DIRAC

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Dirac
A SCIENTIFIC BIOGRAPHY HELGE KRAGH
The right of the University of Cambridge
to print and sell all manner of books
was granted by Henry VIII in 1534. The University has printed and published continuously
since 1584.
CAMBRIDGE UNIVERSITY PRESS Cambridge
New York Port Chester Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia
© Cambridge University Press 1990
First published 1990 Reprinted 1991
Printed in the United States of America

Library of Congress Cataloging-in-Publication Data

Kragh, Helge, 1944Dirac : a scientific biography / Helge Kragh.

p. cm. Bibliography: p.

Includes indexes. ISBN 0-521-38089-8 1. Dirac, P. A. M. (Paul Adrien Maurice), 1902-

2. Physicists - Great Britain - Biography. I. Title.

DC16.D57K73 1990 530'.092 - dc20

M

89-17257

CIP

ISBN 0-521-38089-8 hardback

CONTENTS

Preface 1. Early years 2. Discovery of quantum mechanics 3. Relativity and spinning electrons 4. Travels and thinking 5. The dream of philosophers 6. Quanta and fields 7. Fifty years of a physicist’s life 8. “The so-called quantum electrodynamics” 9. Electrons and ether 10. Just a disappointment 11. Adventures in cosmology 12. The purest soul 13. Philosophy in physics 14. The principle of mathematical beauty
Appendix I. Dirac bibliometrics Appendix II. Bibliography of P. A. M. Dirac Notes and references General bibliography Index of names Index of subjects

page ix 1
14 48 67 87 118 151 165 189 205 223 247 260 275
293 304 315 364 383 387

vii

.

PREFACE
ONE of the greatest physicists who ever lived, P. A. M. Dirac (1902-84) made contributions that may well be compared with those of other, better known giants of science such as Newton, Maxwell, Einstein, and Bohr. But unlike these famous men, Dirac was virtually unknown outside the physics community. A few years after his death, there have already appeared two memorial books [Kursunoglu and Wigner (1987) and Taylor (1987)], a historically sensitive biographical memoir [Dalitz and Peierls (1986)], and a detailed account of his early career in physics [Mehra and Rechenberg (1982+), vol. 4], These works, written by scientists who knew Dirac personally, express physicists’ hom¬ age to a great colleague. In some respects it may be an advantage for a biographer to have known his subject personally, but it is not always or necessarily an advantage. I have never met Dirac.
The present work, though far from claiming completeness, aims to sup¬ plement the volumes mentioned above by providing a more comprehen¬ sive and coherent account of Dirac’s life and contributions to science. Because Dirac was a private person, who identified himself very much with his physics, it is natural to place emphasis on his scientific work, which, after all, has secured his name’s immortality. Most of the chapters (2, 3, 5-6 and 8-11) are essentially accounts of these contributions in their historical context, but a few chapters are of a more personal nature. Taking the view that a scientific biography should deal not only with the portrayed scientist’s successes but also with his failures, I present rela¬ tively detailed accounts (Chapters 8, 9, and 11) of parts of Dirac’s work that are today considered either failures or less important but that nev¬ ertheless commanded his commitment and occupied his scientific life. Other chapters (f, 4, 7, and 12) are almost purely biographical. Chapter 12 attempts a portrait of the person of whom Bohr once remarked, “of all physicists, Dirac has the purest soul.” In addition to describing Dirac’s life and science, I have also, in Chapters 13 and 14, attempted to consider
IX

X

Preface

his views of physics in its more general, philosophical aspect. Two appen¬ dixes, including a bibliography, deal with Dirac’s publications from a quantitative point of view.
During work on this book, I have consulted a number of libraries and archives in search of relevant material and have used sources from the following places: Bohr Scientific Correspondence, Niels Bohr Institute, Copenhagen; Archive for History of Quantum Physics, Niels Bohr Insti¬ tute; Schrodinger Nachlass, Zentralbibliothek fur Physik, Vienna; Bethe Papers, Cornell University Archive, Ithaca; Manuscript Division, Library of Congress, Washington; Dirac Papers, Churchill College, Cam¬ bridge, now moved to Florida State University, Tallahassee; Centre of History of Physics, American Institute of Physics, New York; Ehrenfest Archive, Museum Boerhaave, Leiden; Nobel Archive, Royal Swedish Academy of Science, Stockholm; Sussex University Library; and Standiger Arbeitsausschuss fur die Tagungen der Nobelpreistrager, Lindau. I am grateful for permission to use and quote material from these sources. The many letters excerpted in the text are, if written in English, quoted liter¬ ally; this accounts for the strange English usage found in letters by Pauli, Gamow, Ehrenfest, Heisenberg, and others. I would like to thank the fol¬ lowing people for providing information and other assistance: Karl von Meyenn, Sir Rudolf Peierls, Abraham Pais, Luis Alvarez, Sir Nevill Mott, Silvan Schweber, Helmuth Rechenberg, Olivier Darrigol, Kurt Gottfried, Ulrich Roseberg, Aleksey Kozhevnikov, Richard Eden, Finn Aaserud, and Carsten Jensen. Special thanks to Robert Corby Hovis for his careful editing of the manuscript and many helpful suggestions.

November 1988

Helge Kragh Ithaca, New York

CHAPTER 1
EARLY YEARS
PAUL DIRAC signed his scientific papers and most of his letters P. A. M: Dirac, and for a long time, it was somewhat of a mystery what the initials stood for. Dirac sometimes seemed reluctant to take away that mystery. At a dinner party given for him when he visited America in 1929 - when he was already a prominent physicist - the host decided to find out the first names of his honored guest. At each place around the table, he placed cards with different guesses as to what P.A.M. stood for, such as Peter Albert Martin or Paul Alfred Matthew. Having studied the cards, Dirac said that the correct name could be obtained by a proper combination of the names on cards. After some questioning, the other guests were able to deduce that the full name of their guest of honor was Paul Adrien Maurice Dirac.1
Dirac got his French-sounding name from his father, Charles Adrien Ladislas Dirac, who was Swiss by birth. Charles Dirac was born in 1866 in Monthey in the French-speaking canton Valais, and did not become a British citizen until 1919. At age twenty he revolted against his parents and ran away from home. After studies at the University of Geneva, he left around 1890 for England, where he settled in Bristol. In England Charles made a living by teaching French, his native language, and in 1896 he was appointed a teacher at the Merchant Venturer’s Technical College in Bristol. There he met Florence Hannah Holten, whom he mar¬ ried in 1899. Florence was the daughter of a ship’s captain and was twelve years younger than Charles. The following year they had their first child, Reginald Charles Felix, and two years later, on August 8, 1902, Paul Adrien Maurice was born. At that time, the family lived in a house on Monk Road.2 The third child of the Dirac family was Beatrice Isabelle Marguerite, who was four years younger than Paul.
For many years, Charles Dirac seems to have retained his willful iso¬ lation from his family in Switzerland; they were not even informed of his marriage or first children. However, in 1905 Charles visited his mother
l

2

Dirac: A scientific biography

in Geneva, bringing his wife and two children with him. At that time, Charles’s father had been dead for ten years. Like his brother and sister, Paul was registered as Swiss by birth, and only in 1919, when he was seventeen years old, did he acquire British nationality.
Paul’s childhood and youth had a profound influence on his character throughout his entire life, an influence that resulted primarily from his father’s peculiar lack of appreciation of social contacts. Charles Dirac was a strong-willed man, a domestic tyrant. He seems to have dominated his family and to have impressed on them a sense of silence and isolation. He had a distaste for social contacts and kept his children in a virtual prison as far as social life was concerned. One senses from Paul Dirac’s reminiscences a certain bitterness, if not hatred, toward his father, who brought him up in an atmosphere of cold, silence, and isolation. “Things contrived early in such a way that I should become an introvert,” he once pathetically remarked to Jagdish Mehra.3 And in another interview in 1962, he said, “In those days I didn’t speak to anybody unless I was spo¬ ken to. I was very much an introvert, and I spent my time thinking about problems in nature.”4 When his father died in 1936, Paul felt no grief. “I feel much freer now,” he wrote to his wife.5 In 1962 he said:6

In fact I had no social life at all as a child.... My father made the rule that I should only talk to him in French. He thought it would be good for me to learn French in that way. Since I found I couldn’t express myself in French, it was better for me to stay silent than to talking English. So I became very silent at that time - that started very early ...

Paul also recalled the protocol for meals in the Dirac house to have been such that he and his father ate in the dining room while his mother, who did not speak French well, ate with his brother and sister in the kitchen. This peculiar arrangement, which contributed to the destruction of the social relationship within the family, seems to have resulted from Charles’s strict insistence that only French should be spoken at the dinner table.
Unlike other great physicists - Bohr, Heisenberg, and Schrbdinger, for example - Paul Dirac did not grow up under conditions that were cul¬ turally or socially stimulating. Art, poetry, and music were unknown ele¬ ments during his early years, and discussions were not welcomed in the house on Monk Road. Whatever ideas he had, he had to keep them to himself. Perhaps, as Paul once intimated, Charles Dirac’s dislike of social contacts and the expression of human feelings was rooted in his own childhood in Switzerland. “I think my father also had an unhappy child¬ hood,” Paul said.7
Paul lived with his parents in their home in Bristol until he entered

Early years

3

Cambridge University in 1923. The young Dirac was shy, retiring, and uncertain about what he wanted from the future. He had little to do with other boys, and nothing at all to do with girls. Although he played a little soccer and cricket, he was neither interested in sports nor had any success in them. “He haunted the library and did not take part in games,” recalled one of his own schoolmates. “On the one isolated occasion I saw him handle a cricket bat, he was curiously inept.”8 One incident illus¬ trates the almost pathological antisocial attitude he carried with him from his childhood: In the summer of 1920, he worked as a student apprentice in Rugby at the same factory where his elder brother Reginald was employed. The solitary Paul, who had never been away from home, often met his brother in the town, but when they met, they did not even talk to each other! “If we passed each other in the street,” remarked Paul, “we didn’t exchange a word.”9
Charles Dirac’s bringing-up of his children must have been emotionally crippling for them. Paul’s father resented any kind of social contact, and his mother wished to protect him from girls. As a young man, Paul never had a girlfriend and seems to have had a rather Platonic conception of the opposite sex for a long time. According to Esther Salaman, an author and good friend of Dirac, he once confided to her: “I never saw a woman naked, either in childhood or youth.... The first time I saw a woman naked was in 1927, when I went to Russia with Peter Kapitza. She was a child, an adolescent. I was taken to a girls’ swimming-pool, and they bathed without swimming suits. I thought they looked nice.”10 He was not able to revolt against his father’s influence and compensated for the lack of emotional and social life by concentrating on mathematics and physics with a religious fervor. Paul’s relationship with his father was cold and strained; unable to revolt openly, his subconscious father-hatred manifested itself in isolation and a wish to have as little personal contact with his father as possible. Charles Dirac may have cared for his children and especially for Paul, whose intelligence Charles seems to have been proud of; but the way in which he exercised his care only brought alien¬ ation and tragedy. He was highly regarded as a teacher and was notorious for his strict discipline and meticulous system of punishment. Ambitious on behalf of his children, he wanted to give them as good an education as possible. But his pathological lack of human understanding and his requirement of discipline and submission made him a tyrant, unloving and unloved. Charles Dirac died in 1936; his wife, five years later.
Even more than Paul, his brother Reginald suffered from the way the Dirac children were brought up. Both the lack of social contact and a feeling of intellectual inferiority to his younger brother made Reginald depressed. He wanted to become a doctor, but his father forced him to study engineering, in which he graduated with only a third-class degree

4

Dirac: A scientific biography

in 1919. His life ended tragically in 1924 when, on an engineering job in Wolverhampton, he committed suicide.
Young Paul was first sent to the Bishop Road primary school and then, at the age of twelve, to the school where his father was a teacher, the Merchant Venturer’s College. Unlike most schools in England at the time, this school did not emphasize classics or the arts but concentrated instead on science, practical subjects, and modern languages. Paul did well in school without being particularly brilliant. Only in mathematics did he show exceptional interest and ability. This subject fascinated him, and he read many mathematics books that were advanced for his age. The education he received was a good and modern one, but it lacked the clas¬ sical and humanistic elements that were taught at schools on the Conti¬ nent and at other British schools. Heisenberg, Pauli, Bohr, Weyl, and Schrodinger received a broader, more traditional education than did Dirac, who was never confronted with Greek mythology, Latin, or clas¬ sical poetry. Partly as a result of his early education and his father’s influ¬ ence, his cultural and human perspectives became much narrower than those of his later colleagues in physics. Not that Dirac ever felt attracted to these wider perspectives or would have wanted a more traditional edu¬ cation; on the contrary, he considered himself lucky to have attended the Merchant Venturer’s College. “[It] was an excellent school for science and modern languages,” he recalled in 1980. “There was no Latin or Greek, something of which I was rather glad, because I did not appreciate the value of old cultures.”11
In the compulsory school system, Paul was pushed into a higher class and thus finished when he was only sixteen years old. But this early pro¬ motion was not because he was regarded as extraordinarily brilliant for his age, as he recalled in 1979:12

All the young men had been taken away from the universities to serve in the army. There were some professors left, those who were too old to serve in the army and those who were not physically fit; but they had empty classrooms. So the younger boys were pushed on, as far as they were able to absorb the knowl¬ edge, to fill up these empty classrooms.

Paul had no particular idea about what profession to go into and seems to have been a rather silent and dependent boy who just did as he was told. “I did not have much initiative of my own,” he told Mehra. “[My] path was rather set out for me, and I did not know very well what I wanted.”13 In 1918, Paul entered the Engineering College of Bristol Uni¬ versity as a student of electrical engineering - not because he really wanted to become an engineer, but because this seemed the most natural and smooth career. His elder brother Reginald had also studied at the

Early years

5

Engineering College, which was located in the same buildings that housed the Merchant Venturer’s College. Paul was thus in familiar surroundings.
The lack of initiative and independence that characterized Paul’s per¬ sonality at the time may partly explain why he did not choose to study mathematics, the only subject he really liked. He also believed that as a mathematician he would have to become a teacher at the secondary school, a job he did not want and in which he would almost certainly have been a failure. A research career was not in his mind.
During his training as a student of electrical engineering, Paul came into close contact with mathematics and the physical sciences. He studied all the standard subjects (materials testing, electrical circuits, and electro¬ magnetic waves) and the mathematics necessary to master these and other technical subjects. He enjoyed the theoretical aspects of his studies but felt a vague dissatisfaction with the kind of engineering mathematics he was taught. Although his knowledge of physics and mathematics was much improved at Bristol University, it was, of course, the engineering aspects of and approaches to these subjects that he encountered there. Many topics were not_cpnsidered relevant to the engineer and were not included in the curriculum. For example, neither atomic physics nor Maxwell’s electrodynamic theory was taught systematically. And, of cduTsCsuch'a modern and “irrelevant” subject as the theory of relativity was also absent from the formal curriculum.
During his otherwise rather dull education as an engineeffWe event became of decisive importance to Paul’s later career: the emergence into public prominence of Einstein’s theory of relativity, which was mainly caused by the spectacular confirmation of the general theory made by British astronomers in 1919. In that year, Frank Dyson and Arthur Eddington announced that solar eclipse observations confirmed the bending of starlight predicted by Einstein.14 The announcement created a great stir, and suddenly relativity (at the time fourteen years old) was on everybody’s lips. Dirac, who knew nothing about relativity, was fasci¬ nated and naturally wanted to understand the theory in a deeper way than the newspaper articles allowed. He recalled:15

It is easy to see the reason for this tremendous impact. We had just been living through a terrible and very serious war... . Everyone wanted to forget it. And then relativity came along as a wonderful idea leading to a new domain of thought. It was an escape from the war.... At this time I was a student at Bristol University, and of course I was caught up in this excitement produced by relativ¬ ity. We discussed it very much. The students discussed it among themselves, but had very little accurate information to go on. Relativity was a subject that every¬ body felt himself competent to write about in a general philosophical way. The philosophers just put forward the view that everything had to be considered rel-

6

Dirac: A scientific biography

atively to something else, and they rather claimed that they had known about relativity all along.

In 1920-1, together with some of his fellow engineering students, Dirac attended a course of lectures on relativity given by the philosopher Char¬ lie D. Broad, at the time a professor at Bristol. These lectures dealt with the philosophical aspects of relativity, not with the physical and mathe¬ matical aspects, which Dirac would have preferred. Although he did not appreciate Broad’s philosophical outlook, the lectures inspired him to think more deeply about the relationship between space and time. Ever since that time, Dirac was firmly committed to the theory of relativity, with which he soon became better acquainted. His first immersion in the subject was Eddington’s best-selling Space, Time and Gravitation, pub¬ lished in 1920, and before he completed his subsequent studies in math¬ ematics at Bristol University, he had mastered both the special and gen¬ eral theories of relativity, including most of the mathematical apparatus.
While Paul did very well in the theoretical engineering subjects, he was neither interested in nor particularly good at the experimental and tech¬ nological ones. Probably he would never have become a good engineer, but his skills were never tested. After graduating with first-class honors in 1921, he looked for employment but was unable to find a job. Not only were his qualifications not the best, but at the time the unemployment rate was very high in England because of the economic depression. After some time with nothing to do, Paul was lucky enough to be offered free tuition to study mathematics at Bristol University. He happily accepted.
From 1921 to 1923, Dirac studied mathematics, specializing in applied mathematics. Although he did no research of his own, he studied dili¬ gently and was introduced to the world of pure mathematical reasoning, which was very different in spirit from the engineering approach encoun¬ tered in his earlier studies. The mathematicians at Bristol were not much oriented toward research, but Dirac had excellent teachers in Peter Fraser and H. R. Hasse, who soon recognized his outstanding abilities. Fraser particularly impressed Dirac, who described him many years later as “a wonderful teacher, able to inspire his students with real excitement about basic ideas in mathematics.”16 Both Fraser and Hasse were Cambridge men and thought that Dirac ought to continue for graduate studies at that distinguished university. Dirac completed his examinations at Bristol University with excellent results in the summer of 1923. Thanks to a grant from the Department of Scientific and Industrial Research (DSIR), he was able to enroll at Cambridge in the fall of 1923.
This was not the first time that Dirac visited Cambridge. After gradu¬ ating in engineering in the summer of 1921, at his father’s request he went to the famous university city to be examined for a St. John’s College

Early years

7

Exhibition Studentship. He passed the examination and was offered the studentship, which was worth seventy pounds per year. But since he was unable to raise additional funds and his father was unable or unwilling to support him at Cambridge, he had to return to his parents in Bristol. It was only when he was awarded a DSIR studentship in addition, in 1923, that Dirac was finally able to attend Cambridge.
At Cambridge a new chapter in his life began, leading to his distin¬ guished career as a physicist. He was away from his parents and the scarcely stimulating intellectual environment of Bristol, and at first he was not sure that he was really capable of succeeding in a research career. Cambridge, with its great scientific traditions, was a very different place from Bristol. The twenty-one-year-old Dirac arrived at a university that housed not only established scientists such as Larmor, Thomson, Ruth¬ erford, Eddington, and Jeans, but also rising stars including Chadwick, Blackett, Fowler, Milne, Aston, Hartree, Kapitza, and Lennard-Jones.17 Dirac was admitted to St. John’s College but during some periods lived in private lodgings because there were not enough rooms at the college. During most of 1925, he lived at 55 Alpha Road, only a few hundred meters from St. John’s.
As a research student he had to have a supervisor who would advise on, or determine, the research topic on which he would work. With his limited scientific experience and lack of acquaintance with most of the Cambridge physicists, Dirac wanted to have Ebenezer Cunningham as his supervisor and to pursue research in the theory of relativity. He knew Cunningham from his earlier examination in 1921 and knew that he was a specialist in electromagnetic theory and the author of books and articles on electron theory and relativity.18 Cunningham, who taught at St. John’s College from 1911 to 1946, was only forty-two years old in 1923. He had been a pioneer of relativity in England, but found it difficult to follow the new physics of the younger generation and did not want to take on any more research students. “I just felt they’d run away from me. I was lost,” he said.19 Consequently, Dirac was assigned to Ralph Fowler. This was undoubtedly a happy choice, since Cunningham belonged to the old school of physics whereas Fowler was one of the few British physicists who had an interest and competence in advanced atomic theory. How¬ ever, Fowler’s field was not relativity, and at first Dirac felt disappointed not to have Cunningham as his supervisor.
Fowler was the main exponent of modern theoretical physics at Cam¬ bridge and the only one with a firm grip on the most recent developments in quantum theory as it was evolving in Germany and Denmark. He had good contacts with the German quantum theorists and also particularly with Niels Bohr in Copenhagen. In addition, he was about the only con¬ tact between the theorists and the experimentalists at the Cavendish.

8

Dirac: A scientific biography

However, as a supervisor for research students he was somewhat undis¬ ciplined. He was often abroad, and when at Cambridge he was difficult to find. Alan Wilson, who was a research student under Fowler in 1926— 7, recalled that “Fowler, like the rest of us, worked in his college rooms - in Trinity - and if you wanted to consult him you had to drop in half a dozen times before you could find him in. He lived in Trumpington and did most of his work there.”20 The retiring Dirac probably did not consult Fowler often. Fowler’s main interests were in the quantum theory of atoms and in statistical mechanics, including the application of these fields to astrophysics. In the summer of 1923, Dirac was largely ignorant of atomic theory and statistical physics, fields he found much less inter¬ esting than those he knew most about, electrodynamics and relativity. But as Fowler’s research student, he was forced to learn the new subjects and soon discovered that they were far from uninteresting:21

Fowler introduced me to quite a new field of interest, namely the atom of Ruth¬ erford, Bohr and Somerfeld. Previously I had heard nothing about the Bohr the¬ ory, it was quite an eyeopener to me. I was very much surprised to see that one could make use of the equations of classical electrodynamics in the atom. The atoms were always considered as very hypothetical things by me, and here were people actually dealing with equations concerned with the structure of the atom.
Dirac worked hard to master the new students and to improve his knowl¬ edge of subjects he had learned at Bristol on a level not commensurate with the higher standards of Cambridge. He did well. Most of atomic the¬ ory he learned either from Fowler or by studying research papers in Brit¬ ish and foreign journals available in the Cambridge libraries. He knew sufficient German to read articles in the Zeitschrift fur Physik, the leading vehicle for quantum theory, and to read Arnold Sommerfeld’s authori¬ tative Atombau und Spektrallinien. Within a year, Dirac became fully acquainted with the quantum theory of atoms. As to mathematical meth¬ ods, he scrutinized Whittaker’s Analytical Dynamics, which became the standard reference work for him. From this book, written by a mathe¬ matician and former Cambridge man, he learned the methods of Ham¬ iltonian dynamics and general transformation theory, both of which became guiding principles in his later work in quantum theory. At the same time he improved his knowledge of the theory of relativity by stud¬ ying Eddington’s recently published The Mathematical Theory of Rela¬ tivity, and also by attending Eddington’s lectures, which in one term cov¬ ered special and general relativity and tensor analysis. Occasionally, Dirac had the opportunity to discuss questions with Eddington himself, an experience of which he remarked, “It was really a wonderful thing to

Early years

9

meet the man who was the fountainhead of relativity so far as England was concerned.”22
In addition to studying Eddington’s book and attending his lectures, Dirac also followed Cunningham’s course of lectures on electromagnetic theory and special relativity.23 Forty years later, Cunningham recalled a time when he had worked out a long calculation on the electrical and magnetic components of the radiation field:24

I said to the class one day, I remember, “This is an extraordinarily simple result in the end, but why? Why should it work out like this?” A week later, a young man who had only been in Cambridge a year or two, a year I think, came up to me and said, “Here you are.” That was Dirac.

