Quantum Theory at the Crossroads

Quantum Theory at the Crossroads

arXiv:quant-ph/0609184v1 24 Sep 2006

Re onsidering the 1927 Solvay Conferen e

Guido Ba

iagaluppi Antony Valentini

Cambridge University Press

ISBN: 9780521814218

To the memory of James T. Cushing

Contents

Prefa e Abbreviations Typographi onventions Note on the bibliography Permissions and opyright noti es

1

2

page vii xiii xiv xiv xiv

Part IPerspe tives on the 1927 Solvay onferen e Histori al introdu tion

1.1 Ernest Solvay and the Institute of Physi s 1.2 War and international relations 1.3 S ienti planning and ba kground 1.4 Further details of planning 1.5 The Solvay meeting 1.6 The editing of the pro eedings 1.7 Con lusion Ar hival notes

De Broglie's pilot-wave theory

2.1 2.2

2.3

2.4 2.5

Ba kground A new approa h to parti le dynami s: 192324 2.2.1 First papers on pilot-wave theory (1923) 2.2.2 Thesis (1924) 2.2.3 Opti al interferen e fringes: November 1924 Towards a omplete pilot-wave dynami s: 192527 2.3.1 `Stru ture': Journal de Physique, May 1927 2.3.2 Signi an e of de Broglie's `Stru ture' paper 1927 Solvay report: the new dynami s of quanta Signi an e of de Broglie's work from 1923 to 1927 ii

1 3 3 6 10 16 20 22 24 26 30 30 37 38 43 54 57 61 72 75 84

Contents Ar hival notes

iii 88

3

89 3.1 Summary of Born and Heisenberg's report 90 3.2 Writing of the report 93 3.3 Formalism 94 3.3.1 Before matrix me hani s 94 3.3.2 Matrix me hani s 96 3.3.3 Formal extensions of matrix me hani s 100 3.4 Interpretation 102 3.4.1 Matrix me hani s, Born and Wiener 103 3.4.2 Born and Jordan on guiding elds, Bohr on ollisions105 3.4.3 Born's ollision papers 107 3.4.4 Heisenberg on energy u tuations 109 3.4.5 Transformation theory 111 3.4.6 Development of the `statisti al view' in the report115 3.4.7 Justi ation and overall on lusions 119 Ar hival notes 122

4

S hrödinger's wave me hani s

151

5

From matrix me hani s to quantum me hani s

4.1 4.2 4.3 4.4 4.5 4.6

Planning of S hrödinger's report Summary of the report Parti les as wave pa kets The problem of radiation S hrödinger and de Broglie The oni t with matrix me hani s 4.6.1 Early days 4.6.2 From Muni h to Copenhagen 4.6.3 Continuity and dis ontinuity Ar hival notes

123 124 126 128 133 137 139 140 142 146 150

Part IIQuantum foundations and the 1927 Solvay onferen e Quantum theory and the measurement problem 5.1 5.2

153 What is quantum theory? 153 The measurement problem today 155 5.2.1 A fundamental ambiguity 155 5.2.2 Measurement as a physi al pro ess: quantum theory `without observers'157 5.2.3 Quantum osmology 162 5.2.4 The measurement problem in `statisti al' interpretations of ψ 164

iv

6

Contents

Interferen e, superposition, and wave pa ket ollapse 168 6.1

6.2 6.3 6.4

7

8

9

10

11

12

Probability and interferen e 168 6.1.1 Interferen e in de Broglie's pilot-wave theory 169 6.1.2 Interferen e in the `quantum me hani s' of Born and Heisenberg172 Ma ros opi superposition: Born's dis ussion of the loud hamber177 6.2.1 Quantum me hani s without wave pa ket ollapse?178 Dira and Heisenberg: interferen e, state redu tion, and delayed hoi e182 Further remarks on Born and Heisenberg's quantum me hani s190

Lo ality and in ompleteness

194 194 198 202

Time, determinism, and the spa etime framework

204 204 209 213

7.1 7.2 7.3

8.1 8.2 8.3

Einstein's 1927 argument for in ompleteness A pre ursor: Einstein at Salzburg in 1909 More on nonlo ality and relativity Time in quantum theory Determinism and probability Visualisability and the spa etime framework

Guiding elds in 3-spa e

9.1 9.2

218 Einstein's early attempts to formulate a dynami al theory of light quanta218 The failure of energy-momentum onservation 221

S attering and measurement in de Broglie's pilot-wave theory 227

10.1 10.2 10.3 10.4

S attering in pilot-wave theory 228 Elasti and inelasti s attering: Born and Brillouin, Pauli and de Broglie232 Quantum measurement in pilot-wave theory 244 Re oil of a single photon: Kramers and de Broglie 246

Pilot-wave theory in retrospe t

248 250 258 259 265

Beyond the Bohr-Einstein debate

268 269 272

11.1 11.2 11.3 11.4

Histori al mis on eptions Why was de Broglie's theory reje ted? Einstein's alternative pilot-wave theory (May 1927) Obje tions: in 1927 and today

12.1 The standard histori al a

ount 12.2 Towards a histori al revision

Part IIIThe pro eedings of the 1927 Solvay onferen e

277 H. A. Lorentz † Fifth physi s onferen e

279 281

Contents

v

The intensity of X-ray ree tion (W. L. Bragg )

283 The lassi al treatment of X-ray dira tion phenomena 283 History of the use of quantitative methods 285 Results of quantitative analysis 289 292 Interpretation of measurements of F Examples of analysis 295 The me hanism of X-ray s attering 303 The analysis of atomi stru ture by X-ray intensity measurements308 The refra tion of X-rays 312 Referen es 316 Dis ussion of Mr Bragg's report 318 Notes to the translation 327

Disagreements between experiment and the ele tromagneti theory of radiation (A. H. Compton ) Introdu tion The problem of the ether The emission of radiation The photoele tri ee t Phenomena asso iated with the s attering of X-rays Intera tions between radiation and single ele trons Reliability of experimental eviden e Summary Dis ussion of Mr Compton's report Notes to the translation

329 332 333 335 342 348 352 354 356 372

The new dynami s of quanta (L. de Broglie )

374 I.  Prin ipal points of view 374 382 II.  Probable meaning of the ontinuous waves Ψ III.  Experiments showing preliminary dire t eviden e for the new Dynami s of the ele tron390 Bibliography 397 Dis ussion of Mr de Broglie's report 399 Notes to the translation 407

Quantum me hani s (M. Born and W. Heisenberg )

408 Introdu tion 408 I.  The mathemati al methods of quantum me hani s 410 II.  Physi al interpretation 420 III.  Formulation of the prin iples and delimitation of their s ope429 IV.  Appli ations of quantum me hani s 434 Con lusion 437 Bibliography 438 Dis ussion of Messrs Born and Heisenberg's report 442

vi

Contents Notes to the translation

Wave me hani s (E. S hrödinger )

445

Introdu tion I.  Multi-dimensional theory II.  Four-dimensional theory III.  The many-ele tron problem Dis ussion of Mr S hrödinger's report Notes to the translation

448 448 449 458 462 469 475

General dis ussion of the new ideas presented

477 477 498 501 521 525

Causality, determinism, probability Photons Photons and ele trons Notes to the translation Bibliography

Prefa e

And they said one to another: Go to, let us build us a tower, whose top may rea h unto heaven; and let us make us a name. And the Lord said: Go to, let us go down, and there onfound their language, that they may not understand one another's spee h. Genesis 11: 37

Anyone who has taken part in a debate on the interpretation of quantum theory will re ognise how tting is the above quotation from the book of Genesis, a

ording to whi h the builders of the Tower of Babel found that they ould no longer understand one another's spee h. For when it omes to the interpretation of quantum theory, even the most lear-thinking and apable physi ists are often unable to understand ea h other. This state of aairs dates ba k to the genesis of quantum theory itself. In O tober 1927, during the `general dis ussion' that took pla e in Brussels at the end of the fth Solvay onferen e, Paul Ehrenfest wrote the above lines on the bla kboard. As Langevin later remarked, the Solvay meeting in 1927 was the onferen e where `the onfusion of ideas rea hed its peak'. Ehrenfest's per eptive gesture aptured the essen e of a situation that has persisted for three-quarters of a entury. A

ording to widespread histori al folklore, the deep dieren es of opinion among the leading physi ists of the day led to intense debates, whi h were satisfa torily resolved by Bohr and Heisenberg around the time of the 1927 Solvay meeting. But in fa t, at the end of 1927, a signi ant number of the main parti ipants (in parti ular de Broglie, Einstein, and S hrödinger) remained un onvin ed, and the deep dieren es of opinion were never really resolved. The interpretation of quantum theory seems as highly ontroversial vii

viii

Prefa e

today as it was in 1927. There has also been riti ism  on the part of historians as well as physi ists  of the ta ti s used by Bohr and others to propagate their views in the late 1920s, and a realisation that alternative ideas may have been dismissed or unfairly disparaged. For many physi ists, a sense of unease lingers over the whole subje t. Might it be that things are not as lear- ut as Bohr and Heisenberg would have us believe? Might it be that their opponents had something important to say after all? Be ause today there is no longer an established interpretation of quantum me hani s, we feel it is important to go ba k to the sour es and re-evaluate them. In this spirit, we oer the reader a return to a time just before the Copenhagen interpretation was widely a

epted, when the best physi ists of the day gathered to dis uss a range of views, on erning many topi s of interest today (measurement, determinism, nonlo ality, subje tivity, interferen e, and so on), and when three distin t theories  de Broglie's pilot-wave theory, Born and Heisenberg's quantum me hani s, and S hrödinger's wave me hani s  were presented and dis ussed on an equal footing. * Sin e the 1930s, and espe ially sin e the Se ond World War, it has been ommon to dismiss questions about the interpretation of quantum theory as `metaphysi al' or `just philosophi al'. It will be lear from the lively and wide-ranging dis ussions of 1927 that at that time, for the most distinguished physi ists of the day, the issues were de idedly physi al : Is the ele tron a point parti le with a ontinuous traje tory (de Broglie), or a wave pa ket (S hrödinger), or neither (Born and Heisenberg)? Do quantum out omes o

ur when nature makes a hoi e (Dira ), or when an observer de ides to re ord them (Heisenberg)? Is the nonlo ality of quantum theory ompatible with relativity (Einstein)? Can a theory with traje tories a

ount for the re oil of a single photon on a mirror (Kramers, de Broglie)? Is indeterminism a fundamental limitation, or merely the out ome of oarse-graining over something deeper and deterministi (Lorentz)? After 1927, the Copenhagen interpretation be ame rmly established. Rival views were marginalised, in parti ular those represented by de Broglie, S hrödinger and Einstein, even though these s ientists were responsible for many of the major developments in quantum physi s itself. (This marginalisation is apparent in most histori al a

ounts written throughout the twentieth entury.) From the very beginning,

Prefa e

ix

however, there were some notes of aution: for example, when Bohr's landmark paper of 1928 (the English version of his famous Como le ture) was published in Nature, an editorial prefa e expressed dissatisfa tion with the `somewhat vague statisti al des ription' and ended with the hope that this would not be the `last word on the subje t'. And there were a few outstanding alarm bells, in parti ular the famous paper by Einstein, Podolsky and Rosen in 1935, and the important papers by S hrödinger (in the same year) on the at paradox and on entanglement. But on the whole, the questioning eased in all but a few orners. A general opinion arose that the questions had been essentially settled, and that a satisfa tory point of view had been arrived at, prin ipally through the work of Bohr and Heisenberg. For subsequent generations of physi ists, `shut up and al ulate' emerged as the working rule among the vast majority. Despite this atmosphere, the questioning never ompletely died out, and some very signi ant work was published, for example by Bohm in 1952, Everett in 1957, and Bell in 1964 and 1966. But attitudes hanged very slowly. Younger physi ists were strongly dis ouraged from pursuing su h questions. Those who persisted generally had di ult areers, and mu h of the areful thinking about quantum foundations was relegated to departments of philosophy. Nevertheless, the losing de ade of the twentieth entury saw a resurgen e of interest in the foundations of quantum theory. At the time of writing, a range of alternatives (su h as hidden variables, many worlds,

ollapse models, among others) are being a tively pursued, and the Copenhagen interpretation an no longer laim to be the dominant or `orthodox' interpretation. The modern reader familiar with urrent debates and positions in quantum foundations will re ognise many of the standard points of view in the dis ussions reprodu ed here, though expressed with a remarkable

on ision and larity. This provides a wel ome ontrast with the generally poor level of debate today: as the distinguished osmologist Dennis S iama was fond of pointing out, when it omes to the interpretation of quantum theory `the standard of argument suddenly drops to zero'. We hope that the publi ation of this book will ontribute to a revival of sharp and informed debate about the meaning of quantum theory. * Remarkably, the pro eedings of the fth Solvay onferen e have not re eived the attention they deserve, neither from physi ists nor from

x

Prefa e

historians, and the literature ontains numerous major misunderstandings about what took pla e there. The fth Solvay onferen e is usually remembered for the lash that took pla e between Einstein and Bohr over the un ertainty relations. It is remarkable, then, to nd that not a word of these dis ussions appears in the published pro eedings. It is known that Einstein and Bohr engaged in vigorous informal dis ussions, but in the formal debates re orded in the pro eedings they were relatively silent. Bohr did

ontribute to the general dis ussion, but this material was not published. Instead, at Bohr's request, it was repla ed by a translation of the German version of his Como le ture, whi h appeared in Naturwissens haften in 1928. (We do not reprodu e this well-known paper here.) The appending of this translation to the published pro eedings may be the ause of the

ommon misunderstanding that Bohr gave a report at the onferen e: in fa t, he did not. Born and Heisenberg present a number of unfamiliar viewpoints on erning, among other things, the nature of the wave fun tion and the role of time and of probability in quantum theory. Parti ularly surprising is the seeming absen e of a ollapse postulate in their formulation, and the apparently phenomenologi al status of the time-dependent S hrödinger equation. Born and Heisenberg's `quantum me hani s' seems remarkably dierent from quantum me hani s (in the Dira -von Neumann formulation) as we know it today. De Broglie's pilot-wave theory was the subje t of extensive and varied dis ussions. This is rather startling in view of the laim  in Max Jammer's lassi histori al study The Philosophy of Quantum Me hani s  that de Broglie's theory `was hardly dis ussed at all' and that `the only serious rea tion ame from Pauli' (Jammer 1974, pp. 11011). Jammer's view is typi al even today. But in the published pro eedings, at the end of de Broglie's report there are 9 pages of dis ussion devoted to de Broglie's theory, and of the 42 pages of general dis ussion, 15 ontain dis ussion of de Broglie's theory, with serious rea tions and omments

oming not only from Pauli but also from Born, Brillouin, Einstein, Kramers, Lorentz, S hrödinger and others. Even the well-known ex hange between Pauli and de Broglie has been widely misunderstood. Finally, another surprise is that in his report de Broglie proposed the many-body pilot-wave dynami s for a system of parti les, with the total

onguration guided by a wave in onguration spa e, and not just (as is generally believed) the one-body theory in 3-spa e. De Broglie's theory is essentially the same as that developed by Bohm in 1952, the only

Prefa e

xi

dieren e being that de Broglie's dynami s (like the form of pilot-wave theory popularised by Bell) is formulated in terms of velo ity rather than a

eleration. * This work is a translation of and ommentary on the pro eedings of the fth Solvay onferen e of 1927, whi h were published in Fren h in 1928 under the title Éle trons et Photons. We have not attempted to give an exhaustive histori al analysis of the fth Solvay onferen e. Rather, our main aims have been to present the material in a manner a

essible to the general physi ist, and to situate the pro eedings in the ontext of urrent resear h in quantum foundations. We hope that the book will ontribute to stimulating and reviving serious debate about quantum foundations in the wider physi s

ommunity, and that making the pro eedings available in English will en ourage historians and philosophers to re onsider their signi an e. Part I begins with a histori al introdu tion and provides essays on the three main theories presented at the onferen e (pilot-wave theory, quantum me hani s, wave me hani s). The le tures and dis ussions that took pla e at the fth Solvay onferen e ontain an extensive range of material that is relevant to urrent resear h in the foundations of quantum theory. In Part II, after a brief review of the status of quantum foundations today, we summarise what seem to us to be the highlights of the onferen e, from the point of view of urrent debates about the meaning of quantum theory. Part III of the book onsists of translations of the reports, of the dis ussions following them, and of the general dis ussion. Wherever possible, the original (in parti ular English or German) texts have been used. We have ta itly orre ted minor mistakes in pun tuation and spelling, and we have uniformised the style of equations, referen es and footnotes. (Unless otherwise spe ied, all translations of quotations are ours.) Part I (ex ept for hapter 2) and the reports by Compton, by Born and Heisenberg and by S hrödinger, are prin ipally the work of Guido Ba

iagaluppi. Chapter 2, Part II, and the reports by Bragg and by de Broglie and the general dis ussion in Part III, are prin ipally the work of Antony Valentini. Chapters 2, 10 and 11 are based on a seminar, `The early history of Louis de Broglie's pilot-wave dynami s', given by Antony Valentini at the University of Notre Dame in September 1997, at a onferen e in honour of the sixtieth birthday of the late James T. Cushing.

xii

Prefa e *

To James T. Cushing, physi ist, philosopher, historian and gentleman, we both owe a spe ial and heartfelt thanks. It was he who brought us together on this proje t, and to him we are indebted for his en ouragement and, above all, his example. This book is dedi ated to his memory. Guido Ba

iagaluppi wishes to express his thanks to the Humboldt Foundation, whi h supported the bulk of his work in the form of an Alexander von Humboldt Fors hungsstipendium, and to his hosts in Germany, Carsten Held and the Philosophis hes Seminar I, University of Freiburg, and Harald Atmanspa her and the Institut für Grenzgebiete der Psy hologie und Psy hohygiene, Freiburg, as well as to Ja ques Dubu s and the Institut d'Histoire et de Philosophie des S ien es et des Te hniques (CNRS, Paris 1, ENS) for support during the nal phase. He also wishes to thank Didier Devriese of the Université Libre de Bruxelles, who is in harge of the ar hives of the Instituts Internationaux de Physique et de Chimie Solvay, Université Libre de Bruxelles, for his kindness and availability, and Brigitte Parakenings (formerly Uhlemann) and her sta at the Philosophis hes Ar hiv of the University of Konstanz, for the ontinuous assistan e with the Ar hive for the History of Quantum Physi s. Finally, he should wish to thank Je Barrett for suggesting this proje t to him in Utre ht one day ba k in 1996, as well as Mark van Atten, Jennifer Bailey, Olivier Darrigol, Feli ity Pors, Gregor S hiemann and many others for dis ussions, suggestions,

orresponden e, referen es and other help. Antony Valentini began studying these fas inating pro eedings while holding a postdo toral position at the University of Rome `La Sapienza' (199496), and is grateful to Mar ello Cini, Bruno Bertotti and Dennis S iama for their support and en ouragement during that period. For support in re ent years, he is grateful to Perimeter Institute, and wishes to express a spe ial thanks to Howard Burton, Lu ien Hardy and Lee Smolin. We are both grateful to Tamsin van Essen at Cambridge University Press for her support and en ouragement during most of the gestation of this book, and to Augustus College for support during the nal stages of this work. Guido Ba

iagaluppi Antony Valentini

Lake Maggiore, August 2006

Abbreviations

xiii

Abbreviations AEA: Albert Einstein Ar hives, Jewish National and University Library, Hebrew University of Jerusalem. AHQP: Ar hive for the History of Quantum Physi s. AHQP-BSC: Bohr S ienti Corresponden e, mi rolmed from the Niels Bohr Arkiv, Copenhagen. AHQP-BMSS: Bohr S ienti Manus ripts, mi rolmed from the Niels Bohr Arkiv, Copenhagen. AHQP-EHR: Ehrenfest olle tion, mi rolmed from the Rijksmuseum voor de Ges hiedenis van de Natuurwetens happen en van de Geneeskunde `Museum Boerhaave', Leiden. AHQP-LTZ: Lorentz olle tion, mi rolmed from the Algemeen Rijksar hief, Den Haag. AHQP-RDN: Ri hardson Colle tion, mi rolmed from the Harry Ransom Humanities Resear h Center, University of Texas at Austin. AHQP-OHI: Oral history interview trans ripts. IIPCS: Ar hives of the Instituts Internationaux de Physique et de Chimie Solvay, Université Libre de Bruxelles. Ann. d. Phys. or Ann. der Phys.: Annalen der Physik. Bayr. Akad. d. Wiss. Math. phys. Kl.: Sitzungsberi hte der Mathematis h-Physikalis hen Klasse der Königli h-Bayeris hen Akademie der Wissens haften (Mün hen). Berl. Ber.: Sitzungsberi hte der Preussis hen Akademie der Wissens haften (Berlin). A ad. Roy. Belg. or Bull. A . R. Belg. or Bull. A . roy. de Belgique or Bull. A . roy. Belgique or Bull. A . roy. Belg. or Bull. A . R. Belg., Cl. des S ien es: Bulletin de l'A adémie Royale des S ien es, des Lettres et des Beaux-arts de Belgique. Classe des S ien es. Bull. Natl. Res. Coun.: Bulletin of the National Resear h Coun il (U.S.). Comm. Fenn.: Commentationes Physi o-mathemati ae, So ietas S ientiarum Fenni a. C. R. or C. R. A ad. S . or Comptes Rendus A ad. S i. Paris: Comptes Rendus Hebdomadaires des Séan es de l'A adémie des S ien es (Paris). Gött. Na hr.: Na hri hten der Akademie der Wissens haften in Göttingen. II, Mathematis h-Physikalis he Klasse. J. de Phys. or Jour. de Phys. or Journ. Physique or Journ. d. Phys.: Journal de Physique (until 1919), then Journal de Physique et le Radium. Jour. Frank. Inst.: Journal of the Franklin Institute.

