early, and the GoÈ ttingen quantum-theoreticians took it from the mathematicians, especially, David Hilbert. Also, von Neumann, in his famous proof of `hidden variables' (discussed in the foregoing Section III.3), made use of the same concept.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 721

The preamble (A) started with a de®nition of what Einstein, Podolsky, and Rosen meant by reality, namely:

Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of the concepts we can picture this reality ourselves. (EPR, 1935, p. 777)

Then, they called a theory `satisfactory' if the following two questions could be answered positively: `Is the theory correct?' and `Is the description given by the theory complete?'. By `correct' they meant the `agreement between the conclusions from the theory and human experience,' while they de®ned `complete' by what they stated as a `necessary requirement' in the summary, notably: `Every element of the physical reality must have a counterpart in the physical theory.' (EPR, loc. cit., p. 777) Since `physical reality' had to be derived from experiments, a su½cient de®nition appeared to be the following:

If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quality. (EPR, loc. cit., p. 777)

In order to characterize brie¯y the quantum-mechanical formalism, Einstein, Podolsky, and Rosen considered a state described by the wave function ™ and its eigenvalues for a given quantity A; further, they assumed the commutation relations for canonical pairs of quantities, such as position and momentum, to be be valid, and arrived at two statements (1) and (2), as formulated in their summary (preamble) quoted above. Hence, they quickly concluded in part (B) that the usual statement, `the wave function does contain a complete description of the physical reality of the system in the state to which it corresponds'Ðthough `at ®rst sight entirely reasonable, for the information obtainable from a wave function seems to correspond exactly to what can be measured without altering the state'Ð nevertheless leads to a contradiction if one wants to preserve the above reality condition (EPR, loc. cit., pp. 778±779).
In part (C), EPR constructed their Gedankenexperiment by considering two sytems, each composed of a particleÐEPR spoke of systems I and IIÐwhich were allowed to interact from time t ˆ 0 to t ˆ T, their state being known before t ˆ 0 while it could be calculated for t b T via the SchroÈ dinger equation. This calculation yielded the wave functions for the combined system I ‡ II, from which those of the separated systems were derived according to the standard quantummechanical process of `reduction of the wave packet,' i.e., formally given by

ˆ y ™…x1Y x2† ˆ ™n…x2†un…x1†Y
nˆ1

…650†

722 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

where un…x1† denoted the eigenfunctions of an operator A of the system (particle) I
and ™n…x† the corresponding eigenfunction of the system (particle) II. The measurement of another quantity B might lead to a di¨erent result

ˆ y ™…x1Y x2† ˆ fs…x2†vs…x1†Y
sˆ1

…650 H †

yielding afterward the states vs…x1† and fs…x2† of the systems I and II, respectively. `Thus, it is possible to assign two di¨erent wave functions (in our example, ™k and fr) to the same reality ([i.e.,] the second system after the interaction with the ®rst),' EPR concluded and referred to the fact that `at the time of measurement [of A and

B] the two systems [I and II] no longer interact,' hence `no real change can take

place in the second system in consequence of anything that may be done to the

®rst system' (EPR, loc. cit., p. 779). Now, in the special case that the physical

quantities A and B were taken to be the momentum P and the position Q satisfy-

ing the commutation relations

PQ À QP ˆ 2hpiY

…651†

the following situation emerged: ™…x1Y x2† could be written either as

… ‡y
™…x1Y x2† ˆ
Ày





exp

2pi h

x1p

exp

!

À

2pi h

…x2

À

x0†p

dp

…652†

or

…‡y

™…x1Y x2† ˆ h d…x1 À x†d…x À x2 ‡ x0† dxX

…652 H †

Ày





In case (652), the associated wave functions were up…x1† ˆ exp

2pi h

x1p

and

™p…x2†, corresponding to the operator P with the eigenvalues p1 ˆ p for the particle I and p2 ˆ Àp for the particle II. In case (652H), on the other hand, the wave

functions were vx…x1† ˆ d…x1 À x† and fx…x2† ˆ d…x À x2 ‡ x0†, corresponding to the operator Q with the eigenvalues x1 ˆ x and x2 ˆ x ‡ x0, respectively. `Thus,

by measuring either A or B we are in a position to predict with certainty, and

without in any way disturbing the second system, either the value of the quantity

P (that is pk) or the value of the quantity Q (that is qr),' EPR concluded and

continued:

In accordance with our criterion of reality, in the ®rst case we must consider the
quantity P as being an element of reality, in the second case the quantity Q is an
element of reality. But, as we have seen, both wave functions ™k and fr belong to the same reality. (EPR, loc. cit., p. 780)

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 723
EPR interpreted the result thus obtained as follows in part (D): OriginallyÐ i.e., in part (A)Ðthey had argued that the situation in quantum mechanics should be described either by the assertion (1) or the assertion (2). However, now one had to argue rather:
Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities with noncommuting operators can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of the physical reality given by the wave function is not complete. (EPR, loc. cit.)
One might evade this consequence, EPR continued, by re®ning the de®nition of physical reality, say, by regarding `two or more physical quantities as simultaneous elements of reality only when they can be simultaneously measured or predicted 'Ðwhich would imply that `P and Q are not simultaneously real'ÐEPR added, then `the reality of P and Q depends upon the process of measurement carried out on the ®rst system in any way'; however, they claimed: `No reasonable de®nition of reality could be expected to permit this.' (EPR, loc. cit.) Finally, they expressed the hope which Einstein had cherished for more than a decade, namely, the ®rm belief that another theory may be found for the phenomena of atomic physics, such that a complete description of reality in the sense expressed above will be possible.
In his detailed study, `The EPR Problem in Its Historical Development,' Max Jammer tried to single out the individual contribution of each of the three authors (Jammer, 1985). Evidently, Einstein, as he stated himself repeatedly (e.g., in his letter of 10 November 1945, quoted earlier), conceived the general idea of the EPR-argument.839 Then the work on the paper was shared in equal parts, as Jammer learned especially from interviews with Nathan Rosen: EPR met for several weeks in early 1935 in Einstein's o½ce to discuss the problem; then, `Podolsky was the one who wrote the ®rst draft,' and, as Rosen recalled, `roughly speaking, one can say that Einstein contributed the general point of view and its implications, [and] I found the ™-function (i.e., [the description of ] the ``EPR thought experiment''), and Podolsky composed the paper' (Jammer, loc. cit., p. 142). Thus, Podolsky `who liked to use the language of logic and was good at it' contributed an essential aspect, namely `the completeness argument' which was previously not in the line of Einstein's thinking. (Jammer, loc. cit.)
The EPR paper, which expressed Einstein's unhappiness with the standard interpretation of quantum mechanicsÐor, rather expressed it most explicitlyÐalso made his junior authors known to wider circles, especially the 26-year-old Nathan
839 The background given above has been summarized by Jammer, 1985, pp. 141±144.

724 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Rosen.840 In order to prepare for the understanding of the response to the EPRstudy, let us summarize its contents (with Jammer) as follows:
The Einstein-Podolosky-Rosen [EPR] argument for the incompleteness of quantum mechanics is based . . . on two explicitly formulated and two tacitly assumedÐor only en passent mentionedÐpremises:
1. The reality criterion. ``If without in any way disturbing a system we can predict with certainty . . . the value of a physical quantity, then there exists an element of physical reality corresponding to this physical reality.''
2. The completeness criterion. A physical theory is complete only if ``every element of the physical reality has a counterpart in the physical theory.''
The tacitly assumed arguments are:
3. The locality assumption. If ``at the time of measurement . . . two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the ®rst system.''
4. The validity assumption. The statistical predictions of quantum mechanics are con®rmed by experiment.
. . . The Einstein-Podolsky-Rosen argument then proves that on the basis of the reality criterion 1, assumptions 3 and 4 imply that the quantum mechanics does not satisfy criterion 2, that is, the necessary condition of completeness, and hence provides only an incomplete description of physical reality. (Jammer, 1974, pp. 184±185)
The various points mentioned here soon became the centre of a lively debate among the physicists, ®rst some in the United States, and then the leading ones in Europe.
The publicity began already on Saturday, 4 May 1935Ði.e., before the EPRpaper appeared in The Physical ReviewÐwhen The New York Times carried an extensive report under the provocative headline `Einstein Attacks Quantum Theory,' which was summarized by the sentences: `Professor Einstein will attack science's important theory of quantum mechanics, a theory of which he was sort of grandfather. He concluded that while it [the quantum mechanics] is ``correct'' it
840 Nathan Rosen, born on 22 March 1909, in Brooklyn, New York, received his education at the Massachusetts Institute of Technology (a B.S. in electrochemical engineering in 1929, and an Sc.D. in physics in 1932Ðwith Philip M. Morse as his thesis advisor on quantum chemistry). Then, he held several postdoctoral positions, ®rst at the University of Michigan and Princeton Foundation; from 1934 to 1936, he served as Einstein's assistant at the Institute for Advanced Study in Princeton and instructed Einstein in the details of the properties of wave functions in complex molecular situations. From 1936 to 1938, he worked as a professor of theoretical physics at Kiev State University, and then he returned to MIT; he taught for one year at Black Mountain College in North Carolina and became a member of the faculty of the University of North Carolina at Chapel Hill from 1941 to 1952. During World War II, he worked on uranium-isotope separation. In 1953, Rosen went to Israel and joined the Technion at Haifa as a professor of physics; at the Technion, he established the physics department and the graduate school and retired in 1973; in addition, he served from 1969 to 1971 as Dean of the Engineering School of the newly established Ben Gurion University of the Negev at Beersheba. He died on 18 December 1995, in Haifa.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 725
is not ``complete.'' ' After a non-technical description of the main contents of the paper a statement attributed to Podolsky was added, namely:
Physicists believe that there exist real material things independent of our minds and theories. We construct theories and invent words (such as electron, positron, etc.) in an attempt to explain to ourselves what we know about our external world and help us to obtain further knowledge about it. Before a theory can be considered to be satisfactory it must pass two severe tests. First, the theory must enable us to calculate facts of nature, and these calculations must agree very accurately with observation and experiment. Second, we expect a satisfactory theory, as a good image of objective reality, to contain a counterpart for every element of the physical world. A theory satisfying the ®rst requirement must be called a correct theory while, if it satis®es the second requirement, it may be called a complete theory. (The New York Times, 4 May 1935, p. 11)
The article in the newspaper was followed by a report of an interview with the quantum theorist Edward Uhler Condon, then associate professor at Princeton University, who stressed that the EPR-argument of course depended on `what meaning is to be attached to the word ``reality'' in connection with physics' but concluded that, in spite of Einstein's criticism of quantum-mechanical theories, `I am afraid that thus far the statistical theories have withstood criticism.'
The public stir in The New York Times was completed by the strong statement of Einstein himself in the issue of 7 May (p. 21), who pointed out that `any information upon which the article ``Einstein Attacks Quantum Mechanics'' in your issue of 4 May is based was given without authority.' The newspaper a¨air also terminated the previously friendly collaboration between Einstein and the young Russian-American theoretician Boris Podolsky, who left Princeton shortly thereafter. In a later essay (which will be discussed below), Einstein gave a few indications where his view deviated from that of the unauthorized spokesman (Einstein, 1936). In any case, as Einstein wished, from then on the discussion on the topic was carried on `only in the appropriate forum' of scienti®c journals.841
(c) The Response of the Quantum Physicists, Notably, Bohr and Heisenberg, to EPR (1935)
The very ®rst discussion of the EPR-argument occurred properly in the Physical Review, which published a letter of the Harvard theoretician Edwin C. Kemble, that was dated 25 May 1935, and appeared in the issue of 15 June (Kemble, 1935a). Kemble, a senior and experienced quantum physicist (who wrote a standard textbook on the subject: Kemble, 1937), expressed the opinion that `the argument is not sound'; he had in mind especially the EPR assertion that the sys-
841 The story of Einstein's dissatisfaction with Podolsky, and further details of the early response to the EPR-paper, can be found in Jammer, 1974, pp. 189±194, and especially in Jammer, 1985, pp. 144± 146.