Another student who followed Cunningham’s course in 1923 was John Slater, a postdoctoral research student from Harvard who was in Europe on a traveling fellowship. But, characteri stic of the remoteness of students from each other in Cambridge in those days, it was years later before Slater and Dirac realized they had attended the same course.25 The Bel¬ gian George Lemaitre, who a few years later would revolutionize cos¬ mology, was a research student under Eddington in 1923-4 and also attended some of the same courses as Dirac, but it was a decade later before he and Dirac became acquainted (see also Chapter 11). Still another student who followed some of the same courses as Dirac including those of Fowler, Cunningham, and Eddington - was Llewellyn H. Thomas, who received his B.A. in 1924 and stayed in Cambridge until 1929. He recalled the young Dirac as a quiet man who made no major impression .at Cambridge until he published his papers on quantum mechanics. “He is a man of few words,” Thomas said in 1962. “If you ask him a question, he’d say oh, that’s very difficult. Then a week later he’d come back with the complete answer completely worked out.”2^
Thanks to the stimulating Cambridge environment, Dirac’s scientific perspective became much wider. For the first time, he came in contact with the international research fronts of theoretical physics. As he met more people and established contacts with loose social groups, Dirac gradually became a little less shy and introverted. He recalled attending the combined tea parties and geometry colloquia that took place weekly at the home of Henry Baker, the professor of geometry:27

These tea parties did very much to stimulate my interest in the beauty of math¬ ematics. The all-important thing there was to strive to express the relationships in a beautiful form, and they were successful. I did some work on projective geometry myself and gave one of the talks at one ol the tea parties. This was the first lecture I ever gave, and so of course I remember it very well.

10

Dirac: A scientific biography

Although there were several attractive academic clubs at Cambridge such as the Observatory Club, the Trinity Mathematical Society, and the Cavendish Society - Dirac restricted his interest to two: the V2V (delsquared) Club, which he joined in May 1924, and the Kapitza Club, which he joined in the fall of the same year. In both clubs membership was limited and was decided by election, and meetings took place in the college rooms, often in Dirac’s room in St. John’s. The V2V Club was mainly for mathematical physicists, who presented their own work at the meetings. Most Cambridge theorists were members of this club, which in 1924 included Eddington, Jeffreys, Milne, Chadwick, Hartree, Blackett, Fowler, Stoner, Kapitza, and Dirac. The Kapitza Club, an informal dis¬ cussion club where papers on recent developments in physics were read and discussed at Trinity on Tuesdays, was started in 1922 by the colorful Soviet physicist Peter Kapitza, then a research student under Rutherford at the Cavendish. Experimental physics had predominance in the Kapitza Club, contrary to the theoretical orientation of the V2V Club. After his election to the Kapitza Club in the fall of 1924, Dirac listened to lectures by distinguished foreign guests such as James Franck (October 1924) and Niels Bohr (May 1925). The meetings of the club continued with Kapitza in charge until the summer of 1934, when Kapitza was unable to return to Cambridge from a visit to his homeland (see also Chapter 7). By that time, the club had held 377 meetings, many of them with Dirac as a par¬ ticipant. He remained an active member of both clubs until the war, and in the fall term of 1930 he served as president of the V2V Club.28
Paul Dirac lived a quiet life in Cambridge, totally absorbed in studies and research. Theoretical physics belonged to the Mathematics Faculty, which did not have its own building. There was no tradition of social or professional contact between the few students of theoretical physics, who usually sat alone in their college rooms or in the small library - which also served as a tea room - at the Cavendish Faboratory. It was “a ter¬ ribly isolated business” to be a physics student at Cambridge, Nevill Mott recalled.29 Yet Dirac did not find the isolation terrible at all. Had he wanted to, he could have taken part in what little extramural student life there was; but he did not want to. He deliberately kept away from exter¬ nal activities - whether politics, sports, or girls - that might disturb his studies. According to his recollections:30

At that time, I was just a research student with no duties apart from research, and I concentrated all my energy in trying to get a better understanding of the prob¬ lems facing physicists at that time. I was not interested at all in politics, like most students nowadays. I confined myself entirely to the scientific work, and contin¬ ued at it pretty well day after day, except on Sundays when I relaxed and, if the weather was fine, I took a long solitary walk out in the country. The intention was

Early years

11

to have a rest from the intense studies of the week, and perhaps to try and get a new outlook with which to approach the problem the following Monday. But the intention of these walks was mainly to relax, and I had just the problems maybe floating about in the back of my mind without consciously bringing them up. That was the kind of life that I was leading.
Within an astonishingly short time, Dirac managed to transform himself from a student into a full-fledged scientist. After only half a year at Cam¬ bridge, in March 1924 he was able to submit his first scientific paper to the Proceedings of the Cambridge Philosophical Society, a local but inter¬ nationally recognized periodical.31 This paper dealt with a problem of sta¬ tistical mechanics suggested by Fowler, his supervisor. Neither the prob¬ lem nor the paper was of particular significance. It was merely an exercise, as debut papers often are. Dirac was then determined to become a research physicist and knew that he was good enough to contribute to the advancement of science, but he still did not have any definite ideas about which subject to specialize in. He had a preference for the funda¬ mental and general problems of physics but only had vague ideas about how to deal with these problems in a new way. As a result, his first papers dealt with a rather scattered field of specific problems, mostly in relativ¬ ity, quantum theory, and statistical mechanics (see the bibliography in Appendix II).
Dirac was very productive, publishing seven papers within two years, and succeeded making himself known to the small community of British theoretical physicists. Dirac’s ability to solve difficult theoretical prob¬ lems was soon noticed, both inside and outside Cambridge. Charles Galton Darwin, professor of natural philosophy (physics) at the University of Edinburgh and grandson of the famous naturalist, was told about the bright student by Fowler. Darwin asked Dirac to solve a mathematicalphysics problem with which he had occupied himself, namely, proving that quantizing a dynamical system results in the same answer no matter what coordinates are used. This was just the problem to suit Dirac’s
taste.32 Dirac’s early works appeared in the most recognized British journals
and were communicated by Fowler, Milne, Eddington, and Rutherford. His approach in the papers was to take an already known result, based on established theory, and to criticize it in order to reach a better under¬ standing. If possible, Dirac used relativistic arguments to discuss the results and make them more general:33

There was a sort of general problem which one could take, whenever one saw a bit of physics expressed in a nonrelativistic form, to transcribe it to make it fit in with special relativity. It was rather like a game, which I indulged in at every

12

Dirac: A scientific biography

opportunity, and sometimes the result was sufficiently interesting for me to be able to write up a little paper about it.
One of these little papers dealt with an astrophysical problem: how to calculate the red-shift of solar lines on the assumption that the radiation emitted from the interior of the sun is Compton scattered in the atmo¬ sphere of the sun. This problem was suggested by the mathematician and astronomer Edward A. Milne, who, in the first months of 1925, became Dirac’s supervisor while Fowler was on leave in Copenhagen. Dirac was not particularly interested in astrophysics, but he had followed Milne’s course of lectures on the physics of stellar atmospheres and had obtained a good knowledge of the field.34 He solved the problem suggested by Milne, concluding that the suggested mechanism could not account for the observed red-shift. This result ran against the expectations of Milne. Dirac did not deal with astrophysics again, but his later research covered topics also cultivated by Milne (see Chapter 11).
Another of the early papers dealt with a problem in the theory of rela¬ tivity concerning the definition of velocity.35 The problem had been stated by Eddington in The Mathematical Theory of Relativity, and he took a keen interest in Dirac’s paper before its publication. Eddington suggested various alterations, mainly of an editorial kind, which Dirac was glad to accept. Before communicating Dirac’s paper, Eddington com¬ mented on the manuscript: “[The paper needs] an introductory para¬ graph ... to run something like this ... you will no doubt reword this ... look at these points and let me have it back.”36
To make a long story short, Dirac’s situation in the summer of 1925 was as follows: He had proved to be a talented physicist with a flair for complex theoretical problems and the use of mathematical methods. He had earned himself a name in Cambridge as a promising theorist, but outside Britain he was unknown. His contributions were interesting, but not remarkably so, and not of striking originality. In retrospect, his first seven publications can be seen to have been groundwork for more com¬ plex problems, the nature of which was then still unknown to Dirac. He vaguely felt that he was ready for bigger prey, but it was only after Hei¬ senberg’s pioneering discovery of quantum mechanics that Dirac knew his true hunting ground. Then things happened very quickly and he meta¬ morphosed from a rather ordinary physicist into a natural philosopher whose name could rightly be placed alongside those of Maxwell and New¬ ton. Only ten years after he entered Cambridge University, Dirac received the Nobel Prize in physics.
What was Dirac’s life like before he found quantum mechanics? As mentioned, he lived a modest and undramatic life filled with physics and little else. His introvert character did not change much. Although he was

Early years

13

in contact with several of the Cambridge physicists, and Fowler in par¬ ticular, these contacts did not evolve into friendships. His contacts with other students at Cambridge were almost nil. Dirac spent much of his time alone in libraries and relaxed only on his solitary Sunday walks. “I did my work mostly in the morning,” he wrote. “Mornings I believe are the times when one’s brain power is at its maximum, and towards the end of the day I was more or less dull, especially after dinner.”37
At an early stage of his career, Dirac developed the concise style that was to characterize all of his writings. Conceptual clarity, directness, tech¬ nical accuracy, and logical presentation were virtues he cultivated from an early age. When writing a manuscript for a paper, he would first try to draw up the whole work in his mind. Only then would he write it down on paper in his meticulous handwriting, and this first draft would need few if any corrections. Niels Bohr, whose working habits and mental con¬ stitution were very different from Dirac’s, once remarked: “Whenever Dirac sends me a manuscript, the writing is so neat and free of corrections that merely looking at it is an aesthetic pleasure. If I suggest even minor changes, Paul becomes terribly unhappy and generally changes nothing at all.”38 In the same vein, Igor Tamm recounted an exchange that took place when Bohr read the proofs of one of Dirac’s papers:39

Bohr: “Dirac, why have you only corrected few misprints, and added nothing new to the text? So much time has passed since you wrote it! Haven’t you had any new ideas since then?”
Dirac: “My mother used to say: think first, then write.”

CHAPTER 2
DISCOVERY OF QUANTUM MECHANICS

DIRAC’S scientific life took a dramatic turn in the early fall of 1925, when he became acquainted with the work of Werner Hei¬ senberg in which the fundamental ideas of quantum mechanics were first stated. On July 28, Heisenberg, Dirac’s senior by only eight months, delivered a lecture in Cambridge at a meeting of the Kapitza Club. His subject was “Term-zoology and Zeeman-botany,” that is, the¬ oretical spectroscopy within the framework of the then existing “old” quantum theory of Bohr and Somerfeld. In the lecture Heisenberg did not refer to the new, still unpublished theoretical scheme he had just discov¬ ered. Presumably, Dirac was not in Cambridge at the time and thus missed the opportunity to attend Heisenberg’s lecture.1 However, Fowler was present, and he understood, perhaps from informal discussions with Heisenberg following his lecture, that the young German physicist had recently been able to derive some of the spectroscopic rules in a com¬ pletely new way. In August, Fowler received the proof-sheets of Heisen¬ berg’s new paper. He ran through them and sent them on to Dirac, requesting him to study the work closely. At that time, the end of August, Dirac was in Bristol with his parents.
Heisenberg’s aim in his historic paper was to establish a quantum kine¬ matics that was in close accordance with Bohr’s correspondence principle but that involved only observable quantities.2 For this purpose he con¬ sidered the classical Fourier expansion of an electron’s position coordi¬ nate, for an electron being in its n’th stationary state

-boo

x(n) =

x{n,oc)elTi*n’a)t

a= —co

(2.1)

Here v(n,a) = ocv(n) and x(n,a) = x(n,-a) denote the Fourier frequen¬ cies and amplitudes, respectively, where the condition on the latter guar-

14

Discovery of quantum mechanics

15

antees that x(n) is real. Since x(n) is not directly observable, Heisenberg wanted to replace it with an expression that could be given a more sat¬ isfactory quantum theoretical interpretation. He suggested that v{n,a) be replaced with v(n,n — a), where the latter expression signifies the fre¬ quency corresponding to the quantum transition n^ n - a. The Fourier coefficients x(n,a) were similarly replaced with x(n,n — a), interpreted to be transition amplitudes. Heisenberg argued that only the individual terms - not the summation - in equation (2.1) can be taken over into the quantum domain. These terms are then of a form

x(n,n — cx)eZ7ri*n'n-a)t

which can be arranged in a two-dimensional array (a “Heisenberg array” or, as was recognized a few months later, a matrix).
On extending his analysis to the case of the anharmonic oscillator, Hei¬ senberg was faced with the problem of how to represent a quantity like x2, whose classical expression is

x*(n) = f x(n,a)e*Tl,<'*)‘) ( £ix(n,/3)eirl’lnJI)t

V a

V 0

He showed that for a single x2(n) term, the corresponding term in quan¬ tum theory can be written as

x2(n,m)e2^(n'm)t
The complete x2(n) expression can again be written in an array, each of the terms being related to the x(n) terms by

x\n,m)ell,i‘in’m)t = x(n,k)x{k,m)elTiv(n’m)t
k
For the amplitude factors, this yields

x2(n,m) = y^x(n,k)x(k,m)
k

In the case of a product of two different quantities, x and y, the elements are formed similarly:

xy(n,m) = f^x(n,k)y(k,m)

(2.2)

16

Dirac: A scientific biography

This is Heisenberg’s famous law of multiplication. The fact that in gen¬ eral it is not commutative was noticed by Heisenberg. He found this most disturbing and at first considered it to be a flaw in his theory.
When Dirac first read Fowler’s copy of Heisenberg’s paper, he did not find it very interesting. In his early work in quantum theory, Dirac had stuck to the research program of the Bohr-Sommerfeld theory, which based the theory of atoms on Hamiltonian methods by extensive use of the angle and action variable technique known from classical mechanics. But, as he later realized, this approach was too restricted and not suited to an appreciation of Heisenberg’s work:3

I was very much impressed by action and angle variables. Far too much of the scope of my work was really there; it was much too limited. I see now that it was a mistake; just thinking of action and angle variables one would never have gotten on to the new mechanics. So without Heisenberg and Schrodinger I would never have done it by myself.

It was only when Dirac again studied the proof-sheets, a week or so after he first read them, that he realized that Heisenberg had initiated a revo¬ lutionary approach to the study of atoms. Dirac now occupied himself intensely with Heisenberg’s ideas, trying to master them and also to improve them. He found Heisenberg’s formulation complicated and unclear and was also dissatisfied because it did not take relativity into account. He felt that it should be possible to state the quintessence of Heisenberg’s theory in a Hamiltonian scheme that would conform with the theory of relativity.
After the summer vacation ended, Dirac returned to Cambridge, think¬ ing deeply about Heisenberg’s paper and the strange appearance in it of noncommuting dynamical variables. In order to proceed with his plan for setting up a Hamiltonian version of the new mechanics, he would have to have a classical expression to correspond with the quantity xy — yx appearing implicitly in Heisenberg’s theory. However, Dirac’s first attempt to develop the theory went in another direction, that of extending Heisenberg’s mechanics to systems involving rapidly moving electrons. In an unfinished manuscript, written in early October, Dirac argued that the Heisenberg variables [such as x(n,m)] referred not only to two energy levels but also to the two associated momenta.4 In this case, the variables are connected with what he called “the theory of the uni-directional emis¬ sion of radiation,” that is, with light moving in a particular direction. Elaborating on this idea, Dirac proposed that, for relativistic velocities, the Heisenberg variables x(n,n — a) should be generalized by replacing t by t — z/r, where z is the direction of the light and r its distance from the source. However, Dirac soon sensed that he was on the wrong track and left his paper incomplete. Referring to this episode, he later recalled:

Discovery of quantum mechanics

17

“There was a definite idea which I could work on, and I proceeded to write it up, but I never got very far with it.”5 In this first, abortive work on the new quantum mechanics, Dirac made use of some of his earlier works, in particular a short paper of 1924 in which he had proved the relativistic invariance of Bohr’s frequency condition.6
During his unsuccessful attempt to introduce relativistic arguments into Heisenberg’s theory, Dirac continued to ponder the puzzling non¬ commutativity. He later told how he discovered what became his own key for unlocking the quantum mysteries:7

I went back to Cambridge at the beginning of October, 1925, and resumed my previous style of life, intense thinking about these problems during the week and relaxing on Sunday, going for a long walk in the country alone. The main purpose of these long walks was to have a rest so that I would start refreshed on the fol¬ lowing Monday.... It was during one of the Sunday walks in October, 1925, when I was thinking very much about this uv — vu, in spite of my intention to relax, that I thought about Poisson brackets. I remembered something which I had read up previously in advanced books of dynamics about these strange quantities, Poisson brackets, and from what I could remember, there seemed to be a close similarity between a Poisson bracket of two quantities, u and v, and the com¬ mutator uv — uu.The idea first came in a flash, I suppose, and provided of course some excitement, and then of course came the reaction “No, this is probably wrong.” I did not remember very well the precise formula for a Poisson bracket, and only had some vague recollections. But there were exciting possibilities there, and I thought that I might be getting to some big new idea. It was really a very disturbing situation, and it became imperative for me to brush up my knowledge of Poisson brackets and in particular to find out just what is the definition of a Poisson bracket. Of course, I could not do that when I was right out in the coun¬ try. I just had to hurry home and see what I could then find about Poisson brack¬ ets. I looked through my notes, the notes that I had taken at various lectures, and there was no reference there anywhere to Poisson brackets. The textbooks which I had at home were all too elementary to mention them. There was just nothing I could do, because it was a Sunday evening then and the libraries were all closed. I just had to wait impatiently through that night without knowing whether this idea was really any good or not, but still I think that my confidence gradually grew during the course of the night. The next morning I hurried along to one of the libraries as soon as it was open, and then I looked up Poisson brackets in Whit¬ taker’s Analytical Dynamics, and I found that they were just what I needed.

The quantity which Dirac looked up after his sleepless night was first introduced by the French mathematical physicist Simeon Poisson in 1809. It is defined as

(2.3)

18

Dirac: A scientific biography

where p and q represent any two canonical variables for the system in question, and the summation is over the number of degrees of freedom of the system. Although the idea that Poisson brackets were relevant came to Dirac “rather out of the blue,”8 it obviously stemmed from the fact that Hamiltonian dynamics can be formulated by means of the non¬ commuting Poisson bracket algebra. In particular one has that

[qPQk\ = 0, [ppPk] = 0, [pnq] = 8Jk

(2.4)

where 8jk, the Kronecker delta, has a value of one for j = k and a value of zero otherwise. The connection between Poisson brackets and Heisen¬ berg products conjectured by Dirac that Monday morning in October was the following:

(xy -yx) = — [x,y\

(2.5)

Armed with this idea, Dirac began to write his paper “The Fundamental Equations of Quantum Mechanics,” which became one of the classics of modern physics. The paper received quick publication in Proceedings of the Royal Society, no doubt because of Fowler, who recognized its impor¬ tance. Only three weeks intervened between the receipt of Dirac’s paper by the Royal Society and its appearance in print.
Dirac did not introduce the Poisson bracket formulation at once in his paper; he did so only after deriving the rules of quantum differentiation. In most cases the structure of Dirac’s research publications reflects fairly well the order in which the ideas occurred to him; that is, the “context of justification” roughly agrees with the “context of discovery.” But in this case, Dirac “preferred to set up the theory on this basis where there was some kind of logical justification for the various steps which one made.”9 Let us briefly look at the main results of the paper.
Dirac’s primary aim was to construct algebraic operations of the quan¬ tum variables in agreement with Heisenberg’s theory. In particular, he looked for a process of quantum differentiation, which could give mean¬ ing to quantities like dx/dv, where v and v are quantum variables corre¬ sponding to Heisenberg’s quantum amplitudes (matrices). Dirac found the result

where the a coefficients represent another quantum variable. In con¬ densed notation the formula was just written as

Discovery of quantum mechanics —^ = xa — ax

19 (2.6)

Quantum differentiation of a quantity x, according to Dirac, was then equivalent with “taking the difference of its Heisenberg products with some other quantum variable.” What does equation (2.6) correspond to classically? By means of an argument based on Bohr’s correspondence principle, Dirac proved relation (2.5) and explained, “We make the fun¬ damental assumption that the difference between the Heisenberg products of two quantum quantities is equal to ih/2ir times their Poisson bracket expression.”10
It is remarkable that Dirac’s deduction of equation (2.6) relied heavily on the correspondence principle. This principle played a crucial role in Heisenberg’s road to quantum mechanics, but in general Dirac did not appreciate correspondence arguments; unlike his colleagues in Germany and Denmark, he made almost no use of them. Although, in principle, quantum mechanics made Bohr’s correspondence principle obsolete, or at least far less important, many physicists continued to apply correspon¬ dence arguments after 1925.
With relation (2.5) at his disposal, Dirac could now proceed to for¬ mulate the fundamental laws of quantum mechanics by simply taking them over from classical mechanics in its Poisson bracket formulation. He no longer had need of the correspondence principle which was, so to speak, once and for all subsumed in relation (2.5). The quantum mechan¬ ical commutation relations follow from equations (2.4), yielding

QjQk - QkQ, = PjPk - PkPj = 0

.

ih A

and q,pk - pkq, = — 5jk

(2-7)

From classical theory he further obtained the relation dx/dt = [x,H], where H is the Hamiltonian and is any dynamical variable of p and q ([x,H] denotes the Poisson bracket, not the quantum mechanical com¬ mutator). Translating this into quantum mechanics, he obtained the fun¬ damental equation of motion

dx 2-7T It = m(xH ~ Hx)

<2'8)

and from this he concluded that if AH — HA = 0, the quantum variable
A must be a constant of motion. This result included the law of energy