xiv

Permissions and opyright noti es

Lin ei Rend.: Rendi onti Lin ei. Man hester Memoirs: Man hester Literary and Philosophi al So iety, Memoirs and Pro eedings. Math. Ann. or Mathem. Ann.: Mathematis he Annalen. Naturw. or Naturwiss. or Naturwissens h. or Naturwissens haften: Die Naturwissens haften. Nat. A ad. S i. Pro . or Pro . Nat. A ad. S i. or Pro . Nat. A ad.: Pro eedings of the National A ademy of S ien es (U.S.). Phil. Mag.: Philosophi al Magazine. Phil. Trans. or Phil. Trans. Roy. So .: Philosophi al Transa tions of the Royal So iety of London. Phys. Rev.: Physi al Review. Phys. Zeits. or Phys. Zeits h. or Physik. Zts.: Physikalis he Zeits hrift. Pro . Camb. Phil. So . or Pro . Cambr. Phil. So . or Pro . Cambridge Phil. So .: Pro eedings of the Cambridge Philosophi al So iety. Pro . Phys. So .: Pro eedings of the Physi al So iety of London. Pro . Roy. So . or Roy. So . Pro .: Pro eedings of the Royal So iety of London. Upsala Univ. Årsskr.: Uppsala Universitets Årsskrift. Z. f. Phys. or Zts. f. Phys. or Zeit. f. Phys. or Zeits. f. Phys. or Zeits h. f. Phys. or Zeits hr. f. Phys.: Zeits hrift für Physik.

Typographi onventions The following onventions have been used. Square bra kets [ ℄ denote editorial amendments or (in the translations) original wordings. Curly bra kets { } denote additions (in original types ripts or manus ripts). Angle bra kets < > denote an ellations (in original types ripts or manus ripts).

Note on the bibliography The referen es ited in Parts I and II, and in the endnotes and editorial footnotes to Part III, are listed in our bibliography. The referen es

ited in the original Solvay volume are found in the translation of the pro eedings in Part III.

Permissions and opyright noti es

Part I Perspe tives on the 1927 Solvay onferen e

1 Histori al introdu tion

Quantum re on iliation very [added, deleted℄ unpleasant [deleted℄ tenden y [deleted℄ retrograde [deleted℄ questionable [added, deleted℄ idea [deleted℄ ippant [deleted℄ title leads to misunderstanding. Ehrenfest, on the onferen e plans

1

The onferen e was surely the most interesting s ienti

onferen e I have taken part in so far. 2 Heisenberg, upon re eipt of the onferen e photograph

The early Solvay onferen es were remarkable o

asions, made possible by the generosity of Belgian industrialist Ernest Solvay and, with the ex eption of the rst onferen e in 1912, planned and organised by the indefatigable Hendrik Antoon Lorentz. In this hapter, we shall rst sket h the beginnings of the Solvay onferen es, Lorentz's involvement and the situation in the years leading up to 1927 (se tions 1.1 and 1.2). Then we shall des ribe spe i ally the planning of the fth Solvay onferen e, both in its s ienti aspe ts (se tion 1.3) and in its more pra ti al aspe ts (se tion 1.4). Se tion 1.5 presents the day-by-day progress of the onferen e as far as it an be re onstru ted from the sour es, while se tion 1.6 follows the making of the volume of pro eedings, whi h is the main sour e of original material from the fth Solvay onferen e and forms Part III of this book.

1.1 Ernest Solvay and the Institute of Physi s Ernest Solvay had an extensive re ord of supporting s ienti , edu ational and so ial initiatives, as Lorentz emphasises in a two-page 3

4

Histori al introdu tion

do ument written in September 1914, during the rst months of the rst world war:3 I feel bound to say some words in these days about one of Belgium's noblest

itizens, one of the men whom I admire and honour most highly. Mr Ernest Solvay .... is the founder of one of the most ourishing industries of the world, the soda manufa ture based on the pro ess invented by him and now spread over Belgium, Fran e, England, Germany, Russia and the United States. ....

The fortune won by an a tivity of half a entury has been largely used

by Mr Solvay for the publi benet.

In the rm onvi tion that a better

understanding of the laws of nature and of human so iety will prove one of the most powerful means for promoting the happiness of mankind, he has in many ways and on a large s ale en ouraged and supported s ienti resear h and tea hing.

Part of this a tivity was entred around the proje t of the Cité S ientique, a series of institutes in Brussels founded and endowed by Ernest Solvay and by his brother Alfred Solvay, whi h ulminated in the founding of the Institutes of Physi s and of Chemistry in 1912 and 1913.a This proje t had originally developed through the han e en ounter between Ernest Solvay and Paul Héger, physi ian and professor of physiology at the Université Libre de Bruxelles (ULB), and involved a ollaboration between Solvay, the ULB and the ity of Brussels. In June 1892, it was agreed that Solvay would onstru t and equip two Institutes of Physiology on land owned by the ity in the Par Léopold in Brussels.b There soon followed in 189394 an Institute for Hygiene, Ba teriology and Therapy, funded mainly by Alfred Solvay, and a S hool of Politi al and So ial S ien es, founded by Ernest Solvay in 1894, whi h moved to the Cité S ientique in 1901, and to whi h a S hool of Commer e was added in 1904. The idea for what be ame known as the rst Solvay onferen e in physi s goes ba k to Wilhelm Nernst and Max Plan k,c who around 1910

onsidered that the urrent problems in the theory of radiation and in the theory of spe i heats had be ome so serious that an international meeting (indeed a ` oun il') should be onvened in order to attempt to resolve the situation. The further en ounter between Nernst and Solvay provided the material opportunity for the meeting, and by July 1910, a The following material on the Cité S ientique is drawn mainly from Despy-Meyer and Devriese (1997). b One was to be ome property of the ity and given in use to the ULB, while the other was to be leased for thirty years to and run by Solvay himself.

In the rest of this and in part of the following se tions, we draw on an unpublished

ompilation of the ontents of the Solvay ar hives by J. Pelseneer.4

1.1 Ernest Solvay and the Institute of Physi s

5

Nernst was sending Solvay the detailed proposals. He had also se ured the ollaboration of Lorentz (who was eventually asked to preside), of Knudsen and naturally of Plan k, who wrote: .... anything that may happen in this dire tion will ex ite my greatest interest and .... I promise already my parti ipation in any su h endeavour. For I an say without exaggeration that in fa t for the past 10 years nothing in physi s has so ontinuously stimulated, ex ited and irritated me as mu h as these a quanta of a tion.

Lorentz set up a ommittee to onsider questions relating to the new experimental resear h that had been deemed ne essary during the

onferen e (whi h took pla e between 30 O tober and 3 November 1911). This ommittee in luded Marie Curie, Brillouin, Warburg, Kamerlingh Onnes, Nernst, Rutherford and Knudsen. Lorentz in turn was asked to be the president. Further, at the end of the onferen e, Solvay proposed to Lorentz the idea of a s ienti foundation. Lorentz's reply to Solvay's proposals, of 4 January 1912, in ludes extremely detailed suggestions on the fun tions and stru ture of the foundation, all of whi h were put into pra ti e and whi h an be summarised as follows.6 The foundation would be devoted prin ipally to physi s and physi al

hemistry, as well as to questions relating to physi s from other s ien es. It would provide international support to resear hers (`a Rutherford, a Lenard, a Weiss') in the form of money or loan of s ienti instruments, and it would provide s holarships for young Belgian s ientists (both men and women) to work in the best laboratories or universities, mostly abroad. The question of a link between the foundation and the `Conseil de physique' was left open, but Lorentz suggested to provide meeting fa ilities if Solvay wished to link the two. Lorentz suggested instituting an administrative board ( onsisting of a Solvay family member or appointee, an appointee of the King, and a member of the Belgian s ienti establishment) and a s ienti ommittee (whi h ould initially be the one he had formed during the rst Solvay onferen e). Finally, Lorentz suggested housing the foundation in an annex of one of the existing institutes in the Cité Universitaire. During January, Solvay sent Paul Héger to Leiden to work with Lorentz on the statutes of the foundation, whi h Lorentz sent to Solvay on 2 February. Solvay approved them with hardly any modi ations (only su h as were required by the Belgian legislation of the time). The a Exquisite ending in the original: `.... dass mi h seit 10 Jahren im Grunde ni hts in der Physik so ununterbro hen an-, er-, und aufregt wie diese Wirkungsquanten'.5

6

Histori al introdu tion

foundation, or rather the `Solvay International Institute of Physi s', was o ially established on 1 May 1912, whi h predates by several years the establishment of the omparable Belgian state institutions (Fondation Universitaire: 1920; Fonds National de la Re her he S ientique: 1928). In this onne tion, Lorentz hoped `that governments would understand more and more the importan e of s ienti resear h and that in the long run one will arrive at a satisfa tory organisation, independent of the individual eorts of private persons',7 a sentiment e hoed by Solvay himself.a The institute, whi h Solvay had endowed for thirty years, ould soon boast of remarkable a tivity in supporting s ienti resear h. The numerous re ipients of subsidies granted during the rst two years until the rst world war in luded Lebedew's laboratory, von Laue, Sommerfeld, Fran k and Hertz, W. L. Bragg (who was later to be ome president of the s ienti ommittee), Stark, and Wien. In 1913, an Institute of Chemistry followed suit, organised along similar lines to the Institute of Physi s.

1.2 War and international relations The rst meeting of the s ienti ommittee, for the planning of the se ond Solvay onferen e, took pla e on 30 September and 1 O tober 1912. The onferen e was held the following year, but the a tivities of the institute were soon disrupted by the start of the rst world war, in parti ular the German invasion of Belgium. Immediate pra ti al disruption in luded the fear of requisitions, the di ulty of ommuni ation between the international membership of the s ienti ommittee and, with regard to the publi ation of the pro eedings of the se ond Solvay onferen e, the impossibility of sending Lorentz the proofs for orre tion and the eventual prospe t of German

ensorship.a The war, however, had longer-term negative impli ations for international intelle tual ooperation. In O tober 1914, a group of 93 representatives of German s ien e and ulture signed the manifesto `An die a `Mr Solvay also thinks that it is the role of the state to subsidise and organise s ienti institutions, and he hopes that in thirty years the state will fulll this duty better than it does today.'8 a The pro eedings of the rst Solvay onferen e had had both a Fren h and a German edition. Those of the se ond Solvay onferen e were printed in three languages in 1915, but never published in this form and later mostly destroyed. Only under the hanged onditions after the war, in 1921, were the pro eedings published in a Fren h translation ( arried out, as on later o

asions, by J.-É. Vers haelt).

1.2 War and international relations

7

Kulturwelt!', denying German responsibilities in the war.a Among the signatories were both Nernst and Plan k. This manifesto was partly responsible for the very strong hostility of Fren h and Belgian s ientists and institutions towards renewal of s ienti relations with Germany after the war. No Germans or Austrians were invited to the third Solvay onferen e of 1921. The only ex eption (whi h remained problemati until the last minute) was Ehrenfest, who was Austrian, but who had remained in Leiden throughout the war as Lorentz's su

essor. Similarly, no Germans parti ipated in the fourth Solvay onferen e of 1924. Fren h and Belgian armies had o

upied the Ruhr in January 1923, and the international situation was parti ularly tense. Einstein had (temporarily) resigned from the League of Nations' Committee on Intelle tual Cooperation, and wrote to Lorentz that he would not parti ipate in the Solvay onferen e be ause of the ex lusion of the German s ientists, and that he should please make sure that no invitation was sent.9 Bohr also de lined to parti ipate in the onferen e apparently be ause of the ontinued ex lusion of German s ientists (Moore 1989, p. 157). S hrödinger, however, who was Austrian and working in Switzerland, was invited.a Einstein had distinguished himself by assuming a pa ist position during the war.b Lorentz was pointing out Einstein's ex eptional ase to Solvay already in January 1919: However, in talking about the Germans, we must not lose sight of the fa t that they ome in all kinds of nuan es. A man like Einstein, the great and profound physi ist, is not `German' in the sense one often atta hes to the word today; his judgement on the events of the past years will not dier at all from 11 yours or mine.

a The main laims of the manifesto were: `.... It is not true that Germany is the

ause of this war. .... It is not true that we have wantonly [freventli h℄ infringed the neutrality of Belgium. .... It is not true that the life and property of a single Belgian itizen has been tou hed by our soldiers, ex ept when utter self-defen e required it. .... It is not true that our troops have raged brutally against Leuven. .... It is not true that our ondu t of war disregards the laws of international right. .... It is not true that the struggle against our so- alled militarism is not a struggle against our ulture ....' (translated from Böhme 1975, pp. 479). a Van Aubel (a member of the s ienti ommittee) obje ted strongly in 1923 to the possibility of Einstein being invited to the fourth Solvay onferen e, and resigned when it was de ided to invite him. It appears he was onvin ed to remain on the

ommittee.10 b For instan e, Einstein was one of only four signatories of the ounter-manifesto `Aufruf an die Europäer' (Ni olai 1917). Note also that Einstein had renoun ed his German itizenship and had be ome a Swiss itizen in 1901, although there was some un ertainty about his itizenship when he was awarded the Nobel prize (Pais 1982, pp. 45 and 5034).

8

Histori al introdu tion

In the meantime, after the treaty of Lo arno of 1925, Germany was going to join the League of Nations, but the details of the negotiations were problemati .a As early as February 1926, one nds mention of the prospe t of renewed in lusion of German s ientists at the Solvay

onferen es.13 In the same month, Kamerlingh Onnes died, and at the next meeting of the s ienti ommittee, in early April (at whi h the fth Solvay onferen e was planned), it was de ided to propose both to invite Einstein to repla e Onnes and to in lude again the German s ientists. On 1 April, Charles Lefébure, then se retary of the administrative

ommission, wrote to ommission members Armand Solvay and Jules Bordet,a enquiring about the admissibility of `moderate gures like Einstein, Plan kb and others'16 (Bordet telegraphed ba k: `Germany will soon be League of Nations therefore no obje tion'17 ). On 2 April, Lorentz himself had a long interview with the King, who gave his approval. Thus, nally, Lorentz wrote to Einstein on 6 April, informing him of the unanimous de ision by the members of the ommittee present at the meeting,c as well as of the whole administrative ommission, to invite him to su

eed Kamerlingh Onnes. The Solvay onferen es were to readmit Germans, and if Einstein were a member of the ommittee, Lorentz hoped this would en ourage the German s ientists to a

ept the invitation.18 Einstein was favourably impressed by the positive Belgian attitude and glad to a

ept under the altered onditions.19 Lorentz pro eeded to invite the German s ientists, `not be ause there should be su h a great haste in the thing, rather to show the Germans as soon a Lorentz to Einstein on 14 Mar h 1926: `Things are bad with the League of Nations; if only one ould yet nd a way out until the day after tomorrow'.12 Negotiations provisionally broke down on 17 Mar h, but Germany eventually joined the League in September 1926. a Lefébure was the appointee of the Solvay family to the administrative ommission, and as su h su

eeded Eugène Tassel, who had died in O tober 1922 and had been a long-standing ollaborator of Ernest Solvay sin e 1886. Armand Solvay was the son of Ernest Solvay, who had died on 26 May 1922. Bordet was the royal appointee to the ommission, and had just been appointed in February 1926, following the death of Paul Héger.14 b A

ording to Lorentz, Plan k had always been helpful to him when he had tried to intervene with the German authorities during the war. Further, Plan k had somewhat qualied his position with regard to the Kulturwelt manifesto in an open letter, whi h he asked Lorentz to publish in the Dut h newspapers in 1916. On the other hand, he expli itly ruled out a publi disavowal of the manifesto in De ember 1923.15

Listed as Marie Curie, Langevin, Ri hardson, Guye and Knudsen (with two members absent, W. H. Bragg and Van Aubel).

1.2 War and international relations

9

as possible our good will',20 and sent the informal invitations to Born, Heisenberg and Plan k (as well as to Bohr) in or around June 1926.21 As late as O tober 1927, however, the issue was still a sensitive one. Van Aubel (who had not been present at the April 1926 meeting of the s ienti ommittee) replied in the negative to the o ial invitation to the onferen e.a Furthermore, it was proposed to release the list of parti ipants to the press only after the onferen e to avoid publi demonstrations. Lorentz travelled in person to Brussels on 17 O tober to dis uss the matter.23 Lorentz's own position during and immediately after the war, as a physi ist from one of the neutral ountries, had possibly been rather deli ate. In the text on Ernest Solvay from whi h we have quoted at the beginning of this hapter, for instan e, he appears to be defending the impartiality of the poli ies of the Institute of Physi s in the years leading up to the war. Lorentz started working for some form of re on iliation as soon as the war was over, writing as follows to Solvay in January 1919: All things onsidered, I think I must propose to you not to ex lude formally the Germans, that is, not to lose the door on them forever. I hope that it may be open for a new generation, and even that maybe, in the ourse of the years, one may admit those of today's s holars who one an believe regret sin erely and honestly the events that have taken pla e. Thus German s ien e will be 24 able to regain the pla e that, despite everything, it deserves for its past.