726 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
tem II `cannot be a¨ected by [the observation of I] and must in all cases constitute ``the same physical reality'' ' Kemble, 1935a, p. 973). Kemble argued that `here lies a fallacy, however, for whenever two systems interact for a short time there is a correlation between the subsequent behaviour of one system and that of the other,' and he claimed that the whole question had already been properly treated in `the interpretation of quantum mechanics as a statistical mechanics of assemblages of like systems,' as had been `most clearly formulated by Slater [1929a]' who had invoked the assumption `that the wave functions of the SchroÈ dinger theory have meaning primarily as descriptions of the behaviour of (in®nite) assembleges of identical systems similarly prepared' (Kemble, loc. cit., p. 974). Kemble then showed how to phrase the EPR argument correctly according to this interpretation and concluded: `There seems no reason to doubt the completeness of the quantum-mechanical description of atomic systems within the frame of our present experimental knowledge.' (Kemble, loc. cit.)
The second response to the EPR paper also came from an American author: Arthur E. Ruark's letter dated 2 July was published in the Physical Review issue of 1 September 1935. Ruark principally attacked the conclusion of EPR that the quantities corresponding to both P and Q possess reality, because one should prefer to say `that P and Q could possess reality only if both A and B (not merely one or the other) could be simultaneously measured' (Ruark, 1935, p. 466). He continued:
Whereas Einstein, Podolosky and Rosen say it is not reasonable to suppose the reality of P and Q can depend on the process of measurement carried out on system I, an opponent could reply: (1) that it makes no di¨erence whether the measurements are direct or indirect; (2) that system I is nothing more than an instrument, and the measurement of A makes this instrument un®t for the measurement of B. Such an opponent will feel that the ingenious method of measurement discussed by Einstein, Podolosky and Rosen su¨ers from all the essential di½culties common to measurements which result in disturbing system II. (Ruark, loc. cit., p. 466)
Ruark closed his letter by saying: `It seems to the writer that in the present state of our knowledge the question cannot be decided by reasoning based on accepted principles,' and added: `The arguments which can be advanced on either side seem to be so far from conclusive, and the issue involved appears to be a matter of personal choice or of de®nition.' (Ruark, loc. cit., p. 467) The latter opinion was not shared at all by his European colleagues Bohr, Heisenberg and Pauli.
LeÂon Rosenfeld, Niels Bohr's closest collaborator in the 1930s, recalled that the EPR paper ®rst `came down upon us as a bolt from the blue' (Rosenfeld, 1967, p. 128). Previously, the quantum physicists in Copenhagen had been quite used to Einstein's attacks on quantum mechanics (since 1927, see our discussion in Section II.6). `The situation changed radically, however, on the publication [of this paper],' wrote Jùrgen Kalckar in introducing the `last battle' between Bohr and Einstein on the interpretation of quantum mechanics:

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 727
Not only did it attract the attention of many physicists, but the ensuing discussions aroused interest in the more philosophical aspects of quantum physics far outside the physics community. (Kalckar, in Bohr, 1996, p. 250)
Looking at the `Copenhagen theorists' in more detail, one may recognize two different general attitudes. On the one hand, especially Pauli and HeisenbergÐand, to some extent, Bohr himselfÐwere greatly surprised that Einstein had published statements which appeared to contain just the old and, at times, even `stupid' arguments. On the other hand, Bohr still became rather worried, as Rosenfeld recalled several decades later:
We were then in the midst of groping attempts at exploring the implications of the ¯uctuations of charge and current distributions, which presented us with riddles of a kind we had not met in electrodynamics. A new worry could not come at a less propitious time. Yet, as soon as Bohr heard my report of Einstein's argument, everything else was abandoned; we had to clear up such a misunderstanding at once. We should reply by taking up the same example and showing the right way to speak about it. In great excitement, Bohr immediately started discussing with me the outline of such a reply. Very soon, however, he became hesitant. ``No, this won't do, we must try all over again. . . . we must make it quite clear . . .'' So it went on for a while, with growing wonder at the unexpected subtlety of the argument. Now and then, he would turn to me: ``What can they mean? Do you understand it?'' There would follow some inconclusive exegesis. Clearly, we were farther from the mark than we ®rst thought. Eventually, he broke o¨ with the familiar remark that he ``must sleep on it.'' The next morning, he at once took up the dictation again, and I was struck by a change in the tone of sentences: there was no trace in them of the previous days sharp expression of dissent. As I pointed out to him that he seemed to take a milder view of the case, he smiled: ``That's a sign,'' he said, ``that we are beginning to understand the problem.'' And, indeed, the real problem now began in earnest: day after day, week after week, the whole argument was patiently scrutinized with the help of simpler and more transparent examples. Einstein's problem was reshaped and its solution reformulated with such precision and clarity that weakness in the critic's reasoning became evident. (Rosenfeld, 1967, pp. 128±129)
On 29 June 1935, Bohr wrote a letter to the British journal Nature, in which just before, in the issue of 22 June, a note signed by H. T. F. (i.e., H. T. Flint from the University of London) had drawn attention to the EPR paperÐand sketched his answer to the `criterion of physical reality' of Einstein, Podolsky, and Rosen; in particular, he wrote:
I would like to point out, however, that the named criterion contains an essential ambiguity when it is applied to the problems of quantum mechanics. It is true that in the measurement under consideration any direct mechanical interaction of the system and the measuring agencies is excluded, but a closer examination reveals that the procedure of measurement has an essential in¯uence on the conditions on which the very de®nition of the physical quantities in question rests. Since these conditions must be considered as an inherent element of any phenomenon to which the term ``physical

728 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
reality'' can be unambiguously applied, the conclusion of the above-mentioned authors [EPR] would not appear to be justi®ed. A further development of this argument will be given in an article to be published in the Physical Review. (Bohr, 1935a, p. 65)
This article of Niels Bohr was indeed received by the Physical Review on 13 July 1935, and published in the 15 October issue of the same year (Bohr, 1935b).
Unlike Bohr, Pauli and Heisenberg took the EPR-arguments with much less worry, as was revealed by their correspondence. Thus, Pauli wrote in a letter to Heisenberg, dated 15 June 1935, about `two pedagogical problems where you could perhaps interfere publicly,' addressing with the ®rst an idea of the Italian theorist Gian Carlo Wick on the origin of the proton's magnetic moment, and with the second, especially:
Einstein has once again made a public statement about quantum mechanics, and even in the issue of Physical Review of May 15 (together with Podolsky and Rosen, not a good company by the way). As is well known, that is a disaster whenever it happens. ``Because, thus he concludes most sharply nothing can exist if it ought not to exist. (Weil, so schlieût er messerscharf, nicht sein kann was nicht sein darf.)''
Still I would grant him that if a student in one of his earlier semesters had raised such objections, I would have considered him quite intelligent and promising. Since through this publication there exists a certain danger of confusing the public opinionÐnotably in AmericaÐit might perhaps be advisable to send an answer to the Physical Review which I would like to persuade you to undertake. (See Pauli, 1985, p. 402; English translation in Bohr, 1996, pp. 251±252)
Pauli then outlined in his letter to Heisenberg `the facts demanded by quantum mechanics which cause particular mental troubles to Einstein,' namely essentially `the connection of two systems in quantum mechanics.' After outlining the results obtained by calculation of the systems 1 and 2, he characterized the EPR interpretation as follows:
Now comes the ``deep feeling'' which tells you: ``Since the measurement of 2 does not disturb the particle 1, there must be something called `physical reality,' namely the state of particle 1 per seÐindependently of which measurement one has performed at 2.'' It would be absurd to assume that particle 1 is changed by measurements at 2, i.e., it is transformed from a [given] state into another. In reality, the quantum-mechanical description must attribute characteristics to the particle 1 which contain already all those properties of 1 whichÐafter possible measurements of 2 which do not disturb 1Ðcan be predicted with certainty. (Pauli, loc. cit., p. 403)
Now, the pedagogical response on this argumentation, which Pauli expected Heisenberg to formulate, had to clarify in particular the di¨erence between two di¨erent situations: `(a) Two systems 1 and 2 have no interaction at all (i.e., the interaction energy is missing)Ðin that case the observation of all quantities of 1 yield the same time evolution as if there were no system 2,' and (b) `The total system [1 ‡ 2] is in a state where the partial systems 1 and 2 do not depend on each other

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 729
(separation of an eigenfunction into a product of two eigenfunctions)Ðin that case the expectation values of the quantities F1 of 1 remain, after the performance of measurement of an arbitrary quantity F2 at 2 with known numerical result F2 ˆ …F2†0, the same as without performing a measurement at 2.' According to Pauli, Einstein felt correctly that the composition and separation of systems should play a greater role in considering the foundations of quantum mechanics; since this point happened to be closely connected with Heisenberg's `considerations about [the quantum-mechanical] cut and the possibility to shift it arbitrarily [as he had emphasized it in his talk at the Hanover Naturforscherversammlung (Heisenberg, 1934f )],' Pauli requested Heisenberg to `present [the situation] once in a short [article], not in a popular language but with the use of formulae,' and emphasized:
One must distinguish di¨erent levels of reality (Schichten der RealitaÈt): one R containing all interactions which one can obtain by measurements of 1 and 2, another r (deducible from R) which contains only interactions obtainable by measurements at 1 alone. Then one must show how from the statement (Bekanntgabe) of a measurement's result at 2 a discontinuous change of r (r 3 rA or r 3 rB, etc.) follows (unless the systems of particles were independent); and that necessarily contradictions would arise if one tried to describe these changes without referring to 2Ðsay, by ``hidden properties'' of 1 in a classical or semi-classical manner. (Pauli, loc. cit., p. 404)
In any case, Pauli hoped that Heisenberg would contradict in his answer to the EPR paper the idea which `haunted elderly gentlemen like [von] Laue and Einstein' that the present quantum mechanics was incomplete and must be `completed by statements it does not [yet] contain,' such as `hidden variables'; he (Heisenberg) should especially `make it obvious in an authoritative manner that such a supplement to quantum mechanics is impossible without changing its contents' (Pauli, loc. cit.).
Heisenberg took Pauli's request seriously and soon got down to work on the proposed paper. Meanwhile, he had heard from Copenhagen about Bohr's considerations in response to the EPR-argument; therefore, he concentrated on his manuscript, entitled `Ist eine deterministische ErgaÈnzung der Quantenmechanik moÈglich? (Is a Deterministic Extension of Quantum Mechanics Possible?),' very much on the `Schnitt (cut) problem' and the supposed `incompleteness of quantum theory' (Heisenberg to Pauli, 2 July 1935, in Pauli, 1985, pp. 409±418).842 As Heisenberg would report to Bohr, `the essay was perhaps intended for publication in Naturwissenschaften . . . and thought to contain an answer to von Laue and SchroÈ dinger, especially since I heard from [Arnold] Berliner that soon a similar essay would appear [in that journal] written by SchroÈ dinger'; and further: `In it I
842 It is not certain whether Heisenberg enclosed already the above-mentioned manuscript in his letter of 2 July to Pauli, because he did not mention its existence even in his later letter to Bohr, dated 14 July 1935. We assume that Heisenberg composed it later in July or August; in any case, he sent a copy of the type-written manuscript on 22 August in a letter to Bohr.

730 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
haveÐin order not to write the same as you, because I still cannot do so as wellÐ I have emphasized a little more the formal and logical side of the problem.' (Heisenberg to Bohr, 22 August 1935) That is, Heisenberg, in his paper, mainly tried to reply to the peculiar question addressed by von Laue (1932b, 1934) and also now by Einstein, Podolsky, and Rosen (1935) `whether quantum mechanics may not later, due to new physical experiences, be so supplemented as to become a deterministic theory.'843 Notably, he wrote:
Such a consideration in general assumes, vis-a-vis the experimental successes of quantum mechanics, as a prerequisite that quantum mechanics provides [at present] a correct description of nature. It connects this prerequisite, however, with the hope that the later research will uncover behind statistical connections of quantum mechanics a hitherto hidden net of causal connectionsÐjust as behind the temperature and entropy concepts of heat theory classical mechanics lies hidden. These causal connections should not at all necessarily concern the visualizable (anschaulichen) classical properties of physical systems; rather one concludes from the validity of the indeterminacy relations that the classical concepts do not allow an adequate description of atomic phenomena, that therefore new concepts must be formed which are associated perhaps with the hitherto unknown physical properties of atomic systems. (See Heisenberg's manuscript, reproduced in Pauli, 1985, pp. 409±410)
Heisenberg, in the considerations in his manuscript, wished to demonstrate `that such a deterministic addition to quantum mechanics is impossible, and that one can therefore cherish the hope for a deterministic description of nature only if one considers the most important successes of quantum mechanics to be accidental' (Heisenberg, in Pauli, loc. cit., p. 410). He then demonstrated this claim in three sections, emphasizing at the same time that his manuscript did not contain anything new beyond what could be found in the earlier publications of Bohr, von Neumann, Pauli, and himself.
In Section 1, Heisenberg addressed, in particular, `the noteworthy schism (Zwiespalt)' of the quantum-mechanical description of nature: `On the one hand, it assumes the task of physics to be the lawful description and synopsis of visualizable, objective processes in space and time; on the other hand, it uses for a mathematical representation of physical processes those wave functions in multidimensional con®guration spaces which in no way can be regarded as representative of the objective happenings in space and time such as, say, the coordinates of a mass point in classical mechanics.' (In Pauli, loc. cit., pp. 410±411) This schism, Heisenberg continued, leads to a certain `arbitrariness in applying quantum mechanics': i.e., either the observed atomic system is described by quantum mechanics and the apparatus used for observation obeys the laws of classical physics, or also the apparatus is described by wave functions and only `the observation of the measuring apparatus, e.g., the observation of a line on the photographic plate'
843 Heisenberg's manuscript was found in the Pauli Nachlaû and has been published in Pauli, 1985, pp. 409±418, following Heisenberg's letter to Pauli dated 2 July 1935.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 731

obeys classical laws. This so-called `cut' or `gap' (Schnitt) between the descriptions of quantum mechanics and classical theory could thus be placed arbitrarily, that is: `The quantum-mechanical predictions concerning the result of any experiment do not depend on the position of the cut in question' (Heisenberg, in Pauli, loc. cit., p. 411), as Heisenberg proved explicitly in an example: He took an atomic system A, and considered the existence of several measuring apparatuses B, C, . . . (which provide the observer the ®nal observation) which is treated by quantum mechanics and by classical theory, respectively.
Clearly, the process A must be described by the time-dependent wave function ™A…qAY t†, yielding the probability j™A…qAH Y tH†j2 for the coordinate to assume at time t ˆ tH the value qAH as registered by the apparatus B, C, etc. in accordance with the classical laws. On the other hand, if A ‡ B were treated quantum-mechanically, an application of the time-dependent SchroÈ dinger equation (with HA, HB, and HAB denoting the Hamiltonian operators of the systems A and B and the interaction energy, respectively),





h 2pi

q qt

‡

HA

‡

HB

™…qAY qBY t† ˆ ÀHAB™…qAY qBY t†Y

…653†

provided the wave function ™…qAY qBY t† whichÐsince HAB deviated from zero only for the value qA ˆ qAH Ðmight be expressed as

™…qAY qBY t† ˆ ™A…qAY t†™B…qBY t† ‡ ™A…qAH Y tH†f…qAY qBY tY tH†Y

…654†

where f…qAY qBY tY tH† was independent of the behaviour of the system A before t ˆ tH. Now, the probability of the system B to undergo a change from the original

stateÐi.e., ™B…qBY t†Ðwas given by the integral over the absolute square of the

„indteqrAadcqtiBojn™…tqeAHrmY t

H

on the right-hand side of †f…qAY qBY tY tH†j2; hence, it

Eq. (654) in the variables became proportional to

qA and j™A…qAH Y t

qB, H†j2

and did not depend on the prehistory of the system A. As a consequence, the

quantum-mechanical result in case the cut is transferred beyond B turned out to be

the same as before; similarly, one could transfer the cut beyond C, etc.