20

Dirac: A scientific biography

conservation: if x = H in equation (2.8), then dH/dt = 0; i.e., H is constant.
Having thus set up the general scheme of quantum algebra, Dirac showed that it could be used to give a satisfactory definition of stationary states that agreed with that of the old quantum theory. For such states he derived Bohr’s frequency relation of 1913, Em — En — hr. Since in the old quantum theory the frequency relation, as well as the notion of sta¬ tionary states, both had the status of postulates, it was most satisfying to Dirac that he could now deduce them from his new theory.
The general commutation relations (2.7) were discovered by several physicists in the fall of 1925. Apart from Heisenberg and Dirac, Wolfgang Pauli and Hermann Weyl also proposed, but did not publish, the rela¬ tions, and they also figured prominently in an important paper by Max Born and Pascual Jordan."
After Dirac completed his paper, he sent a handwritten copy to Hei¬ senberg, who congratulated him for the “extraordinarily beautiful paper on quantum mechanics.” In particular, Heisenberg was impressed by its representation of the energy conservation law and Bohr’s frequency con¬ dition. In his letter Heisenberg reported to Dirac the rather disappointing news that most of his results had however already been found in Germany:12

Now I hope you are not disturbed by the fact that indeed parts of your results liave already been found here some time ago and are published independently here in two papers - one by Born and Jordan, the other by Born, Jordan, and me - in Zeitschrift fur Physik. However, because of this your results by no means have become less important [unrichtiger]; on the one hand, your results, espe¬ cially concerning the general definition of the differential quotient and the con¬ nection of the quantum conditions with the Poisson brackets, go considerably further than the just mentioned work; on the other hand, your paper is also writ¬ ten really better and more concisely than our formulations given here.
It must have been disappointing to Dirac to hear about the work of Jor¬ dan and Born in which most of his results had been derived - and more than a month earlier at that.13 In their paper, Born and Jordan for the first time used matrices representing quantum mechanical variables. On this basis they proved the equivalent of Dirac’s equation (2.7), written in matrix notation as
pq - qp = — h 1t 2 TVl

Discovery of quantum mechanics

21

where 1 is the unit matrix. They also proved the frequency condition and the energy conservation law, both of which figured in Dirac’s paper. But they did not make the Poisson bracket connection.
During the fall of 1925 and the following winter, the formulation of the new quantum mechanics initiated by Heisenberg’s paper was attended by stiff competition, primarily between the German physicists (Heisenberg, Jordan, Born, and Pauli) and Dirac in England. The Germans had the great advantage of formal and informal collaboration, while Dirac worked on his own. Even had he wanted to (which he did not), there were no other British physicists with whom he could collaborate on an equal footing. That he lost the competition under such circumstances is no wonder. However, Dirac was satisfied to know that it was possible to develop quantum mechanics independently in accord with his ideas. He was confident that the theory was correct and his method appropriate for further development.
Though handicapped relative to his German colleagues, Dirac, having quick access to the results obtained on the Continent, was better off than most American physicists. The competition in quantum mechanics at the time was given expression by John Slater, who, in a letter to Bohr of May 1926, told somewhat bitterly of his frustration at being beaten in the pub¬ lication race: “It is very difficuft-to work here in America on things that are changing so fast as this [quantum mechanics] is, because it takes us longer to hear what is being done, and by the time we can get at it, prob¬ ably somebody in Europe has already done the same thing.” As an exam¬ ple of this experience, Slater mentioned that he had independently dupli¬ cated most of Dirac’s results: “I had all the results of Dirac, the interpretation of the expressions (pq — qp) in terms of Poisson’s bracket expressions, with applications of that, before his paper came, and was almost ready to send off my paper when his appeared.”14 Born, who vis¬ ited MIT from November 1925 to January 1926, brought with him a copy of the still unpublished Born-Jordan paper, which he showed to Slater. The manuscript to which Slater referred in his letter to Bohr was written at the end of December. Entitled “A Theorem in the Correspondence Principle,” it contained a full account of the Poisson formalism in quan¬ tum theory. However, at that time Dirac’s work had already appeared in Europe.15 Independently of Dirac, and almost at the same time, the Dutch physicist Hendrik Kramers observed the algebraic identity between Pois¬ son brackets and the quantum mechanical commutators, but he did not realize that this identity was of particular significance and merely used it to confirm his belief that quantum mechanical problems always have a classical counterpart.16 Dirac’s conclusion was completely different and immensely more fruitful.

22

Dirac: A scientific biography

At that time Dirac was twenty-three years old. He was still a student, barely known to the Continental pioneers of quantum theory. The Ger¬ man physicists were surprised to learn about their colleague and rival in Cambridge. “The name Dirac was completely unknown to me,” recalled Born. “The author appeared to be a youngster, yet everything was perfect in its way and admirable.”17 A few days after receiving Dirac’s paper, Hei¬ senberg mentioned to Pauli:18

An Englishman working with Fowler, Dirac, has independently re-done the math¬ ematics for my work (essentially the same as in Part I of Born-Jordan). Born and Jordan will probably be a bit depressed about that, but at any rate they did it first, and now we really know that the theory is correct.

Dirac’s reputation in the physics community was soon to change. While in the fall of 1925 he was referred to as just “an Englishman,” within a year he would rise to become a star in the firmament of physics. In Cam¬ bridge Dirac quickly established himself as the local expert in the new quantum theory, lecturing frequently to the Kapitza Club on various aspects of the subject, including his own works.19

At the end of 1925, things were evolving very rapidly in quantum the¬ ory. Heisenberg’s theory was established on a firm basis with the famous Dreimannerarbeit of Heisenberg, Born, and Jordan, and the new theory was now often referred to as “matrix mechanics” or the “Gottingen the¬ ory.” But in spite of, or perhaps because of, the rapid development of the theory, many physicists felt uneasy about it; they wanted to see if it was also empirically fruitful and not merely a strange game with mathemati¬ cal symbols. As recalled by Van Vleck: “I eagerly waited to see if someone would show that the hydrogen atom would come out with the same energy levels as in Bohr’s original theory, for otherwise the new theory would be a delusion.”20
In his next contribution to quantum mechanics, Dirac attacked the problem mentioned by Van Vleck.21 Using an elaborate scheme of action and angle variables, he was able to prove that the transition frequencies for the hydrogen atom are given by the expression

where n is an integer. As Dirac noticed, provided the quantity P is an integral multiple of h/2-ir, this is the same result obtained in the experi¬ mentally confirmed Bohr theory of 1913. However, since he was unable

Discovery of quantum mechanics

23

to prove that P is in fact an integral multiple of h/lir, Dirac’s derivation of the hydrogen spectrum was not complete.
At this place it is appropriate to introduce the symbol h as an abbre¬ viation for h/2ir, where h denotes the usual Planck constant. Dirac first used the symbol in 1930, although in his paper on the hydrogen spec¬ trum and in subsequent work up to 1930 he decided to let the symbol h (“Dirac’s /?”) denote the quantity h/2tt.22 In what follows, h will mean the usual Planck constant.
By January 1926, Dirac had known for some time that a derivation of the hydrogen spectrum had already been obtained by Pauli. He had been so informed by Heisenberg in his letter of November 20 and had received proofs of Pauli’s paper before publication. Although Pauli solved the hydrogen spectrum before Dirac, in fact Dirac’s paper appeared in print before Pauli’s. This was a result not only of fast publication of the Pro¬ ceedings of the Royal Society but also of the fact that Pauli’s paper was subject to considerable delay.23 Dirac said that he “was really competing with him [Pauli] at this time.”24 The fact that Pauli had priority did not matter too much to Dirac, whose primary aim was to test his own scheme of quantum mechanics. Furthermore, Dirac’s derivation was completely different from and much more general than that of Pauli, who made use of a rather special method.
Heisenberg praised Dirac’s work on the hydrogen atom:25

I congratulate you. I was quite excited as I read the work. Your division of the problem into two parts, calculation with ^-numbers on the one side, physical interpretation of ^-numbers on the other side, seems to me completely to corre¬ spond to the reality of the mathematical problem. With your treatment of the hydrogen atom, there seems to me a small step towards the calculation of the transition probabilities, to which you have certainly approached in the meantime. Now one can hope that everything is in the best order, and, if Thomas is correct with the factor 2, one will soon be able to deal with all atom models.

Although Pauli’s work on the hydrogen atom preceded the work of Dirac, Pauli realized that Dirac’s treatment included a general treatment of action and angle variables which he (Pauli) had not yet obtained. In con¬ nection with his efforts to establish a more complete (relativistic) matrixmechanical theory of the hydrogen atom, Pauli had worked hard to for¬ mulate a method for treating action and angle variables. He was therefore a bit disappointed to see that he was, in this respect, superseded by Dirac. In March, after studying Dirac’s paper, he wrote to Kramers: “In the March volume of the Proceedings of the Royal Society there is a very fine work by Dirac, which includes all of the results that I have thought out

24

Dirac: A scientific. biography

since Christmas about the extension of matrix calculus to non-periodic quantities (such as polar angles). I am sorry to have lost so much time working on this, when I could have been doing something else!”26
Dirac’s (and Pauli’s) work on the hydrogen spectrum was further devel¬ oped by the Munich physicist Gregor Wentzel, who also treated the rel¬ ativistic case.27 Wentzel’s approach was rather close to that of Dirac, but although Wentzel knew of Dirac’s paper, he had obtained his main results independently. Still another, and completely different, theory of the hydrogen atom was worked out in Zurich by Erwin Schrodinger on the basis of his new wave mechanics. Schrodinger was not impressed by the works founded on matrix or ^-number mechanics. In June, he wrote to Lorentz: “Dirac (Proc.Roy.Soc.) and Wentzel (Z.f.Phys.) calculate for pages and pages on the hydrogen atom (Wentzel relativistically, too), and finally the only thing missing in the end result is just what one is really interested in, namely, whether the quantization is in ‘half integers’ or ‘integers’!”28
In his work on the hydrogen atom, Dirac did not consider the problem of how to incorporate spin and relativistic corrections, a problem to which he would give a complete solution less than two years later (see Chapter 3). At the beginning of 1926, it was known that in order to find these corrections one would have to calculate the quantum mechanical mean values of 1/r2 and 1/r3; this was a difficult and as yet unsolved mathematical problem. When the young American physicist John H. Van Vleck read Dirac’s paper in the spring of 1926, he realized that the qnumber formalism furnished a means for the calculations. Van Vleck was one of the few physicists who adopted Dirac’s ^-number technique for practical calculations. When he arrived in Copenhagen a few weeks later, he had found the corrections only to learn that the results, based on the methods of matrix mechanics, had just been published by Heisenberg and Jordan.29
Dirac wanted to establish an algebra for quantum variables, or, as he now termed them, ^-numbers (q for “quantum” or, it was said, perhaps for “queer”). He wanted his g-number algebra to be a general and purely mathematical theory that could then be applied to problems of physics. Although it soon turned out that ^-number algebra was equivalent to matrix mechanics, in 1926 Dirac’s theory was developed as an original alternative to both wave mechanics and matrix mechanics. It was very much Dirac’s own theory, and he stuck to it without paying much atten¬ tion to what went on in matrix mechanics. In contrast to his colleagues in Germany, who collaborated fruitfully and also benefited from close connections with local mathematicians (such as Hilbert, Weyl, and Courant), Dirac worked in isolation. He probably discussed his work with Fowler, when he was available, but collaborated neither with him nor

Discovery of quantum mechanics

25

with other British physicists. He relied on a few standard textbooks, in particular Whittaker’s Analytical Dynamics and Baker’s Principles of Geometry, but did not seek the assistance of the Cambridge mathemati¬ cians. Baker’s book proved particularly valuable in connection with the g-number algebra.30
^-numbers are quantum variables that do not satisfy the usual com¬ mutation law for ordinary numbers, or, as Dirac called them, c-numbers (c for “classical”). If (^-numbers represent observable physical quantities, then “in order to be able to get results comparable with experiment from our theory, we must have some way of representing ^-numbers by means of c-numbers, so that we can compare these c-numbers with experimental values.”31 Dirac showed that ^-numbers satisfy Heisenberg’s law of mul¬ tiplication [equation (2.2)]; that is, they can be represented by matrices. However, in the paper of January 1926 he did not say so explicitly and did not, in fact, mention the word “matrix” at all. At the time, he knew, of course, of the Gottingen matrix mechanics, but he seems not yet to have realized that ^-numbers are equivalent to matrices. Dirac was not much impressed by the matrix formulation and believed that his scheme of quantum mechanics was superior in clarity as well as in generality. “It took me quite some time,” he wrote, “to get reconciled to the view that my ^-numbers were not really more general than matrices, and had to have the same limitations that one could prove mathematically in the case of matrices.”32
In the summer of 1926, Dirac published a new and very general version of ^-number algebra, this time presented as a purely mathematical the¬ ory.33 In this paper he did not refer to physics at all. In his attempt to state a general and autonomous theory, he even went so far as not to include Planck’s constant explicitly (that is, he put ih/2tt = 1). The work had little impact on the physics community but seems to have been appreciated by those who cultivated the mathematical aspects of quan¬ tum physics. Jordan, who was such a connoisseur, wrote, “I find this paper . . . very beautiful; for to me the mathematics is just as interesting as the physics!”34 The following are a few of the formulae obtained by Dirac in his quantum algebra.
If q and p are canonically conjugate, any other set of canonical vari¬ ables Q and P can be written by means of the transformations

Qj = bqjb~l and P, = bpp~'

(2.9)

where b is another ^-number and b~' is the quantity defined by bb~x = 1. Similar transformation formulae were given in the Born-Jordan paper in matrix formulation, and they played an important role in the Dreimdnnerarbeit. However, when Dirac first stated them, he did not fully

26

Dirac: A scientific biography

recognize their importance. “These formulae,” he wrote, “do not appear to be of great practical value.”35 Equation (2.9) implies that Q and P are canonical variables if bb~l = 1:

QA ~ pkQj = ih

and QjQk - QkQ, = PA PA = 0

just as stated in his first paper on quantum mechanics. Functions of ^-numbers may be differentiated not only with respect to
the time but with respect to any ^-number. The general definition of qnumber differentiation, as given by Dirac, was as follows: Let the ^-num¬ ber q be conjugate to p, so that qp — pq = ih/2ic, if Q = Q(q), then dQI dq is defined as
dQ dq = Qp - pQ
As to the ^-numbers representing angular momentum, Dirac showed that they satisfy the commutation relations36

Lzx — xLz Lzpx - pxL. LxLy L VLX

(and cyclic permutations)

and
L2LZ — LZL2 = 0, etc.
These relations, too, had been obtained earlier in matrix formulation by Heisenberg, Born, and Jordan.
In the summer of 1926, ^-number algebra was one of several, compet¬ ing schemes of quantum mechanics; the other versions were matrix mechanics (Heisenberg, Born, Jordan), wave mechanics (Schrodinger), and operator calculus (Born, Weiner). Physicists increasingly turned to wave mechanics when calculations had to be made, while g-number alge¬ bra remained almost exclusively Dirac’s personal method. At this time, it was felt that what was needed was a general and unified quantum mechanical formalism - a feeling that Dirac shared. Before dealing with his contributions to this end, we shall briefly survey some other results he obtained in 1926.

Discovery of quantum mechanics

27

Dirac was not, like the young German quantum theorists, raised in the spectroscopic tradition of the old quantum theory. This tradition was very much a Continental one and never received focal interest in England. But as Fowler’s student, Dirac was acquainted with the litera¬ ture and well aware of the connections between the new quantum mechanics and the various spectroscopic subtleties. In continuation of his work on the hydrogen atom, he used his method to throw light on some of the spectroscopic problems that had haunted the old quantum theory.37
In the old quantum theory the magnitude of the angular momentum of a single-electron atom, in units of h/2n, was assumed to be equal to the action variable k. Dirac showed that this is not so in quantum mechanics. If m is the magnitude of the angular momentum, the correct result is

For the total angular momentum of many-electron atoms, he found sim¬ ilar relations. Having established the general formulae for obtaining action and angle variables, Dirac turned to spectroscopic applications. His program was:38
To obtain physical results from the present theory one must substitute for the action variables a set of c-numbers which may be regarded as fixing a stationary state. The different c-numbers which a particular action variable may take form an arithmetical progression with constant difference h, which must usually be bounded, in one direction at least, in order that the system may have a normal state.
For a single-electron atom, he proved that for a given k the z-component of the angular momentum takes values ranging from \k\ — hh/l-K to — | k | + hh/2tt. Furthermore,39
k takes half integral quantum values . . . and thus has the values ± %h, ± %h, ±%h,.. ., corresponding to the S, P, D, . .. terms of spectroscopy. There will thus be 1, 3, 5, . .. stationary states for S, P, D, ... terms when the system has been made non-degenerate by a magnetic field, in agreement with observation for sin¬ glet spectra.
He also proved the selection rules for k and mz, and in particular that S-* S transitions (i.e., from k = to k = — lA) are forbidden. Then he pro¬ ceeded to consider the anomalous Zeeman effect, one of the riddles of the old quantum theory. He showed that ^-number theory is able to repro-

28

Dirac: A scientific biography

duce the correct ^-factor for the energy of the stationary states in a weak magnetic field,40 and also derived formulae for the relative intensities of multiplet lines that agreed with the formulae obtained by using the old quantum theory. Most of the results obtained by Dirac in his paper “On the Elimination of the Nodes in Quantum Mechanics” had been found earlier by the German theorists using the method of matrix mechanics; but Dirac was able to improve on some of the results and deduce them from his own system of quantum mechanics.
Dirac did not introduce the electron’s spin in his treatment of the spec¬ troscopic phenomena. He therefore had to rely on the largely ad hoc assumption of the old quantum theory that the gyromagnetic ratio of the atomic core is twice its classical value. Apart from this, his results did not depend on special assumptions concerning the structure of atoms.
A month later, at the end of April, Dirac considered another empiri¬ cally well-established subject, Compton scattering, and showed that it too followed from his theory.41 In doing so, Dirac extended his formalism to cover relativistic motions, making use of some of his ideas from his unpublished manuscript of October 1925 (see also Chapter 3). As is well known, the basic features of the scattering of high-frequency radiation (e.g., X-rays and gamma rays) on matter were explained in 1923 by Arthur H. Compton with the assumption of light-quanta, or, as they were called by Gilbert Lewis three years later, photons. Compton’s theory con¬ vinced physicists of the reality of light quanta, although some, most nota¬ bly Bohr, continued to consider the concept controversial.
For the sake of argument, Dirac accepted the light quantum hypothesis; but he was not particularly interested in whether electromagnetic radia¬ tion was “really” made up of corpuscles or waves. Dirac was content to get the correct formulae. For the change in wavelength of the radiation, he managed to reproduce Compton’s formula, which expresses conser¬ vation of energy and momentum. As to the intensity of the scattered radi¬ ation, he obtained a result very close to that found by Compton in 1923 but not quite identical with it. “This is the first physical result obtained from the new mechanics that had not been previously known,” Dirac proudly declared.42
Dirac’s treatment of the Compton effect was recognized to be a work of prime importance. In the period 1926-9, the paper was cited at least 33 times and thus became the first of his papers to have a considerable impact on the physics community (see Appendix I). Dirac’s work was generally considered to be very difficult. “We saw a paper by Dirac [on the Compton effect] which was very hard to understand,” Oskar Klein recalled. “I never understood how he did it, but I’ve always admired the fact that he did it because he got the right result.”43 The work was dis¬ cussed in Copenhagen before publication. In March, Sommerfeld visited

Discovery of quantum mechanics

29

Cambridge, where he stayed with Eddington. When Sommerfeld was told about the still unfinished calculations by Dirac, he was much surprised. On Eddington’s initiative, a meeting over a cup of tea with Dirac was arranged.44
Dirac was pleased with his work and felt that he had finally obtained something new. He discussed carefully how his results compared with experiment. In earlier as well as in later papers, Dirac showed little inter¬ est in experimental tests and preferred to emphasize the theoretical sig¬ nificance of his results. This time he was eager to show that his quantum algebra produced a result that fit the data better than earlier theories. He even went to the extreme of illustrating the fit with a diagram.45 When he observed that Compton’s experimental data were all a little below those predicted by his theory, he did not conclude that the theory was incom¬ plete or faulty; no, he concluded that the discrepancy “suggests that in absolute magnitude Compton’s values are 25 per cent cent too small.”46 Dirac had complete confidence in his theory.
When Compton read Dirac’s paper, he was impressed and wrote to Dirac that physicists at the University of Chicago had performed X-ray measurements that nicely confirmed the new theory:47

Mr. P. A. M. Dirac:
You will be interested to know that Messrs. Barrett and Bearden, working here, have completed a set of measurements of the angle of maximum polarization for X-rays of effective wave-length of about .3A, ,2A and .18A, finding in every case an angle of maximum polarization within 1 degree of 90°, in good accord with your theory.
Yours sincerely,
Arthur H. Compton

Later that year Dirac returned to the Compton effect, which he next treated by making extensive use of wave mechanics.48 With the new method, he derived exactly the same expressions that he had found in his first paper on the subject.
In this period of hectic research activity, Dirac found time to write his Ph.D. dissertation, which was completed in May. It consisted mainly of a survey of work he had already published or was about to publish.49 Dirac was completely absorbed in physics and spent most of his time alone in his study room at Cambridge. He had neither time nor desire to become involved in social or other extrascientific activities. In these months, there was much political and social unrest in England, which culminated in the General Strike declared on May 3. The conservative government had established an emergency plan that included a flood of

30

Dirac: A scientific biography

antistrike volunteers. Many of Dirac’s fellow students left their studies for a time to act as antistrike volunteers. One of them was Nevill Mott, who was at the time preparing for the tripos examination.50 Dirac did not want the strike, or anything else, to interfere with his scientific work and did not join the volunteers.51
While still working on his thesis, Dirac was assigned by Fowler to lec¬ ture on quantum theory to the few students of theoretical physics at Cam¬ bridge. The title of Dirac’s course was first announced as “Quantum The¬ ory of Specific Heats” but was changed to “Quantum Theory (Recent Developments).” It was the first course on quantum mechanics ever taught at a British university. The students included A. H. Wilson, B. Swirles, J. A. Gaunt, N. F. Mott, and the American J. R. Oppenheimer. Fowler, who gave another course on quantum theory at the same time as Dirac, attended with his entire class. “Dirac gave us what he himself had recently done, some of it already published, some, I think, not,” recalled one of the attendants of the lectures. “We did not, it is true, form a very sociable group, but for anyone who was there it is impossible to forget the sense of excitement at the new work. I stood in some awe of Dirac, but if I did pluck up courage to ask him a question I always got a direct and helpful answer, with no beating about the bush if I was getting things wrong.”52 Beginning in 1927, Dirac gave a regular course on quantum mechanics and was also assigned other teaching duties. As a teacher and supervisor, Dirac was “unapproachable,” according to Alan Wilson, who was a research student in 1927.53 The slightly younger Mott related the following episode from November 1927, when he had worked out some results that he wanted Dirac and Fowler to see:54

Dirac was there, and Fowler called him and Dirac said timidly that it was all nonsense, and referred me to one of his papers - which is about something quite different. Dirac said that the general theory allowed us to assume_I asked him how he knew, and because I thought that the great man was being stupid, I may have summoned up courage to hector the great. Then I suddenly realised that the great man was timid and that I was being a bully! Funny moment. Fowler sug¬ gested that I should write it all nicely and that Dirac should read it and Dirac said he would - I hope he won’t hate me too much!