It should be noted that Lorentz was not only the s ienti organiser of the Solvay institute and the Solvay onferen es, but also a prime mover behind eorts towards international intelle tual ooperation, through his heavy involvement with the Conseil International de Re her hes, as well as with the League of Nations' Committee on Intelle tual Cooperation, of whi h he was a member from 1923 and president from 1925.b Lorentz's gure and ontributions to the Solvay onferen es are movingly re alled by Marie Curie in her obituary of Lorentz in the pro eedings of the fth Solvay onferen e (whi h opens Part III of this volume). a Lefébure's omment was: `be ause there are Germans! Then why does he stay in the Institute of Physi s?'22 b The Conseil International de Re her hes (founded in 1919) has today be ome the International Coun il for S ien e (ICSU). The Committee on Intelle tual Cooperation (founded in 1922) and the related International Institute of Intelle tual Cooperation (inaugurated in Paris in 1926) were the forerunners of UNESCO.25

10

Histori al introdu tion

1.3 S ienti planning and ba kground What was at issue in the remark that heads this hapter,a s ribbled by Ehrenfest in the margin of a letter from Lorentz, was the proposed topi for the fth Solvay onferen e, namely `the oni t and the possible re on iliation between the lassi al theories and the theory of quanta'.26 Ehrenfest found the phrasing obje tionable in that it en ouraged one to `swindle away the fruitful and suggestive harshness of the oni t by most slimy un lear thinking, quite in analogy with what happened also even after 1900 with the me hani al ether theories of the Maxwell equations', pointing out that `Bohr feels even more strongly than me against this slogan [S hlagwort℄, pre isely be ause he takes it so parti ularly to heart to nd the foundations of the future theory'.27 Lorentz took Ehrenfest's suggestion into a

ount, and dropped the referen e to re on iliation both from the title and from later des riptions of the fo us of the meeting.a The meeting of the s ienti ommittee for the planning of the fth Solvay onferen e took pla e in Brussels on 1 and 2 April 1926. Lorentz reported a few days later to Einstein: As the topi for 1927 we have hosen `The quantum theory and the lassi al theories of radiation', and we hope to have the following reports or le tures:

1 2 3

W. L. Bragg. New tests of the lassi al theory. A. H. Compton. Compton ee t and its onsequen es. C. T. R. Wilson. Observations on photoele trons and ollision ele trons by

the ondensation method.

4 5

L. de Broglie. Interferen e and light quanta. (short note):

Kramers.

Theory of Slater-Bohr-Kramers and analogous

theories.

6

Einstein. New derivations of Plan k's law and appli ations of statisti s to

quanta.

7

Heisenberg. 29 theory.

Adaptation of the foundations of dynami s to the quantum

Another report, by the ommittee's se retary Vers haelt,30 adds, on erning point 5 : `(at least, if Mr Kramers judges that it is still useful)'; a In the original: `Quantenverzoening > }> [?℄ titel wekt misverstand'. Many thanks to Mark van Atten for help with this passage. a To Bohr in June 1926: `.... the oni t between the lassi al theories and the quantum theory .... '; to S hrödinger in January 1927: ` .... the ontrast between the urrent and the earlier on eptions [Auassungen℄ and the attempts at development of a new me hani s'.28

1.3 S ienti planning and ba kground

11

it further lists a few alternative speakers: Compton or Debye for 2 , Einstein or Ehrenfest for 6 , and Heisenberg or S hrödinger for 7.a Thus, the fth Solvay onferen e, as originally planned, was to fo us mainly on the theory of radiation and on light quanta, in luding only one report on the new quantum theory of matter. The shift in fo us between 1926 and 1927 was learly due to major theoreti al advan es (for example by S hrödinger and Dira ) and new experimental results (su h as the Davisson-Germer experiments), and it an be partly followed as the planning of the onferen e progressed. S hrödinger's wave me hani s was one of the major theoreti al developments of the year 1926. Einstein, who had been alerted to S hrödinger's rst paper by Plan k ( f. Przibram 1967, p. 23), suggested to Lorentz that S hrödinger should talk at the onferen e instead of himself, on the basis of his new `theory of quantum states', whi h he des ribed as a development of genius of de Broglie's ideas.31 While it is un lear whether Lorentz knew of S hrödinger's papers by the time of the April meeting,a S hrödinger was listed a week later as a possible substitute for Heisenberg, and Lorentz himself was assuring Einstein at the end of April that S hrödinger was already being onsidered, spe ially as a substitute for the report on the new foundations of dynami s rather than for the report on quantum statisti s.b Lorentz losely followed the development of wave me hani s, indeed

ontributing some essential ritique in his orresponden e with S hrödinger from this period, for the most part translated in Przibram (1967) (see hapter 4, espe ially se tions 4.3 and 4.4, for some more details on this orresponden e). Lorentz also gave a number of olloquia and le tures on wave me hani s (and on matrix me hani s) in the period leading up to the Solvay onferen e, in Leiden, Itha a and Pasadena.32 In Pasadena he also had the opportunity of dis ussing with S hrödinger the possibility that S hrödinger may also give a report at the onferen e, as in fa t he did.c S hrödinger's wave me hani s had also made a great impression on Einstein, although he repeatedly expressed his unease to Lorentz at the use of wave fun tions on onguration spa e (`obs ure',34 `harsh',35 a `Mysterium'36 ), and again during the general dis ussion (p. 488). a For details of the other parti ipants, see the next se tion. a Cf. se tion 4.1. b Note that S hrödinger (1926a) had written on `Einstein's gas theory' in a paper that is an immediate pre ursor to his series of papers on quantisation.

Lorentz was at Cornell from September to De ember 1926, then in Pasadena until Mar h 1927.33 On S hrödinger's Ameri an voyage, see Moore (1989, pp. 23033).

12

Histori al introdu tion

One sees Lorentz's involvement with the re ent developments also in his orresponden e with Ehrenfest. In parti ular, Lorentz appears to have been stru k by Dira 's ontributions to quantum me hani s.a In June 1927, Lorentz invited Dira to spend the following a ademi year in Leiden ,38 and asked Born and Heisenberg to in lude a dis ussion of Dira 's work in their report.39 Finally, in late August, Lorentz de ided that Dira , and also Pauli, ought to be invited to the onferen e, for indeed: Sin e last year, quantum me hani s, whi h will be our topi , has developed with an unexpe ted rapidity, and some physi ists who were formerly in the se ond tier have made extremely notable ontributions.

For this reason I

would be very keen to invite also Mr Dira of Cambridge and Mr Pauli of Copenhagen. .... Their ollaboration would be very useful to us .... I need not

onsult the s ienti ommittee be ause Mr Dira and Mr Pauli were both on 40

a list that we had drawn up last year .... .

Lorentz invited Pauli on 5 September 1927 (Pauli 1979, pp. 4089) and Dira sometime before 13 September 1927.41 On the experimental side, some of the main a hievements of 1927 were the experiments on matter waves. While originally de Broglie was listed to give a report on light quanta, the work he presented was about both light quanta and material parti les (indeed, ele trons and photons!), and Lorentz asked him expli itly to in lude some dis ussion of the re ent experiments speaking in favour of the notion of matter waves, spe i ally dis ussing Elsasser's (1925) proposals, and the experimental work of Dymond (1927) and of Davisson and Germer (1927).42 Thus, in the nal programme of the onferen e, we nd three reports on the foundations of a new me hani s, by de Broglie, Heisenberg (together with Born) and S hrödinger. The talks given by Bragg and Compton, instead, ree t at least in part the initial orientation of the onferen e. Here is how Compton presents the division of labour (p. 329): Professor W. L. Bragg has just dis ussed a whole series of radiation phenomena in whi h the ele tromagneti theory is onrmed.

....

I have been left the

task of pleading the opposing ause to that of the ele tromagneti theory of radiation, seen from the experimental viewpoint.

Bragg fo usses in parti ular on the te hnique of X-ray analysis, as the `most dire t way of analysing atomi and mole ular stru ture' (p. 284), a This orresponden e in ludes for instan e a 15-page ommentary by Lorentz on Dira (1927a).37

1.3 S ienti planning and ba kground

13

the development of whi h, as he had mentioned to Lorentz, was the `line in whi h [he had℄ been espe ially interested'.43 This in ludes in parti ular the investigation of the ele troni harge distribution. At Lorentz's request, he had also in luded a dis ussion of the refra tion of X-rays (se tion 8 of his report), whi h is dire tly relevant to the dis ussion after Compton's report.44 As des ribed by Lorentz in June 1927, Bragg was to report `on phenomena that still somehow allow a

lassi al des ription'.45 A few more aspe ts of Bragg's report are of immediate relevan e for the rest of the onferen e, espe ially to the dis ussion of S hrödinger's interpretation of the wave fun tion in terms of an ele tri harge density (pp. 307, 312, se tion 4.4), and so are some of the issues taken up further in the dis ussion (Hartree approximation, problems with waves in three dimensions), but it is fair to say that the report provides a rather distant ba kground for what followed it. Compton's talk overs the topi s of points 2 and 3 listed above. The expli it fo us of his report is the three-way omparison between the photon hypothesis, the Bohr-Kramers-Slater (BKS) theory of radiation, and the lassi al theory of radiation. Note, however, that Compton introdu es many of the topi s of later dis ussions. For instan e, he dis usses the problem of how to explain atomi radiation (se tion on `The emission of radiation', p. 333), whi h is inexpli able from the point of view of the lassi al theory, given that the `orbital frequen ies' in the atom do not orrespond to the emission frequen ies. This problem was one of S hrödinger's main on erns and one of the main points of oni t between S hrödinger and, for instan e, Heisenberg (see in parti ular the dis ussion after S hrödinger's report and, below, se tions 4.4 and 4.6). Compton's dis ussion of the photon hypothesis relates to the question of `guiding elds' (pp. 331 and 354) and of the lo alisation of parti les or energy quanta within a wave (pp. 339 and 348). These in turn are losely

onne ted with some of de Broglie's and Einstein's ideas (see below

hapter 7, espe ially se tion 7.2, and hapter 9 ), and with de Broglie's report on pilot-wave theory and Einstein's remark about lo ality in the general dis ussion (p. 488). Bohr had been a noted s epti of the photon hypothesis, and in 1924 Bohr, Kramers and Slater had developed a theory that was able to maintain a wave pi ture of radiation, by introdu ing a des ription of the atom based on `virtual os illators' with frequen ies equal to the frequen ies of emission (Bohr, Kramers and Slater 1924a,b).a A stationary state of a As Darrigol (1992, p. 257) emphasises, while the free virtual elds obey the

14

Histori al introdu tion

an atom, say the nth, is asso iated with a state of ex itation of the os illators with frequen ies orresponding to transitions from the energy En . Su h os illators produ e a lassi al radiation eld (a `virtual' one), whi h in turn determines the probabilities for spontaneous emission in the atom, that is, for the emission of energy from the atom and the jump to a stationary state of lower energy. The virtual eld of one atom also intera ts with the virtual os illators in other atoms (whi h in turn produ e se ondary virtual radiation) and inuen es the probabilities for indu ed emission and absorption in the other atoms. While the theory provides a me hanism for radiation onsistent with the pi ture of stationary states ( f. Compton's remarks, p. 335), it violates energy and momentum onservation for single events, in that an emission in one atom is not onne ted dire tly to an absorption in another atom, but only indire tly through the virtual radiation eld. Energy and momentum onservation hold only at a statisti al level. The BKS proposal was short-lived, be ause the Bothe-Geiger and Compton-Simon experiments established the onservation laws for individual pro esses (as explained in detail by Compton in his report, pp. 350 .). Thus, at the time of the planning of the fth Solvay onferen e, the experimental eviden e had ruled out the BKS theory (hen e the above remark: `if Mr Kramers judges that it is still useful').a The short note 5, indeed, dropped out of the programme altogether.b The des ription of the intera tion between matter and radiation, in parti ular the Compton ee t, ontinued to be a problem for Bohr, and

ontributed to the development of his views on wave-parti le dualism and

omplementarity. In his ontribution to the dis ussion after Compton's report (p. 360, the longest of his published ontributions in the Solvay volumea ), Bohr sket hes the motivations behind the BKS theory, the Maxwell equations, i.e. an be onsidered to be lassi al, the virtual os illators and the intera tion between the elds and the os illators are non- lassi al in several respe ts. a Bothe and Geiger had been working on their experiments sin e June 1924 (Bothe and Geiger 1924), and provisional results were being debated by the turn of the year. For two diering views on the signi an e of these results for instan e see Einstein to Lorentz, 16 De ember 1924 (the same letter in whi h he wrote to Lorentz about de Broglie's results)46 and the ex hange of letters between Born and Bohr in January 1925 (Bohr 1984, pp. 3026). By April 1925, Bothe and Geiger had lear- ut results against the BKS theory (Bothe and Geiger 1925a,b; see also the letters between Geiger and Bohr in Bohr 1984, pp. 3524). b On the BKS theory and related matters, see also hapter 3 (espe ially se tions 3.3.1 and 3.4.2), hapter 9, Darrigol (1992, hapter 9), the ex ellent introdu tion by Stolzenburg to Part I of Bohr (1984) and Mehra and Re henberg (1982, se tion V.2). a See below for the fate of his ontribution to the general dis ussion.

1.3 S ienti planning and ba kground

15

on lusions to be drawn from the Bothe-Geiger and Compton-Simon experiments and the further development of his views. Lorentz, in his report of the meeting to Einstein had mentioned `SlaterBohr-Kramers and analogous theories'. This may refer to the further developments (independent of the validity of the BKS theory) that led in parti ular to Kramer's (1924) dispersion theory (and from there towards matrix me hani s), or to Slater's original ideas, whi h were roughly along the lines of guiding elds for the photons (even though the photons were dropped from the nal BKS proposal).a Note that Einstein at this time was also thinking about guiding elds (in three dimensions). Pais (1982, pp. 44041) writes that, a

ording to Wigner, Einstein did not publish these ideas be ause they also led to problems with the onservation laws.b Einstein was asked by Lorentz to ontribute a report on `New derivations of Plan k's law and appli ations of statisti s to quanta' (point 6 ),

learly referring to the work by Bose (1924) on Plan k's law, hampioned by Einstein and applied by him to the theory of the ideal gas (Einstein 1924, 1925a,b). The se ond of these papers is also where Einstein famously endorses de Broglie's idea of matter waves. Einstein thought that his work on the subje t was already too well-known, but he a

epted after Lorentz repeated his invitation.47 On 17 June 1927, however, at about the time when Lorentz was sending detailed requests to the speakers, Einstein informed him in the following terms that he would not, after all, present a report: I re all having ommitted myself to you to give a report on quantum statisti s at the Solvay onferen e. After mu h ree tion ba k and forth, I ome to the

onvi tion that I am not ompetent [to give℄ su h a report in a way that really

orresponds to the state of things. The reason is that I have not been able to parti ipate as intensively in the modern development of quantum theory as would be ne essary for this purpose. This is in part be ause I have on the whole too little re eptive talent for fully following the stormy developments, in part also be ause I do not approve of the purely statisti al way of thinking on whi h the new theories are founded .... Up until now, I kept hoping to be able to ontribute something of value in Brussels; I have now given up that hope. I beg you not to be angry with me be ause of that; I did not take this lightly but tried with all my strength .... (Quoted in Pais 1982, pp. 4312)

Einstein's withdrawal may be related to the following ir umstan es. On a Cf. Slater (1924) and Mehra and Re henberg (1982, pp. 5436). See also Pauli's remark during the dis ussion of de Broglie's report (p. 401). b Cf. Einstein's ontribution to the general dis ussion (p. 486) and the dis ussion below in hapter 9.

16

Histori al introdu tion

5 May 1927, during a meeting of the Prussian A ademy of S ien es in Berlin, Einstein had read a paper on the question: `Does S hrödinger's wave me hani s determine the motion of a system ompletely or only in the sense of statisti s?'48 As dis ussed in detail by Belousek (1996), the paper attempts to dene deterministi parti le motions from S hrödinger's wave fun tions, but was also suddenly withdrawn on 21 May.a The plans for the talks were nalised by Lorentz around June 1927. An extra t from his letter to S hrödinger on the subje t reads as follows: [W℄e hope to have the following reports [Referate℄ (I give them in the order in whi h we might dis uss them): 1. From Mr W. L. Bragg on phenomena that still somehow allow a lassi al des ription (reexion of X-rays by rystals, dira tion and total ree tion of X-rays). 2. From Mr Compton on the ee t dis overed by him and what relates to it. 3. From Mr de Broglie on his theory. I am asking him also to take into a

ount the appli ation of his ideas to free ele trons (Elsasser, quantum me hani s of free ele trons; Dymond, Davisson and Germer, s attering of ele trons). 4. From Dr Heisenberg

or

Prof. Born (the hoi e is left to them) on matrix

me hani s, in luding Dira 's theory. 5. Your report [on wave me hani s℄. Maybe another one or two short ommuni ations [Beri hte℄ on spe ial topi s 50 will be added.

This was, indeed, the nal programme of the onferen e, with Born and Heisenberg de iding to ontribute a joint report.a

1.4 Further details of planning In 192627 the s ienti ommittee and the administrative ommission of the Solvay institute were omposed as follows. a The news of Einstein's ommuni ation prompted an ex hange of letters between Heisenberg and Einstein, of whi h Heisenberg's letters, of 19 May and 10 June, survive.49 The se ond of these is parti ularly interesting, be ause Heisenberg presents in some detail his view of theories that in lude parti le traje tories. Both Einstein's hidden-variables proposal and Heisenberg's rea tion will be des ribed in se tion 11.3. a See se tion 3.2. Note that, as we shall see below, while Bohr ontributed signi antly to the general dis ussion and reported the views he had developed in Como (Bohr 1949, p. 216, 1985, pp. 357), he was unable to prepare an edited version of his omments in time and therefore suggested that a translation of his Como le ture, in the version for Naturwissens haften (Bohr 1928), be in luded in the volume instead. This has given rise to a ommon belief that Bohr gave a report on a par with the other reports, and that the general dis ussion at the onferen e was the dis ussion following it. See for instan e Mehra (1975, p. 152), and Mehra and Re henberg (2000, pp. 246 and 249), who appear further to believe that Bohr did not parti ipate in the o ial dis ussion.

1.4 Further details of planning

17

S ienti ommittee: Lorentz (Leiden) as president, Knudsen (Copenhagen) as se retary, W. H. Bragg (until May 1927, London),a Marie Curie (Paris), Einstein (sin e April 1926, Berlin), Charles-Eugène Guye (Geneva),b Langevin (Paris), Ri hardson (London), Edm. van Aubel (Gent). Administrative ommission: Armand Solvay, Jules Bordet (ULB), Mauri e Bourquin (ULB), Émile Henriot (ULB); the administrative se retary sin e 1922, and thus main orrespondent of Lorentz and others, was Charles Lefébure. The se retary of the meeting was Jules-Émile. Vers haelt (Gent), who had a ted as se retary sin e the third Solvay onferen e.52 The rst provisional list of possible parti ipants (in addition to Ehrenfest) appears in Lorentz's letter to Ehrenfest of 29 Mar h 1926: Einstein, Bohr, Kramers, Born, Heisenberg (Jordan surely more mathemati ian), Pauli, Ladenburg (?), Slater, the young Bragg (be ause of the ` orresponden e' to the lassi al theory that his work has often resulted in), J. J. Thomson, another one or two Englishmen (Darwin?

Fowler?), Léon

Brillouin (do not know whether he has worked on this, he has also already been there a number of times),

Louis

de Broglie (light quanta), one or two

who have on erned themselves with dira tion of X-rays (Bergen Davis?, 53 Compton, Debye, Dira (?)).

Lorentz asked for further suggestions and omments, whi h Ehrenfest sent in a letter dated `Leiden 30 Mar h 1926. Late at night': Langevin, Fowler, Dira , J. Fran[ ℄k (already for the experiments he devised by Hanle on the destru tion of resonan e polarisation through Larmor rotation a and for the work he proposed by Hund on the Ramsauer ee t  undisturbed passage of slow ele trons through atoms and so on), Fermi (for interesting

ontinuation of the experiments by Hanle), Oseen (possibly a wrong attempt at explanation of needle radiation and as sharpwitted riti ), S hrödinger (was perhaps the rst to give quantum interpretation of the Doppler ee t, thus

lose to Compton ee t), Bothe (for Bothe-Geiger experiment on orrelation of Compton quantum and ele tron, whi h destroys Bohr-Slater theory, altogether a ne brain!) (Bothe should be onsidered perhaps

before S hrödinger),

Darwin, Smekal (is indeed a very deserving onnoisseur of quantum nesses, only he writes so

frightfully

mu h).

