Group 6

On the Dirac Theory of the Electron ( 1930-1936)

On the Dirac Theory of the Electron (1930- 1936)

An Annotation by A. Pais, New York

Introduction

Heisenberg once called the 5 years following the 5th Solvay Conference on Physics (held in October 1927 at Brussels) "the golden age of atomic physics" (das goldene Zeitalter der Atomphysik) [1]. The preceding 2 years had witnessed the profound changes brought about by the advent of quantum mechanics, beginning with Heisenberg's own paper of 1925 and followed by Schrodinger's work of 1926 on wave mechanics [2]. Heisenberg, Born, J.ordan, Dirac, and Schrodinger had done much to provide a formal basis for the new mechanics. In March 1927, Heisenberg had stated his uncertainty principle [3]; in September, Bohr had lectured for the first time on complementarity [4]. All those mentioned were among the participants of that 1927 Solvay meeting. As the members of the conference left Brussels, there was a consensus, almost but not quite unanimous, that nonrelativistic quantum mechanics was a well-established discipline. Heisen­berg, the 25-year-old bachelor, returned to Leipzig, where earlier that month he had taken up his new position as the professor for theoretical physics. He and others now took up the "innumerable problems, which, unsoluble before, could be treated and decided by the new methods" [1].

The bliss of the next 5 years was not undivided, however. Early in 1928, Dirac submitted two papers which contain his relativistic equation of the electron [5, 6]. He reported spectacular results: Spin was a necessary consequence, the corect magnetic moment of the electron was obtained, the anomalous Zeeman effect of atoms came out right, the Thomas factor of the electron appeared auto­matically. But there was a serious difficulty: Dirac's equation gave twice as many states as, it seemed, were called for. In this first paper, Dirac took this lightly: "Half the solutions must be rejected as referring to the charge + e of the electron"[?]. A few months later, he realized that such a rejection is easier said than done. In a talk he gave in Leipzig, in June 1928, he noted that the electron could jump from wanted to unwanted states; hence he concluded: "It follows that the current theory is an approximation." ( [8], p. 563: "Folglich ist die gegenwiirtige Theorie eine Anniiherung.")

Even before Dirac's visit to Leipzig, Heisenberg must have been aware of these difficulties. In May 1928, he had written to Pauli: "In order not to be forever irritated by Dirac, I have done so meting else for a change." ([9], p. 443: "Um mich nicht dauernd mit Dirac herumzuiirgern, hab' ich mal was anderes getrieben. ") The something else was the quantum theory of ferromagnetism [1 0]. In Leipzig, Dirac and Heisenberg discussed several aspects of the new theory [11]. Shortly thereafter, Heisenberg wrote to Pauli: "The saddest chapter of

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modern physics is and remains the Dirac theory." ([12]: "Das traurigste Kapitel der modernen Physik ist nach wie vor die Dirac'sche Theorie.") He then mentioned some of his calculations which illustrated the difficulties and added that the magnetic electron had made Jordan "melancholic" (triibsinnig).

Heisenberg's own papers on Dirac's theory were all published in the years 1930-1936. During that period, the interpretation of what constitutes "the" Dirac theory advanced greatly. In order to give perspective to Heisenberg's con­tributions, it is necessary to remind the reader briefly of a few facts and dates [13].

In May 1929, Hermann Weyl suggested: "It is plausible to anticipate that, of the two pairs of components of the Dirac quantity, one belongs to the electron, the other to the proton." ([14], p. 332: "Es ist naheliegend zu erwarten, daft von den beiden Komponentenpaaren der Diracschen Grofte das eine dem Elektron, das andere dem Proton gehort. ")

In December 1929, Dirac pointed out: "One cannot ... simply assert that a negative-energy electron is a proton" ([15], pp. 361- 362). He went on to intro­duce the fundamental idea of a "Lochertheorie", a hole theory: "Let us assume . . . that all the states of negative energy are occupied except perhaps a few of small velocity .... We are ... led to the assumption that the holes in the distri­bution of negative-energy electrons are the protons" ([15], pp. 362-363).