Unknown to other physicists, since December 1925, Erwin Schrodinger in Zurich had worked on a completely new atomic theory in which quan¬ tum phenomena were seen as a kind of wave phenomenon. Schrodinger’s “wave mechanics” developed ideas previously suggested by Louis de Broglie in Paris. The theory made its entry in March 1926, when Schro¬ dinger published the first of a series of monumental papers on quantum mechanics.55 The core of the theory was a differential equation, soon known as the Schrodinger equation:

Discovery of quantum mechanics

31

, , 8ir2m , „
+ —Jf~ (E ~ Em)f = 0

Schrodinger’s theory at once aroused intense interest, and it almost divided the physics community into two camps. Heisenberg and his circle criticized the theory for being inconsistent and conceptually regressive. Schrodinger, on his side, expressed a dislike for matrix mechanics’ Unanschaulichkeit and transcendental algebra,” a dislike he presumably also held with respect to Dirac’s formulation.
At an early stage, Dirac had studied the quantum statistics of Bose and Einstein and also de Broglie’s approach to radiation phenomena. In the summer of 1925, when he gave a talk on the subject to the Kapitza Club, he was sympathetic to de Broglie’s wave theory of matter; he argued that it was equivalent to the light quantum theory of Bose and Einstein.56 But Dirac became occupied with Heisenberg’s new theory and did not think of developing de Broglie’s ideas into a quantum mechanics. Dirac prob¬ ably first heard about Schrodinger’s theory in mid-March, when Sommerfeld visited Cambridge. About a month later, Heisenberg wrote Dirac, wanting to know how his treatment of the hydrogen atom was related to Schrodinger’s method:57
A few weeks ago an article by Schrodinger appeared . . . whose contents to my mind should be mathematically closely connected with quantum mechanics. Have you considered how far Schrodinger’s treatment of the hydrogen atom is connected with the quantum mechanical one? This mathematical problem inter¬ ests me especially because I believe that one can win from it a great deal for the physical significance of the theory.
But Dirac was much too absorbed in his own theory to consider Schro¬ dinger’s theory worthy of careful study:58
I felt at first a bit hostile towards it [Schrodinger’s theory], . . . Why should one go back to the pre-Heisenberg stage when we did not have a quantum mechanics and try to build it up anew? I rather resented this idea of having to go back and perhaps give up all the progress that had been made recently on the basis of the new mechanics and start afresh. I definitely had a hostility to Schrodinger’s ideas to begin which, with persisted for quite a while.
It was only somewhat later, as a result of another letter from Heisenberg, that Dirac’s attitude changed. Right after the publication of Schrodinger’s first paper, many physicists wondered if wave mechanics and matrix mechanics, two theories that were different in style and content yet cov¬ ered the same area of physics, were in fact deeply connected. Schrodinger was the first to prove the formal equivalence between the two theories.

32

Dirac: A scientific biography

But some time before Schrodinger’s paper appeared, the result was known to the German physicists, thanks to an independent proof by Pauli, whose calculations were not published but were circulated quickly to the insiders.59 In a letter to Dirac of May 26, Heisenberg reproduced Pauli’s demonstration of the connection between wave mechanics and matrix mechanics; furthermore, he reported that when relativity was incorporated, the Schrodinger equation would become

VY - jr2 [ (E ~ Epol)2 - mlc*] ^ = 0

(2.10)

where E is the total energy, including the rest mass

As to his general

opinion regarding Schrodinger’s theory, Heisenberg was negative:60

I agree quite with your criticism of Schrodinger’s paper with regard to a wave theory of matter. This theory must be inconsistent as just like the wave theory of light. I see the real progress made by Schrodinger’s theory in this: that the same mathematical equations can be interpreted as point mechanics in a non-classical kinematics and as wave-theory accor. w. Schrod. I always hope that the solution of the paradoxes in quantum theory later could be found in this way. I should very much like to hear more exactly what you have done with the Compton-effect. We all here in Cophenhagen have discussed this problem so much that we are very interested in its quantum mechanical treatment.... I hope very much to see you in Cambridge in July or August. My best regards to Mr. Fowler.

The fact that Schrodinger’s wave mechanics turned out to be mathemat¬ ically equivalent to quantum mechanics caused Dirac’s hostility to van¬ ish. He realized that, computationally, wave mechanics is in many cases preferable. As to the physical interpretation, not to mention the ontology, associated with Schrodinger’s theory, Dirac did not care much: “The question as to whether the [\p] waves are real or not would not be a ques¬ tion which would bother me because I would think upon that as meta¬ physics.”61 Dirac was interested in formulae and simply found wave mechanics suitable for this purpose. Consequently, he began an intense study of the theory, which he soon mastered. This was difficult since the mathematics of wave mechanics, such as the theory of eigenvalues and eigenfunctions, was not part of Dirac’s education and was little known in Cambridge. Dirac’s view of the different formulations of quantum mechanics was essentially pragmatic. He never became a “wave theorist” in the sense of Schrodinger or de Broglie but freely used wave mechanics when he found it useful. Often he mixed it with his own ^-number algebra.
Schrodinger recognized the formal beauty of Dirac’s version of quan-

Discovery of quantum mechanics

33

turn mechanics but preferred to translate its results into the language of wave mechanics and did not, at that time, find himself congenial to Dirac’s way of doing physics, which he found strange and difficult to understand.6" Sometime during 1926, Schrodinger requested one of his students to give a review of one of Dirac’s papers for a seminar in Zurich. The student, Alex von Muralt, tried hard but in vain to understand the paper, and Schrodinger had to give the review himself. He confessed to his depressed student that Dirac’s paper had also caused him great difficulty.63
The first work in which Dirac considered Schrodinger’s theory was the important paper “On the Theory of Quantum Mechanics,” which was completed in late August. While finishing this article, he first met Van Vleck in Cambridge. Van Vleck had participated in a meeting of the Brit¬ ish Association for the Advancement of Science in Oxford during August 4-11. Dirac told him about his new work in quantum mechanics, but Van Vleck, who had not yet studied Schrodinger’s theory, found Dirac’s ideas very difficult to understand.
Dirac started out from the wave equation, which he considered “from a slightly more general point of view,” writing it as

ih d \ - W f=0
2tt dqj

(2.11)

If \p is a solution to (2.11), it can be written as

f
n

where the coefficients are arbitrary constants and \pn are the eigenfunc¬ tions.64 Dirac interpreted | cn \2 as the number of atoms in the n'th quan¬ tum state. The eigenfunctions \pn satisfy the equation afn = anfn, where a is a g-number and an is a c-number. According to Dirac, \pn represents a stateTn“which a has a definite numerical value, an. In the case of a sys¬ tem disturbed by the time-dependent perturbation energy A(p,q,t), the wave equation was written as

(H - W — A)\p = 0

(2.12)

for which the solution is

f = Y1 a"t,

34

Dirac: A scientific biography

where an now depends on the time and \an\2 denotes the number of atoms in state n at time t. The matrix elements corresponding to A are the coefficients of the expansion

A\b„ = Y^Amnin
m
Dirac used this expression to derive a general expression for time-depen¬ dent perturbations, namely

ijt *a= X,(a"A™a*’" ~ a*„A„mam)

which gives the rate of change of the number of atoms in state m. As an important application of the perturbation theory, Dirac considered the emission and absorption of radiation, a subject to which we shall return in Chapter 6. At the time when Dirac wrote his paper, time-dependent perturbation theory had already been developed by Schrodinger, but Dirac was unaware of this.66
In another part of this paper, Dirac examined what subsequently became known as Fermi-Dirac quantum statistics. His point of departure was Heisenberg’s positivistic credo that a fundamental physical theory should enable one to calculate only those quantities that can be measured experimentally. “We should expect this very satisfactory characteristic to persist in all future developments of the theory,” Dirac wrote.66 Dirac adapted Heisenberg’s philosophy to an atom with two electrons in states m and n, respectively, asking if the systems (mri) and (nm) should be counted as one state or two. Since the states are empirically indistinguish¬ able, then “in order to keep the essential characteristic of the theory that it shall enable one to calculate only observable quantities, one must adopt the second alternative that (mri) and (nm) count as only one state.”67 Dirac argued that this restricts the set of eigenfunctions for the whole atom (neglecting electron-electron interactions) to the form

imn = amntm( 1)^(2) + bmn\pm(2)\f/n(l)

(2.13)

where amn and bmn are constants, and ^,„(1) is the eigenfunction of electron number 1, being in state m, etc. He then proved that equation (2.13) is a complete solution only if amn = bmn or amn = —bmn. In the first case the wave function is symmetrical in the two electrons, i.e., ^mn(l,2) = \f/nm(2,1); in the latter case it is antisymmetrical, i.e., \J/mn(l,2) = — ^„m(2,l). Quantum mechanics does not predict which of the two cases is the correct one for electrons, but with the help of Pauli’s exclusion prin-

Discovery of quantum mechanics

35

ciple, Dirac concluded that it must be the antisymmetrical solution, because then the wave function is of the form

implying that if two electrons are in the same state (n = m), then \pmn = 0, which means that there can be no such state. This agrees with the Pauli principle, which was known to hold for electrons, while the other possi¬ bility - the symmetrical case - does not rule out states with n = m.
Dirac further showed that light quanta are described by symmetrical wave functions which thus must be associated with Bose-Einstein statis¬ tics. By a curious (and erroneous) argument he assumed that gas mole¬ cules are governed by antisymmetrical eigenfunctions (because “one would expect molecules to resemble electrons more closely than light quanta”). However, the belief that gas molecules satisfy the same statis¬ tics as electrons was not peculiar to Dirac; it was rather generally assumed in 1926 and was, for example, also a part of Fermi’s early work on quan¬ tum statistics. Using standard statistical methods, Dirac went on to find the energy distribution, the so-called statistics, of molecules. If As denotes the number of states with energy Es, the number of molecules in state s he found to be
exp(a + EJkT) + 1
where k is Boltzmann’s constant, a is another constant, and T is the tem¬ perature. This expression is the basic distribution law for particles obey¬ ing Fermi-Dirac statistics, such as electrons.
Dirac knew that Heisenberg had also applied quantum mechanics to many-particle systems, especially to the helium atom.68 In a letter of April 9, Heisenberg had informed him as follows:69
Since I am in Copenhagen I tried to treat the heliumproblem on the basis of quan¬ tum mechanics. There was an essential difficulty for the explanation of the large distance between Singlet- and Tripletsystem, because this distance could not be explained by interaction of two magnets only. But now I think we have in the helium to deal with a resonance effect of a typical quantum mechanical feature. Really one gets in this way a qualitative explanation of the spectrum with regard as well to the frequencies as to the intensities. And I hope, the quantitative agree¬ ment is only a question of long numerical work.
Heisenberg’s paper appeared a little before Dirac’s and contained the same distinction between symmetrical and antisymmetrical eigenfunc-

36

Dirac: A scientific biography

tions, including the connection to the exclusion principle. The other part of Dirac’s result, the quantum statistics of gas molecules, had also been obtained earlier, by Enrico Fermi in a paper from the spring of 1926.™ When Fermi saw Dirac’s article, he was naturally disturbed that there was no reference to his own work. He wrote at once to Cambridge:71

In your interesting paper “On the theory of Quantum Mechanics” (Proc.Roy.Soc. 112, 661, 1926) you have put forward a theory of the Ideal Gas based on Pauli’s exclusion Principle. Now a theory of the ideal gas that is practical identical to yours was published by me at the beginning of 1926 (Zs.f.Phys. 36, p. 902; Lincei Rend., February 1926). Since I suppose that you have not seen my paper, I beg to attract your attention on it.

The situation was embarrassing to Dirac, who hurried to write a letter of apology to Fermi. Much later, Dirac recalled the situation as follows:72

When I looked through Fermi’s paper, I remembered that I had seen it previously, but I had completely forgotten it. I am afraid it is a failing of mine that my mem¬ ory is not very good and something is likely to slip out of my mind completely, if at the time I do not see its importance. At the time that I read Fermi’s paper, I did not see how it could be important for any of the basic problems of quantum theory; it was so much a detached piece of work. It had completely slipped out of my mind, and when I wrote up my work on the antisymmetric wave functions, I had no recollection of it at all.

Although virtually all of Dirac’s results in “On the Theory of Quantum Mechanics” were thus obtained independently and earlier by other phys¬ icists, the work is justly seen as a major contribution to quantum theory. The new statistics was soon known under the joint names of Fermi and Dirac, although Fermi’s priority was recognized (occasionally the statis¬ tics was referred to as Pauli-Fermi statistics). Incidentally, years later Dirac invented the names fermions and bosons for particles that obey Fermi-Dirac and Bose-Einstein statistics, respectively. The names date from 1945 and are today a part of physicists’ general vocabulary.73 Fol¬ lowing the publication of Dirac’s paper, the new statistics was eagerly taken up and applied to a variety of problems. The first application was made by Dirac’s former teacher, Fowler; as an expert in statistical phys¬ ics, he was greatly interested in the Fermi-Dirac result. Fowler studied a Fermi-Dirac gas under very high pressure, thus beginning a chapter in astrophysics that, a few years later, would be developed into the cele¬ brated theory of white dwarfs by his student Chandrasekhar.74 In Ger¬ many, Pauli and Sommerfeld made other important applications of the new quantum statistics, with which they laid the foundation for the quan¬ tum theory of metals in 1927.75

Discovery of quantum mechanics

37

“On the Theory of Quantum Mechanics” became the most cited of Dirac’s early papers and was studied with interest by both matrix and wave theorists. Although the paper was recognized as an important work, many physicists felt that it was difficult to understand and even cryptic. Schrodinger may be representative in this respect (see also the views of Einstein and Ehrenfest, quoted below). In October, when Dirac was in Copenhagen, Schrodinger told Bohr about his troubles in reading Dirac:76

I found Dirac’s work extremely valuable, because it translates his interesting set of ideas at least partly into a language one can understand. To be sure, there is still a lot in this paper which I find obscure, . . . Dirac has a completely original and unique method of thinking, which - precisely for this reason - will yield the most valuable results, hidden to the rest of us. But he has no idea how difficult his papers are for the normal human being.

After completing his dissertation, Dirac wanted to go abroad to study with some of his Continental peers. At the time, the spring of 1926, he was well acquainted with Heisenberg, and hence it was natural for him to prefer Gottingen as his first destination. Gottingen was the center and birthplace of quantum mechanics, and its physics institute included not only Heisenberg but also Born and Jordan, as well as a number of other talented young physicists. However, on Fowler’s advice Dirac decided first to spend a term at Bohr’s institute in Copenhagen. This was a wise decision, for although Bohr no longer published on the technical aspects of quantum mechanics, he was very active as an organizer and source of inspiration; his flourishing institute was no less a center of quantum phys¬ ics than was Gottingen. Bohr had close contacts to Germany, and Ger¬ man physicists often stayed in Copenhagen. Heisenberg spent much of the period from May 1926 to June 1927 with Bohr, during which time Pauli too was a frequent visitor. Bohr was happy to accept Fowler’s request, and Dirac arrived in September.77 In Copenhagen he met with Heisenberg, Friedrich Hund, Klein, Ehrenfest, and Pauli, and of course with Bohr. “I learned to become closely acquainted with Bohr, and we had long talks together, long talks on which Bohr did practically all of the talking.”78
Although Dirac was now part of an intense intellectual environment in which cooperation and group discussions were much valued, he largely kept to his Cambridge habits of working alone. Not even the friendly atmosphere at Bohr’s institute could break his deep-rooted preference for isolation. According to the Danish physicist Christian Moller, then a young student:79
[Dirac] appeared as almost mysterious. I still remember the excitement with which we [the young students] in those years looked into each new issue of

38

Dirac: A scientific biography

Proc.Roy.Soc. to see if it would include a work of Dirac... . Often he sat alone in the innermost room of the library in a most uncomfortable position and was so absorbed in his thoughts that we hardly dared to creep into the room, afraid as we were to disturb him. He could spend a whole day in the same position, writing an entire article, slowly and without ever crossing anything out.

Dirac was impressed by Bohr’s personality. He later said that “he [Bohr] seemed to be the deepest thinker that I ever met.”80 Although Bohr’s way of thinking was strikingly different from his own, the taciturn Dirac did not avoid being influenced by the thoughts of the talking Bohr. He was certainly influenced by the discussions at the institute, which, in the fall of 1926, concentrated on the physical interpretation of quantum mechan¬ ics. Dirac arrived in Copenhagen shortly after Schrodinger had left. Schrodinger’s meeting with Bohr had resulted in heated discussions about the foundational problems of quantum theory, discussions that continued during the following months. But Dirac was unwilling to par¬ ticipate in the lofty, epistemological debate. He stuck to his equations.
In September 1926, a conceptual clarification of quantum mechanics was felt to be a pressing need. After Born’s probabilistic interpretation of Schrodinger’s theory, the question of how to generalize the probability interpretation and relate it to matrix mechanics came to the forefront. The essential step in this process, leading to a completely general and unified formalism of quantum mechanics, was the transformation theory. This theory was fully developed by Dirac and Jordan at the end of the year. A generalized quantum mechanics was in the air at the time and had already been developed to some extent by Fritz London.81 Pauli, in close contact with Heisenberg in Copenhagen, suggested a probabilistic interpretation also holding in momentum space and related it to the diag¬ onal elements of the matrices; but he was not able to go further.82 The problem occupied Dirac, who thought much about how to interpret wave mechanical quantities in the more general language of quantum mechan¬ ics. At the end of October, Heisenberg reported to Pauli about Dirac’s still immature ideas:83

In order to clarify Schrodinger’s electrical density, Dirac has reflected on it in a very funny way. Question: “What is the quantum mech[anical] matrix of the elec¬ trical density?” Definition of density: I. It is zero everywhere where there is no electron. In equations:

p(x0,y0,z0,t) (x0 - x(t)) = 0

p(- ■ ■

) (ko - k(0) = o

Discovery of quantum mechanics

39

Furthermore, the total charge is e.

f P (x0,y0,z0,t)dx0dy0dz0 = e The solution is (as can rather easily be proved):

Pnm(x0y0z0t) = e\pn\l/*(x0y0z0t)
where \pn and \pm are Schrod[inger]’s normalized functions. This formulation seems really quite attractive to me.

Dirac had his transformation theory ready a month later, inspired by recent works of Heisenberg and Cornelius Lanczos.84 Whereas many of Dirac’s great works were based on ideas that came unexpectedly to him - from “out of the blue” - this theory was the result of a more direct and logical procedure. To follow this procedure appealed to Dirac’s intellect. “This work gave me more pleasure in carrying it through than any of the other papers which I have written on quantum mechanics either before or after,” he later wrote.85
In Copenhagen, Dirac gave a seminar on his theory sometime before he submitted the paper for publication. Oskar Klein recalled the difficulty of following Dirac’s thoughts as expressed at the seminar: “It took us some time to understand the things because he gave some lectures and he wrote all the figures on the blackboard very nicely and he said a few words to them, but they were very, very hard to get.”86 Heisenberg at once reported to Pauli about Dirac’s “extraordinarily grandiose generaliza¬ tion” of the transformaton theory. He was clearly impressed by the gen¬ erality and logical rigor of Dirac’s work, which he judged to be “an extraordinary advance.”87 Let us turn to this extraordinary paper.
Dirac addressed the fundamental problem of what questions can be given an unambiguous answer in quantum theory. “In the present paper a general theory of such questions and the way the answers are to be obtained will be worked out. This will show all the physical information that one can hope to get from the quantum dynamics, and will provide a general method for obtaining it, which can replace all the special assump¬ tions previously used and perhaps go further.”88 In the Gottingen matrix mechanics, canonical transformations of the form

G = bgbA

(2.14)

where g is some dynamical variable and b is a transformation matrix, played an important role. In particular, they served to diagonalize the

40

Dirac: A scientific biography

Hamiltonian. Dirac asked about the significance of the transformation matrix and its relation to Schrodinger’s wave function \p. In order to state the result as generally as possible, he considered continuous matrices, i.e., matrices in which the parameters that label the rows and columns may vary continuously.
If g(a'a") denotes the dynamical variable g expressed in the matrix scheme labeled by the c-numbers a' and a", the transformed matrix G is the variable labeled by another set of indices, say X' and X": G = g(\'\"). Considering transformations between any two matrix schemes (a) and (X), Dirac showed that the canonical transformation (2.14) can be written as the integral

= ff ^(X'X")

(y/a/)da/g(a'a//)(a///X//)da/

In this expression the transformation functions appear as the symbols (y/a') and (a"/\"), which correspond to b(\'a') and b~'(a"X"), respec¬ tively. Any function F(X,a) of conjugate variables X and a was now shown to be transformable into a matrix scheme (a) in which F is diagonal. Dirac proved that the transformation is given by

ih d \

F

(y/a') = F(a') (;y/a')

2tt d\' J

where F(a') are the diagonal elements of the diagonal matrix that repre¬ sents F. This equation is a differential equation in which F on the left side is a differential operator operating on the transformation function (X' /a'). If, as a special case, X' is taken to be equal to q, and F is the Hamil¬ tonian, the equation reads

This is Schrodinger’s wave equation, with the transformation function appearing instead of the usual ^-function. Hence Dirac’s conclusion: “The eigenfunctions of Schrodinger’s wave equation are just the trans¬ formation functions ... that enable one to transform from the (q) scheme of matrix representation to a scheme in which the Hamiltonian is a diag¬ onal matrix.”89
In his work with continuous matrices Dirac introduced an important formal innovation, the famous 5-function. He took its defining properties to be

Discovery of quantum mechanics

41

8(x) = 0 for x 0

and

J+ oo 8(x)dx = 1

— oo

and derived some other properties of 5(x). With the help of this function, or improper function, he showed, for example, that if a dynamical vari¬ able X(X'X") is expressed in some matrix scheme, then its canonical con¬ jugate in the same scheme is

= -ihb'{\' - X")

where 8' means the differential quotient of 8. The commutation relation can then be written as
(Xrj - t|X) (X'X") = ih8(X' - \")
The 5-function had a long prehistory in 1926.90 It was not really invented by Dirac but he introduced it independently, and it was only with his work that it became a powerful tool in physics. Perhaps, as Dirac later stated, the idea of the 5-function grew out of his early engineering train¬ ing.91 At any rate, “Dirac’s 5-function” soon became a standard tool in physics. Originally considered to be merely an elegant and useful nota¬ tion, it has proved to be of extreme importance in virtually all branches of physics. In the realm of pure mathematics, it may be seen as a prede¬ cessor of the theory of distributions created in 1945 by the Swiss mathe¬ matician Laurent Schwartz, whom Dirac later met, in August 1949, when they both lectured at a seminar arranged by the Canadian Mathematical Congress in Vancouver.92
It was not difficult for Dirac to show that his theory comprised Born’s probabilistic interpretation of wave mechanics, according to which | \p |2 determines a probability density. But Dirac’s interpretation was much more general, because it rested on the more general transformation func¬ tions. According to this interpretation, if the dynamical variable a has the initial value a0, then the probability that the system at the time t is in a state between a' and a' + da', is given by

| (a',/a'012da\
In his work with the transformation theory, Dirac came close to for¬ mulating the indeterminacy principle, later given by Heisenberg. Dirac wrote: “One cannot answer any question on the quantum theory which

42

Dirac: A scientific biography

refers to numerical values for both the qr0 and the pr0” And again in his conclusion he said:93

One can suppose that the initial state of a system determines definitely the state of the system at any subsequent time. If, however, one describes the state of the system at an arbitrary time by giving numerical values to the co-ordinates and momenta, then one cannot actually set up a one-one correspondence between the values of these co-ordinates and momenta initially and their values at a subse¬ quent time. All the same one can obtain a good deal of information (of the nature of averages) about the values at the subsequent time considered as functions of the initial values.