Léon Brillouin has published something re ently on matrix physi s, but I 54 have not read it yet.

a W. H. Bragg resigned due to over ommitment and was later repla ed by Cabrera (Madrid).51 b In 1909 the university of Geneva onferred on Einstein his rst honorary degree. A

ording to Pais (1982, p. 152), this was probably due to Guye. Coin identally, Ernest Solvay was honoured at the same time. a For more on the spe ial interest of the Ramsauer ee t, see se tion 3.4.2 below.

18

Histori al introdu tion

At the April meeting (as listed in the report by Vers haelt55 ) it was then de ided to invite: Bohr, Kramers, Ehrenfest, two among Born, Heisenberg and Pauli, Plan k, Fowler, W. L. Bragg, C. T. R. Wilson, L. de Broglie, L. Brillouin, Deslandres, Compton, S hrödinger and Debye. Possible substitutes were listed as: M. de Broglie or Thibaud for Bragg, Dira for Brillouin, Fabry for Deslandres, Kapitza for Wilson, Darwin or Dira for Fowler, Bergen Davis for Compton, and Thirring for S hrödinger.a The members of the s ienti ommitee would all take part ex o io, and invitations would be sent to the professors of physi s at ULB, that is, to Pi

ard, Henriot and De Donder57 (the latter apparently somewhat to Lefébure's hagrin, who, just before the

onferen e started, felt obliged to remind Lorentz that De Donder was `a paradoxi al mind, loud [en ombrant℄ and always ready to seize the word, often with great maladroitness'58 ). Both the number of a tual parti ipants and of observers was to be kept limited,59 partly explaining why it was thought that one should invite only two among Born, Heisenberg and Pauli. The hoi e initially fell on Born and Heisenberg (although Fran k was also onsidered as an alternative).60 Eventually, as noted above, Pauli was also in luded, as was Dira .61 Lorentz was also keen to invite Millikan  and possibly Hall , when he heard that Millikan would be in Europe anyway for the Como meeting (Einstein and Ri hardson agreed).62 However, nothing

ame of this plan. When Einstein eventually withdrew as a speaker, he suggested Fermi or Langevin as possible substitutes (Pais 1982, p. 432). For a while it was not lear whether Langevin (who was anyway a member of the s ienti

ommittee) would be able to ome, sin e he was in Argentina over the summer and due to go on to Pasadena from there. Ehrenfest suggested F. Perrin instead, in rather admiring tones. Langevin was needed in Paris in O tober, however, and was able to ome to the onferen e.63 Finally, the week before the onferen e started, Lorentz extended the invitation to Irving Langmuir,64 who would happen to be in Brussels at the time of the onferen e.a Lorentz sent most of the informal invitations around January 1927.65 In May 1927, he sent to Lefébure the list of all the people he had `provisionally invited',66 in luding all the members of the s ienti ommittee and the prospe tive invitees as listed above by Vers haelt (that is, a A few days later, Guye suggested also Auger as a possible substitute for Wilson.56 a To Langmuir we owe a fas inating `home movie' of the onferen e; see the report in the AIP Bulletin of Physi s News, number 724 (2005).

1.4 Further details of planning

19

as yet without Pauli and Dira ). All had already replied and a

epted, ex ept Deslandres (who eventually replied mu h later de lining the invitation67 ). Around early July, Lorentz invited the physi ists from the university,68 and presumably sent a new invitation to W. H. Bragg, who thanked him but de lined.69 Formal letters of onrmation were sent out by Lefébure shortly before the onferen e.70 Around June 1927, Lorentz wrote to the planned speakers inviting them in the name of the s ienti ommittee to ontribute written reports, to rea h him preferably by 1 September. The general guidelines were: to fo us on one's own work, without mathemati al details, but rather so that `the prin iples are highlighted as learly as possible, and the open questions as well as the onne tions [Zusammenhänge℄ and

ontrasts are laried'. The material in the reports did not have to be unpublished, and a bibliography would be wel ome.71 Compton wrote that he would aim to deliver his manus ipt by 20 August, de Broglie easily before the end of August, Bragg, as well as Born and Heisenberg, by 1 September, and S hrödinger presumably only in the se ond half of September.72 (For further details of the orresponden e between some of the authors and Lorentz, see the relevant hapters below.) The written reports were to be sent to all parti ipants in advan e of the onferen e.a De Broglie's, whi h had been written dire tly in Fren h, was sent by Lorentz to the publishers, Gauthier-Villars in Paris, before he left for the Como meeting. They hoped to send 35 proofs to Lorentz by the end of September. In the meantime, Vers haelt and Lorentz's son had the remaining reports mimeographed by the `Holland Typing O e' in Amsterdam, and Vers haelt with the help of a student added in the formulas by hand, managing to mail on time to the parti ipants at least Compton's and Born and Heisenberg's reports, if not all of them.74 Lorentz had further written to all speakers (ex ept Compton) to ask them to bring reprints of their papers.75 Late during planning, a slight problem emerged, namely an unfortunate overlap of the Brussels onferen e with the festivities for the

entenary of Fresnel in Paris, to be o ially opened Thursday 27 O tober. Lorentz informed Lefébure of the lash writing from Naples after the Como meeting: neither the date of the onferen e ould be hanged nor that of the Fresnel elebrations, whi h had been xed by the Fren h President. The problem was ompounded by the fa t that de Broglie had a Mimeographed opies of Bragg's, Born and Heisenberg's and S hrödinger's reports are to be found in the Ri hardson Colle tion, Harry Ransom Humanities Resear h Center, University of Texas at Austin.73

20

Histori al introdu tion

a

epted to give a le ture to the So iété de Physique on the o

asion.a Lorentz suggested the ompromise solution of a general invitation to attend the elebrations. Those who wished to parti ipate ould travel to Paris on 27 O tober, returning to Brussels the next day, when sessions would be resumed in the afternoon. This was the solution that was indeed adopted.77

1.5 The Solvay meeting The fth Solvay onferen e took pla e from 24 to 29 O tober 1927 in Brussels. As on previous o

asions, the parti ipants stayed at the Htel Britannique, where a dinner invitation from Armand Solvay awaited them.78 Other meals were going to be taken at the institute, whi h was housed in the building of the Institute of Physiology in the Par Léopold;

atering for 5055 people had been arranged.79 The parti ipants were guests of the administrative ommission and all travel expenses within Europe were met.80 From the evening of 23 O tober onwards, three seats were reserved in a box at the Théatre de la Monnaie.81 The rst session of the onferen e started at 10:00 on Monday 24 O tober. A tentative re onstru tion of the s hedule of the onferen e is as follows.82 We assume that the talks were given in the order they were des ribed in the plans and printed in the volume, and that the re eption by the university on the Tuesday ontinued throughout the morning. It is lear that the general dis ussion extended over at least two days, from the fa t that Dira in his main ontribution (p. 491) refers expli itly to Bohr's omments of the day before.a • Monday 24 O tober, morning: W. L. Bragg's report, followed by dis ussion. • Monday 24 O tober, afternoon: A. H. Compton's report, followed by dis ussion. a In Lorentz's letter, the date of de Broglie's le ture is mentioned as 28 O tober, but the o ial invitations state that it was Zeeman who le tured then, and de Broglie the next evening, after the end of the Solvay onferen e. A report on de Broglie's le ture, whi h was entitled `Fresnel's ÷uvre and the urrent development of physi s', was published by Guye in the Journal de Genève of 16 and 18 April 1928.76 a In a letter to Vers haelt, Kramers refers to `the general dis ussion of Thursday', but that in fa t was the day of the Fresnel elebrations in Paris. The photograph of Lorentz in luded in the volume, a

ording to the aption, was also taken on that day. Sin e the elebrations opened only at 8:30pm, it is on eivable that there was a rst dis ussion session on Thursday morning. Pelseneer states, however, that sessions were suspended for the whole day.83

1.5 The Solvay meeting

21

• Tuesday 25 O tober, starting 9:00 a.m.: re eption oered by the ULB.

• Tuesday 25 O tober, afternoon: L. de Broglie's report, followed by dis ussion. • Wednesday 26 O tober, morning: M. Born and W. Heisenberg's report, followed by dis ussion. • Wednesday 26 O tober, afternoon: E. S hrödinger's report, followed by dis ussion. • Thursday 27 O tober, all day: travel to Paris and entenary of Fresnel.a • Friday 28 O tober, morning: return to Brussels.

• Friday 28 O tober, afternoon: general dis ussion.

• Saturday 29 O tober, morning: general dis ussion,b followed by lun h with the King and Queen of the Belgians. • Saturday 29 O tober, evening: dinner oered by Armand Solvay.

The languages used were presumably English, German and Fren h. S hrödinger had volunteered to give his talk in English,c while Born had suggested that he and Heisenberg ould provide additional explanations in English (while he thought that neither of them knew Fren h).86 The phrasing used by Born referred to who should `explain orally the ontents of the report', suggesting that the speakers did not present the exa t or full text of the reports as printed. Multipli ity of languages had long been a hara teristi of the Solvay

onferen es. A well-known letter by Ehrenfest87 tells us of `[p℄oor Lorentz as interpreter between the British and the Fren h who were absolutely unable to understand ea h other. Summarising Bohr. And Bohr responding with polite despair' (as quoted in Bohr 1985, p. 38).d On the last day of the onferen e, Ehrenfest went to the bla kboard and evoked the image of the tower of Babel (presumably in a more a Most of the parti ipants at the Solvay onferen e, with the ex eption of Knudsen, Dira , Ehrenfest, Plan k, S hrödinger, Henriot, Pi

ard and Herzen, travelled to Paris to attend the inauguration of the elebrations, in the grand amphithéatre of the Sorbonne.84 b The nal session of the onferen e also in luded a homage to Ernest Solvay's widow.85

Note that S hrödinger was uent in English from hildhood, his mother and aunts being half-English (Moore 1989, hapter 1). d Both W. H. Bragg and Plan k deplored in letters to Lorentz that they were very poor linguists. Indeed, in a letter explaining in more detail why he would not parti ipate in the onferen e, W. H. Bragg wrote: `I nd it impossible to follow the dis ussions even though you so often try to make it easy for us', and Plan k was in doubt about oming, parti ularly be ause of the language di ulties.88

22

Histori al introdu tion

metaphori al sense than the mere multipli ity of spoken languages), writing: And they said one to another: .... Go to, let us build us .... a tower, whose top may rea h unto heaven; and

let us make us a name ....

And the Lord said: ....

Go to, let us go down, and there onfound their language, that they may not understand one another's spee h. (Genesis 11: 37, reported by Pelseneer, his a emphasis )

Informal dis ussions at the onferen e must have been plentiful, but information about them has to be gathered from other sour es. Famously, Einstein and Bohr engaged in dis ussions that were des ribed in detail in later re olle tions by Bohr (1949), and vividly re alled by Ehrenfest within days of the onferen e in the well-known letter quoted above (see also hapter 12). Little known, if at all, is another referen e by Ehrenfest to the dis ussions between Bohr and Einstein, whi h appears to relate more dire tly to the issues raised by Einstein in the general dis ussion: Bohr had given a very pretty argument in a onversation with Einstein, that one ould not hope ever to master many-parti le problems with threedimensional S hrödinger ma hinery. (more or less!!!!!!):

He said something like the following

a wave pa ket an never simultaneously determine EX-

ACTLY the position and the velo ity of a parti le. Thus if one has for instan e TWO parti les, then they annot possibly be represented in

three-dimensional

spa e su h that one an simultaneously represent exa tly their kineti energy and the potential energy OF THEIR INTERACTION. Therefore......... (What omes after this therefore I already annot reprodu e properly.) In the multidimensional representation instead the potential energy of the intera tion appears totally sharp in the relevant oe ients of the wave equation and one 90 does [?℄ not get to see the kineti energy at all.

1.6 The editing of the pro eedings The editing of the pro eedings of the fth Solvay onferen e was largely

arried out by Vers haelt, who reported regularly to Lorentz and to Lefébure. During the last months of 1927 Lorentz was busy writing up the le ture he had given at the Como meeting in September.91 He then died suddenly on 4 February, before the editing work was omplete. The translation of the reports into Fren h was arried out after the

onferen e, ex ept for de Broglie's report whi h, as mentioned, was a This may have been Ehrenfest's own emphasis. Note that, if not ne essarily present at the sessions, Pelseneer had some onne tion with the onferen e, having taken Lorentz's photograph reprodu ed in the pro eedings.89

1.6 The editing of the pro eedings

23

written dire tly in Fren h. From Vers haelt's letters we gather that by 6 January 1928 all the reports had been translated, Bragg's and Compton's had been sent to the publishers, and Born and Heisenberg's and S hrödinger's were to be sent on that day or the next. Several proofs were ba k by the beginning of Mar h.92 Lorentz had envisaged preparing with Vers haelt an edited version of the dis ussions from notes taken during the onferen e, and sending the edited version to the speakers at proof stage.a In fa t, stenographed notes appear to have been taken, typed up and sent to the speakers, who for the most part used them to prepare drafts of their ontributions. From these, Vers haelt then edited the nal version, with some help from Kramers (who spe i ally ompleted two of Lorentz's

ontributions).93 A opy of the galley proofs of the general dis ussion, dated 1 June 1928, survives in the Bohr ar hives in Copenhagen,94 and in ludes some ontributions that appear to have been still largely unedited at that time.b By January, the editing of the dis ussions was pro eeding well, and at the beginning of Mar h it was almost ompleted. Some ontributions, however, were still missing, most notably Bohr's. The notes sent by Vers haelt had many gaps; Bohr wanted Kramers's advi e and help with the dis ussion ontributions, and travelled to Utre ht for this purpose at the beginning of Mar h.95 At the end of Mar h, Kramers sent Vers haelt the edited version of Bohr's ontributions to the dis ussion after Compton's report (pp. 360 and 370) and after Born and Heisenberg's report (p. 444), remarking that these were all of Bohr's ontributions to the dis ussions during the rst three days of the

onferen e.c In ontrast, material on Bohr's ontributions to the general dis ussion survives only in the form of notes in the Bohr ar hives.d (Some notes by Ri hardson also relate to Bohr's ontributions.97 ) A translation of version of the Como le ture for Naturwissens haften (Bohr 1928) was in luded instead, reprinted on a par with the other reports, and a

ompanied by the following footnote (p. 215 of the published pro eedings): a A

ording to D. Devriese, urator of the IIPCS ar hives, the original notes have not survived. b We have reprodu ed some of this material in the endnotes to the general dis ussion.

Note that Bohr also asked some brief questions after S hrödinger's report (p. 469). Kramers further writes that Bohr suggested to `omit the whole nal Born-Heisenberg dis ussion (Nr. 1823) and equally Fowler's remark 9'. Again, thanks to Mark van Atten for help with this letter.96 d This material is not mi rolmed in AHQP. See also Bohr (1985, pp. 357, 100 and 4789).

24

Histori al introdu tion

This arti le, whi h is the translation of a note published very re ently in

Naturwissens haften,

vol. 16, 1928, p. 245, has been added at the author's

request to repla e the exposition of his ideas that he gave in the ourse of the following general dis ussion. It is essentially the reprodu tion of a talk on the

urrent state of quantum theory that was given in Como on 16 September 1927, on the o

asion of the jubilee festivities in honour of Volta.

The last remaining material was sent to Gauthier-Villars sometime in September 1928, and the volume was nally published in early De ember of that year.98

1.7 Con lusion The fth Solvay onferen e was by any standards an important and memorable event. On this point all parti ipants presumably agreed, as shown by numerous letters, su h as Ehrenfest's letter quoted above (reprodu ed in Bohr 1985), Heisenberg's letter to Lefébure at the head of this hapter, or various other letters of thanks addressed to the organisers after the onferen e:99 I would like to take this opportunity of thanking you for your kind hospitality, and telling you how mu h I enjoyed this parti ular Conferen e. I think it has been the most memorable one whi h I have attended for the subje t whi h was dis ussed was of su h vital interest and I learned so mu h. (W. L. Bragg to Armand Solvay, 3 November 1927) It was the most stimulating s ienti meeting I have ever taken part in. (Max Born to Charles Lefébure, 8 November 1927)

Per eptions of the signi an e of the onferen e diered from ea h other, however. In the o ial history, the fth Solvay onferen e went down (perhaps together with the Como meeting) as the o

asion on whi h the interpretational issues were nally laried. This was presumably a genuine sentiment on the part of Bohr, Heisenberg and the other physi ists of the Copenhagen-Göttingen s hool. We nd it expli itly as early as 1929: In relating the development of the quantum theory, one must in parti ular not forget the dis ussions at the Solvay onferen e in Brussels in 1927, haired by Lorentz.

Through the possibility of ex hange [Ausspra he℄ between the

representatives of dierent lines of resear h, this onferen e has ontributed extraordinarily to the lari ation of the physi al foundations of the quantum theory; it forms so to speak the outward ompletion of the quantum theory .... . (Heisenberg 1929, p. 495)

1.7 Con lusion

25

On the other hand, the onferen e was also des ribed (by Langevin) as the one where `the onfusion of ideas rea hed its peak'.100 From a distan e of almost 80 years, the beginnings of a more dispassionate evaluation should be possible. In the following hapters, we shall revisit the fth Solvay onferen e, fo ussing in parti ular on the ba kground and ontributions relating to the three main `lines of resear h' into quantum theory represented there: de Broglie's pilot-wave theory, Born and Heisenberg's quantum me hani s and S hrödinger's wave me hani s.

26

Notes to pp. 39

Ar hival notes 1 Handwritten remark (by Ehrenfest) in the margin of Lorentz to Ehrenfest, 29 Mar h 1926, AHQP-EHR-23 (in Dut h). 2 Heisenberg to Lefébure, 19 De ember [1927℄, IIPCS 2685 (in German). 3 AHQP-LTZ-12, talk X 23, `Ernest Solvay', dated 28 September 1914 (in English). Cf. also the Fren h version of the same, X 10. (The two pages of the latter are in separate pla es on the mi rolm.)

Historique des Instituts Internationaux de Physique et de Chimie Solvay depuis leur fondation jusqu'à la deuxième guerre mondiale,

4 Pelseneer, J.,

[1962℄, 103 pp., AHQP-58, se tion 1 (hereafter referred to simply as `Pelseneer'). 5 Plan k to Nernst, 11 June 1910, Pelseneer, p. 7 (in German). 6 Lorentz to Solvay, 4 January 1912, Pelseneer, pp. 2026 (in Fren h). See also the reply, Solvay to Lorentz, 10 January 1912, AHQP-LTZ-12 (in Fren h). 7 Lorentz to Solvay, 6 Mar h 1912, Pelseneer, p. 27 (in Fren h). 8 Héger to Lorentz, 16 February 1912, AHQP-LTZ-11 (in Fren h). 9 Einstein to Lorentz, 16 August 1923, AHQP-LTZ-7 (in German). 10 Van Aubel to Lorentz, 16 April, 16 May and 19 July 1923, AHQP-LTZ-11 (in Fren h). 11 Lorentz to Solvay, 10 January 1919, Pelseneer, p. 37 (in Fren h). 12 Lorentz to Einstein, 14 Mar h 1926, AHQP-86 (in German). 13 Lefébure to Lorentz, 12 February 1926, AHQP-LTZ-12 (in Fren h). 14 Cf. Lefébure to Lorentz, 12 February 1926, AHQP-LTZ-12 (in Fren h). 15 Letter by Lorentz, 7 January 1919, Pelseneer, pp. 356 (in Fren h), two letters from Plan k to Lorentz, 1915, AHQP-LTZ-12 (in German), Plan k to Lorentz, Mar h 1916, Pelseneer, pp. 345 (in German), and Plan k to Lorentz, 5 De ember 1923, AHQP-LTZ-9 (in German). 16 From Lefébure [possibly a opy for Lorentz℄, 1 April 1926, AHQP-LTZ-12 (in Fren h). Obituary of Tassel,

L'Éventail,

15 O tober 1922,

AHQP-LTZ-13 (in Fren h). 17 Cf. Lefébure to Lorentz, 6 April 1926, AHQP-LTZ-12 (in Fren h). 18 Lorentz to Einstein, 6 April 1926, AHQP-86 (in German). 19 Einstein to Lorentz, 14 April 1926 and 1 May 1927, AHQP-LTZ-11 (in German). 20 Lorentz to Einstein, 28 April 1926, AHQP-86 (in German). 21 Compare the invitation of Lorentz to Bohr, 7 June 1926, AHQP-BSC-13 (in English), and the replies of Bohr to Lorentz, 24 June 1926, AHQP-LTZ-11 (in English), Plan k to Lorentz, 13 June 1926, AHQP-LTZ-8 (in German), Born to Lorentz, 19 June 1926, AHQP-LTZ-11 (in German), and Heisenberg to Lorentz, 4 July 1926, AHQP-LTZ-12 (in German). 22 Van Aubel to Lefébure, 6 O tober 1927, IIPCS 2545 (in Fren h), with Lefébure's handwritten omment. 23 Lefébure to Lorentz, 14 O tober 1927, IIPCS 2534, and 15 O tober 1927, IIPCS 2536, telegramme Lorentz to Lefébure, 15 O tober 1927, IIPCS 2535, and Lefébure to the King, 19 O tober 1927, IIPCS 2622 (all in Fren h). 24 Lorentz to Solvay, 10 January 1919, Pelseneer, p. 37 (in Fren h). 25 There is a large amount of relevant orresponden e in AHQP-LTZ.