In November 1930, Weyl replied: "However attractive this idea may seem at first it is certainly impossible to hold without introducing other profound mod­ifications .... Indeed according to it [i.e., the hole theory] the mass of a proton should be the same as the mass of an electron; furthermore . . . this hypothesis leads to the essential equivalence of positive and negative electricity under all circumstances . . . . The dissimilarity of the two kinds of electricity thus seems to hide a secret of Nature which lies yet deeper than the dissimilarity of past and future .... I fear that the clouds hanging over this part of the subject will roll together to form a new crisis in quantum physics" ([16], pp. 263, 264, and Preface).

In May 1931, Dirac took note of these and other criticisms [17] and proposed: "A hole, if there were one, would be a new kind of participle, unknown to experi­mental physics, having the same mass and opposite charge to an electron" ([18], p. 61). The positron theory was born.

In September 1932 to March 1933, Carl Anderson discovered the positron -a name he introduced - in cloud chamber pictures of cosmic rays [19].

In retrospect, the positron theory was the greatest triumph of theoretical physics in the 1930s. This may not be so evident from the papers of Heisenberg, to be discussed next. In his own oeuvre, Heisenberg was less concerned with the lowest-order successes of the theory, such as for example the effects of pair formation or annihilation, than with deeper and harder problems: the self-energy of the electron and the photon, vacuum polarization, the scattering of light by light, and other issues of principle and general methodology. Thus he belongs to that quite small band of theoretical physicists who had the courage to pioneer the exploration of those aspects of quantum electrodynamics which were to remain in an uncertain state until the late 1940s, when the renormalization program provided a more systematic and successful basis for tackling them. By that time,

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Heisenberg's main interests had turned to turbulence, superconductivity, shower theory, and other topics removed from the basic issues of quantum electro­dynamics - whose solution he felt could only be achieved by a fundamental theory of all elementary particles [20].

Heisenberg's First Papers on Electron Theory; a Three-Year Hiatus

In 1930, papers began to appear dealing with the self-energy of the electron, the first one by R. Oppenheimer [21], then one by I. Waller [22], then one by Heisenberg (see below). Oppenheimer's paper is memorable for the remark that self-energy effects will cause displacements of spectral lines. Waller calculated the O(e2), the self-energy W for a free electron at rest, and found it to be quadratically divergent:

e2 1 W--·-Jpdp,

he m

where m is the bare mass and the integration is over momenta in the virtual transitions: electron-+ electron + photon-+ electron. Recall that in the year 1930 one was still in the pre-hole period.

In these early self-energy calculations it was supposed that negative-energy states are empty. That assumption was also made by Heisenberg, who considered the self-energy for the case of zero bare mass. (See paper No. 1 below, pp. 106 -115.) His conclusion: "The one-electron problem could be treated correctly without infinite self-energy if there existed solutions of the vacuum electrody­namics without zero-point energy" (l.c., p. 115: "Das Einelektronenprob/em liefte sich . . . korrekt ohne unendliche Se/bstenergie behande/n, wenn es Losungen der Vakuumelektrodynamik ohne Nul/punktenergie giibe") cannot be upheld, since first of all zero-point energy has nothing to do with self-energy, and secondly, the self-energy in Heisenberg's case is in fact equal to zero because the theory with zero bare mass is y5-invariant. (The logarithmic singularity of the electron self-energy in the positron theory was first diagnosed in 1934 [23].)

Prior to the positron theory, Heisenberg wrote one more paper on the elec­tron, this one mainly methodological in character (see paper No. 3 below, pp. 123 -131). The questions he addressed concerned the emission, absorption, and scattering of radiation by a Dirac electron. He described this particle by a four­component c-number wave function 1/1 (of course, at that time one electron was still an old-fashioned one particle state). The radiation field is quantized. Heisen­berg made the point that it is unnecessary to treat these problems by taking "the detour of [using] a quite intransparent Schrodinger equation in an infinite­dimensional space" (l.c., p. 123: "den Umweg iiber eine recht uniibersichtliche Schrodingerg/eichung in einem unendlichdimensionalen Raum"). Instead, he started from the Maxwell equations, considered as operator equations, but with a c-number Dirac charge-current as a source, formally integrated these equations by means of retarded potentials, expanded 1/1 in a power series in e (for his

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purposes, terms of zeroth and first order were sufficient), and, in the final expres­sions, took matrix elements of the products of creation and/ or annihilation oper­ators of the radiation field. This very useful alternative method was applied short­ly afterward to the computation of naturallinewidth by H.B.G. Casimir [24).