Thus, Heisenberg stated that the characteristic property of the application of

the wave-mechanical description of the measuring apparatus was the fact that the

interaction with the atomic system (to be measured) resulted only in transitions of

the coordinate qB at a ®xed value, qAH , of the coordinate qA of the atomic system: and then qBH was just changed to qBHH, or `the total wave function then appears (for a short time after switching on the interaction) as a product of two factors, one of

which being given by the wave function of the observed system A at the moment

the interaction is switched on, while the other represents the reaction of the mea-

suring apparatus B.' (In Pauli, loc. cit., p. 413) This result came out of the peculiar

properties of the quantum-mechanical formalism, and, as a consequence, `the

causal connections of the classical theories used in the measuring apparatus can

be reproduced in quantum mechanics only with that degree of accuracy as the

732 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
visualizable classical characteristics of the measuring apparatus are represented in wave optics'Ðbut `the fundamental indeterminacy created in this way of formulating causal connections is in all practical cases much smaller than the practical uncertainty that must be taken into account for everyÐeven the bestÐ measuring device.' (In Pauli, loc. cit.) Heisenberg concluded the more technical Section 1 with two remarks: (i) the cut cannot be shifted so arbitrarily that certain measuring devices operating like atomic systems (e.g., nuclear systems measuring the neutron ¯ux) are described by classical theory; (ii) since the wave-mechanical formalism per se operates with respect to a causal behaviour like classical theory, and the statistical aspect enters only via the cut, the whole measurement process can represent causal connections in a restricted sense.
In Section 2, Heisenberg investigated `the assumption that the physical systems described statistically by quantum mechanics carry up-to-now unknown physical properties which determine so far only the statistically known behaviour uniquely' (Pauli, loc. cit., p. 414); contrary to the expectations of von Laue and EPR, he showed that `this assumption contradicts the statements of quantum mechanics, especially not only its statistical results but also the de®nite conclusions derived' (in Pauli, loc. cit., p. 415). This impossibility proof of `hidden variables' to establish a causal behaviour was based on the premise that quantum mechanics determined uniquely all properties of the system left of the cut, i.e., either of A, or A ‡ B, or A ‡ B ‡ C, etc. Hence, if extra properties had to be assumed for A in order to turn the statistical statements of the measurement into de®nite results, also changes of the properties of A ‡ B, or A ‡ B ‡ C, etc., must arise, and `every statement about A which was not already contained in the quantum-mechanical connection A ‡ B [or A ‡ B ‡ C, etc.] can contradict the conclusions from this connection' (in Pauli, loc. cit.), thus also the above premise. Heisenberg then illustrated this situation in an example, where he tried to obtain information about complementary quantities of the system A. `For a supplement of the quantum-mechanical statements, the only suitable place was that of the ``cut,'' ' he found, `but this place cannot be ®xed physically, since it is rather the arbitrariness in the choice of the position of the cut that is responsible for the [consistent] application of quantum mechanics'; hence, `Any physical properties so far unknown that must be connected necessarily with a physical system therefore could not serve in principle to supplement quantummechanical statements.' (In Pauli, loc. cit., p. 416) After illustrating this result in the case of the radioactive —-decay (by applying the complementary particle- and wave-pictures, respectively), Heisenberg closed Section 2 with two comments:
It is a decisive feature of quantum mechanics that it permits via its formalism to connect the physical domains foreign, in principle, to our visualization in an organic way with the macroscopic, visualizable domain, such that the results from the formalism can be expressed by visualizable (anschauliche) concepts.
However, quantum mechanics, explicitly presupposesÐlike the argumentation presented hereÐthat at the same place we are ®nally able to turn our interactions into objective entities (unsere Wechselwirkungen zu objektivieren), i.e., allow us to speak about objects and events. Classical physics proves that this can be done for a

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 733
large domain [of experience], and all of experimental science rests on this possibility. (Heisenberg, in Pauli, loc. cit., p. 417)
In Section 3, Heisenberg argued that the philosophical explanations of these conclusions must be traced in the very essence of Nature, orÐas stated by Grete Hermann (1935a)Ð`that a deterministic supplement of quantum mechanics fails because quantum mechanics already allows us to give completely the causes for the occurrence of a given result of measurement.' (Heisenberg, in Pauli, 1985, p. 417) This situation involves the problem to search for the particular feature of nature which forbids us to derive from the uniquely connectedÐone might even say, causalÐformalism of quantum mechanics all (possible) results of measurement, and which creates the statistical connections at the cut. A quantummechanical state, Heisenberg said, is given uniquely by a wave packet moving with a certain velocity at a ®xed space-point plus `further statements about the size and shape of the wave packet, for which there exist no analogues in the classical theory' (Heisenberg, in Pauli, loc. cit., p. 418); he called such a description a `Beobachtungszusammenhang (context of observation)' and emphasized that `the same visualizable events may correspond to di¨erent contexts of observation,' a situation that was not known to occur in classical physics. `The experimental conclusion formulated by quantum mechanics has shown that the observation of a system in general leads from one Beobachtungszusammenhang into another,' Heisenberg explained, and noted:
The causal connection can be followed within a de®nite context of observation, while in the discontinuous transition from one [situation] to the other (especially to a ``complementary'' [one] in the sense of Bohr) only statistical predictions are possible. Hence the possibility of di¨erent, complementary contexts of observation, unknown in the classical theory, becomes responsible for the occurrence of statistical laws. (Heisenberg, in Pauli, loc. cit.)
Finally, Heisenberg questioned whether a future modi®cation of quantum mechanics might give rise to a deterministic supplement, but he ®rmly claimed that experimental evidence so far provided no hint that `the future description of nature will ®t again into the narrow classical scheme of a visualizable and causal description of objective processes in space and time' (in Pauli, loc. cit.).
While no written comment of Pauli on Heisenberg's manuscript has survived among the available documents, Niels Bohr, in a letter dated 15 September 1935, to Heisenberg, asked for a few clari®cations of complementary situations, which Heisenberg tried to provide in his letter of 29 September. Bohr further criticized that he placed too much emphasis on the `shift of the cut,' to which Heisenberg replied as follows:
Why the possibility of shifting the ``cut'' is so particularly important in my opinion, I can most simply explain thus: You say correctly that ``all elements of description are de®ned classically and yet the classical theory leaves no room for quantum-mechanical

734 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
laws.'' This statement appears to physicists used to think formally as a plain contradiction, as I know for instance from talking to Herr von Laue. Hence I thought it to be important to stress the property of the formalism which ensures that no contradiction arises here, and this, it seems to me, lies in the possibility to shift the cut. If this were not so, simply two categories of physical systemsÐclassical and quantummechanical onesÐwould exist, and one could never apply classical concepts to the latter. That's how von Laue sees the situation. I believe that then it might be very di½cult to argue against the hope of a later causal supplement. (Heisenberg to Bohr, 29 September 1935)
In any case, Heisenberg believed that the most direct way to understand why the quantum-mechanical formalism did not at all need new concepts, totally di¨erent from the classical ones, was to make e¨ective use of the possibility to shift the `cut.' He hoped to be able to discuss these questions in greater detail with Bohr in October in Copenhagen, especially the latter's arguments against the `more formal manner of treating quantum theory' and promised not to submit his manuscript for publication prior to these discussions.844
We shall now discuss the o½cial response given by Niels Bohr to the EPR argument, published in the Physical Review issue of 15 October 1935, whichÐ unlike Heisenberg's manuscriptÐworked with very little formalism in the style to which Bohr had become accustomed in the previous 15 years. His answer was contained especially in the comment which he added after he had summarized the conclusion of the EPR-paper, and which read:
Such an argumentation, however, would hardly seem suited to a¨ect the soundness of quantum-mechanical description which is based on a coherent mathematical description covering automatically any procedure of measurement like that indicated.* The apparent contradiction in fact discloses only an essential inadequacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena of the type with which we are concerned in quantum mechanics. Indeed the ®nite interaction between object and measuring agencies conditioned by the very existence of the quantum of action entailsÐbecause of the impossibility of controlling the reaction of the object on the measuring instruments if these are to serve their purposeÐ the necessity of a ®nal renunciation of the classical idea of causality and a radical revision of our attitude towards the problem of physical reality. In fact, as we shall see, a criterion of reality like that proposed by the authors [i.e., EPR] containsÐ however cautious its formulation may appearÐan essential ambiguity when it is applied to the actual problems with which we are here concerned. In order to make the argument to this end as clear as possible, I shall ®rst consider in some detail a few simple examples of measuring arrangements. (Bohr, 1935b, pp. 696±697)
844 We do not know the results of the discussions in Copenhagen in October 1935 on this subject, as they were not mentioned in Heisenberg's letter to Bohr, in which he thanked the latter for the `®ne time' in `your circle' and the `wonderful mixture of leisure and serious thinking.' One reason for not sending his manuscript for publication may also have been the more di½cult situationÐwhich Heisenberg soon experiencedÐthat existed towards modern theoretical physics; in particular, he did not wish to attack people like Planck or von Laue [who were also under attack from Nazi partisans and representatives of `German Physics (Deutsche Physik)'].

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 735

From these lines of argument, the style of Bohr's answer may be recognized. He intended to continue the previous discussions with Einstein (at the ®fth and sixth Solvay Conferences of 1927 and 1930, respectively) by referring to the particular Gedankenexperiments which could be worked out with a minimum of mathematical formalism. To characterize the subordinate position given here to mathematical argumentation, as compared to Heisenberg's procedure, Bohr put the entire formal apparatus essentially into a single footnote, marked by an asterisk (*) (attached to the ®rst sentence in the above quotation). Having emphasized the mathematical completeness of the quantum-mechanical scheme by a sentence, he went on quickly to describe an atomic system consisting of two partial systems (1) and (2), interacting or not, by two pairs of canonical variables, …q1p1† and …q2p2†, which satisfy the commutation rules,

W

‰q1Y

p1Š

ˆ

‰q2Y

p2Š

ˆ

ih 2p

Y

bba

‰q1Y q2Š ˆ ‰ p1Y p2Š ˆ ‰q1Y p2Š ˆ ‰q2Y p1Š ˆ 0X bbY

…655†

A canonical transformation by a simple orthogonal transformation yielded new pairs of conjugate variables, …Q1Y P1† and …Q2Y P2†, de®ned by the equations

W q1 ˆ Q1 cos y À Q2 sin yY p1 ˆ P1 cos y À P2 sin yY a q2 ˆ Q1 sin y ‡ Q2 cos yY p2 ˆ P1 sin y ‡ P2 cos yY Y

…656†

with the angle of rotation y. The analogous commutation relations, with the transformed Q's and P's replacing the original q's and p's in Eq. (655), implied that in the description of the combined system de®nite values could not be assigned to both Q1 and P1, but certainly one could assign such values to Q1 and P2, etc.Ði.e., all variables which commute. Further, from the expressions Q1 and P2, namely,

Q1 ˆ q1 cos y ‡ q2 sin yY P2 ˆ Àp1 sin y ‡ p2 cos yY

…657†

one derived that a subsequent measurement of either q2 or p2 would allow one to predict the value of q1 or p1, respectively. Eqs. (655) to (657) provided all the quantum-mechanical formalism needed by Bohr, who put all his e¨orts in the discussion of the following Gedankenexperiment.
Bohr began by considering the passage of an atomic particle through an arrangement of diaphragms with parallel slits which allow either to detect the position or the momentum of the object accuratelyÐin the ®rst case the diaphragms have to be ®xed rigidly, in the second case not rigidlyÐas was known from previous discussions. Bohr commented: `My main purpose in repeating these