Notes to pp. 918

27

26 Lorentz to Ehrenfest, 29 Mar h 1926, AHQP-EHR-23 (in Dut h), with Ehrenfest's handwritten omments. 27 Ehrenfest to Lorentz, 30 Mar h 1926, AHQP-LTZ-11 (in German). 28 Lorentz to Bohr, 7 June 1926, AHQP-BSC-13, se tion 3 (in English), Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 29 Lorentz to Einstein, 6 April 1926, AHQP-86 (in German and Fren h). 30 Vers haelt to Lefébure, 8 April 1926, IIPCS 2573 (in Fren h). 31 Einstein to Lorentz, 12 April 1926, AHQP-LTZ-11 (in German). 32 See for instan e the atalogue of Lorentz's manus ripts in AHQP-LTZ-11. 33 See also Lorentz to S hrödinger, 21 January and 17 June 1927, AHQP-41, se tion 9 (in German). 34 Einstein to Lorentz, 1 May 1926, AHQP-LTZ-11 (in German). 35 Einstein to Lorentz, 22 June 1926, AHQP-LTZ-8 (in German). 36 Einstein to Lorentz, 16 February 1927, AHQP-LTZ-11 (in German). 37 En losed with Lorentz to Ehrenfest, 3 June [1927, erroneously amended to 1925℄, AHQP-EHR-23 (in Dut h). 38 Lorentz to Dira , 9 June 1927, AHQP-LTZ-8 (in English). 39 Cf. Born to Lorentz, 23 June 1927, AHQP-LTZ-11 (in German). 40 Lorentz to Lefébure, 27 August 1927, IIPCS 2532A/B (in Fren h). [There appears to be a further item also numbered 2532.℄ 41 Dira to Lorentz, 13 September 1927, AHQP-LTZ-11 (in English). 42 Cf. de Broglie to Lorentz, 22 June 1927, AHQP-LTZ-11 (in Fren h), and Ehrenfest to Lorentz, 14 June 1927, AHQP-EHR-23 (in Dut h and German). 43 W. L. Bragg to Lorentz, 7 February 1927, AHQP-LTZ-11 (in English). 44 Cf. also W. L. Bragg to Lorentz, 27 June 1927, AHQP-LTZ-11 (in English). 45 Lorentz to S hrödinger, 17 June 1927, AHQP-41, se tion 9 (in German). 46 Einstein to Lorentz, 16 De ember 1924, AHQP-LTZ-7 (in German). 47 Einstein to Lorentz, 12 April 1926, AHQP-LTZ-11, Lorentz to Einstein, 28 April 1926, AHQP-86, and Einstein to Lorentz, 1 May 1926, AHQP-LTZ-11 (all in German). 48 `Bestimmt S hrödingers Wellenme hanik die Bewegung eines Systems vollständig oder nur im Sinne der Statistik?', AEA 2-100.00 (in German), available on-line at http://www.alberteinstein.info/db/ViewDetails.do?Do umentID=34338 . 49 Heisenberg to Einstein, 19 May 1927, AEA 12-173.00, and 10 June 1927, AEA 12-174.00 (both in German). 50 Lorentz to S hrödinger, 17 June 1927, AHQP-41, se tion 9 (in German). 51 W. H. Bragg to Lorentz, 11 May 1927, AHQP-LTZ-11 (in English). 52 Tassel to Vers haelt, 24 February 1921, AHQP-LTZ-13 (in Fren h). 53 Lorentz to Ehrenfest, 29 Mar h 1926, AHQP-EHR-23 (in Dut h). 54 Ehrenfest to Lorentz, 30 Mar h 1926, AHQP-LTZ-11 (in German). 55 Vers haelt to Lefébure, 8 April 1926, IIPCS 2573 (in Fren h). 56 Guye to Lorentz, 14 April 1926, AHQP-LTZ-8 (in Fren h). 57 Cf. for instan e Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 58 Lefébure to Lorentz, 22 O tober 1927, AHQP-LTZ-12 (in Fren h). 59 Lorentz to Einstein, 28 April 1926, AHQP-86 (in German), Lorentz to Lefébure, 9 O tober 1927, IIPCS, 2530A/B (in Fren h).

28

Notes to pp. 1819

60 Lorentz to Einstein, 28 April 1926, AHQP-86 (in German), Einstein to Lorentz, 1 May 1926, AHQP-LTZ-11 (in German). 61 For the latter, f. also Brillouin to Lorentz, 20 August 1927, AHQP-LTZ-11 (in Fren h). 62 Lorentz to Einstein, 30 January 1927, AHQP-86 (in German), Einstein to Lorentz, 16 February 1927, AHQP-LTZ-11 (in German), Ri hardson to Lorentz, 19 February 1927, AHQP-LTZ-12 (in English). 63 Brillouin to Lorentz, 20 August 1927, AHQP-LTZ-11 (in Fren h), Ehrenfest to Lorentz 18 August 1927, AHQP-EHR-23 (in German). 64 Telegramme Lorentz to Lefébure, 19 O tober 1927, IIPCS 2541 (in Fren h), with Lefébure's note: `Oui'. 65 Lorentz to Brillouin, 15 De ember 1926, AHQP-LTZ-12 (in Fren h), Lorentz to Ehrenfest, 18 January 1927, AHQP-EHR-23 (in Dut h), Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German); ompare various other replies to Lorentz's invitation, in AHQP-LTZ-11, AHQP-LTZ-12 and AHQP-LTZ-13: Brillouin, 8 January 1927 (in Fren h), de Broglie, 8 January 1927 (in Fren h), W. L. Bragg, 7 February 1927 (in English), Wilson, 11 February 1927 (in English), Kramers, 14 February 1927 (in German), Debye, 24 February 1927 (in Dut h), Compton, 3 April 1927 (in English) [late be ause he had been away for two months `in the Orient'℄. 66 Lorentz to Lefébure, 21 May 1927, IIPCS 2521A (in Fren h). 67 Deslandres to Lorentz, 19 July 1927, AHQP-LTZ-11 (in Fren h). 68 See the letters of a

eptan e to Lorentz: De Donder, 8 July 1927, AHQP-LTZ-11, Henriot, 10 July 1927, AHQP-LTZ-12, Pi

ard, 2 O tober 1927, AHQP-LTZ-12 (all in Fren h). 69 W. H. Bragg to Lorentz, 12 and 17 July 1927, AHQP-LTZ-11 (in English). 70 Copies in AHQP-LTZ-12 and IIPCS 2543. Various replies: IIPCS 254451, 25536, 2558, 25603. 71 Cf. Lorentz to S hrödinger, 17 June 1927, AHQP-41, se tion 9 (in German). 72 Compare the answers by Bragg, 27 June 1927, and by Compton, 7 July 1927, both AHQP-LTZ-11 (in English), and the detailed ones by de Broglie, 27 June 1927, AHQP-LTZ-11 (in Fren h) and by Born, 23 June 1927, AHQP-LTZ-11 (in German); also S hrödinger to Lorentz, 23 June 1927, AHQP-LTZ-13 (original with S hrödinger's orre tions), and AHQP-41 se tion 9 ( arbon opy) (in German). 73 Mi rolmed in AHQP-RDN, do uments M-0059 (Bragg, atalogued as `unidentied author'), M-0309 (Born and Heisenberg, with seven pages of notes by Ri hardson) and M-1354 (S hrödinger). 74 See in parti ular the already quoted Lorentz to Lefébure, 27 August 1927, IIPCS 2532A/B (in Fren h), as well as Lorentz to Lefébure, 23 September 1927, IIPCS 2523A/B, and 4 O tober 1927, IIPCS 2528 (both in Fren h), Gauthiers-Villars to Lefébure, 16 September 1927, IIPCS 2755 (in Fren h), Vers haelt to Lorentz, 6 O tober 1927, AHQP-LTZ-13 (in Dut h), de Broglie to Lorentz, 29 August 1927, AHQP-LTZ-8, and 11 O tober 1927, AHQP-LTZ-11 (both in Fren h), and Vers haelt to Lefébure, 15 O tober 1927, IIPCS 2756 (in Fren h). 75 Lorentz to Ehrenfest, 13 O tober 1927, AHQP-EHR-23 (in Dut h). See also Born to Lorentz, 11 O tober 1927 AHQP-LTZ-11 (in German), de

Notes to pp. 1924

29

Broglie to Lorentz, 11 O tober 1927, AHQP-LTZ-11 (in Fren h), Plan k to Lefébure, 17 O tober 1927, IIPCS 2558, and Ri hardson to Lefébure, IIPCS 2561. 76 `Une rise dans la physique moderne I & II', IIPCS 275051 (in Fren h). 77 Lorentz to Lefébure, 29 September 1927, IIPCS 2523A/B, Brillouin to Lorentz, 11 O tober 1927, AHQP-LTZ-11, Fondation Solvay to Lefébure, IIPCS 2582. Invitation: IIPCS 2615 and 2619, AHPQ-LTZ-8. Replies: IIPCS 2617 and 2618. (All do uments in Fren h.) 78 IIPCS 2530A/B, 2537. 79 IIPCS 2533, 2586A/B/C/D/E (the proposed menus from the Taverne Royale), 2587A/B. 80 Lorentz to S hrödinger, 21 January 1927, AHQP-41, se tion 9 (in German). 81 IIPCS 2340. 82 For the time and pla e of the rst session, see IIPCS 2523A/B. For the re eption at ULB, see IIPCS 2540 and 2629. There is a seating plan for the lun h with the royal ouple, IIPCS 2627. For the dinner with Armand Solvay, see IIPCS 2533, 2624 and 2625. See also Pelseneer, pp. 4950. 83 See Kramers to Vers haelt, 23 Mar h 1928, AHQP-28 (in Dut h), and Pelseneer, p. 50. 84 IIPCS 2621. 85 See AHQP-LTZ-12 (draft), and presumably IIPCS 2667. 86 S hrödinger to Lorentz, 23 June 1927, AHQP-LTZ-13 and AHQP-41, se tion 9 (in German), Lorentz to S hrödinger, 8 July 1927, AHPQ-41, se tion 9 (in German), Born to Lorentz, 23 June 1927, AHQP-LTZ-11 (in German). 87 Ehrenfest to Goudsmit, Uhlenbe k and Dieke, 3 November 1927, AHQP-61 (in German). 88 W. H. Bragg to Lorentz, 17 July [1927℄, AHQP-LTZ-11 (in English), Plan k to Lorentz, 2 February 1927, AHQP-LTZ-8 (in German). 89 See Pelseneer, pp. 5051. 90 Ehrenfest to Kramers, 6 November 1927, AHQP-9, se tion 10 (in German). The words `bekommt man' [?℄ are very faint. 91 Lorentz to Lefébure, 30 De ember 1927, IIPCS 2670A/B (in Fren h). 92 Vers haelt to Lefébure, 6 January 1928, IIPCS 2609, 2 Mar h 1928, IIPCS 2610 (both in Fren h). 93 Lefébure to Lorentz, [after 27 August 1927℄, IIPCS 2524 (in Fren h), Bohr to Kramers, 17 February 1928, AHQP-BSC-13 (in Danish), Brillouin to Lorentz, 31 De ember 1927, AHQP-LTZ-8 (in Fren h), and Kramers to Vers haelt, 28 Mar h 1928, AHQP-28, se tion 4 (in Dut h). 94 Mi rolmed in AHQP-BMSS-11, se tion 5. 95 Bohr to Kramers, 17 and 27 February 1928, AHQP-BSC-13 (both in Danish). 96 Kramers to Vers haelt, 28 Mar h 1928, AHQP-28, se tion 4 (in Dut h). 97 In luded with the opy of Born and Heisenberg's report in AHQP-RDN, do ument M-0309. 98 Gauthier-Villars to Vers haelt, 6 De ember 1928, IIPCS 2762 (in Fren h), and Vers haelt to Lefébure, 11 [De ember℄ 1928, IIPCS 2761 (in Fren h). 99 IIPCS 2671 (in English), IIPCS 2672 (in German). 100 Quoted in Pelseneer, p. 50.

2 De Broglie's pilot-wave theory

2.1 Ba kground At a time when no single known fa t supported this theory, Louis de Broglie asserted that a stream of ele trons whi h passed through a very small hole in an opaque s reen must exhibit the same phenomena as a light ray under the same onditions. Prof. C. W. Oseen, Chairman of the Nobel Committee for Physi s, presentation spee h, 12 De ember 1929 (Oseen 1999)

In September 1923, Prin e Louis de Brogliea made one of the most astonishing predi tions in the history of theoreti al physi s: that material bodies would exhibit the wave-like phenomena of dira tion and interferen e upon passing through su iently narrow slits. Like Einstein's predi tion of the dee tion of light by the sun, whi h was based on a reinterpretation of gravitational for e in terms of geometry, de Broglie's predi tion of the dee tion of ele tron paths by narrow slits was made on the basis of a fundamental reappraisal of the nature of for es and of dynami s. De Broglie had proposed that Newton's rst law of motion be abandoned, and repla ed by a new postulate, a

ording to whi h a freely moving body follows a traje tory that is orthogonal to the surfa es of equal phase of an asso iated guiding wave. The resulting `de Broglian dynami s'  or pilot-wave theory as de Broglie later alled it  was a new approa h to the theory of motion, as radi al as Einstein's interpretation a The de Broglies had ome to Fran e from Italy in the seventeenth entury, the original name `Broglia' eventually being hanged to de Broglie. On his father's side, de Broglie's an estors in luded dukes, prin es, ambassadors, and marshals of Fran e. Nye (1997) onsiders the oni t between de Broglie's pursuit of s ien e and the expe tations of his aristo rati family. For a biography of de Broglie, see Lo hak (1992).

30

2.1 Ba kground

31

of the traje tories of falling bodies as geodesi s of a urved spa etime, and as far-rea hing in its impli ations. In 1929 de Broglie re eived the Nobel Prize, `for his dis overy of the wave nature of ele trons'. Strangely enough, however, even though de Broglie's predi tion was

onrmed experimentally a few years later, for most of the twentieth

entury single-parti le dira tion and interferen e were routinely ited as eviden e against de Broglie's ideas: even today, some textbooks on quantum me hani s assert that su h interferen e demonstrates that parti le traje tories annot exist in the quantum domain (see se tion 6.1). It is as if the dee tion of light by the sun had ome to be widely regarded as eviden e against Einstein's general theory of relativity. This remarkable misunderstanding illustrates the extent to whi h de Broglie's work in the 1920s has been underestimated, misrepresented, and indeed largely ignored, not only by physi ists but also by historians. De Broglie's PhD thesis of 1924 is of ourse re ognised as a landmark in the history of quantum theory. But what is usually remembered about that thesis is the proposed extension of Einstein's wave-parti le duality from light to matter, with the formulas E = hν and p = h/λ (relating energy and momentum to frequen y and wavelength) being applied to ele trons or `matter waves'. Usually, little attention is paid to the fa t that a entral theme of de Broglie's thesis was the onstru tion of a new form of dynami s, in whi h lassi al (Newtonian or Einsteinian) laws are abandoned, and repla ed by new laws a

ording to whi h parti le velo ities are determined by guiding waves, in a spe i manner that unies the variational prin iples of Maupertuis and Fermat. Nor, indeed, have historians paid mu h attention to de Broglie's later and more

omplete form of pilot-wave dynami s, whi h he arrived at in a paper published in May 1927 in Journal de Physique, and whi h he then presented in O tober 1927 at the fth Solvay onferen e. Unlike the other main ontributors to quantum theory, de Broglie worked in relative isolation, having little onta t with the prin ipal resear h entres in Berlin, Copenhagen, Göttingen, Cambridge and Muni h. While Bohr, Heisenberg, Born, S hrödinger, Pauli and others visited ea h other frequently and orresponded regularly, de Broglie worked essentially alone in Paris.a In Fran e at the time, while pure a In his typed `Replies to Mr Kuhn's questions' in the Ar hive for the History of Quantum Physi s,1 de Broglie writes (p. 7): `Between 1919 and 1928, I worked very mu h in isolation ' (emphasis in the original). Regarding his Ph.D. thesis de Broglie re alled (p. 9): `I worked very mu h alone and almost without any ex hange of ideas'.

32

De Broglie's pilot-wave theory

mathemati s was well represented, there was very little a tivity in theoreti al physi s. In addition, after the rst world war, s ienti relations with Germany and Austria were interrupted.a All this seems to have suited de Broglie's rather solitary temperament. De Broglie's isolation, and the fa t that Fran e was outside the mainstream of theoreti al physi s, may a

ount in part for why so mu h of de Broglie's work went relatively unnoti ed at the time, and has remained largely ignored even to the present day. For some seventy years, the physi s ommunity tended to believe either that `hidden-variables' theories like de Broglie's were impossible, or that su h theories had been disproven experimentally. The situation

hanged onsiderably in the 1990s, with the publi ation of textbooks presenting quantum me hani s in the pilot-wave formulation (Bohm and Hiley 1993; Holland 1993). Pilot-wave theory  as originated by de Broglie in 1927, and elaborated by Bohm 25 years later (Bohm 1952a,b)  is now a

epted as an alternative (if little used) formulation of quantum theory. Fo ussing for simpli ity on the nonrelativisti quantum theory of a system of N (spinless) parti les with 3-ve tor positions xi (i = 1, 2, ..., N ), it is now generally agreed that, with appropriate initial onditions, quantum physi s may be a

ounted for by the deterministi dynami s dened by two dierential equations, the S hrödinger equation N

i~

∂Ψ X ~2 2 − ∇ Ψ+VΨ = ∂t 2mi i i=1

(1)

for a `pilot wave' Ψ(x1 , x2 , ..., xN , t) in onguration spa e, and the de Broglie guidan e equation mi

dxi = ∇i S dt

(2)

for parti le traje tories xi (t), where the phase S(x1 , x2 , ..., xN , t) is lo ally dened by S = ~ Im ln Ψ (so that Ψ = |Ψ| e(i/~)S ). This, as we shall see, is how de Broglie presented his dynami s in 1927. Bohm's presentation of 1952 was somewhat dierent. If one takes the time derivative of (2), then using (1) one obtains Newton's law of motion for a

eleration mi x ¨i = −∇i (V + Q) ,

(3)

a For more details on de Broglie's situation in Fran e at the time, see Mehra and Re henberg (1982a, pp. 57884).