In 1934, this method was extended to the q-number description of 'If by Heisenberg (paper No.4, pp. 132 -154) and by V.F. Weisskopf [23]. In the post­war years, this approach was cast in modern from by C.N. Yang and D. Feldman [25] and independently by G. Kallen [26]. In general, one deals with a set of coupled integral equations for the Maxwell and the Dirac fields. This method is in every respect equivalent to Hamiltonian techniques.

In Heisenberg's papers on the Dirac theory, there was a hiatus of more then 3 years (between February 1931 and June 1934). Several reasons may be given for this fact: The discovery of the neutron in 1932 drew his attention to the nucleus and led to his important series of papers on nuclear physics of 1932-1933. Fur­thermore, Heisenberg (and others) needed time to digest not only the hole theory and the discovery of the positron (1931 -1933), but also Fermi's theory of /3-radioactivity, which had come out in early 1934. All these topics are touched on in the letters of that period exchanged between Heisenberg and Pauli. However, this correspondence shows that, from the summer of 1933 until early 1936, no physics topic preoccupied Heisenberg more than the hole theory. In July 1933, he wrote to Pauli: "In the hole theory the concept of 'particle density' is just as problematical as in the light quantum theory," and added: "That the hole theory will lead to many kinds of horrors as long as the self-energy cannot be put in order, that I quite believe". ([27]: "In der Lochertheorie ist eben der Begriff der 'Teilchendichte' ebenso problematisch, wie in der Lichtquantentheorie" ... "DajJ die Lochertheorie noch zu mancherlei ScheujJ/ichkeiten fiihren wird, solange die Selbstenergie nicht in Ordnung gebracht werden kann, das glaub' ich gern" .) Pauli, writing to Heisenberg in September, revealed how gingerly the hole theory was approached: "At this time my attitude toward the hole theory is like Bohr's and your own, neither completely disapproving nor negative." ([28): "Meine Haltung zur Lochertheorie ist nunmehr, ebenso wie bei Bohr und Dir, keine vollig ablehnende oder negative.") At that time, the complexities of the hole theory had by no means been fully diagnosed, however.

It may in fact be said that positron theory as a serious discipline started in October 1933 with Dirac's address to the 7th Solvay Conference [29]. Dirac began by noting that it is obvious how to treat noninteracting particles in the hole theory: In order to obtain the physical energy density and charge density of a set of free electrons and positrons, subtract the energy density and charge density of the completely filled free energy negative-energy states. (We now call this a zeroth-order subtraction.) Then he raised the all-important question of how and what to subtract in realistic situations, where the particles interact with each other and with external electromagnetic fields. He did not give a general answer but presented a calculation, the first of its kind, of the polarization of the vacuum in the presence of a static external source e(x) [30]. He noted that (I paraphrase) the presence of e induces an additional charge density oe(x) owing to the virtual creation and annihilation of electron-positron pairs and found to O(e2) [31)

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oe = ~ {ae--1- (___!!____)2L1e+O(L1L1e>},

lie 151C me (1)

where a is a logarithmically divergent integral. Consequently, a new infinity had entered the theory, the charge renormalization term. The first two terms in Eq. (1) suffice if e is, spatially, slowly varying. (In 1935, the case of a generally varying static e was treated by E.A. Uehling [32], the case of time-dependent external sources by R. Serber [33].)

In making these calculations, Dirac introduced a new tool (refined in a sub­sequent paper [34]), an off-diagonal density matrix R defined by

(k',x'IR ik",x") = L 'llk•(x') 1/tk"(x") , (2) occ

where (x', k') and (x", k") are distinct sets of space-time-spin variables and where the summation goes over all occupied states. 'II and its adjoint 'II tare one­particle c-number Dirac wave functions in the presence of external fields (if any), supposed to be approximately determined by the Hartree-Fock self-consistent field method. The summation goes over all occupied states, which of course include the infinite sea of negative-energy states. The evident purpose of intro­ducing an off-diagonal distance, x 11 = x~ -x~', was to obtain finite expressions for R which, however, contain terms singular in x11 as x11 -+ 0. These singularities depend only on x11 x 11; we are dealing with a light-cone expansion. The singular terms, Dirac proposed in essence, must first be subtracted; then one lets x11 -+ 0 and so obtains physically relevant answers. His method (Dirac noted [34]) needed further refinement to take rigorous account of the exclusion principle.