736 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
simple . . . considerations, is to emphasize that in the phenomena concerned we are not dealing with an incomplete description characterized by the arbitrary picking out of di¨erent elements of physical reality at the cost of sacri®cing other such elements, but a rational discrimination between essentially di¨erent experimental arrangements and procedures which are suited either for an unambiguous use of the idea of space location, or for a legitimate application of the conservation theorem of momentum.' (Bohr, loc. cit., p. 699) On the one hand, there was the `freedom of handling the measuring instruments, characteristic of the very idea of experiment'; on the other hand, quantum theory, because of `the impossibility of accurately controlling the reaction of the object to the measuring apparatus, i.e., the transfer of momentum in case of position measurements, and the displacement in case of momentum measurements,' implied `the renunciation in each experimental arrangement of one or the other of the two aspects of the description of physical phenomenaÐthe combination of which characterizes the method of classical physics.' (Bohr, loc. cit.) Bohr continued:
Just in this last respect any comparison between quantum mechanics and ordinary statistical mechanics . . . is essentially irrelevant. Indeed we have in each experimental arrangement suited for the study of proper quantum phenomena not merely to do with an ignorance of the value of certain physical quantities, but with the impossibility of de®ning these quantities in an unambiguous way. (Bohr, loc. cit.)
After these preliminary remarks, Bohr reproduced the EPR Gedankenexperiment on the interaction of two particles:
at least in principle, by a simple experimental arrangement, comprising a rigid diaphragm with two parallel slits, which are very narrow compared with their separation, and through each of which one particle with given initial momentum passes independently of the other. If the momentum of this diaphragm is measured accurately before as well as after the passing of the particles, we shall in fact know the sum of the components perpendicular to the slits of the momenta of the two escaping particles, as well as the di¨erence of their initial positional coordinates in the same direction; while of course the conjugate quantities, i.e., the di¨erence of the components of their momenta, and the sum of the positional coordinates, are entirely unknown. (Bohr, loc. cit.)
At this point, Bohr added a footnote which explained how the experiment thus proposed was theoretically described by the transformation of the variables according to Eqs. (656) with the particular rotational angle y ˆ pa2; further he emphasized that the wave function (652) of EPR corresponded `to the special choice of P2 ˆ 0 and the limiting case of two in®nitely narrow slits' (Bohr, loc. cit., footnote). `In this arrangement it is therefore clear that a subsequent single measurement either of the position or of the momentum of one of the particles will automatically determine the position or momentum, respectively, of the other particle with any desired accuracy,' he continued and further admitted: `As

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 737
pointed out by the named authors [i.e., EPR], we are therefore faced at this stage with a completely free choice whether we want to determine the one or the other of the latter quantities by a process which does not directly interfere with the particle concerned.' (Bohr, loc. cit., p. 699) However, Bohr interpreted this `freedom of choice' just as `a discrimination between di¨erent experimental procedures [in quantum mechanics] which allow of the unambiguous use of complementary classical concepts,' (Bohr, loc. cit., p. 700) and then went on to explain the well-known situation in atomic theory which required a quite di¨erent interpretation than proposed by EPR. In particular, he summarized the quantum-theoretical position as follows:
From our point of view we now see that the wording of the . . . criterion of physical reality proposed by Einstein, Podolosky and Rosen contains an ambiguity as regards the meaning of the expression ``without in any way disturbing the system.'' Of course there is in a case like that just considered no question of a mechanical disturbance of the system under investigation during the last critical stage of the measuring procedure. But even at this stage there is essentially the question of an in¯uence on the very conditions which de®ne the possible types of predictions regarding the future behaviour of the system. Since these conditions constitute an inherent element of the description of any phenomenon to which the term ``physical reality'' can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that the quantum-mechanical description is essentially incomplete. (Bohr, loc. cit., p. 699)
Quantum mechanics rather `may be characterized as a rational utilization of all possibilities of unambiguous interpretation of measurements, compatible with the ®nite and uncontrollable interaction between the objects and the measuring instruments in the ®eld of quantum theory,' Bohr emphasized (Bohr, loc. cit., our italics)Ði.e., only the recognition of this fact in atomic physics, in his opinion, `provides room for new physical laws' characterized by `the notion of complementary aims' (Bohr, loc. cit.).
In the discussion of Bohr's experiment, the time played only a secondary role, but certainly also the consideration of the time and energy measurements which had been emphasized by EPR could be discussed according to the rules of the fundamental quantum-mechanical complementarity. To Bohr, the essential point seemed to be `the necessity of discriminating in each experiment between those parts of the physical system considered which constitute the objects under investigation'; their necessity `may indeed be said to form a principal distinction between classical and quantum and quantum-mechanical descriptions of physical phenomena,' Bohr concluded, explaining:
While, however, in classical physics, the distinction between object and measuring agencies does not entail any di¨erence in the character of the description of the phenomena concerned, its fundamental importance in quantum theory, as we have seen, has its root in the indispensable use of classical concepts in the interpretation of all

738 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
proper measurements, even though the classical theories do not su½ce in accounting for the new types of regularities with which we are concerned in atomic physics. (Bohr, loc. cit., p. 701)
Hence, `there can be no question of any unambiguous interpretation of the symbols of quantum mechanics other than that embodied in the well-known rules . . . which have found their general expression through the transformation theorems.' These theorems secured the correspondence of quantum mechanics with the classical theory and excluded `any imaginable inconsistency in the quantummechanical description, connected with a change of the place where the discrimination is made between object and measuring agencies.' Bohr concluded his paper by announcing in a footnote a further study `where the writer will in particular discuss a very interesting paradox suggested by Einstein concerning the application of gravitation theory to energy measurements, and the solution of which o¨ers an especially instructive illustration of the generality of the argument of complementarity,' and further: `On the same occasion a more thorough discussion of space-time measurements in quantum theory will be given with all necessary developments and diagrams of experimental arrangements, which had been left out in this article, where the main stress is laid on the dialectic aspect of the question at issue.' (Bohr, loc. cit., pp. 701±702) However, this detailed paper intended to extend the complementarity philosophy further never appeared.
(d) Erwin SchroÈdinger Joins Albert Einstein: The Cat Paradox (1935±1936)
Unlike Albert Einstein, Erwin SchroÈ dinger had regularly published since 1927 his thoughts about quantum mechanics and its interpretation (e.g., SchroÈ dinger, 1928; 1929b, c; 1932b). From the very beginning, he had shared with Einstein the uneasiness, ®rst concerning certain resultsÐsuch as the uncertainty or indeterminacy relationsÐand later the `unvisualizable (unanschauliche)' consequences of quantum mechanics. In fact, he often discussed these questions with Einstein when they were together in Berlin, and they both left after the Nazis took over the government of Germany. While Einstein, after spending several months in Europe (in the remote and secluded Villa `Savoyarde' in Le Coq-sur-mer, the resort town near Ostende on the Belgian coast), settled down for good at the Institute for Advanced Study in Princeton, SchroÈ dinger ®rst went in summer 1933 as a Fellow of Magdalen College at Oxford, and did not really know whether he should stay in England in the following years. On 17 May 1935, he wrote to Albert Einstein: `The feeling grows that I hold no position and depend on the generosity of others,' and added, `When I came here I thought I could do something valuable for teaching, but one did not care about that here. And further, I think that in truth I must tell myself that in reality I am staying here for a very nice old man [Augustus Love] to die or become disabled and that one calls upon me to be his successor.' He therefore hoped, as he reported to Einstein further, to obtain a position in

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 739
Austria, namely, the chair of Professor Michael Radakovic in Graz.845 Three weeks later, SchroÈ dinger took up his correspondence again with Einstein, and entered into a lively discussion of the contents of the paper of Einstein, Podolsky, and Rosen:846
Dear Einstein, I have much rejoiced that in your just published paper in Physical Review, you have publicly gotten to the heart (oȨentlich beim Schla½tchen erwischt hast) the dogmatic quantum mechanics, about which we have discussed so much in Berlin. May I add a few things to it? At ®rst they look like objections; but they concern only points which I wish had been formulated more clearly. (SchroÈ dinger to Einstein, 7 June 1935)
SchroÈ dinger thus began his letter to Einstein, in which he analyzed the procedure of proof in the EPR paper. In particular, he argued: `In constructing a contradiction, it does not su½ce in my opinion that for the identical preparation of a pair of systems the following may occur: one de®nite single measurement of the ®rst system yields for the second a certain value A, another a certain value B, and the simultaneous reality of A and B is excluded because of general reasons.' Identical preparation would not always lead to the same result, but may yield in one case the value AH for the quantity A, in a second one the value AHH for the same quantity, and in a third the value BH for the di¨erent quantity B. Thus, in order to establish a genuine contradiction, one should rather require for a pair of systems the existence of two quantities A and B whose reality is mutually excluded, and further:
1. One method of measurement exists which yields for the quantity A for a wave function always a sharply de®ned (though not always the same) value, hence I can say without actually performing the experiment: in case of the given wave function, A possesses reality, independently of its value.
845 O½cially, SchroÈ dinger was at Oxford on leave of absence from the University of Berlin, which was extended until he requested his Emeritierung in early 1935. In Austria, where he had looked for a permanent position since summer 1933 (and also asked for the restoration of his Austrian citizenship), it took until September 1936 when the SchroÈdingers could move to Graz. Two years later, after the annexation (Anschluû) of Austria with the Third Reich, SchroÈdinger lost this position andÐafter a transitory period again at Magdalen College in OxfordÐhe received in December 1938 a professorship at the University of Ghent in Belgium. Upon the outbreak of World War II, SchroÈdinger had to leave Belgium (having become o½cially a `hostile alien'). The Irish politician and prime minister, Eamon de Valera, who had always been an amateur mathematician and had hoped to establish an `Institute for Advanced Studies' in Dublin, invited SchroÈ dinger (who had taken refuge at the Ponti®cal Academy of Sciences at the Vatican) to meet with him in Geneva, Switzerland (where he, de Valera, was attending a meeting of the League of Nations); as a result of their meeting, de Valera advanced the schedule of the founding of the Institute for Advanced Studies in Dublin, and invited SchroÈdinger to join it as a Senior Professor of Theoretical Physics in October 1939 (being made the Director of the Theoretical Physics Division in November 1940).
846 The Einstein±SchroÈ dinger correspondence between 1935 and 1947 on the interpretation of quantum mechanics has not been included in the correspondence collection edited by Karl Przibram (SchroÈdinger et al., 1963). We thank Robert Schulmann for providing us with the contents.

740 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

2. Another method of measurement should at least occasionally give the quantity B a sharp value (always for the same wave function, of course). (SchroÈ dinger to Einstein, loc. cit.)

In general, SchroÈ dinger continued, there exists only one way of expanding a function of two variables (or groups of variables) in a bilinear series,

ˆ y ™…x1Y x2† ˆ cn™n…x2†un…x1†Y
nˆ1

…658†

such that both un…x1† and ™n…x2† form a normalized orthogonal system. Now, if two of the coe½cients cn assume identical absolute value, the expansion (658) ceases to be uniquely de®ned, and in the EPR case, all cn were taken to be equal: `Hence you can rotate [by a canonical transformation of the quantum-mechanical system] in an arbitrary manner, even from the ``Q-position'' into the ``Pposition.'' ' Apart from suggesting this sharper formulation, however, he agreed with the EPR conclusions and considered the unsatisfactory situation as arising from the inability of `the orthodox scheme [of quantum mechanics] to describe the separation process' of the two systems.
Einstein replied to SchroÈ dinger's letter on 19 June, being `very pleased' with this support. `The real situation lies in the fact that physics is a kind of ``metaphysics,'' ' he wrote, and further: `Physics describes ``reality,'' but we do not know what ``reality'' is, as we know it only through physical description!' The latter might be `complete' or `incomplete,' as he explainedÐleaving out the `erudition' of Podolsky (who had redacted the paper but spoilt, in Einstein's opinion, the presentation of the argument)Ðin the example of two boxes having collapsible lids and a sphere which may be found by `observation,' i.e., by opening the lid of a box:

Now I describe a state as follows: The probability to ®nd the sphere in the ®rst box is 1/2. Is this a complete description? [Answer] No. A complete description is: the ball is in the ®rst box (or it is not there). This must look like the characterization of a complete description. [Answer] Yes. Before I open the lid, the ball is not in either of the two boxes. Its being in a certain box comes about only by opening the lid. In this way, only the statistical character of the experienced world, or the empirical structure of its law (Gesetzlichkeit) arises. The state before opening [the lid] can be completely characterized by the number 1/2, whose meaning manifests itself in the process of observation only as a statistical statement. The statistics arises only by introducing insu½ciently known factors, foreign to the system considered, through the observation. (Einstein to SchroÈ dinger, 19 June 1935)

Einstein then argued that one might not be able to distinguish between the two conclusions mentioned above, unless one called for help upon an `additional principle,' the `principle of separability,' and stated explicitly: `The second box plus everything concerning its contents is independent of what happens in the ®rst box ([both are] separated partial systems).' Thus, `if one sticks to the principle of

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 741
separability, one excludes the second [he called it ``SchroÈ dinger-like''] interpretation and retains only the ®rst [``Born's''], according to which the above description of the state, however, is an incomplete description of reality, or the real state, respectively.' (Einstein to SchroÈ dinger, loc. cit.)
Einstein admitted that his example represented the quantum-mechanical situation only in an imperfect manner, although it stressed the `essential feature' of whether the normalized wave function ™ can be uniquely associated with the real state of the system, and the statistical character of the results of measurement emerges exclusively from the process of measurement. If it were so, he would call the situation a complete description of reality by the theory; if not, it would be incomplete. SchroÈ dinger responded to Einstein on 13 July: `Your letter shows that I completely agree with you concerning the opinion about the existing theory . . . I now take pleasure and use your note to challenge with it the most di¨erent, intelligent people: London, Teller, Born, Pauli, Szilard, Weyl.' That is, he had asked these colleagues (representing the orthodox viewpoint of quantum mechanics) personally (if available in Great Britain, e.g., London, Teller, and Born) or by letters (Pauli, Weyl) about their stand on this question. Now, SchroÈ dinger reported in particular that the `most relevant' answer came from Pauli, `who at least admits that the use of the word ``state'' for the ™-function is very suspicious (anruÈchig).' (SchroÈ dinger to Einstein, 13 July 1935, p. 1) Evidently, he referred to a letter, in which PauliÐthough claiming that `one cannot, as the old conservative gentlemen wish to do, declare the statistical statements of quantum mechanics (wave mechanics) as correct and nevertheless put a hidden causal mechanism behind it'Ð had (after explaining to SchroÈ dinger Bohr's reply to the EPR argument) contemplated about the question whether a `pure case' (described by a given wave function) might be called a `state (Zustand )':
A pure case [of a system] A represents a whole situation, in which the results of certain measurements at A (to the maximal extent) can be predicted with certainty. If one calls this a ``state,'' I do not mindÐbut then it does follow that a change of the state AÐi.e., of what is predictable about AÐlies also, di¨erently from the in¯uence of a direct perturbation of A itselfÐi.e., also after the isolation of AÐ, in the free choice of the experimentalist. (Pauli to SchroÈ dinger, 9 July 1935, in Pauli, 1985, p. 420)
`The great di½culty to reach an understanding with the orthodox people,' SchroÈ dinger went on to write in his letter to Einstein on 13 July 1935, `has induced me to try to attempt an analysis of the present situation of the interpretation ab ovo [i.e., from the very beginning]. Whether and what I shall publish of it, I do not know; but for me this is the best way to clarify matters for myself.' (SchroÈ dinger to Einstein, 13 July 1935, p. 2)
In this analysis, he wrote to Einstein, several points in the current foundations of quantum mechanics occurred to him as `strange (komisch).' The ®rst such point was that the new quantum theory, which deviated so strongly from the previous one by the statements of indeterminacy, acausality, and many more speci®c ones,