However, in some contrast to Born and Jordan, Dirac did not consider the probabilistic interpretation as something inherent in the quantum mechanical formalism, but rather as something that relied on assump¬ tions that could be criticized. He ended his paper: “The notion of prob¬ abilities does not enter into the ultimate description of mechanical proc¬ esses; only when one is given some information that involves a probability ... can one deduce results that involve probabilities.”94 This view appealed to Heisenberg, who preferred it to the more deeply entrenched probabilistic physicalism of Born and Jordan. In February, Heisenberg wrote to Pauli: “One can say, as Jordan does, that the laws of nature are statistical. But one can also say, with Dirac (and this seems to me substantially more profound), that all statistics are produced by our experiments.”95 Pauli agreed. He felt that statistical notions should not enter the fundamental equations of a really satisfactory physical theory.96
Almost simultaneously with Dirac, Pascual Jordan published another transformation theory that matched Dirac’s in generality and scope and essentially contained the same results.97 Jordan’s detailed and mathemat¬ ically intricate work was independent of Dirac’s and was based on a strictly probabilistic view. Since at least in a formal sense the two theories were equivalent, physicists soon talked of “the Dirac-Jordan transfor¬ mation theory.” Dirac knew about Jordan’s work before it was published but not before he submitted his own paper. On Christmas Eve in 1926, he outlined the essence of his tranformation theory in a letter to Jordan and argued that the two theories were equivalent:98

Dr Heisenberg has shown me the work you sent him, and as far as I can see it is equivalent to my own work in all essential points. The way of obtaining the results may be rather different though-In your work you consider transfor¬ mations from one set of dynamical variables to another, instead of a transfor¬ mation from one scheme of matrices representing the dynamical variables to another scheme representing the same dynamical variables, which is the point of

Discovery of quantum mechanics

43

view adopted throughout my paper. The mathematics would appear to be the same in the two cases however.

In the letter, Dirac did not introduce the 5-function explicitly but men¬ tioned that the quantity J(X'/a')da(a'/X") would equal zero when ^ X" and equal “a certain kind of infinity” when X' = X".
At the beginning of February 1927, Dirac took leave of Bohr and parted for Gottingen by train. On his way he stopped in Hamburg, where a local meeting of the German Physical Society took place on February 5-6. In Hamburg he met with many of Germany’s best physicists. The day before the meeting began, Dirac chaired an informal seminar on quantum mechanics." From Hamburg he went on to Gottingen in the company of other physicists who had joined the Hamburg meeting.
Dirac’s months in Gottingen were divided between research and fur¬ ther education in some of the mathematical methods of physics with which he had not become well acquainted during his studies in Cam¬ bridge. Among other things, Dirac went to a course of lectures on group theory given by the distinguished mathematician Hermann Weyl. Group theory, extensively introduced in quantum mechanics by Weyl and Eugene Wigner during 1927-9, was then a very new subject in theoretical physics, but Dirac did not find it particularly appealing. Many physicists objected to the Gruppenpest fashion because of the abstract and unfamil¬ iar character of group theory.100 Dirac was not frightened by abstract mathematics but felt that group theory was largely unnecessary for phys¬ ical applications. He always preferred to do without group theoretical methods and thought that, instead of adapting quantum mechanics to the mathematical structure of group theory, one should consider groups as merely a part of ordinary quantum mechanics. In January 1929, Dirac gave a talk to the Kapitza Club in Cambridge entitled “Quantum Mechanics without Group Theory,” and slightly later he had this to say about the subject:101

Group theory is just a theory of certain quantities that do not satisfy the com¬ mutative law of multiplication, and should thus form a part of quantum mechan¬ ics, which is the general theory of all quantities that do not satisfy the commuta¬ tive law of multiplication. It should therefore be possible to translate the methods and results of group theory into the language of quantum mechanics and so obtain a treatment of the exchange phenomena which does not presuppose any knowl¬ edge of groups on the part of the reader.

An important part of Dirac’s stay abroad, in Gottingen as in Copen¬ hagen, was the people he met and the contacts he made. In Gbttingen

44

Dirac: A scientific biography

were, of course, Born, Weyl, Jordan, and Heisenberg, although Heisen¬ berg spent much of his time with Bohr in Copenhagen. Dirac seems to have impressed the people in Gbttingen, the Germans as well as the vis¬ iting foreigners. “Dirac is at Gottingen and is the real master of the situ¬ ation,” wrote the American physicist Raymond T. Birge with regard to the experiences of his former student Edward Condon, who visited Gbt¬ tingen during the same period as Dirac. “When he [Dirac] talks, Bom just sits and listens to him open-mouthed. That Dirac thinks of absolutely nothing but physics.”102 Dirac also met the American John Robert Oppenheimer, with whom he established a lasting friendship. Oppenheimer, Dirac’s junior by two years, had studied at the Cavendish and then moved on to Gottingen in order to prepare his doctoral dissertation under Born. He knew Dirac from Cambridge, but it was only in Gottin¬ gen that their relationship evolved into a friendship. Together they took walks in the surroundings of Gottingen, including an expedition to the Harz Mountains.
Scientifically, Dirac’s stay in Gbttingen was very fruitful. It resulted, in particular, in important work on the quantum theory of radiation, to which we shall return in Chapter 6. Another very important development during the period was Heisenberg’s new theory of the physical interpre¬ tation of quantum mechanics, the core of which was the famous indeter¬ minacy relations.103 Dirac was not directly involved in this theory, which was mainly worked out during Heisenberg’s stay in Copenhagen; but Dir¬ ac’s work on the transformation theory and his discussions with Heisen¬ berg in Copenhagen were instrumental in forming Heisenberg’s views. As far as the quantitative aspects are concerned, the indeterminacy principle grew out of the Dirac-Jordan transformation theory; with regard to the no less important qualitative and philosophical aspects, it was indebted to Heisenberg’s long discussions with Bohr. As mentioned, Dirac was technically close to the indeterminacy relations in December 1926, but he felt no pressing need to formulate the insight of his transformation theory in a principle of coordinated measurement. Perhaps Dirac was, at any rate, of too unphilosophical a mind to proceed in the direction fol¬ lowed by Heisenberg.
Although Dirac’s primary occupation was not with the measurement problem, he was, of course, interested in the new principle of indetermi¬ nacy. In May, he wrote to Heisenberg, raising by means of Gedankenexperimente various objections to the limited accuracy of coordinated measurements in quantum mechanics.104 Apparently he was not at that time convinced of the universal validity of the indeterminacy relations. Heisenberg had no difficulty in countering Dirac’s objections and explained to him how he and Bohr had analyzed the impossibility of

Discovery of quantum mechanics

45

measuring both the position and velocity of an electron at the same time by means of the so-called gamma-microscope.
Dirac left Gottingen at the end of June, spending a few days in Holland on his way back to England. Invited by Paul Ehrenfest, he went to Leiden together with Oppenheimer. At that time, Holland had a strong tradition in theoretical physics, as witnessed by names such as Kramers, Uhlenbeck, Fokker, and Ehrenfest. The dean of Dutch physics was seventythree-year-old Hendrik Lorentz, who was still remarkably active and fol¬ lowed the new developments in physics with interest. Lorentz was impressed by Dirac’s work and wanted him to come to Leiden for the two terms of 1927-8: “I think I have well understood your general trend of thought, admiring the beauty of your method and your remarkable deduction of Schrodinger’s wave equation,” he wrote to Dirac in June 1927.105

Dirac at the Kammerlingh-Onnes Laboratory in Leiden, 1927. In the front row, left to right: G. Uhlenbeck, H. Honl, F. Florin, unidentified, A. D. Fokker, H. A. Kramers, and S. A. Goudsmit. Behind them, left to right: K. F. Niessen, P. A. M. Dirac, J. R. Oppenheimer, L. Polak, T. Ehrenfest, P. Ehrenfest, and H. R. Woltjer. The two persons in the third row are unidentified. Reproduced with permission of AIP Niels Bohr Library.

46

Dirac: A scientific biography

Dirac did not accept Lorentz’s offer, but he did stay in Leiden for a few days and also spent a day in Utrecht visiting Kramers. For some time, Ehrenfest had recognized the originality of Dirac’s physics, which he admired but found hard to understand. He had wanted Dirac to come to Leiden earlier, in October 1926, and was now happy to get the opportu¬ nity to meet the young Englishman.106 After Dirac had agreed to visit Lei¬ den, Ehrenfest wrote to him that the Dutch physicists had discussed “the last three Diracian crossword puzzles: Physical interpretation, emission and absorption, dispersion. We spent many, many hours going over a few pages of your work before we understood them! And many points are still as dark to us as the most moonless night!”107 One of the features in Dir¬ ac’s scheme of quantum mechanics, to which Ehrenfest objected, was what he called the “time illness.” After Dirac’s stay in Leiden in the sum¬ mer of 1927, Ehrenfest wrote to Uhlenbeck about his opinion of the cur¬ rent status of quantum mechanics: “In fact, the Dirac theory works not at all with the time, although indeed with a q and p which are algebraically ill,” he wrote. “Dirac denies that one should ask when something hap¬ pens. One should be content knowing with what probability something happens. . .. [Dirac’s] apparatus is exaggeratedly blind.”108 Evidently Ehrenfest studied Dirac’s work very hard. In correspondence with Max Planck, he offered his view on Dirac’s quantum theory, suggesting that it might be clarified by making an analogy between ^-number algebra and the more familiar tensor algebra. Planck found Ehrenfest’s interpretation helpful and decided to study Dirac once again during his summer vacation.109
After returning to Cambridge following his journey to the Continent, Dirac was faced with a problem of a more mundane nature, that of remaining within the academic institution. His fellowship was running out, but in November 1927 he received a new fellowship at St. John’s College. Apparently he had told Oppenheimer that he would take a rest from quantum mechanics during the summer vacation, a promise that he was not quite able to keep. Oppenheimer, then back in the United States, wrote to Dirac in November: “I have just heard that you received the fellowship. My very best felicitations. There has been no direct news of what you have been doing for a long time. Did you keep your promise to stop quantum mechanics over the summer? I should very much appre¬ ciate it if you would let me know what you have got.”110
In the fall of 1927, the so-called Copenhagen interpretation of quantum mechanics had emerged as a powerful paradigm in physics. It was based on Heisenberg’s indeterminacy relations, Bohr’s complementarity prin¬ ciple, and a strictly acausal and probabilistic interpretation of the sub¬ atomic domain. On the formal side, the Copenhagen interpretation became based on von Neumann’s axiomatic Hilbert space theory, a

Discovery of quantum mechanics

47

mathematically advanced development of the Dirac-Jordan transfor¬ mation theory. As is well known, leading physicists such as Schrodinger and Einstein did not accept the Copenhagen doctrines. At the Solvay Conference of October 1927, the discussion centered on the interpreta¬ tion of quantum mechanics, and the highlight was Bohr’s successful defense of the Copenhagen interpretation in face of the objections raised by Einstein and others. Dirac was invited to participate in this Solvay Conference, which indicates that he was then recognized as one of the world’s top physicists. We shall return to the 1927 Solvay Conference, and Dirac’s stand in the long and lasting debate on the interpretation of quantum mechanics, in Chapters 4 and 13.

CHAPTER 3
RELATIVITY AND SPINNING ELECTRONS

WHEN Dirac went to the Solvay Conference in October 1927, he could look back at two years of successful involvement in quantum mechanics. Twenty-five years old, he was a physicist of international repute. But in spite of all his productivity and originality, most of his results had also been obtained by other physicists who, more often than not, had published their works a little before Dirac. He felt that he still lived in the shadow of Heisenberg and the other German theorists and that he had not yet produced a deep and really novel theory, a theory nobody else had thought of. These ambitions were fulfilled by the end of the year with Dirac’s celebrated relativistic theory of the elec¬ tron, one of the great landmarks in the history of science.
Dirac was, from an early age, fascinated by the theory of relativity. He believed, then as later, that a physical theory could be really fundamental only if it lived up to the standards of Lorentz invariance set by Einstein. His first move in quantum mechanics, in October 1925, had been an attempt to make Heisenberg’s theory conform to the theory of relativity. Not successful at this, he left the subject for a while. He returned to it half a year later in connection with his theory of the Compton effect (see also Chapter 2). In his paper on this subject, he proposed to treat time as a quantum variable, arguing that “the principle of relativity demands that the time shall be treated on the same footing as the other variables, and so it must therefore be a ^-number.”1 Guided by classical Hamiltonian mechanics, he showed that what he called the “quantum time” would be the variable conjugated to - W, where W is the energy function:

tW — Wt = ih

(3.1)

Dirac argued that the relation (3.1) was also suggested by the following formal relativistic argument: The commutation relations [xpp] = ih (j = 1,2,3,) ought also to hold for the fourth components of the correspond-

48

Relativity and spinning electrons

49

ing four-vectors, x4 = ict and p4 = iW/c; in that case one has [xA,p4] = ih, which is the same as (3.1). The commutation relation (3.1) had earlier been postulated by Pauli, but he had only stated it privately, in a letter to Heisenberg.2
In order to prepare for his main task, the calculation of the Compton effect, Dirac wrote down the classical-relativistic Hamiltonian equations of motion. For a free particle, the Hamiltonian is

p2 — W2/c2 = —mlc2

(3.2)

For a charged particle in an external electromagnetic field, the corre¬ sponding equation is

(3.3)
where A is the vector potential, 0 the scalar potential, and e the charge of the particle (i.e., — e for an electron). Dirac used equation (3.3) in his theory of Compton scattering, in which he considered the electromag¬ netic field of the incident radiation to be given by a vector in the direction of the y-axis, that is, A = (0,^,0) and 0 = 0.
In his first paper on the Compton effect, Dirac did not refer to Schrodinger’s new wave mechanics and hence did not attempt to formulate a relativistic wave equation. He did so in his second paper on the Compton effect, but in the meantime the problem had been discussed by several other physicists. To make possible a better appreciation of Dirac’s con¬ tributions, a brief review of this development may be helpful.3
On Schrodinger’s original road to wave mechanics, relativistic consid¬ erations were of crucial importance.4 In fact, he first derived a relativistic eigenvalue equation, which he did not publish, mainly because he real¬ ized that it did not reproduce the hydrogen spectrum with acceptable accuracy. This first, relativistic Schrodinger equation had the form

(3.4)

for an electron of rest mass m0 moving in the Coulomb field of a hydro¬ gen nucleus (Schrodinger did not use the symbol h). The equation gave a fine structure for the hydrogen spectrum, but not the correct one. The hydrogen fine structure was first given a theoretical explanation in 1915 by Sommerfeld, who worked with a relativistic extension of Bohr’s

50

Dirac: A scientific biography

atomic theory. In his celebrated work, Sommerfeld found that the energy levels of the hydrogen atom were given by the expression

Wn>k = moc2

_a*_ (n — k — V&2 — a2)2/

(3.5)

where a is the fine structure constant (equal to e/hc), n the principal quantum number, and k the azimuthal quantum number. Sommerfeld’s fine structure formula, or rather its first-order approximation

m0e4

W,n,k

2 h2

(3.6)

was soon verified experimentally by Paschen and other physicists. When the Bohr-Sommerfeld theory was replaced by quantum mechanics, the fine structure formula became a test case for the new theory: if quantum mechanics was to establish its empirical validity, it ought to reproduce equation (3.5) or, at least, equation (3.6). It was this test that Schrodinger’s early relativistic equation (3.4) did not pass. Consequently, Schrodinger reported only the non-relativistic approximation of equation (3.4) in his first publication on wave mechanics.
Independently of Schrodinger, the relativistic second-order equation was found in the spring of 1926 by Oskar Klein, who was the first to pub¬ lish it. During the next half year, it was investigated by several other physicists, including Fock, Gordon, de Broglie, Schrodinger, and Kudar, and eventually became known as the Klein-Gordon equation (KG equa¬ tion in what follows). In addition to the eigenvalue equation (3.4), the time-dependent KG equation corresponding to equation (3.4) was studied:

h2^ + lieh %-+<?

at

dt r

r2

mlc4 e2 t = 0 (3.7)

The fact that Schrodinger had abandoned his relativistic wave equation to avoid disagreement with experiment was commented on extensively by Dirac. Schrodinger once explained the story to Dirac, probably in 1933 during their stay in Stockholm to receive the Nobel Prize. Later in his life, Dirac considered the story to fit well with his general view of progress in theoretical physics, and he often used it as an illustrative example of how disagreement between theory and experiment should be handled. The disagreement between equation (3.4) and the hydrogen spectrum was, Dirac said,5

Relativity and spinning electrons

51

.. . most disappointing to Schrodinger. It was an example of a research worker who is hot on the trail and finding all his worst fears realized. A theory which was so beautiful, so promising, just did not work out in practice. What did Schro¬ dinger do? He was most unhappy. He abandoned the thing for some months, as he told me. . . . Schrodinger had really been too timid in giving up his first rela¬ tivistic equation. . . . Klein and Gordon published the relativistic equation which was really the same as the equation which Schrodinger had discovered previously. The only contribution of Klein and Gordon in this respect was that they were sufficiently bold not to be perturbed by the lack of agreement of the equation with observations.

According to Dirac, then, Schrodinger should have stuck to his beautiful relativistic theory and not worried too much over its disagreement with experiment; but because of simple psychological reasons - fear that his entire theory could collapse - he was mentally unable to do so. As Dirac, expounding another pet idea of his, further stated in 1971,6

It is a general rule that the originator of a new idea is not the most suitable person to develop it because his fears of something going wrong are really too strong and prevent his looking at the method from a purely detached point of view in the way that he ought to.

After this digression, let us return to the situation in 1926. As mentioned, the relativistic version of wave mechanics was investi¬
gated by several researchers in the summer and fall of 1926. Although it was not possible to make the eigenvalue equation fit the fine structure of hydrogen, in other respects the theory looked quite promising. In partic¬ ular, it proved successful in handling the Compton effect wave mechan¬ ically. Also, from a more aesthetic point of view, its four-dimensional form was appealing, especially since it proved possible to define charge and current densities that were parts of a four-vector satisfying the con¬ tinuity equation. Whereas in usual, non-relativistic wave mechanics the charge and current densities were given respectively by

p = e\i'\2 = exhP*

(3.8)

and

J

=

eh (^*V^ 2 mi

-

w*)

(3.9)

52

Dirac: A scientific biography

in the KG theory the corresponding expressions turned out to be

(3.10)

and

7 = —: -2zmi

- W)

(3.11)

Some physicists, especially in Germany, adopted another approach in their attempts to reconcile quantum mechanics and relativity. Rather than follow the Klein-Gordon approach, which was based on wave mechanics, they tried to include relativistic effects as perturbations, or corrections, to the non-relativistic theory. This method led to a partial success in the spring of 1926, when Jordan and Heisenberg, developing ideas due to Pauli, were able to derive the fine structure formula in the form (3.6). In doing so, they added to the usual Hamiltonian not only a perturbation term describing the relativistic correction to the kinetic energy but also a term referring to the spin of the electron. However, in spite of its empirical success in accounting for the doublet riddles of spec¬ troscopy, the Jordan-Heisenberg theory was not entirely satisfactory. Since relativity was added as a first-order correction, the theory was not genuinely relativistic, leading to equation (3.6) but not to equation (3.5); also, the spin effect was introduced in an ad hoc manner, being grafted to the theory rather than explained. An entirely satisfactory theory would not only be able to account for the doublet phenomena but would also explain them in the sense of deducing them from the basic principles of relativity and quantum mechanics.
During 1926, Dirac was mainly occupied with developing the formal basis of quantum mechanics. Apparently he did not consider the problem of finding a relativistic wave equation a pressing one at the time. Or per¬ haps he found it a rather trivial problem after Schrodinger had published his theory. When Dirac first made use of wave mechanics in August 1926, he did consider the Schrodinger equation in its relativistic form.7 For a free gas molecule (and also for an electron), he wrote it as

(p2 — W2I& + mlc2)\p = 0 With the substitutions

(3.12)

p = —ihV and

(3.13)

this yielded

Relativity and spinning electrons

J_
V2 C dt2

i = 0

53 (3.14)

which is just the KG equation for a free particle. Although Dirac was one of the first to state this equation, he seems not to have considered it of much importance. In his further treatment of the quantum statistics of ideal gases, he did not use equation (3.12) but used only its non-relativistic approximation.
When Dirac investigated the Compton effect wave mechanically three months later, he also started with the KG equation, but again he used only an approximation.8 At that time, Klein and Gordon had already treated the Compton effect, but Dirac stressed that his approach was inde¬ pendent and different. “The wave equation,” he wrote, referring to the KG equation, “is used merely as a mathematical help for the calculation of the matrix elements, which are then interpreted in accordance with the assumptions of matrix mechanics.”9 While staying at Bohr’s institute (where Klein was present) in November 1926, Dirac seems for a short time to have considered the KG theory as a serious candidate for a rela¬ tivistic quantum mechanics. He even played with Klein’s idea of a five¬ dimensional theory embracing both quantum mechanics and general rel¬ ativity.10 But he was soon diverted by other problems, in particular the transformation theory.
When Einstein visited Ehrenfest in Leiden in the autumn of 1926, they used the occasion to discuss Dirac’s (first) paper on the Compton effect. On October 1, Ehrenfest wrote to Dirac:11

Einstein is currently in Leiden (until Oct. 9th). In the few days we have left, he, Uhlenbeck and I are struggling together for hours at a time studying your work, for Einstein is eager to understand it. But we are hitting at a few difficulties, which - because the presentation is so short - we seem absolutely unable to overcome.

Ehrenfest asked Dirac to explain to him a number of problems, one of which was:

Why do you write the Hamilton equation in the form:

and not:

W2ld — p\ — pi — pi = m2d me2 Vl — (pi + Pi + p^i/rtfc1 - W.

Does it make a difference?

54

Dirac: A scientific biography

And at the end of his letter:

Please forgive me if some of the questions rest on gross misunderstandings. But despite all efforts we cannot get through them! Of course it would be especially nice if you were to come to Leiden yourself while Einstein is still here. Unfortu¬ nately we can only cover the cost of your ship ticket here and back. But you would be our guest here in Holland! It would be wonderful if you yourself arrived here at the same time as your letter!!!!!