Insofar as theoretical physics is concerned, the autumn of 1933 marks, I would say, the end of innocence. Gone were the days in which one massive par­ticle by itself was a pure state; virtual pairs mix in. The correct, though by now somewhat archaic notation of an infinite sea of filled negative-energy states and the correspondingly more complex formalism were hard to assimilate even by such masters as Heisenberg and Pauli. In early February 1934, Pauli received a copy of Dirac's new paper [34], then wrote a letter Heisenberg signed: "Yours (drowned in Dirac's formulae) W. Pauli" ([35]: "Dein (in Diracs Formeln ertrun­kener) W. Pauli"). Heisenberg replied: "I regard the Dirac theory ... as learned trash which no one can take seriously." ([36]: "lch halte die Diracsche Theorie ... fur einen gelehrten Mist, den kein Mensch ernst nehmen kann. ")

Yet, both men took the issues very seriously and would not drop them. During the next 2 years, there are numerous letters back and forth filled with detailed discussions of technical questions. Heisenberg's attitude in those years is perhaps best characterized in a letter of April 1935: "With regard to quantum electrodynamics, we are still at the stage in which we were in 1922 with regard to quantum mechanics. We know that everything is wrong. But in order to find the direction in which we should depart from what prevails, we must know the con­sequences of the prevailing formalism better than we do". ([37]: "Wir sind ... in bezug auf Quantenelektrodynamik noch in dem Stadium, in dem wir beziiglich der Quantenmechanik 1922 waren. Wir wissen da.P alles falsch ist. Aber um die Richtung zufinden, in der wir das bisherige verlassen sollen, miissen wir die Kon-

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sequenzen des bisherigen Formalismus vie/ besser wissen, a/s wires tun.") With the advantage of hindsight, one may not agree in detail with Heisenberg's dictum, yet his statement, which illuminates so well the ruggedness of his ap­proach, commands respect.

In defense of Heisenberg's "everything is wrong", it should be added that not only did the theoretical aspects of Dirac's theory present great mathematical complexity and severe problems of interpretation, but in addition, it was not at once clear how well the theory compared with experiment. In particular, cosmic rays presented problems: The Klein-Nishina formula did not seem to fit the data - as Heisenberg knew very well (see paper No. 1 of Group 8 below) - and the rays did not seem to be absorbed enough. There was no muon yet!

Papers on the Positron Theory

Heisenberg went to work on the positron theory right after the Solvay Conference, interrupted only briefly by a trip to Stockholm (to receive the Nobel prize). Pauli, sceptical about "limit acrobatics" (Limes-Akrobatik) [38], served as his sounding board and critic. At one point, Heisenberg suggested that they publish jointly on this subject. Pauli initially considered this a good idea and drafted an outline for a paper [39]. Nothing came of this, however, because meanwhile Pauli had developed an idea of his own: Heisenberg's paper on sub­traction physics was submitted in June 1934 (see paper No. 4); the next month, Pauli and W eisskopf completed their article on the quantum field theory of spin­less fields [40].

Heisenberg's paper (No.4, pp. 132-154, with a short correcting addendum, No. 6, p. 161) consists of two parts. In the first, he elaborated the Dirac density matrix approach in the Hartree-Fock approximation and obtained the impprtant result that subtractions are compatible with the conservation laws for electric charge (see Eq. (20), p. 140) and for energy momentum (see Eq. (25), p. 141). He also rederived (and corrected) Dirac's Eq. (1), but was not clear about its inter­pretation: "[The second term in Eq. (1)] has no physical meaning .... The 'polarization of the vacuum' first becomes a physical problem for external densities which vary in time ... " (I.e., p. 145: "hat keine physikalische Bedeu­tung . . . . Zu einem physikalischen Problem wird die 'Polarisation des Vaku­ums' erst bei zeitlich veriinderlichen iiu.Peren Dichten").