2.1 Ba kground

33

where Q≡−

X ℏ2 ∇2 |Ψ| i 2m |Ψ| i i

(4)

is the `quantum potential'. Bohm regarded (3) as the law of motion, with (2) added as a onstraint on the initial momenta, a onstraint that Bohm thought ould be relaxed (see se tion 11.1). For de Broglie, in ontrast, the law of motion (2) for velo ity had a fundamental status, and for him represented the uni ation of the prin iples of Maupertuis and Fermat. One should then distinguish between de Broglie's rst-order (velo itybased) dynami s of 1927, and Bohm's se ond-order (a

eleration-based) dynami s of 1952. In this hapter, we shall be on erned with the histori al origins of de Broglie's 1927 dynami s dened by (1) and (2). Some authors have referred to this dynami s as `Bohmian me hani s'. Su h terminology is misleading: it disregards de Broglie's priority, and misses de Broglie's physi al motivations for re asting dynami s in terms of velo ity; it also misrepresents Bohm's 1952 formulation, whi h was based on (1) and (3). These and other histori al mis on eptions on erning de Broglie-Bohm theory will be addressed in se tion 11.1. The two equations (1), (2) dene a deterministi (de Broglian or pilotwave) dynami s for a single multiparti le system: given an initial wave fun tion Ψ(x1 , x2 , ..., xN , 0) at t = 0, (1) determines Ψ(x1 , x2 , ..., xN , t) at all times t; and given an initial onguration (x1 (0), x2 (0), ..., xN (0)), (2) then determines the traje tory (x1 (t), x2 (t), ..., xN (t)). For an ensemble of systems with the same initial wave fun tion Ψ(x1 , x2 , ..., xN , 0), and with initial ongurations (x1 (0), x2 (0), ..., xN (0)) distributed a

ording to the Born rule 2

P (x1 , x2 , ..., xN , 0) = |Ψ(x1 , x2 , ..., xN , 0)| ,

(5)

the statisti al distribution of out omes of quantum measurements will agree with the predi tions of standard quantum theory. This is shown by treating the measuring apparatus, together with the system being measured, as a single multiparti le system obeying de Broglian dynami s, so that (x1 , x2 , ..., xN ) denes the `pointer position' of the apparatus as well as the onguration of the measured system. Given the initial

ondition (5) for any multiparti le system, the statisti al distribution of parti le positions at later times will also agree with the Born rule 2 P = |Ψ| . Thus, the statisti al distribution of pointer positions in any experiment will agree with the predi tions of quantum theory, yielding

34

De Broglie's pilot-wave theory

the orre t statisti al distribution of out omes for standard quantum measurements. In his 1927 Solvay report, de Broglie gave some simple appli ations of pilot-wave theory, with the assumed initial ondition (5). He applied the theory to single-photon interferen e, to atomi transitions, and to the s attering (or dira tion) of ele trons by a rystal latti e. But a detailed demonstration of equivalen e to quantum theory, and in parti ular a pilot-wave a

ount of the general quantum theory of measurement, was not provided until the work of Bohm in 1952. How did de Broglie ome to propose this theory in 1927? In this

hapter, we tra e de Broglie's work in this dire tion, from his early work leading to his do toral thesis of 1924 (de Broglie 1924e, 1925), to his ru ial paper of 1927 published in Journal de Physique (de Broglie 1927b), and ulminating in his presentation of pilot-wave theory at the fth Solvay onferen e. We examine in detail how de Broglie arrived at this new form of parti le dynami s, and what his attitude towards it was. Later, in hapter 10, we shall onsider some of the dis ussions of de Broglie's theory that took pla e at the onferen e, in parti ular the famous (and widely misunderstood) lash between de Broglie and Pauli. De Broglie's dynami s has the striking feature that ele trons and photons are regarded as both parti les and waves. Like many s ienti ideas, this mingling of parti le-like and wave-like aspe ts had pre ursors. In Newton's Opti ks (rst published in 1704), both wave-like and parti lelike properties are attributed to light. Newton's so- alled ` orpus ular' theory was formulated on the basis of extensive and detailed experiments ( arried out by Grimaldi, Hooke, and Newton himself) involving what we would now all interferen e and dira tion. A

ording to Newton, light orpus les  or light `Rays' as he alled thema  generate `Waves of Vibrations' in an `Aethereal Medium', mu h as a stone thrown into water generates water waves (Newton 1730; reprint, pp. 3479). In addition, Newton supposed that the waves in turn ae t the motion of the orpus les, whi h `may be alternately a

elerated and retarded by the Vibrations' (p. 348). In parti ular, Newton thought that the ee t of the medium on the motion of the orpus les was responsible for the phenomena of interferen e and dira tion. He writes, for example (p. 350): And doth not the gradual ondensation of this Medium extend to some dis-

a The opening denition of the Opti ks denes `Rays' of light as `its least Parts'.

2.1 Ba kground

35

tan e from the Bodies, and thereby ause the Inexions of the Rays of Light, whi h pass by the edges of dense Bodies, at some distan e from the Bodies?

Newton understood that, for dira tion to o

ur, the motion of the light

orpus les would have to be ae ted at a distan e by the dira ting body  `Do not Bodies a t upon Light at a distan e, and by their a tion bend its Rays .... ?' (p. 339)  and his proposed me hanism involved waves in an inhomogeneous ether. Further, a

ording to Newton, to a

ount for the oloured fringes that had been observed by Grimaldi in the dira tion of white light by opaque bodies, the orpus les would have to exe ute an os illatory motion `like that of an Eel' (p. 339): Are not the Rays of Light in passing by the edges and sides of Bodies, bent several times ba kwards and forwards, with a motion like that of an Eel? And do not the three Fringes of olour'd Light above-mention'd arise from three su h bendings?

For Newton, of ourse, su h non-re tilinear motion ould be aused only by a for e emanating from the dira ting body. It is interesting to note that, in the general dis ussion at the fth Solvay onferen e (p. 510), de Broglie ommented on this very point, with referen e to the `emission' (or orpus ular) theory, and pointed out that if pilot-wave dynami s were written in terms of a

eleration (as done later by Bohm) then just su h for es appeared: In the orpus ular on eption of light, the existen e of dira tion phenomena o

uring at the edge of a s reen requires us to assume that, in this ase, the traje tory of the photons is urved. The supporters of the emission theory said that the edge of the s reen exerts a for e on the orpus le. Now, if in the new me hani s as I develop it, one writes the Lagrange equations for the photon, one sees appear on the right-hand side of these equations a term .... [that℄ .... represents a sort of for e of a new kind, whi h exists only .... where there is interferen e. It is this for e that will urve the traje tory of the photon when its wave

ψ

is dira ted by the edge of a s reen.

The striking similarity between Newton's qualitative ideas and pilotwave theory has also been noted by Berry, who remarks that during interferen e or dira tion the de Broglie-Bohm traje tories indeed `wriggle like an eel' (Berry 1997, p. 42), in some sense vindi ating Newton. A mathemati al pre ursor to de Broglian dynami s is found in the early nineteenth entury, in Hamilton's formulation of geometri al opti s and parti le me hani s. As de Broglie points out in his Solvay report (pp. 377 and 383), Hamilton's theory is in fa t the short-wavelength limit of pilot-wave dynami s: for in that limit, the phase of the wave fun tion

36

De Broglie's pilot-wave theory

obeys the Hamilton-Ja obi equation, and de Broglie's traje tories redu e to those of lassi al me hani s. A physi al theory of light as both parti les and waves  in ee t a revival of Newton's views  emerged again with Einstein in 1905. It is less well known that, after 1905, Einstein tried to onstru t theories of lo alised light quanta oupled to ve tor elds in 3-spa e. As we shall see in hapter 9, Einstein's ideas in this vein show some resemblan e to de Broglie's but also dier from them. It should also be mentioned that, in the autumn of 1923 (the same year in whi h de Broglie rst elaborated his ideas), Slater tried to develop a theory in whi h the motion of photons was guided by the ele tromagneti eld. It appears that Slater rst attempted to onstru t a deterministi theory, but had trouble dening an appropriate velo ity ve tor; he then

ame to the on lusion that photons and the ele tromagneti eld were related only statisti ally, with the photon probability density being given by the intensity of the eld. After dis ussing his ideas with Bohr and Kramers in 1924, the photons were removed from the theory, apparently against Slater's wishes (Mehra and Re henberg 1982a, pp. 5426). Note that, while de Broglie applied his theory to photons, he made it lear (for example in the general dis ussion, p. 509) that in his theory the guiding `ψ -wave' was distin t from the ele tromagneti eld. In the ase of light, then, the idea of ombining both parti le-like and wave-like aspe ts was an old one, going ba k indeed to Newton. In the

ase of ordinary matter, however, de Broglie seems to have been the rst to develop a physi al theory of this form. It is sometimes laimed that, for the ase of ele trons, ideas similar to de Broglie's were put forward by Madelung in 1926. What Madelung proposed, however, was to regard an ele tron with mass m and wave fun tion ψ not as a pointlike parti le within the wave, but as a ontinuous uid spread over spa e with mass density m |ψ|2 (Madelung 1926a,b). In this `hydrodynami al' interpretation, mathemati ally the uid velo ity

oin ides with de Broglie's velo ity eld; but physi ally, Madelung's theory seems more akin to S hrödinger's theory than to de Broglie's. Finally, before we examine de Broglie's work, we note what appears to be a re urring histori al opposition to dualisti physi al theories

ontaining both waves and parti les. In 180103, Thomas Young, who by his own a

ount regarded his theory as a development of Newton's ideas,a removed the orpus les from Newton's theory and produ ed a purely una See, for example, Bernard Cohen's prefa e to Newton's Opti ks (1730, reprint).

2.2 A new approa h to parti le dynami s: 192324

37

dulatory a

ount of light. In 1905, Einstein's dualist view of light was not taken seriously, and did not win widespread support until the dis overy of the Compton ee t in 1923. In 1924, Bohr and Kramers, who regarded the Bohr-Kramers-Slater theory as a development of Slater's original idea, insisted on removing the photons from Slater's theory of radiation.a And in 1926, S hrödinger, who regarded his work as a development of de Broglie's ideas, removed the traje tories from de Broglie's theory and produ ed a purely undulatory `wave me hani s'.b

2.2 A new approa h to parti le dynami s: 192324 In this se tion we show how de Broglie took his rst steps towards a new form of dynami s.a His aim was to explain the quantum phenomena known at the time  in parti ular the Bohr-Sommerfeld quantisation of atomi energy levels, and the apparently dual nature of radiation  by unifying the physi s of parti les with the physi s of waves. To a

omplish this, de Broglie began by extending Einstein's wave-parti le duality for light to all material bodies, by introdu ing a `phase wave' a

ompanying every material parti le. Then, inspired by the opti alme hani al analogy,b de Broglie proposed that Newton's rst law of motion should be abandoned, and repla ed by a new prin iple that unied Maupertuis' variational prin iple for me hani s with Fermat's variational prin iple for opti s. The result was a new form of dynami s in whi h the velo ity v of a parti le is determined by the gradient of the phase φ of an a

ompanying wave  in ontrast with lassi al me hani s, where a

elerations are determined by for es. (Note that de Broglie's phase φ has a sign opposite to the phase S as we would normally dene it now.) This new approa h to dynami s enabled de Broglie to obtain a wavelike explanation for the quantisation of atomi energy levels, to explain the observed interferen e of single photons, and to predi t for the rst time the new and unexpe ted phenomenon of the dira tion and interferen e of ele trons. a In 1925, Born and Jordan attempted to restore the photons, proposing a sto hasti theory reminis ent of Slater's original ideas; it appears that they were dissuaded from publi ation by Bohr. See Darrigol (1992, p. 253) and se tion 3.4.2. b Cf. se tion 4.5. a An insightful and general a

ount of de Broglie's early work, up to 1924, has been given by Darrigol (1993). b Possibly, de Broglie was also inuen ed by the philosopher Henri Bergson's writings

on erning time, ontinuity and motion (Feuer 1974, pp. 20614); though this is denied by Lo hak (1992).

38

De Broglie's pilot-wave theory

As we shall see, the theory proposed by de Broglie in 192324 was, in fa t, a simple form of pilot-wave dynami s, for the spe ial ase of independent parti les guided by waves in 3-spa e, and without a spe i wave equation.

2.2.1 First papers on pilot-wave theory (1923) De Broglie's earliest experien e of physi s was losely tied to experiment. During the rst world war he worked on wireless telegraphy, and after the war his rst papers on erned X-ray spe tros opy. In 1922 he published a paper treating bla kbody radiation as a gas of light quanta (de Broglie 1922). In this paper, de Broglie made the unusual assumption that photons had a very small but non-zero rest mass m0 . He was therefore now applying Einstein's relations E = hν and p = h/λ (relating energy and momentum to frequen y and wavelength) to massive parti les, even if these were still only photons. It seems that de Broglie made the assumption m0 6= 0 so that light quanta ould be treated in the same way as ordinary material parti les. It appears that this paper was the seed from whi h de Broglie's subsequent work grew.a A

ording to de Broglie's later re olle tions (L. de Broglie, AHQP interview, 7 January 1963, p. 1),2 his rst ideas on erning a pilot-wave theory of massive parti les arose as follows. During onversations on the subje t of X-rays with his older brother Mauri e de Broglie,b he be ame onvin ed that X-rays were both parti les and waves. Then, in the summer of 1923, de Broglie had the idea of extending this duality to ordinary matter, in parti ular to ele trons. He was drawn in this dire tion by onsideration of the opti al-me hani al analogy; further, the presen e of whole numbers in quantisation onditions suggested to him that waves must be involved. This last motivation was re alled by de Broglie (1999) in his Nobel le ture of 1929: .... the determination of the stable motions of the ele trons in the atom involves whole numbers, and so far the only phenomena in whi h whole numbers were involved in physi s were those of interferen e and of eigenvibrations. That

a In a olle tion of papers by de Broglie and Brillouin, published in 1928, a footnote added to de Broglie's 1922 paper on bla kbody radiation and light quanta remarks: `This paper .... was the origin of the ideas of the author on wave me hani s' (de Broglie and Brillouin 1928, p. 1). b Mauri e, the sixth du de Broglie, was a distinguished experimental physi ist, having done important work on the photoele tri ee t with X-rays  experiments that were arried out in his private laboratory in Paris.

2.2 A new approa h to parti le dynami s: 192324

39

suggested the idea to me that ele trons themselves ould not be represented as simple orpus les either, but that a periodi ity had also to be assigned to them too.

De Broglie rst presented his new ideas in three notes published (in Fren h) in the Comptes Rendus of the A ademy of S ien es in Paris (de Broglie 1923a,b, ), and also in two papers published in English  one in Nature (de Broglie 1923d), the other in the Philosophi al Magazine (de Broglie 1924a).a The ideas in these papers formed the basis for de Broglie's do toral thesis. The paper in the Philosophi al Magazine reads, in fa t, like a summary of mu h of the material in the thesis. Sin e the thesis provides a more systemati presentation, we shall give a detailed summary of it in the next subse tion; here, we give only a brief a

ount of the earlier papers, ex ept for the ru ial se ond paper, whose on eptual ontent warrants more detailed ommentary.b The rst ommuni ation (de Broglie 1923a), entitled `Waves and quanta', proposes that an `internal periodi phenomenon' should be asso iated with any massive parti le (in luding light quanta). In the rest frame of a parti le with rest mass m0 , the periodi phenomenon is assumed to have a frequen y ν0 = m0 c2 /h. In a frame where the parti le has uniform velo ity p v , de Broglie onsiders the two frequen ies ν and ν1 , where 2 ν = ν0 / 1 − v 2 /c2 is the frequen y p ν = mc /h asso iated p with the 2 2 relativisti mass in rease m = m0 / 1 − v /c and ν1 = ν0 1 − v 2 /c2 is the time-dilated frequen y. De Broglie shows that, be ause ν1 = ν(1− v 2 /c2 ), a ` titious' wave of frequen y ν and phase velo ity vph = c2 /v (propagating in the same dire tion as the parti le) will remain in phase with the internal os illation of frequen y ν1 . De Broglie then onsiders an atomi ele tron moving uniformly on a ir ular orbit. He proposes that orbits are stable only if the  titious wave remains in phase with the internal os illation of the ele tron. From this ondition, de Broglie derives the Bohr-Sommerfeld quantisation ondition. The se ond ommuni ation (de Broglie 1923b), entitled `Light quanta, dira tion and interferen e', has a more on eptual tone. De Broglie begins by re alling his previous result, that a moving body must be asso iated with `a non-material sinusoidal wave'. He adds that the parti le velo ity v is equal to the group velo ity of the wave, whi h de Broglie here alls `the phase wave' be ause its phase at the lo ation of the a It seems possible that Comptes Rendus was not widely read by physi ists outside Fran e, but this ertainly was not true of Nature or the Philosophi al Magazine. b We do not always keep to de Broglie's original notation.

40

De Broglie's pilot-wave theory

parti le is equal to the phase of the internal os illation of the parti le. De Broglie then goes on to make some very signi ant observations about dira tion and the nature of the new dynami s that he is proposing. De Broglie asserts that dira tion phenomena prove that light quanta

annot always propagate in a straight line, even in what would normally be alled empty spa e. He draws the bold on lusion that Newton's rst law of motion (the `prin iple of inertia') must be abandoned (p. 549): The light quanta [atomes de lumière℄ whose existen e we assume do not always propagate in a straight line, as proved by the phenomena of dira tion. It then seems

ne essary

to modify the prin iple of inertia.

De Broglie then suggests repla ing Newton's rst law with a new postulate (p. 549): We propose to adopt the following postulate as the basis of the dynami s of the free material point: `At ea h point of its traje tory, a free moving body follows in a uniform motion the

ray

of its phase wave, that is (in an isotropi

medium), the normal to the surfa es of equal phase'.

The dira tion of light quanta is then explained sin e, as de Broglie notes, `if the moving body must pass through an opening whose dimensions are small ompared to the wavelength of the phase wave, in general its traje tory will urve like the ray of the dira ted wave'. In retrospe t, de Broglie's postulate for free parti les may be seen as a simplied form of the law of motion of what we now know as pilot-wave dynami s  ex ept for the statement that the motion along a ray be `uniform' (that is, have onstant speed), whi h in pilot-wave theory is true only in spe ial ases.a De Broglie notes that his postulate respe ts

onservation of energy but not of momentum. And indeed, in pilot-wave theory the momentum of a `free' parti le is generally not onserved: in ee t (from the standpoint of Bohm's Newtonian formulation), the pilot wave or quantum potential a ts like an `external sour e' of momentum (and in general of energy too).b The abandonment of something as a From Bohm's se ond-order equation (3) applied to a single parti le, for timeindependent V it follows that d( 12 mv2 + V + Q)/dt = ∂Q/∂t (the usual energy

onservation formula with a time-dependent ontribution Q to the potential). In free spa e (V = 0), the speed v is onstant if and only if dQ/dt = ∂Q/∂t or v · ∇Q = 0 (so that the `quantum for e' does no work), whi h is true only in spe ial ases. b Again from (3), in free spa e the rate of hange of momentum p = mv (where v = ∇S/m) is dp/dt = −∇Q, whi h is generally non-zero. Further, in general (3) implies d( 21 mv2 + V )/dt = −v · ∇Q, so that the standard ( lassi al) expression for energy is onserved if and only if v · ∇Q = 0. If, on the other hand, one denes 1 mv2 + V + Q as the `energy', it will be onserved if and only if ∂Q/∂t = 0, whi h 2

2.2 A new approa h to parti le dynami s: 192324

41

elementary as momentum onservation is ertainly a radi al step by any standards. On the other hand, if one is willing  as de Broglie was  to propose a fundamentally new approa h to the theory of motion, then the loss of lassi al onservation laws is not surprising, as these are really properties of lassi al equations of motion. De Broglie then makes a remarkable predi tion, that any moving body (not just light quanta) an undergo dira tion: .... any moving body ould in ertain ases be dira ted. A stream of ele trons passing through a small enough opening will show dira tion phenomena. It is in this dire tion that one should perhaps look for experimental onrmation of our ideas.