Ehrenfest’s question concerning the form of the relativistic Hamiltonian equation later proved to be highly relevant with respect to the wave equa¬ tion of the electron. In 1930, Dirac gave the following argument, which could have been his response to Ehrenfest’s question four years earlier:12

Equation (3) [the second equation in Ehrenfest’s letter], although it takes into account correctly the variation of the mass of the particle with its velocity, is yet unsatisfactory from the point of view of relativity, because it is very unsymmetrical between W and the p’s, so much so that one cannot generalize it in a relativ¬ istic way to the case when there is a field present. . . . Equation (4) [the first of Ehrenfest’s equations] is not completely equivalent to equation (3) since, although every solution of (3) is also a solution of (4), the converse is not true.
However, in October 1926, the problem of a relativistic wave equation as an alternative to the equation of Klein and Gordon was scarcely in Dirac’s mind.
It was Dirac’s preoccupation with the general principles of quantum mechanics, and the transformation theory in particular, that led him to realize that the formal structure of the Schrbdinger equation [i.e., the form (H — ihd/dt)\l = 0] had to be retained even in a future unification of quantum theory and relativity. Since the KG equation is of second order in d/dt, it seemed to Dirac to be in conflict with the general for¬ malism of quantum mechanics. A similar conclusion was reached by Pauli and his Hungarian assistant Johann Kudar at the end of 1926. “Herr Pauli,” reported Kudar to Dirac, “regards the relativistic wave equation of second order with much suspicion.”13 As an alternative, the Hamburg physicists tended to consider the first-order version

(he VmflC4 - V2)^ = (W - V)f

(3.15)

as more reasonable. Although Pauli, in a letter to Schrbdinger, admitted that this equation was “mathematically rather unpleasant” because of its square root operator, he found it to be “in itself sensible.” “On the whole it seems to me that an appropriate formulation of quantum mechanics will only be possible when we succeed in treating space and time as equal to one another,” he added.14 Dirac would have agreed. Heisenberg also

Relativity and spinning electrons

55

recognized what he called “the profound meaning of the linearity of the Schrodinger equations” and for that reason considered equations of the KG type to be without prospect.15 But it was only Dirac who managed to harvest the rich fruit of this insight. Before we follow Dirac’s road to the relativistic equation, a look at the theory of spin may be helpful.
At the end of 1926, it was widely accepted that spin and relativity were intimately related. However, it proved impossible to incorporate spin in the KG theory, and the nature of the relations among spin, relativity, and quantum mechanics remained a problem. In December 1926, Heisenberg and Dirac made a bet in Copenhagen as to when the spin phenomenon would be properly explained. Heisenberg wrote to Pauli, “I made a bet with Dirac that the spin phenomena, like the structure of the nucleus, will be understood in three years at the earliest; while Dirac maintains that we will know for sure about the spin within three months (counting from the beginning of December).”16 Who won the bet is debatable, but with regard to the spin, at least, Dirac was closer to the mark than Heisen¬ berg.17 Dirac himself began to think about the spin problem shortly after the bet, probably inspired by discussions with Pauli, who visited Bohr’s institute in January 1927. According to his memoirs, Dirac got the idea of representing spin by three spin variables independently of Pauli.18 But he did not follow up this idea, which was developed by Pauli in his important quantum mechanical theory of spin of May 1927.19 Pauli pro¬ posed a two-component wave function for the electron in order to accom¬ modate spin, which was represented by a new kind of quantum variables, namely, the 2X2 “Pauli matrices”

(1 o)’ ff-v (i o)’ and(T

(3.16)

which were related to the spin vector by

5 = Z2 ha

(3.17)

With this approach the energy eigenvalue equation became two coupled equations of the type

ihm=H{9b~ihik’")'l/

<3'18)

At about the same time, Darwin independently obtained a spin theory that was equivalent to Pauli’s but was expressed in the language of wave
mechanics.20 The Pauli-Darwin theory was welcomed by physicists because it man-

56

Dirac: A scientific biography

aged to represent spin quantum mechanically, avoiding the ad hoc fea¬ ture of the earlier Heisenberg-Jordan theory. On the other hand, it did not solve new empirical problems and did not contribute to the solution of the profound problem of integrating quantum mechanics with relativ¬ ity. As in the earlier theory, Pauli and Darwin were forced to include rel¬ ativistic effects only as a first-order correction to the non-relativistic Hamiltonian. This was sufficient from an empirical point of view, but from an aesthetic point of view, as adopted by Dirac, it was a blemish which indicated that the theory was only provisional. Both Pauli and Darwin recognized that “one must require from a final theory that it is formulated relativistically invariant from the outset, and also allows cal¬ culation of higher corrections.”21 During 1927, all attempts to improve on the theory of Pauli and Darwin proved fruitless.
Dirac did not try to develop the spin theory beyond the limits set by Pauli and Darwin, but he was very interested in the theory. While wave mechanicians like Schrodinger preferred Darwin’s approach, Dirac was much in favor of Pauli’s method, which, he argued, was in better agree¬ ment with the general theory of quantum mechanics. In lectures given at Cambridge in the fall of 1927, Dirac praised Pauli’s theory:22

It consists in abandoning from the beginning any attempt to follow the classical theory. One does not try to take over into the quantum theory the classical treat¬ ment of some model, which incorporates the empirical facts, but takes over the empirical facts directly into the quantum theory. This method provides a very beautiful example of the general quantum theory, and shows that this quantum theory is no longer completely dependent on analogies with the classical theory, but can stand on its own feet.

At that time Dirac was closely acquainted with the spin theory, including its applications to spectra. In his lecture notes he pointed out its deficien¬ cies too, such as its failure to agree to better than first order with Sommerfeld’s fine structure formula. Although preoccupied with spin, Dirac did not attempt to obtain a more satisfactory theory. The problem of get¬ ting a relativistic wave equation was in his mind, but he did not yet con¬ nect this problem with spin. On the contrary, Dirac thought that his still unborn theory would describe a spinless particle, supposed to be the sim¬ plest kind of particle; only after the theory for such a hypothetical particle was established did he expect that spin could be incorporated.
During the Solvay Congress in October 1927, Dirac mentioned to Bohr his concern about a relativistic wave equation:23

Then Bohr answered that the problem had already been solved by Klein. I tried to explain to Bohr that I was not satisfied with the solution of Klein, and I wanted

Relativity and spinning electrons

57

to give him reasons, but I was not able to do so because the lecture started just then and our discussion was cut short. But it rather opened my eyes to the fact that so many physicists were quite complacent with a theory which involved a radical departure from the basic laws of quantum mechanics, and they did not feel the necessity of keeping to these basic laws in the way that I felt.

After his return from Brussels, Dirac concentrated on the problem of for¬ mulating a first-order relativistic theory of the electron. Within two months he had solved the whole matter.24
A few days before Christmas 1927, Darwin went to Cambridge. He was completely surprised to learn of Dirac’s new theory, about which he reported to Bohr:25

I was at Cambridge a few days ago and saw Dirac. He has now got a completely new system of equations for the electron which does the spin right in all cases and seems to be “the thing.” His equations are first order, not second, differential equations! He told me something about them but I have not yet even succeeded in verifying that they are right for the hydrogen atom.

As usual, Dirac had worked alone, almost secretly. He thrived best in this way and very seldom discussed his ideas with other physicists. Mott, who was at the time as close to Dirac as anyone, recalled that “all Dirac’s dis¬ coveries just sort of fell on me and there they were. I never heard him talk about them, or he hadn’t been in the place chatting about them. They just came out of the sky.”26 “The Quantum Theory of the Electron,” probably Dirac’s greatest contribution to physics, was received by Pro¬ ceedings of the Royal Society on January 2, 1928.
Dirac’s theory was a product of his emerging general philosophy of physics. He wanted the theory to be founded on general principles rather than on any particular model of the electron. In contrast to Pauli, Schrodinger, and Darwin, who all imagined that the problem of integrating spin and relativity would probably require some sophisticated model of the electron, Dirac was not at all interested in model-building. “The ques¬ tion remains as to why Nature should have chosen this particular model for the electron instead of being satisfied with a point-charge,” he pointed out at the beginning of his paper.27 Consequently, he considered the elec¬ tron to be a point charge.
Dirac’s point of departure was that “we should expect the interpreta¬ tion of the relativistic quantum theory to be just as general as that of the non-relativistic theory.”28 In full compliance with his general outlook on physics, he was guided by two invariance requirements: first, the spacetime properties of the equation should transform according to the theory of relativity; and second, the quantum properties should transform

58

Dirac: A scientific biography

according to the transformation theory of quantum mechanics. Dirac rec¬ ognized that the latter requirement excluded the KG theory. Only if the wave equation is linear in d/dt is the probability interpretation secured. This reasoning suggests the starting procedure

/ft 77 = c \Jnf& + p\ + p\ + pit ot

(3.19)

for a free electron. This is the same equation Pauli had considered in private communications [see equation (3.15)]. Of course, it faces the same mathematical difficulties because of the square root operator, which seems to yield a differential equation of infinite order. But, Dirac rea¬ soned, it would be a promising start if the square root could be arranged in a linear form in the momenta. Then he faced a purely mathematical problem: How can a square root of four quantities possibly be linearized? At this point he could have consulted the mathematicians; the German algebraists might have supplied him with an answer.29 But Dirac was not one to ask for assistance. He worked out the problem in his own way, by “playing around with mathematics,” as he said.30 He found a clue in an identity he had noticed when he “played” with the Pauli spin matrices, namely

VFi + Pi + pi = g\P\ + (T2P2 + V3P3

(3.20)

At that stage, Dirac may have tried to use the quantity a • p as the Ham¬ iltonian in a wave equation; that is, he may have considered

a - pi =

(3.21)

where \p is a two-component wave function. This equation is Lorentzinvariant, contains a spin of one-half, and is of first order in the time derivative. But since it does not contain a mass term, it does not apply to an electron.31 Hence Dirac had to reconsider the possible significance of equation (3.19): If it could be generalized to four squares instead of two, it would obviously indicate a solution; for then a linearization of the type wanted

Vpi + p\ + p\ + (mQc)2 = a,p, + 012P2 + a3p3 + a4m0c (3.22)

could be provided. But were there coefficients with this property, and if so, what did they look like? Dirac argued that the linear wave equation, as provisionally given by equations (3.19) and (3.22), has to contain the

Relativity and spinning electrons

59

KG equation as its square. In this way he was able to deduce the following set of conditions for the coefficients in equation (3.22):

= 0
<*1 = 1

(m A v)

^ 23)

Dirac knew that a set of similar conditions are fulfilled by the spin matri¬ ces, of which there are, however, only three. So he naturally tried to take olj = <7j and sought for another 2X2 candidate for a4. However, such a candidate does not exist, and Dirac realized that working with 2X2 matrices just would not do. Then he got again one of those invaluable ideas out of the blue: “I suddenly realized that there was no need to stick to quantities, which can be represented by matrices with just two rows and columns. Why not go to four rows and columns?”32 This idea solved the problem, and he found the explicit form of the a matrices:

aj U °) / 0 a. \

and

1 0 0 1

0 0

00 \

1 «4 =

0 0

Vo 0

-1 0

-10/

where j = x, y, or z, and <j, are the Pauli matrices. With the linearization successfully carried out, the ice was broken. The
next stage - to formulate the wave equation for a free electron - was easy. Equations (3.19) and (3.22) immediately yielded

(W/c + a ■ p + aAm0c)ip = 0

(3.24)

which is known as the Dirac equation. Dirac reduced a physical problem to a mathematical one, and mathe¬
matics forced him to accept the use of 4 X 4 matrices as coefficients. This again forced him to accept a four-component wave function \p = (^1,1^2,^3,^4). Though logical enough, this was a bold proposal since there was no physical justification for the two extra components. The conclu¬ sion rested upon Dirac’s confidence in the power of mathematical rea¬ soning in the realm of physics. Indeed, Dirac’s theory' of the electron is a beautiful example of what Wigner has called “the unreasonable effective¬ ness of mathematics in the natural sciences.”33 If Dirac had followed an empiricist logic of science, he would not have introduced such “unphys¬ ical” terms as 4 X 4 matrices. As Darwin acknowledged, in comparing Dirac’s work with his own attempt: “Dirac’s success in finding the accu-

60

Dirac: A scientific biography

rate equations shows the great superiority of principle over the previous empirical method.”34
At this stage the equation was only an inspired guess. Dirac had to prove that it was logically, as well as empirically, satisfactory. It was con¬ structed to conform with the principles of quantum mechanics, and since Dirac could prove its Lorentz invariance, the equation met the formal requirements. But what about its application to experimental reality? To check this, a free electron would not do; it had to be capable of interact¬ ing. Dirac placed the electron in an electromagnetic field, using the stan¬ dard procedure of replacing W with (W — e4>) and p with (p — e/c A); that is, he used the Hamiltonian given by equation (3.2). A little manip¬ ulation of the a matrices then converted equation (3.24) into

(W_$
+ P\ \c c

• (p — ~cA j j + p3m0c | \p = 0 (3.25)

where pb p3, and a = {a\,a2^A are new 4X4 matrices derived from the old a matrices. By a further transformation Dirac was then able to show that this differential operator, if squared, included the KG operator and. in addition, the term

c
where B = V X A is the magnetic field. If divided by 2m, this term rep¬ resents an additional energy of the electron corresponding to a magnetic moment eha/lmc. “This magnetic moment is just that assumed in the spin electron model” (i.e., in Pauli’s theory), Dirac wrote.35 Without introducing the spin in advance, Dirac was thus able to deduce the correct spin from the first principles upon which his equation was built. This was a great and unexpected triumph:36
I was not interested in bringing the spin of the electron into the wave equation, did not consider the question at all and did not make use of Pauli’s work. The reason for this is that my dominating interest was to get a relativistic theory agree¬ ing with my general physical interpretation and transformation theory. ... It was a great surprise for me when I later on discovered that the simplest possible case did involve the spin.
Of course, it was an exaggeration for Dirac to say that he did not make use of Pauli’s theory; he did, as we have seen, when playing around with the spin matrices. But the use he made of the spin matrices was heuristic only.

Relativity and spinning electrons

61

When Dirac’s theory appeared, its strength lay at the conceptual and methodological level, not at the empirical level. In fact, at first the theory did not yield even one result or explain even one experimental fact not already covered by earlier theories. Dirac showed in his paper that the new theory in its first approximation led to the same energy levels for the hydrogen atom as those given by the theories of Darwin and Pauli; that is, he deduced the approximate fine structure formula. But he did not attempt to go further, either by including higher corrections or by looking for an exact solution that would, hopefully, yield the exact fine structure formula. To derive this formula, which was still unexplained by quantum mechanics, would have gone far to credit the new theory. One may there¬ fore wonder why Dirac did not attack the problem with more determi¬ nation. According to his recollections, he did not even attempt to find an exact solution but looked for an approximation from the start:37

I was afraid that maybe they [the higher order corrections] would not come out right. Perhaps the whole basis of the idea would have to be abandoned if it should turn out that it was not right to the higher orders, and I just could not face that prospect. So I hastily wrote up a paper giving the first order of approximation and showing it to that accuracy; at any rate, we had agreement between the theory and experiment. In that way I was consolidating a limited amount of success that would be something that one could stand on independently of what the future would hold. One very much fears the need for some consolidated success under circumstances like that, and I was in a great hurry to get this first approximation published before anything could happen which might just knock the whole thing on the head.

However, there are reasons to believe that Dirac’s retrospection, based on his hope-and-fear moral, is not quite correct. When he created the the¬ ory, he was guided by a strong belief in formal beauty and had every rea¬ son to be confident that his theory was true. It seems unlikely that he really would have feared that the theory might break down when applied to the hydrogen atom. After all, the Sommerfeld formula had never been tested beyond its first or second approximation; if relativistic quantum mechanics did not reproduce that formula exactly, it could justifiably be argued that it was not exactly true. Dirac’s hurry in publication may have been motivated simply by competition, the fear of not being first to pub¬ lish. Several other physicists were working hard to construct a relativistic spin theory, a fact of which Dirac must have been aware. Naturally he felt that the credit belonged to him. He did not want to be beaten in the race, a fate he had experienced several times already. If agreement with the fine structure formula had the crucial importance that Dirac later asserted, one would expect that he would have attempted to derive the exact fine structure after he submitted his paper for publication. He did

62

Dirac: A scientific biography

not. I think Dirac was quite satisfied with the approximate agreement and had full confidence that his theory would also provide an exact agree¬ ment. He simply did not see any point in engaging in the complicated mathematical analysis required for the exact solution.
Other physicists who at the time tried to construct a relativistic spin theory included Hendrik Kramers in Utrecht; Eugene Wigner and Pascual Jordan in Gottingen; and Yakov Frenkel, Dmitri Iwanenko, and Lev Landau in Leningrad. Kramers obtained an approximate quantum description of a relativistic spinning electron in terms of a second-order wave equation and later proved that his equations were equivalent to Dirac’s equation. When he got news of Dirac’s theory, he was deeply dis¬ appointed, and this feeling evolved into a continuing frustration with regard to Dirac’s physics. It is unknown in what direction Jordan and Wigner worked (they never published their work), but it seems to have been toward a relativistic extension of Pauli’s spin theory. “We were very near to it,” Jordan is supposed to have said, “and I cannot forgive myself that I didn’t see that the point was linearization.”38 Although disap¬ pointed, Jordan recognized the greatness of Dirac’s work. “It would have been better had we found the equation but the derivation is so beautiful, and the equation so concise, that we must be happy to have it.”39 Frenkel, Iwanenko, and Landau engaged in laborious tensor calculations and suc¬ ceeded in working out theories that in some respects were similar to Dirac’s. But apart from being published after Dirac’s work, they lacked, like Kramers’s theory, the beauty and surprising simplicity that charac¬ terized Dirac’s theory.40 Still, there can be little doubt that had Dirac not published his theory in January 1928, an equivalent theory would have been published by other physicists within a few months. Dirac later said that if he had not obtained the wave equation of the electron, Kramers would have.41
The news of Dirac’s new equation spread rapidly within the small com¬ munity of quantum theorists. The key physicists knew about it before publication. In Gottingen they learned about the theory from a letter Dirac sent to Born, and Bohr was informed by Fowler (and earlier by Darwin), who, as a Fellow of the Royal Society, had communicated the paper to the ProceedingsThe reception was enthusiastic. Leon Rosenfeld, who at the time was working with Born in Gottingen, recalled that the deduction of the spin “was regarded as a miracle. The general feeling was that Dirac had had more than he deserved! Doing physics in that way was not done! ... It [the Dirac equation] was immediately seen as the solution. It was regarded really as an absolute wonder.”43 From Leipzig Heisenberg wrote to Dirac about his collaboration with Pauli on quan¬ tum electrodynamics, and added: “I admire your last work about the spin in the highest degree. I have especially still for questions: do you get the

Relativity and spinning electrons

63

Sommerfeld-formula in all approximations? Then: what are the currents in your theory of the electron?”44 A month later, Ehrenfest reported his opinion to the Russian physicist Joffe: “I find Dirac’s latest work on elec¬ tron spin just splendid. Tamm has explained it all to us very well. He is continuing to work on this.”45
Within two weeks following submission of the paper, Walter Gordon in Hamburg was able to report to Dirac that he had derived the exact fine structure formula from the new equation and that Heisenberg’s first ques¬ tion could thus be answered affirmatively. Reporting the main steps in the calculation, Gordon wrote: “I should like very much to learn if you knew these results already and if not, you think I should publish them.”46 A little later, Darwin got the same result. Darwin was impressed by Dirac’s genius but found the theory very difficult to understand unless he transcribed it to a more conventional, wave mechanical formalism. As he told Bohr: “I continue to find that though Dirac evidently knows all about everything the only way to get it out of his writings is to think of it all for oneself in one’s own way and afterwards to see it was the same thing.”47 The fact that Dirac’s equation yielded exactly the same formula for the hydrogen atom that Sommerfeld had found thirteen years earlier was another great triumph. It also raises the puzzling question of how Sommerfeld’s theory, based on the old Bohr theory and without any notion of spin, could give exactly the same energy levels as Dirac’s the¬ ory. But this was a historical curiosity that did not bother the physicists.48
A month after the publication of his first paper on the relativistic elec¬ tron, Dirac completed a sequel in which he investigated various conse¬ quences of the theory for the behavior of spectral lines. In a letter to Jor¬ dan he reported the main results:49

I have worked out a few more things for atoms with single electrons. The spectral series should be classified by a single quantum number; taking positive and neg¬ ative integral values (not zero) instead of the two, k and j, of the previous theory. The connection between j values and the usual notation is given by the following
scheme:

j= -1 1 —2 2 — 3^ 3_^y.

S

P

D

F

j and -O' + 1) form a spin doublet. One finds for the selection rule, j-*j± 1 or j -► —j and for the g-value in a weak magnetic field g = j/j + %. The magnetic quantum number m satisfies — \j | + )4 < m < \j\ + %.

That is, Dirac showed that all the doublet phenomena were contained in his equation.