742 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
had not changed at all in one peculiar aspect, namely, the fact that: `The only real thing in the world [of science], the result of the measurement, can be explained by it only totally classically just as the measurement of a property in a classical model.' Hence, di¨erently from the situation in electrodynamics, where the new (Maxwell) theory had created new concepts (e.g., the ®eld strengths, etc.) to be measured, in quantum mechanics, `one so-to-say measures happily further (angeblich lustig weiter) the same concepts as before [in the classical theory], because supposedly our language is not able at all to grasp something else.' (SchroÈ dinger to Einstein, loc. cit.) Even the totally new `probability' statements in the quantummechanical calculation referred in SchroÈ dinger's opinion just to classical concepts instead of dealing with the new properties of the atomic systems.847 Second, to determine the obviously continuous ™-function by a ®nite or discrete set of `suitably chosen and ideally accurate measurements' appeared to be quite an unbelievable `hocus-pocus' (SchroÈ dinger to Einstein, loc. cit., p. 3). Third, he strongly criticized a statement of Paul Dirac's, according to which `canonical variables may have as eigenvalues all real numbers, from minus in®nity to plus in®nity' as being unbelievable and practically not veri®able by measurements. This point had been clearly noticed already by John von Neumann (in his 1932 book on the mathematical foundations of quantum mechanics) when he declared his own description of the quantum-mechanical measurement process as `at least for the moment, the mathematically most practicable.' `I believe that here our Johnny has already indicated sharply (den Meiûel angesetzt) where a reformulation is needed,' SchroÈ dinger commented and added:
One had actually lost the classical model. One did not ®nd a new one but hit upon the biggest di½culties opposing [the construction of ] any model at all. Hence one says: Hey, we just retain the classical one, declare that all its properties are measurable in principle, and add in a wise, philosophical manner that these measurements represent the only reality, and everything else is metaphysics. Then the monstrosity of our statements concerning the model does not disturb us. We do have recanted itÐand therefore we are allowed to use it all the more happily. The mistake [of this standpoint] is the following: if one wants to adopt this highly philosophical viewpoint, one must declare really feasible measurements, or idealizations of these, to be the ``only reality.'' (SchroÈ dinger to Einstein, loc. cit., pp. 4±5)
Einstein replied to SchroÈ dinger on 8 August 1935, and said: `You are practically the only person with whom I like to argue, because all the other fellows (Kerle) do not view the theory from the facts but only view the facts from the theory; they cannot escape from the once adopted net of concepts but can only toss about it nicely ( possierlich darin herumzappeln).' He immediately proceeded to stress the di¨erence in their respective criticisms of the quantum-mechanical situation (`We represent the sharpest contrasts, he noted.'). While Einstein himself
847 At this point, SchroÈ dinger evidently forgot about the spin property, certainly a nonclassical concept.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 743
preferred to have the ™-function describe not the state of one system but an ensemble of systems, SchroÈ dinger did consider ™ as the representation of reality (and he also wished, as we have discussed above, to abolish the connection with the concepts of ordinary mechanics). However, SchroÈ dinger's interpretation would fail to describe the macroscopic experience, as Einstein now illustrated by the example of a pile of gun powder in a chemically labile state. SchroÈ dinger disagreed: `Since long I have left the stage behind me that the ™-function can somehow be viewed as the description of reality,' he wrote back immediately and reported:
In a longer essay, which I have just written, I discuss an example very similar to your exploding powder barrel. I just put the emphasis there to bring into play an uncertainty which according to our present understanding is really of the ``Heisenberg type'' and not of the ``Boltzmann type.'' A Geiger counter is enclosed in a steel chamber connected with a tiny amount of uraniumÐso little that in the next hour one atomic decay is as probable as improbable. An amplifying relay makes sure that the ®rst atomic decay crashes a little retort containing hydrocyanic acid. This andÐ cruellyÐa cat are contained in the steel chamber. After an hour, then, in the ™function of the total systemÐsit venia verbo (excuse my words)Ða living and dead cat are smeared out in equal parts. (SchroÈ dinger to Einstein, 19 August 1935)
Although he did not follow the mathematical details of SchroÈ dinger's letter, Einstein was quite pleased with the example of the cat, which showed `that we agree completely with respect to the character of the present theory,' because:
A ™-function, in which a living and a dead cat enter [simultaneously], cannot just be considered to describe a real state. This example precisely hints at the fact that it is reasonable to attribute the ™-function to a statistical ensemble, which embraces equally well a system with a living cat as well as a dead one. (Einstein to SchroÈ dinger, 4 September 1935)
SchroÈ dinger, on the other hand, had submitted his essay already around 12 August to the German journal Naturwissenschaften.848 It was entitled `Die gegenwaÈrtige Situation in der Quantenmechanik (The Present Status of Quantum Mechanics),' and SchroÈ dinger organized in it in a quite detailed manner his ideas in 15 sections, which were published in three issues of the journal between 29 November and 13 December 1935 (SchroÈ dinger, 1935a).
SchroÈ dinger started his essay by explaining the nature of a `classical model' with its `determining characteristics (BestimmungsstuÈcke)'Ði.e., the `model con-
848 SchroÈ dinger had previously often published in this journal on various topics, including epistemological questions (1929a). As he wrote to Einstein on 19 August 1935, he had previously exchanged letters with Arnold Berliner, the long-time editor of Naturwissenschaften. Just recently, Berliner had informed him that he was ®red as the editor and was only allowed to serve as an advisor, but he had requested SchroÈ dinger still to send him papers for `his' journal. Contrary to his prior intentions, SchroÈ dinger let the paper appear in GermanyÐagainst Einstein's protestÐas his last contribution until the end of the Third Reich. When Berliner was ordered years later to leave his home in Berlin, he committed suicide on 22 March 1942.

744 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
stants,' such as energy, momentum, angular momentum, etc., of which a complete set ®xed the physical state of the model (§1). `The turning point of today's quantum mechanics constitutes a dogma . . . stating that models with characteristics which are determined uniquely, like the classical ones, do not correspond to nature,' SchroÈ dinger stated at the beginning of §2 on `The Statistics of Model Variables in Quantum Mechanics' (SchroÈ dinger, 1935a, p. 808). The new models referred to the classical ones but emphasized restrictionsÐnamely, the `mutual determination'Ðin the following way: (i) The classical concept of state is lost, since at most half of the complete set of characteristics can be associated with ®xed numerical values, while the others remain completely indeterminate (in certain cases, like the Rutherford atomic model, all of them appear to be uncertain, i.e., restricted by the indeterminacy relations). (ii) As not all the variables can be determined at a given instant of time, they won't be determined also at a later instant; hence, the principle of causality fails. That is, quantum mechanics replaces the causal relations by a particular statistics: It allows one to compute, from the maximal number of completely determined characteristics, the `statistical distribution' of every variable at a given instant of time and at any later instants.
While the new theory thus declared the classical model as being unable to represent the mutual connection of the characteristics (BestimmungsstuÈcke)Ðthus renouncing the very reason why the model was inventedÐit assumed, on the other hand, that the classical model still remained a suitable tool to inform us as to which type of measurements can be carried out in principle on a given object in nature. `This would seem to those who invented the picture [i.e., the classical model] as an unprecedented overstraining of their paradigm (Denkmodell ), a frivolous anticipation of the future development,' SchroÈ dinger concluded (SchroÈ dinger, loc. cit., p. 809).
The probability predictions of quantum mechanics, SchroÈ dinger explained in the next section (§3), were quite sharp, even `sharper than any real measurement could ever provide'; but the classical concepts (like angular momentum or energy) were used only `to force the contents with some e¨ort into the Spanish boots of a probability statement' or: `According to the wording [of quantum mechanics], all statements refer to the classical model; but the valuable statements connected with it are little visualizable, and its visualizable characteristics possess only little value.' Thus:
The classical model plays the role of Proteus in quantum mechanics. Each of its determining characteristics may, under suitable circumstances, become the object of interest and gain a certain reality; but all of them can never do soÐonce there are certain characteristics, next time there are others, especially at most always half of a complete set of dynamical variables provide a clear picture of the instantaneous state of the system under consideration. (SchroÈ dinger, loc. cit., p. 810)
The question now arose about the `reality' of the otherÐthe uncertainÐ variables, and SchroÈ dinger discussed two alternatives. One alternative endowed all of them with reality but did not permit a simultaneous knowledge (of all), similar

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 745
to the statistical description of molecular systems in the late nineteenth century (§4). SchroÈ dinger then demonstrated in several examples of quantum-mechanical variables that they could not be described by `ideal ensembles': `At no instant of time, an aggregate of classical model states exists described by the ensemble of quantum-theoretical results.' (SchroÈ dinger, loc. cit., p. 811) The other alternative, namely, the assumption that the undetermined characteristics possessed noÐor just a `washed out (verschwommene)'Ðreality, seemed to be acceptable only at ®rst inspection (§5). One may, of course, use the tool of the ™-function to describe as clearly as in the classical case the degree of the `washing-out' of all variables; however, `serious doubts arise if one realizes that the indeterminacy seizes coarsely touchable and visible objects where the concept of washing-out simply turns out to be wrong' (SchroÈ dinger, loc. cit.). For example, in dealing with the radioactive —-decay, it was possible to describe the interior of the atom by washed-out variables; yet the observation of the emitted —-rays revealed de®nite tracks in a Wilson cloud chamber or clear scintillation spots on a screen. `One can even construct quite burlesque cases,' SchroÈ dinger continued, such as:
A cat is captured in a steel chamber together with the following infernal machinery (which one must protect from the direct grip of the cat): in a Geiger counter there exists a tiny amount of a radioactive substance, so little that in the course of an hour perhaps one atom decays, and with equal probability it does not decay; if it decays, the counter clicks and operates via a relay a small hammer such that it shatters a little retort containing hydrocynic acid. On leaving the system to itself for an hour, one may still say that the cat is still alive if no atom has decayed meanwhile; the very ®rst decay would have poisoned it. Then the ™-function of the total system would describe the situation by claiming that it contains the living and the dead cat mixed or smeared out in equal parts. (SchroÈ dinger, loc. cit., p. 812)
The typical feature of such examples was that an indeterminacy in the atomic domain caused an indeterminacy which might be sensed macroscopically (or `grob sinnlich'); this fact `hinders us in accepting in such a naive manner a ``washed-out model'' as a picture of reality,' SchroÈ dinger said in conclusion of §5 of his essay.
As the lesson to be derived, SchroÈ dinger opened §6 on `The Conscious Change of the Epistemological Point of View;' one could adopt the ruling dogma of the quantum theorists, namely:
One tells us that no di¨erence has to be made between the real state of an object of nature and what I know about it, or better, what I can learn to know about it with [all] e¨orts. One says that only perception, observation, measurement are actually real. Thus, once I have obtained at a given instant the best possible knowledge about the state of the physical object that can be achieved according to the laws of nature, I may refute any question about the ``real state'' which goes further as lacking in sense (gegenstandslos), if I am convinced that no additional observation can enlarge my knowledgeÐat least not without reducing it by the same amount (namely, by changing the state). (SchroÈ dinger, loc. cit., p. 823)

746 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
Consequently, only observations had to be considered real, and all our physical cognition was based on measurements that might be performed in principle, or it was the theory which determined where nature posed the `ignorabimus limit'Ði.e., the limit beyond which we can never proceed to know. However, SchroÈ dinger did not like this limitation really and therefore went on to analyze the quantummechanical situation and to suggest ways out of the ruling dogma.
The ™-function, he argued in §7, acts as `a catalogue of expectation.' Quantum mechanics told us that, although its time-evolution occurs by a partial di¨erential equation according to the `classical causal model,' any measurement causes `a peculiar, rather sudden change,' `a break with the naive realism' and the causal law. Consequently, a quantum-mechanical theory of measurement (§8) arose stating: `A variable possesses in general no determined value before I measure it,' or `the [act of ] measuring does not mean to determine the value which it possesses,' but rather:
An interaction between two systems [called ] measured object and measuring device, achieved on a given plan, is called measurement of the ®rst system if the value of a directly perceptible variable property of the second system (a pointer position) in an immediate repetition of the process (with the same measured object which should not be a¨ected meanwhile by other in¯uences) will always be reproduced within certain limits of error. (SchroÈ dinger, loc. cit., p. 824)
Therefore, in quantum mechanics, one had to distinguish between two types of statistics, the error statistics of the measurement and the theoretically predicted statistics.
The ™-function evidently described the state of a system insofar as `di¨erent ™functions denote di¨erent states' and `the same ™-function describes the same state of the system,' SchroÈ dinger noted in § 9 (SchroÈ dinger, loc. cit., p. 825). Now, he tried to construct (in §10) a new theory of measurement, based on an `objective description of the interaction between the measured object and the measuring instruments' (SchroÈ dinger, loc. cit., p. 826). He then noted the result (already reported above as the consequence of Heisenberg's unpublished manuscript): `The best possible knowledge of the whole [system] does not necessarily imply the same knowledge about its parts.' Hence, he concluded the `insu½ciency of the ™-function as a substitute of the model' (SchroÈ dinger, loc. cit., p. 827). Instead, the following result was obtained for the observed object: `An organized catalogue of expected data of the object has been split into a conditional disjunction of catalogues of expectation values. (Der Erwartungskatalog des Objektes hat sich in eine konditionale Disjunktion von Erwartungskatalogen aufgespalten.') (SchroÈ dinger, loc. cit.) Finally, SchroÈ dinger concluded that before one inspects the result of the measurement, the discontinuous jump characterizing quantum mechanics occurs: The original ™function then disappears and a new one reappears (connected with the former by a discontinuous change). Actually, the interaction between two systems (or bodies) `correlates (entangles)' the expectation catalogues of data of the individual systems, as he found in §11. Then, in §12, he discussed the EPR case, which he gen-