Next, de Broglie puts his proposals in a general on eptual and histori al perspe tive. Con erning the role of the phase wave, he writes (p. 549): We therefore on eive of the phase wave as guiding the movements of energy, and this is what an allow the synthesis of waves and quanta.

Here, for the rst time, de Broglie hara terises the phase wave as a `guiding' wave. De Broglie then remarks that, histori ally speaking, the theory of waves `went too far' by denying the dis ontinuous stru ture of radiation and `not far enough' by not playing a role in dynami s. For de Broglie, his proposal has a lear histori al signi an e (p. 549, itali s in the original): The new dynami s of the free material point is to the old dynami s (in luding that of Einstein) what wave opti s is to geometri al opti s. Upon ree tion one will see that the proposed synthesis appears as the logi al ulmination of the omparative development of dynami s and of opti s sin e the seventeenth

entury.

In the se ond part of this note, de Broglie onsiders the explanation of opti al interferen e fringes. He assumes that the probability for an atom to absorb or emit a light quantum is determined by `the resultant of one of the ve tors of the phase waves rossing ea h other there [se roisant sur lui℄' (pp. 54950). In Young's interferen e experiment, the light quanta passing through the two holes are dira ted, and the probability of them being dete ted behind the s reen will vary from point to point, depending on the `state of interferen e' of the phase waves. De Broglie is true only for spe ial ases (in parti ular for stationary states, sin e for these |Ψ| is time-independent).

42

De Broglie's pilot-wave theory

on ludes that there will be bright and dark fringes as predi ted by the wave theories, no matter how feeble the in ident light. This approa h to opti al interferen e  in whi h interfering phase waves determine the probability for intera tion between photons and the atoms in the dete tion apparatus  is elaborated in de Broglie's thesis (see below). Soon after ompleting his thesis (apparently), de Broglie abandoned this idea in favour of a simpler approa h, in whi h the interfering phase waves determine the number density of photon traje tories (see se tion 2.2.3). In de Broglie's third ommuni ation (de Broglie 1923 ), entitled `Quanta, the kineti theory of gases and Fermat's prin iple', part 1 onsiders the statisti al treatment of a gas of parti les a

ompanied by phase waves. De Broglie makes the following assumption: The state of the gas will then be stable only if the waves orresponding to all of the atoms form a system of stationary waves.

In other words, de Broglie onsiders the stationary modes, or standing waves, asso iated with a given spatial volume. He assumes that ea h mode ` an transport zero, one, two or several atoms', with probabilities determined by the Boltzmann fa tor.a A

ording to de Broglie, for a gas of nonrelativisti atoms his method yields the Maxwell distribution, while for a gas of photons it yields the Plan k distribution.b In part 2 of the same note, de Broglie shows how his new dynami al postulate amounts to a uni ation of Maupertuis' prin iple of least a tion with Fermat's prin iple of least time in opti s. Let us re all that, in the me hani al prin iple of Maupertuis for parti le traje tories, Z b δ mv · dx = 0 , (6) a

the ondition of stationarity determines the parti le paths. (In (6) the energy is xed on the varied paths; at the end points, ∆x = 0 but ∆t need not be zero.) While in the opti al prin iple of Fermat for light rays, Z b δ dφ = 0 , (7) a

the stationary line integral for the phase hange  the stationary `opti al a As remarked by Pais (1982, pp. 4356), in this paper de Broglie `evaluated independently of Bose (and published before him) the density of radiation states in terms of parti le (photon) language'. b As shown by Darrigol (1993), de Broglie made some errors in his appli ation of the methods of statisti al me hani s.

2.2 A new approa h to parti le dynami s: 192324

43

path length'  provides a ondition that determines the path of a ray

onne ting two points, in spa e (for the time-independent ase) or in spa etime. Now, a

ording to de Broglie's basi postulate: `The rays of the phase waves oin ide with the dynami ally possible traje tories' (p. 632). The rays are des ribed by Fermat's prin iple (for the ase of a dispersive medium), whi h de Broglie shows oin ides with Maupertuis' prin iple, as follows: writing the element of phase hange as dφ = 2πνdl/vph, where dl is an element of path and vph = c2 /v is the phase velo ity, and using the relation ν = E/h = mc2 /h, the element of phase

hange may be rewritten as (2π/h)mvdl, so that (7) oin ides with (6). As de Broglie puts it (p. 632): In this way the fundamental link that unites the two great prin iples of geometri al opti s and of dynami s is brought fully to light.

De Broglie remarks that some of the dynami ally possible traje tories will be `in resonan e with the phase wave', and that these orrespond to R Bohr's stable orbits, for whi h νdl/vph is a whole number. Soon afterwards, de Broglie introdu es a ovariant 4-ve tor formulation of his basi dynami al postulate (de Broglie 1924a,b). He denes n), where n ˆ is a unit ve tor in the a 4-ve tor wµ = (ν/c, −(ν/vph )ˆ dire tion of a ray of the phase wave, and assumes it to be related to the energy-momentum 4-ve tor pµ = (E/c, −p) by pµ = hwµ . De Broglie notes that the identity of the prin iples of Maupertuis and Fermat then follows immediately. We shall dis uss this in more detail in the next subse tion.

2.2.2 Thesis (1924) He has lifted a orner of the great veil. Einstein, ommenting on de Broglie's thesis

a

De Broglie's do toral thesis (de Broglie 1924e) was mostly based on the above papers. It seems to have been ompleted in the summer of 1924, and was defended at the Sorbonne in November. The thesis was published early in 1925 in the Annales de Physique (de Broglie 1925).b a Letter to Langevin, 16 De ember 1924 (quoted in Darrigol 1993, p. 355). b An English translation of extra ts from de Broglie's thesis appears in Ludwig (1968). A omplete translation has been done by A. F. Kra klauer ( urrently online at http://www.ensmp.fr/ab/LDB-oeuvres/De_Broglie_Kra klauer.htm ). All translations here are ours.

44

De Broglie's pilot-wave theory

When writing his thesis, de Broglie was well aware that his theory had gaps. As he put it (p. 30):a .... the main aim of the present thesis is to present a more omplete a

ount of the new ideas that we have proposed, of the su

esses to whi h they have led, and also of the many gaps they ontain.

De Broglie begins his thesis with a histori al introdu tion. Newtonian me hani s, he notes, was eventually formulated in terms of the prin iple of least a tion, whi h was rst given by Maupertuis and then later in another form by Hamilton. As for the s ien e of light and opti s, the laws of geometri al opti s were eventually summarised by Fermat in terms of a prin iple whose form is reminis ent of the prin iple of least a tion. Newton tried to explain some of the phenomena of wave opti s in terms of his orpus ular theory, but the work of Young and Fresnel led to the rise of the wave theory of light, in parti ular the su

essful wave explanation of the re tilinear propagation of light (whi h had been so lear in the orpus ular or `emission' theory). On this, de Broglie

omments (p. 25): When two theories, based on ideas that seem entirely dierent, a

ount for the same experimental fa t with equal elegan e, one an always wonder if the opposition between the two points of view is truly real and is not due solely to an inadequa y of our eorts at synthesis.

This remark is, of ourse, a hint that the aim of the thesis is to ee t just su h a synthesis. De Broglie then turns to the rise of ele trodynami s, relativity, and the theory of energy quanta. He notes that Einstein's theory of the photoele tri ee t amounts to a revival of Newton's orpus ular theory. De Broglie then sket hes Bohr's 1913 theory of the atom, and goes on to point out that observations of the photoele tri ee t for X- and γ -rays seem to onrm the orpus ular hara ter of radiation. At the same time, the wave aspe t ontinues to be onrmed by the observed interferen e and dira tion of X-rays. Finally, de Broglie notes the very re ent orpus ular interpretation of Compton s attering. De Broglie on ludes his histori al introdu tion with a mention of his own re ent work (p. 30): ....

the moment seemed to have arrived to make an eort towards unifying

the orpus ular and wave points of view and to go a bit more deeply into the true meaning of the quanta. That is what we have done re ently ....

a Here and below, page referen es for de Broglie's thesis orrespond to the published version in Annales de Physique (de Broglie 1925).

2.2 A new approa h to parti le dynami s: 192324

45

De Broglie learly regarded his own work as a synthesis of earlier theories of dynami s and opti s, a synthesis in reasingly for ed upon us by a

umulating experimental eviden e. Chapter 1 of the thesis is entitled `The phase wave'. De Broglie begins by re alling the equivalen e of mass and energy implied by the theory of relativity. Turning to the problem of quanta, he remarks (pp. 323): It seems to us that the fundamental idea of the quantum theory is the impossibility of onsidering an isolated quantity of energy without asso iating a

ertain frequen y with it. This onne tion is expressed by what I shall all the quantum relation:

energy = h × frequency where

h

is Plan k's onstant.

To make sense of the quantum relation, de Broglie proposes that (p. 33) .... to ea h energy fragment of proper mass phenomenon of frequen y

ν0

m0

there is atta hed a periodi

su h that one has:

hν0 = m0 c2 ν0

being measured, of ourse, in the system tied to the energy fragment.

De Broglie asks if the periodi phenomenon must be assumed to be lo alised inside the energy fragment. He asserts that this is not at all ne essary, and that it will be seen to be `without doubt spread over an extensive region of spa e' (p. 34). De Broglie goes on to onsider p the apparent ontradi tion between the 2 ν = mc /h = ν / 1 − v 2 /c2 and the time-dilated frequen y frequen y 0 p 2 2 ν1 = ν0 1 − v /c . He proposes that the ontradi tion is resolved by the following `theorem of phase harmony' (p. 35): in a frame where the moving body has velo ity v , the periodi phenomenon tied to the moving body and with frequen y ν1 is always in phase with a wave of frequen y ν propagating in the same dire tion as the moving body with phase velo ity vph = c2 /v . This is shown by applying the Lorentz transformation to a rest-frame wave sin (ν0 t0 ), yielding a wave h i
p sin ν0 t − vx/c2 / 1 − v 2 /c2 (8) p of frequen y ν = ν0 / 1 − v 2 /c2 and phase velo ity c2 /v . Regarding the nature of this wave de Broglie says that, be ause its velo ity is greater than c, it annot be a wave transporting energy: rather, `it represents the spatial distribution of the phases of a phenomenon; it is a phase wave ' (p. 36). De Broglie shows that the group velo ity of the phase

46

De Broglie's pilot-wave theory

wave is equal to the velo ity of the parti le. In the nal se tion of

hapter 1 (`The phase wave in spa etime'), he dis usses the appearan e of surfa es of onstant phase for dierently moving observers, from a spa etime perspe tive. Chapter 2 is entitled `Maupertuis' prin iple and Fermat's prin iple'. The aim is to generalise the results of the rst hapter to non-uniform, non-re tilinear motion. In the introdu tion to hapter 2 de Broglie writes (p. 45): Guided by the idea of a deep unity between the prin iple of least a tion and that of Fermat, from the beginning of my investigations on this subje t I was led to

assume

that, for a given value of the total energy of the moving body

and therefore of the frequen y of its phase wave, the dynami ally possible traje tories of the one oin ided with the possible rays of the other.

De Broglie dis usses the prin iple of least a tion, in the dierent forms given by Hamilton and by Maupertuis, and also for relativisti parti les in an external ele tromagneti eld. He writes Hamilton's prin iple as Z Q pµ dxµ = 0 δ (9) P

(µ = 0, 1, 2, 3, with dx0 = cdt), where P , Q are points in spa etime and pµ is the anoni al energy-momentum 4-ve tor, and notes that if p0 is

onstant the prin iple be omes Z B δ pi dxi = 0 (10) A

(i = 1, 2, 3), where A, B are the orresponding points in spa e  that is, Hamilton's prin iple redu es to Maupertuis' prin iple. De Broglie then dis usses wave propagation and Fermat's prin iple from a spa etime perspe tive. He onsiders a sinusoidal fun tion sin φ, where the phase φ has a spa etime-dependent dierential dφ, and writes the variational prin iple for the ray in spa etime in the Hamiltonian form Z Q δ (11) dφ = 0 . P

De Broglie then introdu es a 4-ve tor eld wµ on spa etime, dened by dφ = 2πwµ dxµ ,

(12)

where the wµ are generally fun tions on spa etime. (Of ourse, this implies that 2πwµ = ∂µ φ, though de Broglie does not write this expli itly.) De Broglie also notes that dφ = 2π(νdt − (ν/vph )dl) and

2.2 A new approa h to parti le dynami s: 192324

47

wµ = (ν/c, −(ν/vph )ˆ n), where n ˆ is a unit ve tor in the dire tion of propagation; and that if ν is onstant, the prin iple in the Hamiltonian form Z Q δ wµ dxµ = 0 (13) P

redu es to the prin iple in the Maupertuisian form Z B wi dxi = 0 , δ

(14)

A

or

δ

Z

B A

ν dl = 0 , vph

(15)

whi h is Fermat's prin iple. De Broglie then dis usses an `extension of the quantum relation' (that is, an extension of E = hν ). He states that the two 4-ve tors pµ and wµ play perfe tly symmetri al roles in the motion of a parti le and in the propagation of a wave. Writing the `quantum relation' E = hν as w0 = h1 p0 , de Broglie proposes the generalisation wµ =

1 pµ , h

(16)

so that dφ = 2πwµ dxµ =

2π pµ dxµ . h

Fermat's prin iple then be omes Z B pi dxi = 0 , δ

(17)

(18)

A

whi h is the same as Maupertuis' prin iple. Thus, de Broglie arrives at the following statement (p. 56): Fermat's prin iple applied to the phase wave is identi al to Maupertuis' prin iple applied to the moving body; the dynami ally possible traje tories of the moving body are identi al to the possible rays of the wave.

He adds that (p. 56): We think that this idea of a deep relationship between the two great prin iples of Geometri al Opti s and Dynami s ould be a valuable guide in realising the synthesis of waves and quanta.

48

De Broglie's pilot-wave theory

De Broglie then dis usses some parti ular ases: the free parti le, a parti le in an ele trostati eld, and a parti le in a general ele tromagneti eld. He al ulates the phase velo ity, whi h depends on the ele tromagneti potentials. He notes that the propagation of a phase wave in an external eld depends on the harge and mass of the moving body. And he shows that the group velo ity along a ray is still equal to the velo ity of the moving body. For the ase of an ele tron of harge e and velo ity v in an ele trostati potential ϕ, de Broglie writes down the following expressions for the frequen y ν and phase velo ity vph of the phase wave (p. 57): ν = (mc2 + eϕ)/h , vph = (mc2 + eϕ)/mv (19) p (where again m = m0 / 1 − v 2 /c2 ). He shows that vph may be rewritten as the free value c2 /v multiplied by a fa tor hν/(hν − eϕ) that depends on the potential ϕ. The expressions (19) formed the starting point for S hrödinger's work on the wave equation for de Broglie's phase waves (as re onstru ted by Mehra and Re henberg (1987, pp. 4235), see se tion 2.3). While de Broglie does not expli itly say so in his thesis, note that from the denition (12) of wµ , the generalised quantum relation (16) may be written in the form pµ = ℏ∂µ φ .

(20)

This is what we would now all a relativisti guidan e equation, giving the velo ity of a parti le in terms of the gradient of the phase of a pilot wave (where here de Broglie denes the phase φ to be dimensionless). In other words, the extended quantum relation is a rst-order equation of motion. In the presen e of an ele tromagneti eld, the anoni al momentum pµ ontains the 4-ve tor potential. For a free parti le, with pµ = (E/c, −p), the guidan e equation has omponents ˙ p = −ℏ∇φ , E = ℏφ,

where the spatial omponents may also be written as m0 v mv = p = −ℏ∇φ . 1 − v 2 /c2

(21)

(22)

For a plane wave of phase φ = ωt − k · x, we have E = ℏω, p = ℏk .

(23)

Thus, de Broglie's uni ation of the prin iples of Maupertuis and

2.2 A new approa h to parti le dynami s: 192324

49

Fermat amounts to a new dynami al law, (16) or (20), in whi h the phase of a guiding wave determines the parti le velo ity. This new law of motion is the essen e of de Broglie's new, rst-order dynami s. Chapter 3 of de Broglie's thesis is entitled `The quantum onditions for the stability of orbits'. De Broglie reviews Bohr's ondition for ir ular orbits, a

ording to whi h the angular R 2πmomentum of the ele tron must be a multiple of ℏ, or equivalently 0 pθ dθ = Hnh (pθ onjugate to θ). He also reviews Sommerfeld's generalisation, H P3 pi dqi = ni h (integral ni ) and Einstein's invariant formulation i=1 pi dqi = nh (integral n). De Broglie then provides an explanation for Einstein's ondition. The traje tory of the moving body oin ides with one of the rays of its phase wave, and the phase wave moves along the traje tory with a onstant frequen y (be ause the total energy is onstant) and with a variable speed whose value has been al ulated. To have a stable orbit, laims de Broglie, the length l of the orbit must be in `resonan e' with the wave: thus l = nλ in the ase of onstant wavelength, and H (ν/vph )dl = n (n integral) generally. De Broglie notes that this is pre isely the integral appearing in Fermat's prin iple, whi h has been shown to be equal to the integral giving the Maupertuisian a tion divided by h. The resonan e ondition is then identi al to the required stability

ondition. For the simple ase of ir ular orbits in the Bohr atom de Broglie shows, using H vph = νλ and h/λ = m0 v , that the resonan e

ondition be omes m0 vdl = nh or m0 ωR2 = nℏ (with v = ωR), as originally given by Bohr.a (Note that the simple argument ommonly found in textbooks, about the tting of whole numbers of wavelengths along a Bohr orbit, originates in this work of de Broglie's.) De Broglie thought that his explanation of the stability or quantisation

onditions onstituted important eviden e for his ideas. As he puts it (p. 65): This beautiful result, whose demonstration is so immediate when one has a

epted the ideas of the pre eding hapter, is the best justi ation we an give for our way of atta king the problem of quanta.

Certainly, de Broglie had a hieved a on rete realisation of his initial intuition that quantisation onditions for atomi energy levels ould arise from the properties of waves. In his hapter 4, de Broglie onsiders the two-body problem, in para De Broglie also laims to generalise his results from losed orbits to quasi-periodi (or multi-periodi ) motion: however, as shown by Darrigol (1993), de Broglie's derivation is faulty.

50

De Broglie's pilot-wave theory

ti ular the hydrogen atom. He expresses on ern over how to dene the proper masses, taking into a

ount the intera tion energy. He dis usses the quantisation onditions for hydrogen from a two-body point of view: he has two phase waves, one for the ele tron and one for the nu leus. The subje t of hapter 5 is light quanta. De Broglie suggests that the

lassi al (ele tromagneti ) wave distribution in spa e is some sort of time average over the true distribution of phase waves. His light quantum is assigned a very small proper mass: the velo ity v of the quantum, and the phase velo ity c2 /v of the a

ompanying phase wave, are then both very lose to c. De Broglie points out that radiation is sometimes observed to violate re tilinear propagation: a light wave striking the edge of a s reen dira ts into the geometri al shadow, and rays passing lose to the s reen deviate from a straight line. De Broglie notes the two histori al explanations for this phenomenon  on the one hand the explanation for dira tion given by the wave theory, and on the other the explanation given by Newton in his emission theory: `Newton assumed [the existen e of℄ a for e exerted by the edge of the s reen on the orpus le' (p. 80). De Broglie asserts that he an now give a unied explanation for dira tion, by abandoning Newton's rst law of motion (p. 80): .... the ray of the wave would urve as predi ted by the theory of waves, and the moving body, for whi h the prin iple of inertia would no longer be valid, would suer the same deviation as the ray with whi h its motion is bound up [solidaire℄ ....