64

Dirac: A scientific biography

The first occasion at which Dirac himself presented his theory to the German physicists was when he delivered a lecture to the Leipziger Universitatswoche during June 18-23, 1928.50 This was the first in a series of annual symposia on current research in physics, and it was arranged by Debye and Heisenberg, the new professors of physics at Leipzig. Dirac gave a survey of his new theory and called attention to a further argument for the linear wave equation. He showed that the charge density associ¬ ated with the KG theory, equation (3.10), is not positive definite: Since the KG equation is of second order in the time derivative, when ip(t0) is unknown, (dip/dt)l=lQ is undetermined and therefore \p(t > t0) is also unde¬ termined; and since p is a function of \p and d\p/dt, knowledge of p(t0) leaves p(t > t0) undetermined, so that the electrical charge JpdV may attain any value. “The principle of charge conservation would thus be violated. The wave equation must consequently be linear in d/dtfi asserted Dirac.51 That Dirac’s new equation did not face the same diffi¬ culty was explicitly shown by Darwin, who gave expressions for the charge and current densities.52 In Dirac’s theory the probabilities and charge densities took the same form as in the non-relativistic theory, that is, \\p\2 and e\\p\2.
Dirac’s theory of the electron had a revolutionary effect on quantum physics. It was as though the relativistic equation had a life of its own, full of surprises and subtleties undreamed of by Dirac when he worked it out. During the next couple of years, these aspects were uncovered. The mathematics of the equation was explored by von Neumann, Van der Waerden, Fock, Weyl, and others, and the most important result of this work was the spinor analysis, which built upon a generalization of the properties of the Dirac matrices. Dirac had not worried about the math¬ ematical nature of his four-component quantities; at first, it took the mathematical physicists by surprise to learn that the quantities were nei¬ ther four-vectors nor tensors. Other theorists attempted to incorporate the Dirac equation into the framework of general relativity or, as in the case of Eddington, to interpret it cosmologically. Producing generaliza¬ tions of the Dirac equation, most of them without obvious physical rel¬ evance, became a pastime for mathematical physicists. Although in the early part of 1928 it seemed that the theory had no particular predictive power or empirical surplus content, it soon turned out to be fruitful for the experimentalists too. In particular, it proved successful in the study of relativistic scattering processes, first investigated by Mott in Cam¬ bridge and by Klein, Nishina, and Moller in Copenhagen.
By the early thirties, the Dirac equation had become one of the corner¬ stones of physics, marking a new era of quantum theory. Its undisputed status was more a result of its theoretical power and range than of its empirical confirmation. In fact, several of the predictions that followed

Relativity and spinning electrons

65

from Dirac’s theory appeared to disagree with experiment. For example, in 1930 Mott predicted, on the basis of Dirac’s theory, that free electrons could be polarized, a result that not only was contradicted by experiment but also ran counter to Bohr’s intuition.53 The negative outcome of the experiments threatened to discredit not only Mott’s theory but also, by implication, Dirac’s. In the mid-1930s, the feeling was widespread in some quarters that “the Dirac equation needs modification in order to account successfully for the absence of polarization.”54 But although the Dirac equation was confronted with this and other apparent failures, its authority remained intact.55 Even in the spectrum of hydrogen - a show¬ piece for the Dirac equation - anomalies turned up. During the thirties, improved experiments showed a small but significant discrepancy between the experimentally determined fine structure of the hydrogen lines and that predicted by the Dirac theory. In 1934, William V. Hous¬ ton and Y. M. Hsieh at Caltech made careful calculations of the Balmer lines which forced them to conclude that “the theory, as we have used it, is inadequate to explain the observations.”56 However, although the dis¬ crepancies between theoretical predictions and observed values were con¬ firmed by several other studies, the exact validity of Dirac’s theory was not seriously questioned until 1947.
Other difficulties faced Dirac’s theory in connection with cosmic radi¬ ation and the new field of nuclear physics. For example, the theory was believed also to apply to protons, for which it predicted a magnetic moment of one nuclear magneton. When Otto Stern and Otto Frisch suc¬ ceeded in measuring that quantity in 1933, their result was almost three times as large as predicted.57 But this anomaly also was unable to seri¬ ously discredit Dirac’s theory, which was, after all, a theory of electrons. Most physicists concluded that the theory just did not apply to nuclear particles. Bohr was of the opinion that “we cannot ... expect that the characteristic consequences of Dirac’s electron theory will hold unmodi¬ fied for the positive and negative protons.”58 As a last example of the difficulties that confronted Dirac’s theory, we may mention the so-called Meitner-Hupfeld effect, an anomalously large scattering found when high-energy gamma rays were scattered in heavy elements. Although the Meitner-Hupfeld effect, first reported in the spring of 1930, disagreed with the predictions of Dirac’s electron theory, the theory was not found guilty; instead, the anomaly was thought to arise from intranuclear
electrons.59 The real difficulties for the theory were connected with the physical
interpretation of its mathematical structure, in particular, with the nega¬ tive-energy solutions. This problem, which in its turn led to new and amazing discoveries, will be considered in more detail in Chapter 5. Let it suffice to mention here that Dirac had already noticed the difficulty in

66

Dirac: A scientific biography

his first paper, where it caused him to label the theory “still only an approximation.” In his Leipzig address he commented briefly on a related difficulty, namely, that the theory allows transitions from charge + e to — e. Dirac had no answer to this problem, which was already much discussed in the physics community, except the following vague remark: “It seems that this difficulty can only be removed through a fundamental change in our previous ideas, and may be connected with the difference between past and future.”60 Heisenberg seems to have been a bit disap¬ pointed that Dirac had not yet come up with a solution. Deeply worried over the situation, Heisenberg wrote to Jordan after Dirac’s lecture: “Dirac has lectured here only on his current theory, giving a pretty foun¬ dation for it that the differential equations must be linear in d/dxHe has not been able to solve the well-known difficulties... .”61 And a month later he wrote to Bohr: “I am much more unhappy about the question of the relativistic formulation and about the inconsistency of the Dirac the¬ ory. Dirac was here and gave a very fine lecture about his ingenious the¬ ory. But he has no more of an idea than we do about how to get rid of the difficulty e -* —e.. .”62

CHAPTER 4
TRAVELS AND THINKING
THE relativistic theory of the electron made Dirac in great demand at physics conferences and centers around the world, and in general increased his status as a scientist. In early 1928, Klein visited Cambridge and took back to Copenhagen the latest news of Dirac’s ideas. Two months later, Schrodinger went to Cambridge and gave a talk to the Kapitza Club on the “Physical Meaning of Quantum Mechanics.” In April, Dirac spent a few weeks at Bohr’s institute and then traveled on to Leiden, where he discussed physics with Ehrenfest and other Dutch physicists and also gave a few lectures.1 Dirac had begun working on a book on quantum mechanics, to be published two years later as The Principles of Quantum Mechanics, and he took the oppor¬ tunity to test the first chapters on his audience in Leiden. Ehrenfest was much impressed by Dirac but found it difficult to understand the papers of this British wizard of quantum mechanics. He was therefore delighted when on one occasion Dirac was asked a question to which he, just for once, could not give an immediate and precise answer. “Writing very small he [Dirac] made some rapid calculations on the blackboard, shield¬ ing his formulae with his body. Ehrenfest got quite excited: ‘Children,’ he said, ‘now we can see how he really does his work.’ But no one saw much; Dirac rapidly erased his tentative calculations and proceeded with an ele¬ gant exposition in his usual style.”2 After spending about a month with Ehrenfest, Dirac continued to Leipzig to attend the Universitatswoche and to discuss the latest developments in physics with Heisenberg. After Leipzig the tour went on to Gottingen, where Dirac stayed until the beginning of August. In Gottingen he met, among others, the twentyfour-year-old Russian physicist George Gamow, who had just succeeded in explaining alpha-radioactivity on the basis of quantum mechanics. In all these places - Copenhagen, Leiden, Leipzig, and Gottingen - Dirac lectured on his electron theory.
Among the physicists in Leiden was Igor Tamm, a thirty-two-year-old
67

68

Dirac: A scientific biography

visitor from Moscow State University. In a letter to a relative, he reported being eager to meet Dirac, of whom he had been told the strang¬ est things: “It is now definite that on April 23 Dirac will come here for three months. So I shall be able to learn something from that new physics’ great genius. Though they say that Dirac is not a great one for words and you have to try very hard to start up a conversation with him. He seems to talk only with children, and they have to be under ten. .. ,”3 Dirac lived up to Tamm’s great expectations, and the two became closely acquainted. “The criteria to go by now are Dirac’s,” wrote Tamm. “And compared to him I am just a babe in arms. Of course, it is still more stupid to measure oneself by a man of genius. ... With great patience Dirac is teaching me the way I should go about things; I am very proud that we have come to be friends.”4 During their stay in Holland, Tamm taught Dirac to ride a bicycle, and together they went for long tours. “I have not forgotten the cycling that I learnt in Leiden,” Dirac later wrote from Cambridge. “I have already cycled about 2000 km in the neigh¬ bourhood of Cambridge.”5 When Dirac went to Leipzig, Tamm accom¬ panied him. On his return to Moscow, Tamm told about his experiences abroad:6

While abroad, I lived for five months in Holland and for two in Germany. What pleased me most was coming together with Dirac. He and I kept company for three months and came to be very close. Dirac is a true man of genius. Do not smile that it sounds high-flown; I really mean it. I know that when I grow old I’ll be telling my grandchildren with pride about that acquaintance of mine.
In the summer of 1928, Heisenberg was professor at Leipzig, a position he had obtained at only twenty-six years of age. Dirac received his first offer of a chair at the same age, when Milne, who had occupied the pro¬ fessorship in applied mathematics at Manchester University since 1924, was elected a professor of mathematics at Oxford. Dirac was asked in July if he was interested in the vacant chair, but he declined the offer.7 He wanted to continue his own style of life and be free to cultivate the spe¬ cialized research in which he was an expert. Neither the prestige of a pro¬ fessorship nor the prospect of much-improved material and economic conditions tempted him:8

I feel greatly honoured by being considered a possible successor to Prof. Milne, but I am afraid I cannot accept the appointment. My work is of too specialised a nature to be satisfactorily carried on outside a great centre such as Cambridge where there are others interested in the same subject, and my knowledge of and interest in mathematics outside my own special branch are too small for me to be competent to undertake the duties of a Professor of applied mathematics.

Travels and Thinking

69

A few months later, he was asked by A. H. Compton to accept a new chair of theoretical physics to be created at the University of Chicago.9 The offer was rewarding and so was the salary of $6,000 a year. Dirac declined.
Bohr wanted Dirac to come to Copenhagen again in September to par¬ ticipate in a conference, but Dirac decided to go to the Soviet Union instead. To Klein he wrote:10

I am afraid I shall not be able to come to Copenhagen in September as I intended. I am going to the Physical Congress in Russia (on the Volga) and I expect to return to England via Constantinople. I very much regret that I shall not be meeting you and Prof. Bohr, and I hope to be able to visit Copenhagen next year. I have not met with any success in my attempts to solve the ± e difficulty. Heisenberg (whom I met in Leipzig) thinks the problem will not be solved until one has a theory of the proton and electron together. I shall be leaving for Leningrad on about August 2nd. Best wishes to you and Prof. Bohr, and Nishina if he is still with you.
In the decade 1925-35, there was relatively close contact between West¬ ern and Soviet science. Soviet authorities wanted to develop the national science rapidly and stimulated contact with the West, which included the employment of foreign scientists in Russia, the arrangement of interna¬ tional conferences, publishing in foreign journals, and sending Soviet sci¬ entists abroad to Western institutions. Around 1930, theoretical physics in the USSR, and particularly in Leningrad, could compete with that in any Western nation. Several of the best Russian theorists, including Tamm, Frenkel, Landau, Iwanenko, and Fock, worked in the same areas as Dirac and looked forward to meeting him."
The sixth All-Union Conference on physics took place in August and September 1928, arranged by Abraham Joffe, president of the Russian Association of Physicists. Apart from the Russian hosts, several Western physicists participated: Brillouin from France, G. N. Lewis from the United States, Dirac and Darwin from Great Britain, and Born, Pringsheim, Scheel, Pohl, Ladenburg, and Debye from Germany. In the com¬ pany of Born and Pohl, Dirac went from Gottingen to Leningrad, which he found to be the most beautiful city he had ever seen, and only joined the congress in Moscow some days after it opened on August 4. A week later, the congress went to Nisjni-Nowgorod, from where it continued on board a steamer along the Volga river, visiting Gorki, Kazan, Saratow, and Tiflis.12 During the Volga trip, Dirac bathed in the Volga and learned to eat caviar and watermelons. He lectured on his theory of the electron and after the congress traveled alone through the Caucasus to Batoum and the Black Sea coast. While in the Caucasus, Dirac participated in an excursion that took him to a height of about 3,000 meters, which was, he reported to Tamm,13

70

Dirac: A scientific biography

On the Volga. Dirac in company with Yakov Frenkel (next to him) and Alfred Lande at the 1928 physics congress in Russia. Reproduced with permission of AIP Niels Bohr Library.
a good deal higher than my previous record.... I spent three days in Tiflis, mostly resting and making up for lost sleep, and then went to Batoum to try to get a boat for Constantinople.... From Constantinople I took a ship to Marseilles, visiting Athens and Naples on the way, and then came home across France and ended a most pleasant holiday.
He arrived back in Cambridge after half a year’s traveling around Europe. This was not a comfortable life, but lack of comfort never bothered Dirac.
After his return to Cambridge, Dirac resumed his solitary style of life, spending most of his time working in his college room or in one of the libraries. He completed a work on statistical quantum mechanics and started preparing his lecture notes for a book, which would become his great textbook on quantum mechanics. In January 1929, he met again with Gamow, who, after a stay in Copenhagen, spent about a month in

Travels and Thinking

71

Cambridge. Later in the year, Dirac was appointed to the more secure (insured for three years) position of University Lecturer at St. John’s Col¬ lege, a post that carried a basic annual stipend of two hundred pounds. Knowing that other universities had offered Dirac a position, the college was anxious to give him the best conditions in order to retain him. Dirac was therefore given very little teaching and administrative work and could continue to spend most of his time on research.
After half a year in Cambridge, Dirac next went to the United States in order to spend two months as a visiting professor at the University of Wisconsin in Madison, a job for which he was paid $1,800. American theoretical physics was in a state of transformation, which in the course of a few years would make the United States the leading nation in the field. But in the late twenties, American universities were still relatively weak in theoretical physics and made considerable use of visiting physi¬ cists from Europe. Dirac did not know much about physics in the United States, and what he knew did not impress him. When asked by Edward Condon in 1927 if he would like to visit America, he had replied, “There are no physicists in America.”14 Earlier invitations to visit America, issued by A. H. Compton in Chicago and his brother, Karl T. Compton, at Princeton, had also failed to tempt Dirac.15
Dirac arrived in New York on March 20, 1929, and proceeded the fol¬ lowing day to Princeton where he met with the mathematician Oswald Veblen.16 From there he went to Wisconsin to be welcomed by Van Vleck, who had recently been appointed professor of mathematical phys¬ ics at the university. As part of the terms for his position, Van Vleck was to invite a foreign theorist to visit the university each year; Dirac was the first, to be followed by Wentzel and Fowler. He lectured on quantum mechanics, mostly his transformation theory, and alone or together with Van Vleck went for walks in nearby Minnesota. During this first stay in America, Dirac was associated with the University of Wisconsin and later with the summer school of the University of Michigan; during the spring vacation, he gave lectures at the University of Iowa. The content of the lectures Dirac planned to give in Madison was the subject of a letter he wrote to Van Vleck in December 1928; through this letter we also learn that he had by then begun to write his great quantum mechanics text:17

I think that in my lectures at Madison the best place to begin is with the trans¬ formation theory. A good knowledge of this is necessary for all the later devel¬ opments. I could deal with this subject assuming my audience have only an ele¬ mentary knowledge of Heisenberg’s matrices and Schrodinger’s wave equation, or alternatively I could present it in a way which makes no reference to previous forms of the quantum theory, following a new method which I am now incorpo¬ rating in a book. This alternative would perhaps be a little more difficult for non-

72

Dirac: A scientific biography

mathematical students and would need more time. After dealing with the trans¬ formation theory I propose to apply it to problems of emission and absorption, the quantization of continuing media, and the relativity theory of the electron.
Dirac’s introversive style and his interest in abstract theory were rather foreign to the scientists at the University of Wisconsin. They recognized his genius but had difficulties in comprehending his symbolic version of quantum theory.18 The Americans also found him a bit of a strange char¬ acter. A local newspaper, the Wisconsin State Journal, wanted to inter¬ view the visiting physicist from Europe and assigned this task to a humorous columnist known as “Roundy.” His encounter with Dirac is quoted here in extenso because it not only reveals some characteristic features of Dirac’s personality but also is an amusing piece of journalism:19

I been hearing about a fellow they have up at the U. this spring - a mathematical physicist, or something, they call him - who is pushing Sir Isaac Newton, Einstein and all the others off the front page. So I thought I better go up and interview him for the benefit of the State Journal readers, same as I do all the other top notchers. His name is Dirac and he is an Englishman. He has been giving lectures for the intelligentsia of the math and physics department - and a few other guys who got in by mistake.
So the other afternoon I knocks at the door of Dr. Dirac’s office in Sterling Hall and a pleasant voice says “Come in.” And I want to say here and now that this sentence “come in” was about the longest one emitted by the doctor during our interview. Ele sure is all for efficiency in conversation. It suits me. I hate a talka¬ tive guy.
I found the doctor a tall youngish-looking man, and the minute I see the twinkle in his eye I knew I was going to like him. His friends at the U. say he is a real fellow too and good company on a hike - if you can keep him in sight, that is.
The thing that hit me in the eye about him was that he did not seem to be at all busy. Why if I went to interview an American scientist of his class - supposing I could find one - I would have to stick around an hour first. Then he would blow in carrying a big briefcase, and while he talked he would be pulling lecture notes, proof, reprints, books, manuscripts, or what have you, out of his bag. But Dirac is different. He seems to have all the time there is in the world and his heaviest work is looking out of the window. If he is a typical Englishman it’s me for England on my next vacation!
Then we sat down and the interview began. “Professor,” says I, “I notice you have quite a few letters in front of your last name. Do they stand for anything in particular?”
“No,” says he. “You mean I can write my own ticket?” “Yes,” says he.
“Will it be all right if I say that P. A. M. stands for Poincare Aloysius Mussolini?”

Travels and Thinking

73

“Yes,” says he. “Fine,” says I, “We are getting along great! Now doctor will you give me in a few words the low-down on all your investigations?” “No,” says he. “Good,” says I. “Will it be all right if I put it this way - ‘Professor Dirac solves all the problems of mathematical physics, but is unable to find a better way of figuring out Babe Ruth’s batting average’?” “Yes,” says he. “What do you like best in America?” says I. “Potatoes,” says he. “Same here,” says I. “What is your favorite sport?” “Chinese chess,” says he. That knocked me cold! It sure was a new one to me! Then I went on: “Do you go to the movies?” “Yes,” says he. “When?” says I. “In 1920 - perhaps also 1930,” says he. “Do you like to read the Sunday comics?” “Yes,” says he, warming up a bit more than usual. “This is the most important thing yet Doctor,” says I. “It shows that me and you are more alike than I thought. And now I want to ask you something more: They tell me that you and Einstein are the only two real sure-enough high-brows and the only ones who can really understand each other. I wont ask you if this is straight stuff for I know you are too modest to admit it. But I want to know this - Do you ever run across a fellow that even you cant understand?” “Yes,” says he. “This will make great reading for the boys down to the office,” says I. “Do you mind releasing to me who he is?” “Weyl,” says he. The interview came to a sudden end just then, for the doctor pulled out his watch and I dodged and jumped for the door. But he let loose a smile as we parted and I knew that all the time he had been talking to me he was solving some prob¬ lem no one else could touch. But if that fellow Professor Weyl ever lectures in this town again I sure am going to take a try at understanding him! A fellow ought to test his intelligence once in a while.

While Dirac stayed in Madison, Heisenberg, invited by Compton, lec¬ tured at the University of Chicago. Just after Dirac had arrived in Mad¬ ison, Heisenberg had written from Cambridge, Massachusetts: “Next Fri¬ day I will arrive in Chicago; since we then are not more than 200 miles apart from another, I would like to establish the connection between
us.”20 At the end of May, Dirac completed his lectures in Wisconsin and then
traveled west alone as a tourist, visiting the Grand Canyon, Yosemite National Park, Pasadena, and Los Angeles.21 Then he went east again, to

74

Dirac: A scientific biography

Ann Arbor, where he lectured at the University of Michigan’s summer school in physics. In early August, he dined with Heisenberg in Chicago and then left by train for the West Coast. The two physicists met again in Berkeley, where Dirac lectured on the quantum mechanics of manyelectron systems and Heisenberg talked on ferromagnetism and quantum electrodynamics.
Since both of the scientists had been invited to lecture in Japan, Hei¬ senberg had suggested as early as February 1928 that they return to Europe together, going westward via Japan,22 and he made the necessary arrangements for their travel.23 On August 16, 1929, they left San Fran¬ cisco on a Japanese steamer that brought them to Hawaii four days later. According to one account, they decided to visit the Univcisity of Hawaii in Honolulu to pass some of the time until the boat resumed its voyage. They introduced themselves to the chairman of the physics department, but apparently quantum mechanics had not yet reached Hawaii. The chairman told the two youthful visitors that if they would like to attend some of the physics lectures at the university they would be welcome to do so!24 Another story from this journey concerns Dirac’s dislike of reporters. As the boat approached Yokohama in Japan, a reporter wanted to interview the two famous physicists, but Dirac evaded him and with Heisenberg’s help avoided being interviewed. When the reporter, who did not know which of the passengers was Dirac, ran into Heisenberg and Dirac, he said to Heisenberg “I have searched all over the ship for Dirac, but I cannot find him.” Instead of identifying Dirac, who was standing right beside him, Heisenberg offered to answer the reporter’s questions about Dirac. “So I just stood there, looking in another direction, pretend¬ ing to be a stranger and listening to Heisenberg describing me to the reporter,” Dirac later recounted.25 One would expect that Heisenberg and Dirac used much of their time onboard together to discuss questions of physics, but the tour was primarily intended to be a holiday, so they hardly talked physics at all.26 In fact, Heisenberg spent most of his time practicing table tennis.
Heisenberg and Dirac arrived in Japan on August 30 and spent about a month there, partly as tourists and partly giving lectures to the Japanese physicists. Of these Dirac knew Yoshio Nishina, their main host, well. Nishina had recently returned to Japan after several years’ stay with Bohr in Copenhagen. He had visited Dirac in Cambridge in November 1928 and had then mentioned the possibility of having him visit Japan. Nish¬ ina was delighted that his efforts succeeded and only regretted that Hei¬ senberg and Dirac’s stay was so short.27 Another of the Japanese hosts was Hantaro Nagaoka, an older physicist, who as far back as 1904 had proposed a Saturnian atomic model and had since then contributed much to the growth of Japanese physics. Dirac and Heisenberg lectured at Tok-

Travels and Thinking

75

yo’s Institute for Physical and Chemical Research and at Kyoto’s Impe¬ rial University. The lectures were successful and did much to stimulate the young Japanese physicists, some of whom (e.g., Yukawa, Tomonaga, Sakata, and Inui) would later make important contributions to theoreti¬ cal physics. The content of the Tokyo lectures was later published as a book/8 From the Miyako Hotel in Kyoto, Dirac informed Tamm about his further travel plans.29

I shall leave Japan for Moscow on Sept. 21st. I shall take the Northern route from Vladivostok, which does not go through China at all, as this is the only way now open. I shall leave Vladivostok in the early morning of Oct. 3rd at about 8 o’clock. I cannot remember the exact time of my arrival in Moscow and have left my time¬ table in Tokyo, but as there is now only one train a week from Vladivostok I expect you will be able to find it out without difficulty. I am afraid I will not be able to stay in Moscow for more than about two days (perhaps till the evening of Oct. 5th) as the term in Cambridge begins soon after.
After their Japanese intermezzo, Dirac and Heisenberg separated. Hei¬ senberg returned to Germany via Shanghai, Hong Kong, India, and the Red Sea while Dirac followed the route oulined in his letter to Tamm. Originally he had planned to go through Manchuria, but on Nishina’s advice he decided to avoid the area. At the time, Nishina reported, there were political troubles at the Chinese-Soviet border and danger that the Russians might close it.30 From Vladivostok Dirac went on the TransSiberian Railroad to Moscow via Chabarowsk and Tchita. From the Rus¬ sian capital he went to Leningrad by train, and from there to Berlin by airplane, which was an unusual way of traveling at that time.
Right after his return to Cambridge, Dirac began to prepare for two lectures he had agreed to present in Paris in December, and it was also in November 1929 that he worked out his revolutionary idea of negativeenergy electrons (see Chapter 5). In the following years, he was primarily occupied with developing this idea and improving the theory of quantum electrodynamics. As far as his many travels allowed him, he continued to give lectures on quantum mechanics in Cambridge and also found time to deal with other subjects that did not belong to his own specialized field of research. In seminars at Cambridge he dealt with recent developments in quantum molecular theory, the theory of magnetism, and other areas in which he never published.31 In February 1930, Dirac was elected a Fel¬ low of the Royal Society (FRS), a title of the highest prestige. To a British scientist, being able to use the initials FRS ranks almost with winning a Nobel Prize. Dirac was now, among other things, allowed to communi¬ cate papers to the Proceedings, his favorite vehicle of publication.
In the period 1928-30, he also published important works on statistical