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 747
eralized in the next section §13 by consideringÐbesides the measurement of momentum and positionÐalso that of the other variables, such as p2 ‡ q2 or p2 ‡ a2q2 (with a an arbitrary positive constant). He arrived at the unsatisfactory conclusion:
But how the numerical values of all these variables of one system are mutually connected, we do not know at all, though the system must possess for each of them a quite de®nite readiness [or acceptability], because we may, if we wish, get to know it [i.e., the numerical value] exactly at the auxiliary [i.e., second] system and always ®nd it substantiated by direct measurement. (SchroÈ dinger, loc. cit., p. 846±847)
Evidently, this situationÐwhere one does not know about the relations between the values of the variablesÐdid not exist in classical mechanics. But quantum mechanics still exhibited another peculiarity, which SchroÈ dinger discussed in §14: The correlations or entanglements of the system are connected with a `sharply de®ned time.' Such a distinction of time, however, seemed to SchroÈ dinger to be quite inconsistent, because `the numerical value (Maûzahl ) of time is like that of every other variable the result of an observation' and one may ask the question why `one is permitted to attribute to the measurement with a clock an exceptional position' (SchroÈ dinger, loc. cit., p. 848). The exceptional role of the time measurement would especially create di½culties with the relativistic formulation of quantum mechanics (§15). He ®nally wrote at the end of his comprehensive analysis:
Perhaps the simple procedure which the nonrelativistic [SchroÈ dinger called it ``unrelative''] theory possesses [for describing the quantum-mechanical correlations] is as yet only a comfortable trick which has however obtained, as we have seen, an immensely large in¯uence on our fundamental view towards nature. (SchroÈ dinger, loc. cit., p. 849)
Although SchroÈ dinger indicated certain hints as to how relativistic quantum mechanics might eventually change the situation again, he was not really able to o¨er a solution of the problem of interpretation; still, he hoped that the situation presented by quantum mechanics would not be the ®nal word in this question.
(e) Reality and the Quantum-Mechanical Description (1935±1936)
The responses in the scienti®c literature following the articles of Einstein, Podolsky, and Rosen (1935), Bohr (1935b), and SchroÈ dinger (1935a) showed mainly that these authors followed, as SchroÈ dinger would say, the usual `Lehrmeinung (dogma).' Thus, Wendell Hinkle Furry of Harvard University, in a `Note on the Quantum-Mechanical Theory of Measurement,' submitted in November 1935, analyzed more general examples than the one treated by EPR with the methods of measurement theory (Furry, 1936a). He put his ®nger on the point where EPR and the orthodox quantum theoristsÐrepresented especially by Heisenberg, von Neu-

748 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
mann, and PauliÐdisagreed: The former assumed the interaction to exist only at an instant of time and then applied the usual probability evaluation (method A); the latter, however, made use of the full quantum-mechanical formalism (method B). By investigating the position and momentum measurement of a heavy atomic particle (say, a proton) by the process of scattering it with a lighter one (say, an electron), Furry concluded that `assumption A is consistent with quantummechanics,' especially:
Both by mathematical arguments and by discussion of a conceptual experiment, we have seen that the assumption that a system when free from mechanical interference necessarily has independently real properties is contradicted by quantum mechanics. This conclusion means that a system and the means used to observe it are to be regarded as related in a more subtle and intimate way than was assumed in classical theory. (Furry, loc. cit., p. 399)
In a later letter, dated 2 March 1936, and published soon afterward also in Physical Review, Furry addressed SchroÈ dinger's examples and his discussion of measurement theory and rejected the latter (Furry, 1936b).849
Thus, among the quantum physicists, Einstein and SchroÈ dinger appeared indeed to be `lone wolves' who defended epistemological views that deviated from those of the community of experts in modern atomic theory. But to what amounted their views which they had expressed in the discussion of their Gedankenexperiments discussed above? Many analyses of the science-theoretical and cognition-theoretical contents have been published since 1935, especially in the decades after 1950 when the subject of quantum-theoretical interpretation would receive renewed interest by the stimulating e¨orts of David Bohm and others.850 The main idea brought into play in 1935 seems to have been what Einstein and SchroÈ dinger called `realism.' Toward the end of his life, Einstein characterized it in a letter as follows:
It is basic for physics that one assumes a real world independently of any act of perception [our italics]. But this we do not know. We take it only as a programme in
849 Henry Margenau of Yale University and Hugh C. Wolfe of the City College of New York arrived at similar conclusions in contributions submitted in November and December 1935 to the Physical Review (Margenau, 1936; Wolfe, 1936). Margenau especially wanted to abolish a postulate usually assumed in quantum mechanics, i.e.: `When a measurement is performed on a physical system, then immediately after the measurement the state of the system is known with certainty.' (Margenau, 1936, p. 241) He claimed that it was unnecessary to assume this. Einstein, to whom Margenau sent a copy of the manuscript, pointed out `that the formalism of quantum mechanics requires inevitably the postulate: ``If a measurement performed upon a system yields a value m, then the same measurement performed immediately afterwards yields again the value m with certainty.'' ' (Margenau, 1958, p. 29) This exchange with Einstein entered into Margenau's later paper (Margenau, 1937), where he distinguished between `state preparation' and `measurement' (see Jammer, 1974, p. 224 ¨.).
850 We shall return to this discussion later in the Epilogue. Here, we just wish to refer to two quite detailed accounts of the problem of which we have made some use below, namely, Max Jammer's book The Philosophy of Quantum Mechanics (1974, Chapter 6, pp. 181±251) and Arthur Fine's book The Shaky Game: Einstein's Realism and the Quantum Theory (1986; second edition, 1996).

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 749
our scienti®c endeavours. This programme is, of course, prescienti®c and our ordinary language is already based on it. (Einstein to M. Laserna, 8 January 1955; quoted in Fine, 1986, p. 95)
Einstein had addressed the connection between physics and the `real world' quite early in the quantum-mechanical discussion, but he placed the ®rst detailed statements on this issue in a paper entitled `Physik und RealitaÈt (Physics and Reality),' which appeared in print in the March 1936 issue of the Journal of the Franklin Institute (Einstein, 1936). In this article, Einstein also amended certain formulations of the EPR-paper and explained the proper meaning of its contents.851
In Einstein's opinion, science was a re®nement of everyday thinking of the `real external world'; while the latter rests exclusively on the sense impressions, science must proceed further in the `setting of a ``real world.'' ' He wrote:
The ®rst step is the formation of the concept of bodily objects of various kinds. . . . The second step is to be found in the fact that, in our thinking (which determines our expectation), we attribute to this concept of the bodily object a signi®cance, which is to a high degree independent of the sense impression which originally gives rise to it. This is what we mean when we attribute to the bodily object ``a real existence.'' The justi®cation of such a setting rests exclusively on the fact that, by means of such concepts and mental relations between them, we are able to orient ourselves in the labyrinth of sense impressions. (Einstein, loc. cit.; English translation, pp. 349±350)
Having established the criterion of a `real external world,' Einstein demanded `its comprehensibility' by assuming the existence of relations between the concepts: such special relations, namely, the theorems expressing `statements about reality' constituted the laws of nature (Einstein, loc. cit., p. 352). `Science concerns the totality of primary concepts, i.e., concepts directly connected with sense experiences, and theorems connecting them,' Einstein continued, and added: `The aim of science is, on the one hand, a comprehension, as complete as possible, of the connection between the sense experiences in their totality, and, on the other hand, the accomplishment of this aim by the use of a minimum of primary concepts.' (Einstein, loc. cit.) He then talked about several stages in the development of science: The `®rst layer' retains the primary concepts and relations; a `secondary system' also involves concepts of the `secondary layer' which are not directly connected with sense experiences, but it is logically more complete, as it possesses a `higher logical unity.' `Thus the story goes on until we have arrived at a system of the greatest conceivable unity, and of the greatest poverty of concepts of the logical foundations, which are still compatible with the observations made by our own senses,' Einstein concluded these general historical comments and stated: `We do not know whether or not this ambition will ever result in a de®nite system.' (Einstein, loc. cit., p. 353) He rather thought that the answer was negative, `however,
851 Max Jammer called this paper `Einstein's credo concerning the philosophy of physics' (Jammer, 1974, p. 230).

750 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

one will never give up the hope that this greatest of all aims can really be attained to a very high degree' (Einstein, loc. cit.).
With this background preparation, Einstein proceeded `to demonstrate what paths the constructive human mind has entered, in order to arrive at a basis of physics which is logically as uni®ed as possible' (Einstein, loc. cit., p. 354). Thus, he ®rst discussed in some detail `mechanics and the attempts to base all physics on it' in §2 of the paper, `the ®eld concept' of electrodynamics (in §3), and the theory of relativity (in §4). In §5, he turned to `quantum theory and the fundamentals of physics,' which he introduced by saying:

The theoretical physicists of our generation are expecting the erection of a new theoretical basis of physics which would make use of fundamental concepts greatly different from those of the ®eld theory considered up to now. The reason is that it has been found necessary to useÐfor the mathematical representation of the so-called quantum phenomenaÐnew sorts of methods of consideration. (Einstein, loc. cit., p. 371)

Einstein then outlined what he considered to be the essence of wave mechanics (emphasizing limiting connections with classical mechanics) and stressed its wide application to `such a heterogeneous group of phenomena of experience' and added:

In spite of this, however, I believe that the theory is up to beguile us into error in our search for a uniform basis for physics, because, in my belief, it is an incomplete representation of real things although it is the only one which can be built out of the fundamental concepts of force and material points (quantum corrections to classical laws). The incompleteness of the representation is the outcome of the statistical nature (incompleteness) of the laws. (Einstein, loc. cit., 374)

Einstein supported his opinion concerning the incomplete representation of quantum theory by asking the particular question whether the ™-function describes `a real condition of a mechanical system' (Einstein, loc. cit.). For that purpose, he selected a periodic system which, according to quantum mechanics, possessed discrete energy states E1, E2, etc. Now, if the system in the lowest state (E1) were perturbed during a ®nite time by a small force, the wave function could be written as

ˆ y ™ ˆ cr™rY
rˆ1

…659†

with jc1j being nearly unity and jc2j, jc3j, etc., very small quantities. But, he argued, that ™ cannot `describe a real condition of the system,' because this should
have an energy exceeding E1 by a small amount; hence, it would lie between E1 and E2, which is excluded by quantum theory. `Our -function . . . represents rather a statistical description in which the cr represent probabilities of the individual

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 751
energy values,' Einstein continued and suggested: `The ™-function does not in any way describe a condition which could be that of a single system; it relates rather to many systems, to ``an ensemble of systems'' in the sense of statistical mechanics.' (Einstein, loc. cit., p. 375) In Einstein's opinion, such an interpretation removed the `paradox recently demonstrated by myself and two collaborators,' but `what happens to a single system remains . . . entirely eliminated from the representation by the statistical manner of consideration' (Einstein, loc. cit., pp. 376±377). `But now I ask,' he continued:
Is there really any physicist who believes that we shall ever get an inside view of these important alterations in the single systems, in their structure and their causal connections, and this regardless of the fact that these single happenings have been brought so close to us, thanks to the inventions of the Wilson [cloud] chamber and the Geiger counter? To believe this is logically possible without contradiction; but it is so very contrary to my scienti®c instinct that I cannot forego to search for a more complete conception. (Einstein, loc. cit.)
Quantum mechanics, he admitted, `has seized hold of a beautiful element of truth,' and he did not doubt `that it will be a test stone for any future theoretical basis.' `However, I do not believe that quantum mechanics will be the starting point in the search for this basis,' as it seemed to Einstein `entirely justi®able seriously to consider the question as to whether the basis of all ®eld physics cannot by any means be put into harmony with the facts of quantum theory' (Einstein, loc. cit., p. 378).852
Like Einstein in America, SchroÈ dinger in England also continued to think about the interpretation of quantum mechanics beyond his essay to the Naturwissenschaften. In two papers, sent in August 1935 and April 1936 (communicated by Max Born and Paul Dirac, respectively) to the Proceedings of the Cambridge Philosophical Society, he investigated the `probability relations between separated systems,' which the EPR-paper had shown to constitute a central point at which the classical and the quantum-mechanical treatments di¨ered (SchroÈ dinger, 1935b; 1936). Indeed, SchroÈ dinger called `the characteristic trait of quantum mechanics, the one that enforces its entire departure from the classical lines of thought,' the following:
When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual in¯uence the systems separate again, and then they cannot any longer be described in the same way as before, viz. by endowing each of them with a representative of its own. (SchroÈ dinger, 1935b, p. 555)
That is, the ™-function describing the two systems became entangled by the interaction such that afterward only an experiment, rather than any previous knowl-
852 The suggestions made by Einstein in the direction of bringing `the basis of ®eld physics into harmony with the facts of quantum theory' in §6 (entitled `Relativity Theory and Corpuscles') did not go beyond some indication of how to obtain a singularity-free representation of electric corpuscles.