De Broglie's words here deserve emphasis. As is also very lear in his se ond paper of the pre eding year (see se tion 2.2.1), de Broglie regards his explanation of parti le dira tion as based on a new form of dynami s in whi h Newton's rst law  the prin iple that a free body will always move uniformly in a straight line  is abandoned. At the same time, de Broglie re ognises that one an always adopt a lassi al-me hani al viewpoint if one wishes (pp. 8081): .... perhaps one ould say that the wall exerts a for e on it [the moving body℄ if one takes the urvature of the traje tory as a riterion for the existen e of a for e.

Here, de Broglie re ognises that one may still think in Newtonian terms, if one ontinues to identify a

eleration as indi ative of the presen e of a for e. Similarly, as we shall see, in 1927 de Broglie notes that his pilotwave dynami s may if one wishes be written in Newtonian form with a quantum potential. But de Broglie's preferred approa h, throughout his

2.2 A new approa h to parti le dynami s: 192324

51

work in the period 192327, is to abandon Newton's rst law and base his dynami s on velo ity rather than on a

eleration. After onsidering the Doppler ee t, ree tion by a moving mirror, and radiation pressure, all from a photon viewpoint, de Broglie turns to the phenomena of wave opti s, noting that (p. 86): The stumbling blo k of the theory of light quanta is the explanation of the phenomena that onstitute wave opti s.

Here it be omes apparent that, despite his understanding of how nonre tilinear parti le traje tories arise during dira tion and interferen e, de Broglie is not sure of the details of how to explain the observed bright and dark fringes in dira tion and interferen e experiments with light. In parti ular, de Broglie did not have a pre ise theory of the assumed statisti al relationship between his phase waves and the ele tromagneti eld. Even so, he went on to make what he alled `vague suggestions' (p. 87) towards a detailed theory of opti al interferen e. De Broglie's idea was that the phase waves would determine the probability for the light quanta to intera t with the atoms onstituting the equipment used to observe the radiation, in su h a way as to a

ount for the observed fringes (p. 88): .... the probability of rea tions between atoms of matter and atoms of light is at ea h point tied to the resultant (or rather to the mean value of this) of one of the ve tors hara terising the phase wave; where this resultant vanishes the light is undete table; there is interferen e. One then on eives that an atom of light traversing a region where the phase waves interfere will be able to be absorbed by matter at ertain points and not at others. This is the still very qualitative prin iple of an explanation of interferen e .... .

As we shall see in the next se tion, after ompleting his thesis de Broglie arrived at a simpler explanation of opti al interferen e fringes. The nal se tion of hapter 5 onsiders the explanation of Bohr's frequen y ondition hν = E1 − E2 for the light emitted by an atomi transition from energy state E1 to energy state E2 . De Broglie derives this from the assumption that ea h transition involves the emission of a single light quantum of energy E = hν (together with the assumption of energy onservation). De Broglie's hapter 6 dis usses the s attering of X- and γ -rays. In his hapter 7, de Broglie turns to statisti al me hani s, and shows how the on ept of statisti al equilibrium is to be modied in the presen e of phase waves. If ea h parti le or atom in a gas is a

ompanied by a phase wave, then a box of gas will be ` riss- rossed in all dire tions'

52

De Broglie's pilot-wave theory

(p. 110) by the waves. De Broglie nds it natural to assume that the only stable phase waves in the box will be those that form stationary or standing waves, and that only these will be relevant to thermodynami equilibrium. He illustrates his idea with a simple example of mole ules moving in one dimension, onned to an interval of length l. In the nonrelativisti limit, ea h phase wave has a wavelength λ = h/m0 v and the `resonan e ondition' is l = nλ with n integral. Writing v0 = h/m0 l, one then has v = nv0 . As de Broglie notes (p. 112): `The speed will then be able to take only values equal to integer multiples of v0 '. (This is, of

ourse, the well-known quantisation of momentum for parti les onned to a box.) De Broglie then argues that a velo ity element δv orresponds to a number δn = (m0 l/h)δv of states of a mole ule ( ompatible with the existen e of stationary phase waves), so that an element m0 δxδv of phase spa e volume orresponds to a number m0 δxδv/h of possible states. Generalising to three dimensions, de Broglie is led to take the element of phase spa e volume divided by h3 as the measure of the number of possible states of a mole ule, as assumed by Plan k. De Broglie then turns to the photon gas, for whi h he obtains Wien's law. He laims that, in order to get the Plan k law, the following further hypothesis is required (p. 116): If two or several atoms [of light℄ have phase waves that are exa tly superposed, of whi h one an therefore say that they are transported by the same wave, their motions an no longer be onsidered as entirely independent and these atoms an no longer be treated as separate units in al ulating the probabilities.

In de Broglie's approa h, the stationary phase waves play the role of the elementary obje ts of statisti al me hani s. De Broglie denes stationary waves as a superposition of two waves of the form i i x x sin h  sin h  and , (24) 2π νt + + φ0 2π νt − + φ0 cos cos λ λ

where φ0 an take any value from 0 to 1 and ν takes one of the allowed values. Ea h elementary wave an arry any number 0, 1, 2, ... of atoms, and the probability of arrying n atoms is given by the Boltzmann fa tor e−nhν/kT . Applying this method to a gas of light quanta, de Broglie

laims to derive the Plan k distribution.a De Broglie's thesis ends with a summary and on lusions (pp. 1258). The seeds of the problem of quanta have been shown, he laims, to be a Again, as shown by Darrigol (1993), de Broglie's appli ation of statisti al me hani s ontains some errors.

2.2 A new approa h to parti le dynami s: 192324

53

ontained in the histori al `parallelism of the orpus ular and wave-like

on eptions of radiation'. He has postulated a periodi phenomenon asso iated with ea h energy fragment, and shown how relativity requires us to asso iate a phase wave with every uniformly moving body. For the ase of non-uniform motion, Maupertuis' prin iple and Fermat's prin iple ` ould well be two aspe ts of a single law', and this new approa h to dynami s led to an extension of the quantum relation, giving the speed of a phase wave in an ele tromagneti eld. The most important onsequen e is the interpretation of the quantum onditions for atomi orbits in terms of a resonan e of the phase wave along the traje tories: `this is the rst physi ally plausible explanation proposed for the Bohr-Sommerfeld stability onditions'. A `qualitative theory of interferen e' has been suggested. The phase wave has been introdu ed into statisti al me hani s, yielding a derivation of Plan k's phase volume element, and of the bla kbody spe trum. De Broglie has, he laims, perhaps ontributed to a uni ation of the opposing on eptions of waves and parti les, in whi h the dynami s of the material point is understood in terms of wave propagation. He adds that the ideas need further development: rst of all, a new ele tromagneti theory is required, that takes into a

ount the dis ontinuous stru ture of radiation and the physi al nature of phase waves, with Maxwell's theory emerging as a statisti al approximation. The nal paragraph of de Broglie's thesis emphasises the in ompleteness of his theory at the time: I have deliberately left rather vague the denition of the phase wave, and of the periodi phenomenon of whi h it must in some sense be the translation, as well as that of the light quantum. The present theory should therefore be

onsidered as one whose physi al ontent is not entirely spe ied, rather than as a onsistent and denitively onstituted do trine.

As de Broglie's on luding paragraph makes lear, his theory of 1924 was rather abstra t. There was no spe ied basis for the phase waves (they were ertainly not regarded as `material' waves); nor was any parti ular wave equation suggested. It should also be noted that at this time de Broglie's waves were real-valued fun tions of spa e and time, of the form ∝ sin(ωt − k · x), with a real os illating amplitude. They were not omplex waves ∝ ei(k·x−ωt) of uniform amplitude. Thus, de Broglie's `phase waves' had an os illating amplitude as well as a phase. (De Broglie seems to have alled them `phase waves' only be ause of his theorem of phase harmony.) Note also that, in his treatment of parti les

54

De Broglie's pilot-wave theory

in a box, de Broglie superposes waves propagating in opposite dire tions, yielding stationary waves whose amplitudes os illate in time. In his thesis de Broglie does not expli itly dis uss dira tion or interferen e experiments with ele trons, even though in his se ond ommuni ation of 1923 (de Broglie 1923b) he had suggested ele tron dira tion as an experimental test. A

ording to de Broglie's later re olle tions (L. de Broglie, AHQP Interview, 7 January 1963, p. 6),3 at his thesis defen e on 25 November 1924: Mr Jean Perrin, who haired the ommittee, asked me if my ideas ould lead to experimental onrmation. I replied that yes they ould, and I mentioned the dira tion of ele trons by rystals. Soon afterwards, I advised Mr Dauvillier .... to try the experiment, but, absorbed by other resear h, he did not do it. I do not know if he believed, or if he said to himself that it was perhaps very un ertain, that he was going to go to a lot of trouble for nothing  it's possible. .... But the following year it was dis overed in Ameri a by Davisson and Germer.

2.2.3 Opti al interferen e fringes: November 1924 On 17 November 1924, just a few days before de Broglie defended his thesis, a further ommuni ation of de Broglie's was presented to the A ademy of S ien es: entitled `On the dynami s of the light quantum and interferen e', and published in the Comptes Rendus, this short note gave a new and improved a

ount of opti al interferen e in terms of light quanta (de Broglie 1924d). De Broglie began his note by re alling his unsatisfa tory dis ussion of opti al interferen e in his re ent work on the quantum theory (p. 1039): .... I had not rea hed a truly satisfying explanation for the phenomena of wave opti s whi h, in prin iple, all ome down to interferen e. I limited myself to putting forward a ertain onne tion between the state of interferen e of the waves and the probability for the absorption of light quanta by matter. This viewpoint now seems to me a bit arti ial and I tend towards adopting another, more in harmony with the broad outlines of my theory itself.

As we have seen, in his thesis de Broglie was unsure about how to a

ount for the bright and dark fringes observed in opti al interferen e experiments. He did not have a theory of the ele tromagneti eld, whi h he assumed emerged as some sort of average over his phase waves. To a

ount for opti al fringes, he had suggested that the phase waves somehow determined the probability for intera tions between photons and the atoms in the apparatus. Now, after ompleting his thesis, he

2.2 A new approa h to parti le dynami s: 192324

55

felt he had a better explanation, that was based purely on the spatial distribution of the photon traje tories. De Broglie's note ontinues by outlining his `new dynami s', in whi h the energy-momentum 4-ve tor of every material point is proportional to the ` hara teristi ' 4-ve tor of an asso iated wave, even when the wave undergoes interferen e or dira tion. He then gives his new view of interferen e fringes (p. 1040): The rays predi ted by the wave theories would then be in every ase the possible traje tories of the quantum. In the phenomena of interferen e, the rays be ome on entrated in those regions alled `bright fringes' and be ome diluted in those regions alled `dark fringes'.

In my rst explanation of

interferen e, the dark fringes were dark be ause the a tion of fragments of light on matter was zero there; in my urrent explanation, these fringes are dark be ause the number of quanta passing through them is small or zero.

Here, then, de Broglie explains bright and dark fringes simply in terms of a high or low density of photon traje tories in the orresponding regions. When de Broglie speaks of the traje tories being on entrated and diluted in regions of bright and dark fringes respe tively, he presumably had in mind that the number density of parti les in an interferen e zone should be proportional to the lassi al wave intensity, though he does not say this expli itly. We an dis ern the essen e of the more pre ise and omplete explanation of opti al interferen e given by de Broglie three years later in his Solvay report: there, de Broglie has the same photon traje tories, with a number density spe ied as proportional to the amplitude-squared of the guiding wave (see pp. 384 f., and our dis ussion in se tion 6.1.1). In his note of November 1924, de Broglie goes on to illustrate his proposal for the ase of Young's interferen e experiment with two pinholes a ting as point sour es. De Broglie ites the well-known fa ts that in this ase the surfa es of equal phase are ellipsoids with the pinholes as fo i, and that the rays (whi h are normal to the ellipsoidal surfa es) are on entrated on hyperboloids of onstru tive interferen e where the

lassi al intensity has maxima. He then notes (p. 1040): Let let

r1 and r2 be the distan es from a point in spa e ψ be the fun tion 12 (r1 + r2 ), whi h is onstant on

to the two holes and ea h surfa e of equal

phase. One easily shows that the phase velo ity of the waves along a ray is equal to the value it would have in the ase of free propagation divided by the derivative of

ψ

taken along the ray; as for the speed of the quantum, it will

be equal to the speed of free motion multiplied by the same derivative.

De Broglie gives no further details, but these are easily re onstru ted.

56

De Broglie's pilot-wave theory

For an in ident beam of wavelength λ = 2π/k , ea h pinhole a ts as a sour e of a spheri al wave of wavelength λ, yielding a resultant wave proportional to the real part of ei(kr1 −ωt) ei(kr2 −ωt) + . r1 r2

(25)

If the pinholes have a separation d, then at large distan es (r1 , r2 >> d) from the s reen the resultant wave may be approximated as   ei(kr−ωt) kd 2 (26) cos θ r 2 where θ is the (small) angular deviation from the normal to the s reen. The amplitude shows the well-known interferen e pattern. As for the phase φ = kr − ωt, with r = 12 (r1 + r2 ), the surfa es of equal phase are indeed the well-known ellipsoids. Further, the phase velo ity is given by vph =

|∂φ/∂t| ω 1 c = = , |∇φ| k |∇r| |∇r|

(27)

while de Broglie's parti le velo ity  given by (22)  has magnitude v=

ℏ |∇φ| k ℏ |∇φ| = = c2 |∇r| = c |∇r| 2 m (ℏω/c ) ω

(28)

(where m is the relativisti photon mass), in agreement with de Broglie's assertions. At the end of his note, de Broglie omments that this method may be applied to the study of s attering. In November 1924, then, de Broglie understood how interfering phase waves would ae t photon traje tories, ausing them to bun h together in regions oin iding with the observed bright fringes. Note that in this paper de Broglie treats his phase waves as if they were a dire t representation of the ele tromagneti eld. In his dis ussion of Young's interferen e experiment, he has phase waves emerging from the two holes and interfering, and he identies the interferen e fringes of his phase waves with opti al interferen e fringes. However, he seems quite aware that this is a simpli ation,a remarking that `the whole theory will be ome truly lear only if one manages to dene the stru ture of the light wave'. De Broglie is still not sure about the pre ise relationship between his phase waves and light waves, a situation that persists even until the a De Broglie may have thought of this as analogous to s alar wave opti s, whi h predi ts the orre t opti al interferen e fringes by treating the ele tromagneti eld simply as a s alar wave.

2.3 Towards a omplete pilot-wave dynami s: 192527

57

fth Solvay onferen e: there, while he gives (pp. 384 f.) a pre ise a

ount of opti al interferen e in his report, he points out (p. 509) that the onne tion between his guiding wave and the ele tromagneti eld is still unknown.

2.3 Towards a omplete pilot-wave dynami s: 192527 On 16 De ember 1924, Einstein wrote to Lorentz:a A younger brother of the de Broglie known to us [Mauri e de Broglie℄ has made a very interesting attempt to interpret the Bohr-Sommerfeld quantization rules (Paris Dissertation, 1924).

I believe that it is the rst feeble ray of light

to illuminate this, the worst of our physi al riddles.

I have also dis overed

something that supports his onstru tion.

What Einstein had dis overed, in support of de Broglie's ideas, appeared in the se ond of his famous papers on the quantum theory of the ideal gas (Einstein 1925a). Einstein showed that the u tuations asso iated with the new Bose-Einstein statisti s ontained two distin t terms that

ould be interpreted as parti le-like and wave-like ontributions  just as Einstein had shown, many years earlier, for bla kbody radiation. Einstein argued that the wave-like ontribution should be interpreted in terms of de Broglie's matter waves, and he ited de Broglie's thesis. It was largely through this paper by Einstein that de Broglie's work be ame known outside Fran e. In the same paper, Einstein suggested that a mole ular beam would undergo dira tion through a su iently small aperture. De Broglie had already made a similar suggestion for ele trons, in his se ond ommuni ation to the Comptes Rendus (de Broglie 1923b). Even so, in their report at the fth Solvay onferen e, Born and Heisenberg state that in his gas theory paper Einstein `dedu ed from de Broglie's daring theory the possibility of dira tion of material parti les' (p. 426), giving the in orre t impression that Einstein had been the rst to see this onsequen e of de Broglie's theory. It seems likely that Born and Heisenberg did not noti e de Broglie's early papers in the Comptes Rendus. In 1925 Elsasser  a student of Born's in Göttingen  read de Broglie's thesis. Like most others outside Fran e, Elsasser had heard about de Broglie's thesis through Einstein's gas theory papers. Elsasser suspe ted that two observed experimental anomalies ould be explained a Quoted in Mehra and Re henberg (1982a, p. 604).

58

De Broglie's pilot-wave theory

by de Broglie's new dynami s. First, the Ramsauer ee t  the surprisingly large mean free path of low-velo ity ele trons in gases  whi h Elsasser thought ould be explained by ele tron interferen e. Se ond, the intensity maxima observed by Davisson and Kunsman at ertain angles of ree tion of ele trons from metal surfa es, whi h had been assumed to be aused by atomi shell stru ture, and whi h Elsasser thought were

aused by ele tron dira tion. Elsasser published a short note sket hing these ideas in Die Naturwissens haften (Elsasser 1925). Elsasser then tried to design an experiment to test the ideas further, with low-velo ity ele trons, but never arried it out. A

ording to Heisenberg's later re olle tion, Elsasser's supervisor Born was s epti al about the reality of matter waves, be ause they seemed in oni t with the observed parti le tra ks in loud hambers.a On 3 November 1925, S hrödinger wrote to Einstein: `A few days ago I read with the greatest interest the ingenious thesis of Louis de Broglie .... '.a S hrödinger too had be ome interested in de Broglie's thesis by reading Einstein's gas theory papers, and he set about trying to nd the wave equation for de Broglie's phase waves. As we have seen (se tion 2.2.2), in his thesis de Broglie had shown that, in an ele trostati potential ϕ, the phase wave of an ele tron of harge e and velo ity v would have (see equation (19)) frequen y ν = p (mc2 + V )/h and phase 2 velo ity vph = (mc + V )/mv , where m = m0 / 1 − v 2 /c2 and V = eϕ is the potential energy. These expressions for ν and vph , given by de Broglie, formed the starting point for S hrödinger's work on the wave equation. S hrödinger took de Broglie's formulas for ν and vph and applied them to the hydrogen atom, with a Coulomb eld ϕ = −e/r.b Using the formula for ν to eliminate v , S hrödinger rewrote the expression for the phase velo ity vph purely in terms of the frequen y ν and the ele tronproton distan e r: vph =

hν q m0 c

1 (hν/m0

c2

− V /m0

2 c2 )

(29) −1

a For this and further details on erning Elsasser, see Mehra and Re henberg (1982a, pp. 6247). See also the dis ussion in se tion 3.4.2. a Quoted in Mehra and Re henberg (1987, p. 412). b Here we follow the analysis by Mehra and Re henberg (1987, pp. 4235) of what they all S hrödinger's `earliest preserved [unpublished℄ manus ript on wave me hani s'. Similar reasoning is found in a letter from Pauli to Jordan of 12 April 1926 (Pauli 1979, pp. 31520).

2.3 Towards a omplete pilot-wave dynami s: 192527

59

(where V = −e2 /r). Then, writing de Broglie's phase wave as ψ = ψ(x, t), he took the equation for ψ to be the usual wave equation ∇2 ψ =

1 ∂2ψ 2 ∂t2 vph

(30)

with phase velo ity vph . Assuming ψ to have a time dependen e ∝ e−2πiνt , S hrödinger then obtained the time-independent equation ∇2 ψ = −

4π 2 ν 2 ψ 2 vph

(31)

with vph given by (29). This was S hrödinger's original (relativisti ) equation for the energy states of the hydrogen atom. As is well known, S hrödinger found that the energy levels predi ted by (31)  that is, the eigenvalues hν  disagreed with experiment. He then adopted a nonrelativisti approximation, and found that this yielded the orre t energy levels for the low-energy limit. It is in fa t straightforward to obtain the orre t nonrelativisti limit. Writing E = hν − m0 c2 , in the low-energy limit |E| /m0 c2