76

Dirac: A scientific biography

quantum theory and atomic theory. Following earlier work by von Neu¬ mann, Dirac in October 1928 examined statistical quantum mechanics when applied to a Gibbsean ensemble.32 For such an ensemble he showed that there exists a close analogy between the classical and quantum equa¬ tions. Just before leaving for Wisconsin, he published an important work on the calculation of atomic properties of many-electron systems, and he followed it up the next year with a further developed theory.33 These works were in atomic theory proper, a branch that attracted many of the best physicists but one in which Dirac had not previously shown much interest. In his work of 1929, Dirac studied the exchange interaction of identical particles, which he related to the permutations of the coordi¬ nates. He introduced permutations as dynamical variables (operators) and built up a vector model of spin that he applied to the interaction of two or more electrons in an atom. Dirac’s theory was extended by Van Vleck, who applied it to complex spectra and ferromagnetism.34 The cal¬ culation of atomic properties of atoms with many electrons had previ¬ ously been worked out with various approximation methods, in particu¬ lar by Thomas (1926), Hartree (1927), and Fermi (1928). Douglas Hartree’s method of the so-called self-consistent field was given a better theoretical basis by Vladimir Fock in early 1930, but the Hartree-Fock method proved too complicated to be of much practical use for systems with very many electrons. In his work from 1930, Dirac supplied Tho¬ mas’s model with a theoretical justification and proposed a calculational improvement that yielded a better approximation. Dirac’s theory of 1929 included a general method for calculating atomic and molecular energy levels, which was, however, generally overlooked. Just after Dirac’s paper appeared, Slater published his important work on the wave function determinant method, which was at once adopted by the atomic physi¬ cists.35 Most physicists and chemists preferred Slater’s wave mechanical theory to Dirac’s more abstract version. Dirac’s works of 1929-30 were major contributions to atomic and, it turned out, solid-state physics, as eponymized in such terms as “the Fock-Dirac atom” and “the ThomasFermi-Dirac method.” However, in Dirac’s career they were mere parentheses.
Since the fall of 1927, Dirac had given a course of lectures on quantum mechanics at Cambridge.36 The content of these lectures formed the basis of his celebrated book The Principles of Quantum Mechanics, the first edition of which was published in the summer of 1930. The book was written at the request of Oxford University Press - and not, remarkably, Cambridge University Press - which was preparing a series of mono¬ graphs in physics. The general editors of the series were two of Dirac’s friends, the Cambridge physicists Fowler and Kapitza. The author and science journalist James Crowther arranged the publication of Dirac’s

Travels and Thinking

77

book for Oxford University Press. “When I first called on Dirac,” he recalled, “he was living in a simply furnished attic in St. John’s College. He had a wooden desk of the kind which is used in schools. He was seated at this, apparently writing the great work straight off.”37 Dirac started writing Principles in 1928, but because of his travels progress was slow. In January 1929, he wrote to Tamm:38

The book is progressing with a velocity of about 10“8 Frenkel. I have started writ¬ ing it again in what I hope is the final form and have written about 90 pages. I shall try hard to finish it before going to America. It is to be translated into Ger¬ man. Have you seen Weyl’s book on “Gruppentheorie und Quantenmechanik”? It is very clearly written and is far the most connected account of quantum mechanics that has yet appeared, although it is rather mathematical and therefore not very easy.

Van Vleck, at the time on sabbatical leave from the University of Wis¬ consin, visited Cambridge in March 1930 and was allowed to read the proof-sheets of Dirac’s text. “What I have read so far I like very much,” he wrote Dirac.39
Principles became a success. It went through several editions and trans¬ lations and is still widely used.40 Its Russian translation, in particular, became very popular; while the first English edition sold two thousand copies, Printsipy Kvantovoi Mekhaniki sold three thousand copies in a few months. In the thirties, Principles was the standard work on quantum mechanics, almost achieving a position like that which Sommerfeld’s Atombau und Spektrallinien had before quantum mechanics. The nuclear physicist Philip A. Morrison recalled, with some exaggeration, that “everybody who had ever looked at books had a copy of Dirac.”41 Unlike most other textbooks, Principles was not only of use to students in courses on quantum mechanics but was probably studied as much by experienced physicists, who could find in it a concise presentation of the mathematical principles of quantum mechanics, principles that were likely to be of eternal validity. When Heisenberg received the fourth edi¬ tion of Principles in 1958, he gave Dirac the following fine compliment: “I have in the past years repeatedly had the experience that when one has any sort of doubt about difficult fundamental mathematical problems and their formal representation, it is best to consult your book, because these questions are treated most carefully in your book.”42
The book expressed Dirac’s personal taste in physics and possessed a style unique to its author. Regarded as a textbook, it was and is remark¬ ably abstract and not very helpful to the reader wanting to obtain physical insight into quantum mechanics. Unlike most other modern textbooks, Principles is strictly ahistorical and contains very few references and no

78

Dirac: A scientific biography

illustrations at all. Neither is there any bibliography or suggestions for further reading. The first edition did not even include an index.
Principles was based on what Dirac called “the symbolic method,” which “deals directly in an abstract way with the quantities of fundamen¬ tal importance (the invariants, &c., of the transformations).” This method, Dirac said, “seems to go more deeply into the nature of things.”43 In accordance with the symbolic method, he wanted to present the general theory of quantum mechanics in a way that was free from physical interpretation. “One does not anywhere specify the exact nature of the symbols employed, nor is such specification at all necessary. They are used all the time in an abstract way, the algebraic axioms that they satisfy and the connexion between equations involving them and physi¬ cal conditions being all that is required.”44 Thanks to the wide distribu¬ tion of the book, Dirac’s interpretation of quantum mechanics was dis¬ seminated to a whole generation of physicists, who through it learned about the formal aspects of the Copenhagen School’s views of the mea¬ surement process and the nature of quantum mechanical uncertainty.
The Copenhagen spirit and the vague idealism associated with Princi¬ ples did not go unnoticed by Soviet commissars, who supplied the Rus¬ sian translation with a word of warning; although Dirac’s book was valu¬ able to Russian physics and thus to the Soviet Union, ideologically it was all wrong:45

The publishers are well aware that there is contained in this work a whole series of opinions, both explicit and implicit, which are totally incompatible with Dia¬ lectical Materialism. But it is precisely the necessity for a smashing attack on the theoretical front against idealism, against mechanism, and against a whole series of eclectic doctrines, that makes it the duty of the publisher to provide Soviet scientists with the concrete material that plays a crucial part in the foundation of these theories in order that, critically assimilated, their material may be employed on the front for the fight for Dialectical Materialism.

Still, one may safely assume that Russian physicists studied Principles to absorb the abstract theory of quantum mechanics, not to launch a smash¬ ing blow against bourgeois idealism.
Most physicists welcomed Dirac’s exposition and praised it for its ele¬ gance, directness, and generality. To Einstein it was the most logically perfect presentation of quantum mechanics in existence.46 Some years before Principles appeared, Eddington had praised Dirac’s method for its symbolism and its emancipation from visualizable models, a character¬ istic that was also valid with respect to the book:47

If we are to discern controlling laws of Nature not dictated by the mind it would seem necessary to escape as far as possible from the cut-and-dried framework into

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which the mind is so ready to force everything that it experiences. I think that in principle Dirac’s method asserts this kind of emancipation.

Although Dirac preferred an abstract or symbolic approach to physics, a kind of pictorial model appeared frequently in his works. But these mod¬ els had very little in common with the traditional models of classical physics. Dirac used models, metaphors, and pictures to think about pre¬ mature physical concepts and to transform vague ideas into a precise mathematical formalism. “One may,” he stated in Principles, “extend the meaning of the word, ‘picture’ to include any way of looking at the fun¬ damental laws which makes their selfconsistency obvious. With this extension, one may gradually acquire a picture of atomic phenomena by becoming familiar with the laws of the quantum theory.”48
When Pauli reviewed Principles in 1931, he recommended it strongly. But he also pointed out that Dirac’s symbolic method might lead to “a certain danger that the theory will escape from reality.” Pauli complained that the book did not reveal the crucial fact that quantum mechanical measurement requires real, solid measuring devices that follow the laws of classical physics and is not a process that merely involves mathemat¬ ical formulae.49 This was an important point in Bohr’s conception of the measurement process in quantum mechanics, a conception that Pauli shared. According to Bohr and his disciples, the classical nature of the measuring apparatus is crucial, but this point was not appreciated by Dirac.
For all its qualities, Principles was not a book easily read or one that suited the taste of all physicists. It reflected Dirac’s aristocratic sense of physics and his neglect of usual textbook pedagogy. Ehrenfest studied it very hard, only to find it “ein greuliches Buch” that was difficult to under¬ stand. “A terrible book - you can’t tear it apart!” he is said to have exclaimed.50 In their reviews of Principles, both Oppenheimer and Felix Bloch emphasized its generality and completeness. Oppenheimer com¬ pared the book with Gibb’s Elementary Principles in Statistical Mechan¬ ics (1902) and warned that it was too difficult and abstract to be a suitable text for beginners in quantum theory.51
In the second edition of Principles, published in 1935, Dirac revised the book considerably. Apart from correcting some mistakes, he added a chapter on field theory and in general presented his subject in a slightly less abstract form. The major change was in his use of the concept of a “state,” a notion central to Dirac’s exposition: whereas in 1930 he had used the word in its relativistic sense, referring to the conditions of a dynamical system throughout space-time, he now argued that “state” should refer to conditions in a three-dimensional space at one instant of time. This means that Dirac’s theory was built on a non-relativistic con-

80

Dirac: A scientific biography

cept, a fact he saw as indicating serious problems in quantum mechanics rather than in his exposition of it. The second edition was reviewed by Darwin, who was happy to notice that there had been no change in the author’s refusal to waste time over rigorous discussions of unimportant difficulties. He paraphrased Dirac’s relaxed attitude to mathematical rigor with the words, “Though something seems a bit wrong, it can’t be really serious, and with reasonable precautions there is no danger.”52 I shall have more to say about Dirac’s view on mathematics and rigor in Chapter 14.
At the time of the publication of Principles, Dirac’s ideas about the more philosophical aspects of quantum mechanics were fully developed. He did not change them much during his later career. We shall now deal in a more systematic way with Dirac’s position around 1930 in the appar¬ ently everlasting debate over the interpretation of quantum mechanics (see also Chapter 13).
By and large, Dirac shared the positivist and instrumentalist attitude of the Copenhagen-Gottingen camp, including its belief that quantum mechanics is devoid of ontological content. He thought that the value of quantum mechanics lay solely in supplying a consistent mathematical scheme that would allow physicists to calculate measurable quantities. This, he claimed, is what physics is about; apart from this, the discipline has no meaning. Instrumentalism was part of Dirac’s lectures on quan¬ tum mechanics from 1927 onward. In the introductory remarks to the lecture notes he wrote:53

The main feature of the new theory is that it deals essentially only with observable quantities, a very satisfactory feature. One may introduce auxiliary quantities not directly observable for the purpose of mathematical calculation; but variables not observable should not be introduced merely because they are required for the description of the phenomena according to ordinary classical notions . . . The the¬ ory enables one to calculate only observable quantities . . . and any theories which try to give a more detailed description of the phenomena are useless.

This does not mean that Dirac was ever a positivist in any reflective sense; he just did not care about ontological problems or problems of wider philosophical significance. From 1926 on through the thirties, he was rightly regarded as belonging to, or at least being an ally of, the Copenhagen school. In agreement with the views of Bohr, Jordan, Hei¬ senberg, and other members of the Copenhagen circle, Dirac taught that the indeterminacy relations were not the result of an incompleteness of quantum mechanics but instead expressed a fundamental feature of nature. He wrote, for example:54

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There is ... an essential indeterminacy in the quantum theory, of a kind that has no analogue in the classical theory, where causality reigns supreme. The quantum theory does not enable us in general to calculate the result of an observation, but only the probability of our obtaining a particular result when we make the obser¬ vation. [This] lack of determinacy in the quantum theory should not be consid¬ ered as a thing to be regretted.

Still, there were differences between the views of Dirac and those of the more orthodox Copenhageners. As to quantum mechanical indetermi¬ nacy, Dirac put emphasis in a different place than did his colleagues in the Copenhagen school:55

One of the most satisfactory features of the present quantum theory is that the differential equations that express the causality of classical mechanics do not get lost, but are retained in symbolic form, and indeterminacy appears only in the application of these equations to the results of observations.
Neither did Dirac join Bohr, Heisenberg, and Jordan in their strict rejec¬ tion of microscopic causality. Instead of completely abandoning causal¬ ity, he wanted to revise the concept so that it still applied to undisturbed atomic states. “Causality will still be assumed to apply to undis¬ turbed systems and the equations which will be set up to describe an undisturbed system will be differential equations expressing a causal con¬ nexion between conditions at one time and conditions at a later time.”56 However, in spite of his slightly unorthodox views, Dirac never showed any interest in the opposition waged against Bohr’s views by Einstein, Schrodinger, or de Broglie, nor was he aroused by later theories involving hidden variables. Basically, he was not very interested in the interpreta¬ tion debate and did not feel committed to argue either for the Copenha¬ gen philosophy or for opposite views. When questions about the objec¬ tivity and completeness of quantum mechanics became much discussed in the thirties (in, for example, the Bohr-Einstein-Podolsky-Rosen debate), Dirac was silent.
In 1936, Born portrayed what he called Dirac’s I’art pour Tart attitude to physics:57
Some theoretical physicists, among them Dirac, give a short and simple answer to this question [concerning the existence of an objective nature]. They say: the existence of a mathematically consistent theory is all we want. It represents every¬ thing that can be said about the empirical world; we can predict with its help unobserved phenomena, and that is all we wish. What you mean by an objective world we don’t know and don’t care.

82

Dirac: A scientific biography

No doubt, this is a fair characterization of Dirac’s position. This instru¬ mentalist and aristocratic attitude is also recognizable in Dirac’s early works. At the 1927 Solvay Congress, all the key contributors to the new atomic theory (with the exception of Jordan) met. For the first time, Dirac met Einstein, a man he admired more than any other physicist. But Einstein’s views on physics differed much from Dirac’s, and consequently Einstein reacted negatively to Dirac’s early contributions to quantum mechanics. “I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful,” Einstein told Ehrenfest in August 1926. In another letter he wrote, “I don’t understand Dirac at all (Compton effect).”58
The 1927 Solvay Congress is famous for the debate that occurred there between Einstein and Bohr over the interpretation of quantum mechan¬ ics. Of this Dirac later recalled: “In this discussion at the Solvay Confer¬ ence between Einstein and Bohr, I did not take much part. I listened to their arguments, but I did not join in them, essentially because I was not very much interested. I was more interested in getting the correct equa¬ tions.”59 However, he participated in the discussion following the report of Born and Heisenberg, and after Bohr’s address he gave a more elabo¬ rate account of his own view. He agreed with Bohr that classical deter¬ minism had to be abandoned. Quantum physics, he said, consists essen¬ tially in relating two sets of numbers: one referring to an isolated system, and the other to the system when perturbed. In order to measure the sys¬ tem, the observer forces it into a certain state by means of a perturbation and, “It is only the numbers describing these acts offree will which can be taken as initial numbers for a calculation in the quantum theory. Other numbers describing the initial state of the system are essentially unob¬ servable and are not revealed in the quantum theoretical treatment.”60 He further analyzed the nature of the measurement process as follows:61

This theory [quantum mechanics] describes the state of the world at any given moment by a wave function \f/ which normally varies according to a causal law in such a way that its initial value determines its value at any later moment. It may happen, however, that at a given moment rb \p may be expanded into a series of the type \p = X-Cnin in which the \pn's are wave functions of such a kind that they are unable to interfere mutually at a moment later than Should that be the case, the state of the world in moments further removed from t, will be described not
by xp, but by one of the \pn's. One could say that it is nature that chooses the par¬ ticular \pn that is suitable, since the only information given by the theory is that the probability that any one of the \pn's will get selected is | c„ |2. Once made, the
choice is irrevocable and will affect the entire future state of the world. The value of n chosen by nature can be determined by experiment and the results of any experiment are numbers that describe similar choices of nature.

Travels and Thinking

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Dirac’s interpretation was supported by Born, who mentioned that it was in perfect agreement with the still unpublished work of von Neumann. Heisenberg, although he agreed in general with Dirac’s exposition, objected to his statement that nature makes a choice when something is observed. It is not nature, but the observer, who chooses one of the pos¬ sible eigenfunctions, Heisenberg maintained.62 That is, Heisenberg tended to conceive of nature as a product of the free will of the human observer. Against this subjectivistic notion, Dirac held a more moderate position: While the observer decides what type of measurement to make, and thus fixes which set of eigenfunctions is relevant, nature chooses the particular eigenfunction that is to signify the result of the measurement.63
However, one should not exaggerate the difference of opinion between Dirac and Heisenberg. At least after 1927, Dirac seemed in most respects to be in line with the Copenhagen school, including its tendency toward subjectivism. This was part of the message of the preface to Principles, which contained a rather full exposition of how he conceived of the phi¬ losophy of the new physics. Borrowing phrases from Eddington’s philos¬ ophy, Dirac stated his position as follows:64

[Nature’s] fundamental laws do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies. The formulation of these laws requires the use of the mathematics of transformations. The impor¬ tant things in the world appear as the invariants (or more generally the nearly invariants, or quantities with simple transformation properties) of these transfor¬ mations. . . . The growth of the use of transformation theory, as applied first to relativity and later to the quantum theory, is the essence of the new method in theoretical physics. Further progress lies in the direction of making our equations invariant under wider and still wider transformations. This state of affair is very satisfactory from a philosophical point of view, as implying an increasing recog¬ nition of the part played by the observer in himself introducing the regularities that appear in his observations, and a lack of arbitrariness in the ways of nature, but it makes things less easy for the learner of physics.

While Dirac had much in common with Heisenberg, the situation was somewhat different with respect to Bohr, the other leader of the Copen¬ hagen school. Dirac was a mathematically minded physicist who did not really understand Bohr’s insistence on the primary' of physical - not to mention philosophical - considerations over mathematical formalism. According to Heisenberg, Bohr “feared ... that the formal mathematical structure would obscure the physical core of the problem, and in any case, he was convinced that a complete physical explanation should absolutely precede the mathematical formulation.”65 Bohr’s entire “nur die Fulle

84

Dirac: A scientific biography

fiihrt zur Klarheit” philosophy, as it most cogently manifested itself in the complementarity principle, was foreign to Dirac’s mind.66 “I didn’t alto¬ gether like it,” Dirac said in 1963, referring to the complementarity prin¬ ciple. He argued that “it doesn’t provide you with any equations which you didn’t have before.”67 For Dirac, this was reason enough to dislike the idea. Although it did not appeal to Dirac, the principle of comple¬ mentarity may have influenced his way of thinking. It may be argued that Dirac’s emphasis on invariance transformations, as stated in the preface to Principles and elsewhere, emerged as a response to Bohr’s notion of complementarity.68
The complementarity principle was first introduced in Bohr’s Como address of September 1927, although the principle was only published in April 1928.69 Dirac helped Bohr with the proofs of the article although, as Bohr realized, Dirac did not fully agree with its content. It seems that Dirac, in accordance with his statement at the Solvay Congress, criticized Bohr for making too much room for subjectivism in quantum mechanics. This difference of opinion can be glimpsed from a letter Bohr wrote to Dirac shortly before his Como address appeared in print:70

I do not know, however, whether you are quite in sympathy with the point of view, from which I have tried to represent the paradoxes of the quantum the¬ ory. ... Of course I quite appreciate your remarks that in dealing with observa¬ tions we always witness through some permanent effects a choice of nature between the different possibilities. However, it appears to me that the perma¬ nency of results of measurements is inherent in the very idea of observation; whether we have to do with marks on a photographic plate or with direct sensa¬ tions the possibility of some kind of remembrance is of course the necessary con¬ dition for making any use of observational results. It appears to me that the per¬ manency of such results is the very essence of the ordinary causal space-time description. This seems to me so clear that I have not made a special point of it in my article. What has been in my mind above all was the endeavour to represent the statistical quantum theoretical description as a natural generalisation of the ordinary causal description and to analyze the reasons why such phrases like a choice of nature present themselves in the description of the actual situation. In this respect it appears to me that the emphasis on the subjective character of the idea of observation is essential.

As we shall soon see (in Chapters 5 and 6), the intuitions of Bohr and Dirac about the future development of quantum theory differed much during the thirties. However, even though he disagreed with Bohr in many respects, Dirac continued to admire him greatly. In Dirac’s opin¬ ion, Bohr occupied the same position with regard to atomic theory that Newton did with respect to macroscopic mechanics.71

Travels and Thinking

85

In October 1930, Dirac participated in his second Solvay Congress, which was, as usual, held in Brussels. The meeting’s main theme was solid-state physics and magnetism, but the conference eventually became better known as the site of the second round of the Bohr-Einstein debate. Dirac gave no lecture at the conference but discussed problems in the interpretation of quantum mechanics with Bohr. This was evidently a subject that at the time interested Dirac, although he kept a low profile in public with regard to this interest. After returning to Cambridge, he wrote to Bohr and continued the oral discussion begun in Brussels; the letter is reproduced in extenso:72

Dear Professor Bohr,

30-11-30.

I would like to thank you for your very interesting talks to me in Brussels about uncertainty relations. I have been thinking over the last problem about coherence for two light quanta that are emitted in quick succession from an atom A.

I think it is quite certain that the two light quanta will both fall on the same point of the screen S, even when one measures accurately the initial and final momen¬ tum of the atom A. One must look at the question from the point of view of the many-dimensional wave function to get a definite answer. If X,xhx2 denote the positions of the atom and the two light-quanta, then just after the first emission we shall have a 6-dimensional wave function that vanishes everywhere except where X — X\ is very small. Just after the second emission we shall have a nine¬ dimensional wave function that vanishes everywhere except where both X - x,
and X — x2 are very small. We can now express the total wave function \p as the
sum of a number of terms

P ~ Pi +

+ • • ■

such that each term \pr vanishes except when x, and x2 have definite numerical values just after the second emission, these two values being nearly equal to each other, but different for different r. Each \pr will now give both light-quanta falling on the same point of the screen, this point being different for the different \p's.
In looking over the question of the limit to the accuracy of determination of position, due to the limit c to the velocity of the shutter, I find I cannot get your result, and I am afraid I have missed an essential point somewhere. I should be

86

Dirac: A scientific biography

very glad if you would kindly repeat the argument to me briefly. There are several people in Cambridge who are interested in the question and would like to know exactly how it goes.
Yours sincerely
P. A. M. Dirac

After this lengthy digression on Dirac’s general views on quantum the¬ ory, we now proceed to consider his most important contribution to physics during the period, the theory of anti-particles.