752 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

edge of the states can disentangle them; however, this measurement then also exhibited the strange features revealed by the EPR-analysis. In particular, SchroÈ dinger showed that the `paradox' stated by Einstein, Podolsky, and Rosen was `the rule and not the exception' in quantum mechanics by proving a general mathematical theorem: The function g…xY y† representing the state of the composite system after the two subsystems have separated again is not a product of two functions containing only the variables x and y of the individual systems separately. However, if one performs a measurement on the second system yielding the state fn…y†, then g…xY y† becomes

ˆ g…xY y† ˆ cngn…x† fn…y†Y
n

…660†

with the function gk…x† and the probability coe½cient ck determined by the equations

… gkÅx†gk…x† dx ˆ 1

…661†

and

…

ckgk…x† ˆ

f

à k

…

y†g…xY

y†

dyX

…662†

In general, the gk…x† thus obtained will not be orthogonal to each other, but under suitable conditions for the fk, namely, that they satisfy the homogeneous linear integral equation,

… f …y† ˆ l K… yY yH† f … yH† dyHY

…663†

with the eigenvalue l and the Hermitean kernel K…yY yH†,
… K… yY yH† ˆ gÅxY yH†g…xY y† dxY

…663a†

they will be orthogonal. The `biorthogonal development' of g…xY y† due to Eq. (660) then provided
SchroÈ dinger the `true insight' into the di½cult problem of quantum-mechanical entanglement, as he found:
If there are no coincidences among the jckj2 (excluding also the case that more than one of them vanish) the relevant fk's form a well determined and complete set and so do the gk's. Then one can say that the entanglement consists in that one and only one variable (or set of commuting variables) of one system is uniquely determined by a

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 753

de®nite observable (or set of observables) of the other system. This is the general case. We shall now turn to the opposite extreme, which is the Einstein-Podolsky-Rosen case. It could be characterized by all jckj2 being equal and all possible developments being biorthogonal. Every observable (or set, etc.) of one system is determined by an observable (or set, etc.) of the other one. (SchroÈ dinger, loc. cit., p. 558)
Since a di½culty arose with normalizing the sum jckj2 to unity in the latter, degenerate case, SchroÈ dinger chose a di¨erent procedure there. He considered two systems, denoted by position and momentum variables x1, p1 and x2, p2, and observed that the following variables x and p of the total system g,

x ˆ x1 À x2 and p ˆ p1 ‡ p2Y

…664†

commuted; hence, they satis®ed

xg ˆ xHg and pg ˆ pHgY

…665†

with eigenvalues xH and pH, respectively. Consequently, the value of the variable x1, namely, x1H , could be deduced from measuring the value x2H of the variable x2 and, similarly, the value p1H from p2H . Further, SchroÈ dinger claimed that more knowledge might be obtained about system 1 from measurements in system 2, say, from the measurement of energy or any other variable. Thus, `the two families of observables, relating to the ®rst and the second system, respectively, are linked by at least one match between two de®nite members, one of either family,' where `the word match is short of stating that the values of the two observables in question determine each other uniquely and therefore (since the actual labelling is irrelevant) can be taken to be equal' (SchroÈ dinger, loc. cit., p. 563).
In his second paper on the probability relations between separated systems, SchroÈ dinger tried to avoid the `match' linking the two families of observables of systems 1 and 2, i.e., the conclusion that `the experimenter even with the indirect method, which avoids touching the system [1] itself, controls its future state in very much the same way as it is well known in the case of direct measurement.' (SchroÈ dinger, 1936, p. 446) This match, which Einstein and he had demonstrated as characterizing the standard quantum mechanics, seemed to him to be the greatest hindrance toward a more satisfactory theory describing what both (he and Einstein) meant by physical reality. To achieve this purpose, SchroÈ dinger now became involved in a detailed discussion of quantum-mechanical mixtures, obtaining the result `that in general a sophisticated experimenter can, by a suitable device which does also involve measuring non-commuting variables, produce a non-vanishing probability of driving the system into any state he chooses, whereas with the ordinary direct measurement at least the states orthogonal to the original ones are excluded' (SchroÈ dinger, loc. cit.). In particular, he described the case of two systems as a special example of a mixture, and after the corresponding calculation was performed, he concluded:

754 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into de®nite constituents by a de®nite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the ®rst system are just realized by all measuring programmes that can be carried out on the second one. In general every state of the ®rst system can be given a ®nite chance by a suitable choice of the programme. (SchroÈ dinger, loc. cit., p. 452)
In fact, SchroÈ dinger hoped to eliminate the experimenter's in¯uence on the state of the system which he does not measure by an additional assumption, notably, by assuming:
that the knowledge of the precise relation between the complex constants ck [occurring in the wave function g…xY y† of the combined systems according to Eq. (660)] has been entirely lost in consequence of the process of separation. This would mean that not only the parts, but the whole system, would be in the situation of a mixture, not a pure state. It would not preclude the possibility of determining the state of the ®rst system by suitable measurements of the second one or vice versa. But it would utterly eliminate the experimenter's in¯uence on the state of that system which he does not to touch. (SchroÈ dinger, loc. cit., p. 451)
SchroÈ dinger agreed that the description thus proposed was `very incomplete,' but he called it `a possible one, until I am told either why it is devoid of meaning or with which experiment it disagrees' (SchroÈ dinger, loc. cit., pp. 451±452). For the moment, he remained convinced that the conclusions `unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying nonrelativistic quantum mechanics beyond its legitimate range' (SchroÈ dinger, loc. cit., p. 452).853
Having reported about the e¨orts of the `conservative' Einstein±SchroÈ dinger camp, let us now shift to the opposite camp and report about the further development of the arguments, especially those of Niels Bohr and Werner Heisenberg. In the analysis of Bohr's reply to the EPR-argument, Mara Beller and Arthur Fine have emphasized the fact that Niels Bohr had turned around the original complaintÐthat quantum mechanics was incomplete because it did not endow the two quantities, position and momentum, with equal realityÐand rather argued that this `de®ciency' spoke in favour of the consistency and theoretical soundness of the new quantum theory; in particular, they claimed that Bohr had overlooked two extra assumptions made by Einstein, Podolsky, and Rosen, namely, ®rst, `that the same ``reality'' pertains to the unmeasured component [i.e., variable] of the twoparticle systems,' and second, the assumption of `a principle of separation according to which, after the two particles are far enough apart, the measurement of particle 1 does not e¨ect the reality that pertains to particle 2' (Beller and Fine,
853 As Jammer has pointed out, the study of Furry (1935a) proceeded along with involving much the same mathematical steps as SchroÈ dinger used in his paper (1936), though he arrived at rather different conclusions.

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 755
1994, p. 8).854 Was Bohr only following a positivistic attitude when he tried to work out ambiguities in the arguments of his opponents and thus hoped to persuade them about his own standpoint, and did he even apply the `improper' assumption and interpretation of the EPR-arguments (as Beller and Fine claimed)? The simplest historically substantiated answer is Yes, to a certain extent Bohr accepted positivistic arguments, as the brief correspondence between him and Philipp Frank (quoted by Beller and Fine, loc. cit., pp. 19±20) showed. But one can also easily notice di¨erences in the opinions between Bohr, the author of complementarity, and Frank, the positivist from Prague, who presented their respective views quite clearly at the `Second International Congress for the Unity of Science (Zweiter Internationaler Kongress fuÈr Einheit der Wissenschaft),' held in Copenhagen from 21 to 26 June 1936, where both spoke on the same day (22 June). Bohr's talk, entitled `KausalitaÈt und KomplementaritaÈt (Causality and Complementarity)' addressed the problem in quite general terms (Bohr, 1937a). He argued that the mathematical formalism of quantum mechanics allowed an unambiguous representation of the experimental facts but did not admit the classical causal representation of the quantum phenomena; rather:
The renunciation of the causal ideal in atomic physics is founded conceptually alone on the fact that we were not able, because of the inevitable interaction between experimental objects and measuring devices . . . anymore to talk about the independent behaviour of a physical object. Finally, an arti®cial word like ``complementarity'' which does not belong to the concepts of daily life and therefore cannot be attributed any visualizable content with the help of the usual concepts, just serves to remind us of the completely new epistemological situation in physics. (Bohr, loc. cit., p. 298)
Frank, on the other hand, spoke explicitly on `Philosophische Deutungen und Miûdeutungen der Quantentheorie (Philosophical Interpretations and Misinterpretations of Quantum Theory),' (Frank, 1937). He especially identi®ed as `the essential misinterpretation' what he called `the passage through the ``real'' metaphysical world (Durchgang durch die ``reale'' metaphysische Welt).' (Frank, loc. cit., p. 306). By analyzing the situation in quantum mechanics, he identi®ed several misinterpretations as arising from the use of classical concepts to describe atomic phenomena and stated:
Quantum mechanics talks neither about particles, whose position and velocity exist but cannot be observed accurately, nor about particles with inde®nite position and velocity, but about measuring devices, in the description of which the phrases ``position of a particle'' and ``velocity of a particle'' cannot be used simultaneously . . . Measuring devices, of which one is described by the expression ``position of a particle'' and the other by ``velocity,'' or more accurately ``momentum,'' are called complementary descriptions.
854 Beller and Fine referred to the fact that Furry (1936a) had noticed these extra assumptions and had answered them properly in Bohr's sense.

756 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
If one sticks to this terminology, one will never run tinto the danger of a metaphysical conception of the physical complementarity. Because here it is evident that nothing has been said about a ``real world,'' neither about its nature, nor about the possibility to recognize it. (Frank, loc. cit., pp. 308±309)
In their lectures at Copenhagen in 1936, Bohr and Frank did not talk explicitly about positivism, although Frank clearly stated a positivistic formulation of the principle of complementarity while Bohr said nothing of that sort. The same cautious use of philosophical doctrines was made in the manuscript communicated by Moritz Schlick on `Quantentheorie und Erkennbarkeit der Natur (Quantum Theory and the Perceptibility of Nature,' which was read after his sudden death at the Congress (Schlick, 1937). Schlick, the founder of the Vienna Circle, especially emphasized in his last contribution the fact that the restriction enforced by the new theory on the classical concepts like `position' and `momentum,' and also on the causal description of natural phenomena, or on the physical description of biological phenomena, would not lead to a limitation in principle of our cognition of nature. `The whole question constitutes a beautiful example of the important fundamental theorem of the consequential empiricism as represented by the Vienna school, namely that nothing in the world cannot be recognized in principle,' he concluded, and added:
There exist, though, many questions which may never be answered because of practical, technical reasons; however, in principle, a problem does not yield a solution in just a single case, namely that it's no problem at all, i.e., one is dealing with a wrongly posed question. The limit of cognition exists where there is nothing which can be grasped. Where quantum theory places a limit on causal experience, it does not mean that the further, still existing laws must remain unknown; it rather means that further laws do not exist and cannot be found, because the quest for them would make no sense. (Schlick, loc. cit., p. 326)
Bohr would return to the problem of reality and completeness of quantum mechanics also in later years, especially in the short address on `The Causal Problem in Atomic Physics,' which he delivered at the Conference of the Institute of Intellectual Cooperation, held from 30 May to 3 June 1938, in Warsaw (Bohr, 1939a). On that occasion, as on previous and later ones, he embedded the topic deeply into his views on complementarity, as he replied to a remark of John von Neumann in the discussion of his talkÐnamely, that these views might be elegantly phrased in the language of formal logic:855
We must also notice that the question of the logical forms which are best adapted to quantum theory is in fact a practical problem, concerned with the choice of the most
855 A logical formulation, di¨erent from the one given by von Neumann, was given in 1936 by Max Strauû of Berlin (1936a, b).

IV.2 The Debate on the Completeness of Quantum Mechanics and Its Description of Reality 757
convenient manner in which to express the new situation that arises in this domain. [Personally he (Bohr) compelled himself ] to keep the logical forms of daily life to which actual experiments were necessarily con®ned. The aim of the idea of complementarity was to allow of keeping the logical forms while procuring the extension necessary for including the new situation relative to the problem of observation in atomic physics. (Bohr, loc. cit., pp. 38±39)
Bohr may have argued that the same applies to any philosophical doctrine, be it positivism or realism: It was ®ne as long as it accounted appropriately for the empirical facts. But Einstein's realism would not do, though he repeated it at various times:
1930: Physics is an attempt at the conceptual construction of a model of the real world, as well as its lawful structure. 1940: Some physicists, among them myself, cannot believe that we must abandon, actually and forever, the idea of direct representation of physical reality, in space and time; or that we must accept the view that events in nature are analogous to a game. 1950: Summing up we may characterize the framework of physical thinking . . . as follows: There exists a physical reality independent of substantiation and perception. It can be completely comprehended by a theoretical construction which describes phenomena in space and time. . . . The laws of nature imply complete causality. . . . Will this credo survive forever? It seems to me that a smile is the best answer. (See Fine, 1986, p. 97, for the selection of quotations from Einstein)
Again and again the `modern' quantum theorists would argue with Einstein about it, and we shall return to some aspects of these discussions in the Epilogue.
In the second half of the 1930's, Heisenberg did not seem to be involved in any arguments with Einstein and SchroÈ dinger on the principles of the physical interpretation of quantum mechanics. In fact, he had to survive far less intellectual than rather serious political attacks on his own person and defend simultaneously all modern theories, not just his own, against some dangerous enemies in Germany. On 13 December 1935, Johannes Stark spoke at the inauguration ceremony of the `Philipp Lenard Institut' at the University of Heidelberg. Stark, the previous pioneer of quantum physics and Nobel laureate of physics in 1919, strongly criticized `the conception and methods of the ``Einstein physics'' which are most widely spread in Germany,' and stated explicitly:
Upon the sensation and advertisement of Einstein's relativity then followed the matrix theory of Heisenberg and the so-called wave mechanics of SchroÈ dinger, one as obscure as the other. (See Menzel, 1936, p. 27)
Stark denounced these theories as `Jewish' or `degenerate' physics, in contrast to what his political ally Lenard called `German' or `Aryan' physics and would de®ne in his programmatic book Deutsche Physik only vaguely opposed to the `peculiar physics of the Jews,' characterized by its `internationalism' and lack of `under-

758 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics
standing of truth' (Lenard, 1936, pp. ix±x). Both, the retired old professor Lenard and the younger, active president of the Physikalisch-Technische Reichsanstalt Stark, continued to conduct a malicious battle especially against Heisenberg. In particular, Stark in¯uenced (and probably wrote himself ) in July 1937 the article ` ``Weiûe Juden'' in der Physik (``White Jews'' in Physics)' in the newspaper Das Schwarze Korps of the powerful Nazi organization SS (Stark, 1937). After calling the already accomplished expulsion of Jewish scientists from Germany just a `partial victory,' the article demanded to attack those scientists who were not racially but intellectually Jews, naming Heisenberg `Statthalter des ``einsteinschen Geistes'' (Keeper of the Einstein Spirit) . . . who, like Jews, had to disappear.' The theories which Stark wished to condemn, he described less polemically in a contribution to the Physikalische Zeitschrift (of which he was then an editor), entitled `Widerspruch zwischen Erfahrung und dogmatischer Atomtheorie (Contradiction between Experience and Dogmatic Atomic Theory)':
If I criticize in the following the dogmatic atomic theory, I wish to refute ®rst of all . . . the accusation that I am hostile toward any theory. On the contrary, I respect greatly the realistic (wirklichkeitgetretene) theories which, like Maxwell's theory, represent the results of experimental research in the exact language of mathematics; similarly the theories which apply experimentally substantiated laws to special cases, as it happens in the elasticity theory and hydrodynamics. I am, however, suspicious toward dogmatic theories which build whole schemes of mathematical formulae upon not proven or arbitrary assumptions, and I reject such dogmatic theories emphatically which contradict experiences or do not represent them completely. The latter situation pertains to modern dogmatic atomic and quantum theories. (Stark, 1938, p. 190)
For a man like Heisenberg who did not wish to ¯ee his native country and rather hoped to continue to do research in and teaching of modern physical theories in Germany, these open political attacks were extremely dangerous. He tried to obtain the assistance of colleagues, and he underwent extremely di½cult interrogation in the main quarter of the frightful political police (the Gestapo) of the Third Reich. These attacks ended when Heinrich Himmler, the chief of the SS, wrote a letter to Heisenberg on 21 July 1938, informing him that he (Himmler) disapproved of the article in Das Schwarze Korps; and on the very same day, Himmler sent another letter to his deputy Reinhard Heydrich stating `that one cannot a¨ord to kill [!] Heisenberg.'856 In this tense situation, Heisenberg could not, and would not, argue anymore with scienti®c opponents like Einstein, Planck, and SchroÈ dinger, about details of the interpretation of quantum mechanics. In Germany, at least, the whole state of modern physical theories was at stake, and Heisenberg was glad to get publishedÐthough with considerable delayÐan article, composed in 1940 and dealing with his positive evaluation of modern
856 For details, see Beyerchen, 1977, Chapter 8, and Rechenberg, 1992.

IV.3 New Elementary Particles in Nuclear and Cosmic-Ray Physics (1929±1937)

759

theoretical physics, in the o½cial journal of the Reichsdozentenbund in which he especially emphasized that:

Relativity and quantum theory have so far been substantiated by experience. Since they are at the moment the only theories which provide precise statements about the most recent ®elds of physicsÐatomic physics, nuclear physics, cosmic-ray physics, etc.Ðthey will also continue to remain as the basis for research in these ®elds, especially so long as a contradiction might show up between these theories and experience. The scienti®c battle against these theories could only be conducted by proving such contradictions experimentally. Philosophical essays do not change anything and tell [us] nothing about facts, hence they do not contribute to the decision about the usefulness of theories. Arguments di¨erent from scienti®c ones or scienti®c methods in a scienti®c con¯ict are not consistent with the dignity of German research. (Heisenberg, 1943a, p. 212)
By then the climate of scienti®c discussion had stabilized to the point that Heisenberg had to be satis®ed to publish these obvious remarks in favour of the `obscure, dogmatic atomic theories.' The whole episode characterized quite clearly the decline of physics due to the actions of the government of the Third Reich.

IV.3 New Elementary Particles in Nuclear and Cosmic-Ray Physics (1929±1937)
(a) Introduction: `Pure Theory' Versus `Experiment and Theory'
In looking back on his life as a physicist, Victor F. Weisskopf complained at the Erice Summer School in 1971 that, when he began his studies in physics at the University of Vienna, he came upon the scene three years too late:
I came to the university in 1926 after quantum mechanics was invented, and, of course, I needed a few years to learn physics. That meant that I could not start active work before 1929±1930, and all the fundamental developments in quantum mechanics were made between 1925 and 1930. . . .
Those fellows [of the previous years] such as [Hans] Bethe, [Rudolf ] Peierls, [Felix] Bloch, and [Walter] Heitler were lucky. Every Ph.D. thesis at that time opened a new ®eld. Peierls worked on heat conduction and opened one part of solid-state physics. Bethe wrote his Ph.D. paper on electron di¨raction of crystals and opened up another part of solid-state physics. [Walter] Heitler and [Fritz] London opened up quantum chemistry, [Gregor] Wentzel the theory of the photoe¨ect. (Weisskopf, 1972, pp. 1 and 4)
And yet, even Weisskopf had quite a satisfactory start as a productive physicist, because he could work on his doctoral thesis beginning in 1928 at the University of GoÈ ttingen as a member of Max Born's famous Institute of Theoretical Physics, with brilliant young scholars like Pascual Jordan, Walter Heitler, Gerhard Herz-

760 Chapter IV The Conceptual Completion and the Extensions of Quantum Mechanics

berg, and Eugene Wigner around him.857 It was Wigner who provided him important guidance in his ®rst steps in research. As Weisskopf recalled:

I was especially interested in the question of radiation damping, the natural width of spectral lines. I dabbled around alone and tried to ®nd exponential solutions to electrodynamics. I did not get far because I was too young and inexperienced. I asked the great Wigner for help. . . . Of course, he helped me right away; together we wrote a paper on the natural width of spectral lines, a paper that contained for the ®rst time a divergent integral. I tried to convince Wigner that the integral could be made to vanish. Wigner said, ``No, no, it is in®nite.'' I didn't believe him, but he was right, of course. This paper, part of which later became my thesis, was the ®rst paper in which the divergent integral appeared. They have not been resolved; they are still there after 40 years. (Weisskopf, loc. cit., p. 4)

Although it would seem that Weisskopf somewhat exaggerated his own situa-

tion, for divergent integrals had already appeared in the Heisenberg±Pauli work

on quantum electrodynamics,858 the two papers which he wrote with Wigner and

submitted on 2 May and 12 August 1930, to Zeitschrift fuÈ r Physik marked quite

a worthy entrance into the ®eld of theoretical physics for a not-yet 22-year-old

student of Max Born (Weisskopf and Wigner, 1930a, b). Weisskopf and Wigner

departed from the previous results (of Paul Ehrenfest and others), describing the

intensity J…n† of radiation (frequency n) emitted by an oscillator of the quantum

frequency nBAH in the vicinity of that eigenfrequency as

J…n†

dn

ˆ

gBAH

4 1 2

2 gA ‡4p…n

À

nBAH

5À1 †2 Y

…666†

857 Victor Weisskopf was born on 19 September 1908, in Vienna, where he received his education in a gymnasium and entered the University of Vienna to study physics for the ®rst two years under the guidance of Hans Thirring. Following Thirring's advice, he left Vienna in 1928 to continue his studies in GoÈ ttingen, where he also attended an inspiring lecture course of Paul Ehrenfest (who, in 1929, substituted for Max Born during his illness). Weisskopf matured in the company of ®ne fellow students, such as Max DelbruÈck, Maria Goeppert-Mayer, and Edward Teller, and graduated in 1931 with a thesis on the line-width of spectral lines (which he completed mostly under the guidance of Eugene Wigner). From GoÈttingen, he went on to Leipzig (to work with Heisenberg, 1931±1932), Berlin (to work with SchroÈdinger, 1932±1933), Kharkov (1933 with Landau), Copenhagen (with Bohr), and Cambridge (with Dirac), being supported in his later studies by a grant from the Rockefeller Foundation. In fall 1933, Wolfgang Pauli in Zurich hired Weisskopf as a successor to his assistant Hendrik Casimir (who had returned to Leyden after the death by suicide of his mentor Paul Ehrenfest). In spring 1936, Weisskopf visited Bohr in Copenhagen again, and there he married Ellen Tvede. Bohr assisted him in obtaining an instructorship at the University of Rochester in 1937, where (in 1940) he was promoted to an assistant professorship. As a U.S. citizen (since 1942), he joined (in 1943) the American (Manhattan) atomic bomb project in Los Alamos (under J. Robert Oppenheimer), where he assisted Hans Bethe in directing the Theory Division. In 1946, after the war, Weisskopf obtained a full professorship at MIT in Cambridge, Massachusetts; his career there was interrupted by several foreign obligations, such as the position of Director General at the European high-energy centre (CERN) at Geneva from 1960 to 1965.
Weisskopf began to work on quantum ®eld theory from the early 1930s; in 1936, he moved on to work in nuclear physics, but in the 1950s, he returned to research on high-energy physics. (For details of Weisskopf 's life and work, see Weisskopf, 1972; Rechenberg, 1978; and von Meyenn, 1985).
858 See Heisenberg and Pauli, 1929, especially, p. 53, and Heisenberg and Pauli, 1930, p. 184.

IV.3 New Elementary Particles in Nuclear and Cosmic-Ray Physics (1929±1937)

761

with

gBAH ,

denoting

the

line-width

in

question,

and

 gA ˆ

€

 gBAH

denoting

the

sum

BH

of all line-widths connected with the state A (i.e., the reciprocal of its lifetime).

Now, they reproduced Eq. (666) primarily with the help of Paul Dirac's radiation

theory (Dirac, 1927b, c); however, in deriving this result, an in®nite integral oc-

curred (see Weisskopf and Wigner, 1930a, Eq. (17), p. 63), which was handled in a

rather handwaving manner in order to obtain a consistent result (see Weisskopf

and Wigner, loc. cit., footnote (*) on pp. 64±65). It was this observation which

Weisskopf later remembered as the ®rst occurrence of an in®nity in quantum

electrodynamics. In the course of the next few years, as we shall report in Section

IV.5, Weisskopf would have to deal with more singularities in quantum ®eld

theory and become a real expert in handling them.

By 1930, relativistic quantum ®eld theory existed in two versions, one by

Heisenberg and Pauli, and the other by Enrico Fermi. Stimulated by Paul

Dirac's papers of 1927, Fermi had begun to write a series of short notes in spring

1929 (Fermi, 1929a, b, c; 1930d; 1931). Moreover, he taught the theory of quantum

electrodynamicsÐdeveloped in these notesÐin his lecture courses to his collabo-

rators in Rome as well as abroad, e.g., in April 1929 at the Institut Henri PrincareÂ

in Paris and at the 1930 Summer School of Theoretical Physics at the University of

Michigan in Ann Arbor, from which an extensive article resulted that was pub-

lished in Reviews of Modern Physics (Fermi, 1932a).858a While the ®rst three notes

suggested a quantum-theoretical reformulation of classical electrodynamics and a

subsequent application to explain interferfence fringes (Fermi, 1929a, b, c), the

fourth one (Fermi, 1930d) pointed out the di¨erence with the meanwhile published

papers of Heisenberg and Pauli (1929, 1930). As Fermi stated in his later review

article:

A general theory of the electromagnetic ®eld was constructed by Heisenberg and Pauli by a method in which the values of the electromagnetic potentials in all the points of space are considered as variables. Independently the writer proposed another method of quantization starting from a Fourier analysis of the potentials. Though Heisenberg and Pauli's method puts in evidence much more clearly the properties of relativistic invariance and is in many respects more general, we prefer to use . . . the method of the writer, which is more simple and more analogous to the method used in the theory of radiation [i.e., by Dirac, 1927b, c]. (Fermi, 1932a, p. 125)

The new method actually consisted of expanding both the scalar potential V and the vector potential U at a given time into a Fourier series [see Fermi, 1929a, Eqs. (3) and (4)], notably,

V

ˆ

r 8‡pc

ˆ
s

Qs



cos

2p—s Á ls

X

‡

 ˜s

…666a†

858a See the introduction to Fermi's papers on quantum electrodynamics by Edoardo Amaldi, in Fermi, 1962a, p. 305.