The second part of Heisenberg's paper is entitled "Quantentheorie der We/­lenjelder" (Quantum Theory of Wave Fields). It is motivated by "the necessity of formulating the fundamental equations of the theory in a way which goes beyond the Hartree[-Fock] approximation procedure" (p. 132: "die Notwendigkeit, die Grundgleichungen der Theorie in einer Uber das Hartreesche Approximations­verfahren hinausgehenden Weise zu formulieren"). He continues to employ Eq. (2) for the density matrix, but now takes 1/1 to be the standard q-number representation of the Dirac field [41] and replaces the summation by an expecta­tion value. An iterative procedure is developed along the lines of Dirac's 1931 paper [18]. Pauli realized at once that this was an advance: "In principle and in respect to physics, your ansatz has proved workable, I believe, so that the route

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is given for liberating Dirac's ansatz from the assumption of the applicability of the Hartree-Fock method and for calculating self-energies." ([42]: "Im Prinzip und in physika/ischer Hinsicht ist, glaube ich, Dein Ansatz als durchfuhrbar er­wiesen und damit der Weg gegeben, wie der Diracsche Ansatz von der Voraus­setzung der Anwendbarkeit der Hartree-Fock-Methode befreit werden kann und wie donn ouch Selbstenergien berechnet werden ktJnnen" .) [43] Indeed, in this second part, Heisenberg gives for the first time the foundation for the quantum electrodynamics of the full Dirac-Maxwell set of equations in the way we know it today [44]. Independently, Furry and Oppenheimer had the same idea [30], but Heisenberg pushed it much further. Heisenberg also emphasized that his form of the density matrix makes manifest "the invariance of the theory under a change of sign of the elementary charge" (l.c., p. 142).

Heisenberg's first application of the new method dealth with the photon self­energy. It contains an instructive error. It was found that (to O(e2)) this energy diverges even before the limit xJJ-+ 0 is taken (No. 4, Eqs. (62)- (69), pp. 152-154). Shortly thereafter, it was shown that this impossible answer was due to an inappropriately performed contact transformation [45]. In later years, the photon self-energy (it must be equal to zero) would also cause problems if mani­fest covariance and gauge invariance were not carefully maintained at all stages of the calculation.

A month after having completed his major paper, Heisenberg presented a short note on charge fluctuations (No.5, pp. 155 -160). Fluctuation phenomena in quantum mechanics had been dear to him since the "Dreimtinnerarbeit" [46], in which a section is devoted to a problem raised by Einstein in 1909, the energy fluctuations in a subvolume of a cavity filled with radiation in equilibrium [47]. Heisenberg returned to this question in 1931, showing that these fluctuations become infinitely large if the subvolume has sharp boundaries (see paper No. 2, pp. 116-122). Ongoing discussions [48] as to whether this result signifies that energy fluctuations are not defined in the correct way are interesting, but not pertinent here. What does matter is Heisenberg's observation that this calcula­tion yields finite answers if the boundary of the subvolume is smoothed out over a finite thickness. He obtained similar results in July 1934 for the charge fluctua­tions due to the virtual creation and annihilation of pairs, when he computed (in present terminology) the vacuum expectation value of

S S e(x)e(x')dx dx' , v v

where e = lflt If/ is the charge density operator and where v denotes the volume of spatial integration. Boundary divergences again appear if the boundary is sharp. Finite answers are again obtained if the boundary is smoothed out over a thick­ness b [49].

A year and a half passed before Heisenberg completed his next and last paper on positron theory. In the meantime, he continued to think about this area of problems, however. In April 1935, "sticking to my old custom of improving unclear thoughts by letters to you ... " ([50]: "um meiner a/ten Gelegenheit treu zu bleiben, unklare Gedanken durch Briefe an Dich zu verbessern"), he sent Pauli a 12-page memorandum about questions of observability in quantum field

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theory, expressing his hunch that such studies might lead to fundamental novel­ties [51]. He also had set his students H. Euler and B. Kockel to work on the scat­tering of light by light, a consequence of the positron theory first noted (in October 1933) by 0. Halpern (and for some time called Halpern scattering) [52]. In February 1935, Euler and Kockel reported their results [53]. They had treated the regime A. ll!> h/mc (A. is the photon wavelength in the center-of-momentum system) and found that, to O(e2), the scattering matrix element is finite [54]. Their answers correspond to a cross section a- (e2/hc) 4 (hlmc) 8 A. - 6. They expressed their results in terms of an effective Lagrangian which acquires a gauge invariant, scalar addition of the fourth order in the electromagnetic field.

The main purpose of Heisenberg's December 1935 paper, written jointly with Euler (No.7, pp. 162-180), was to find still higher-order terms in the effective Lagrangian, induced by static, homogenous, external fields, and in the absence of real pair formation [55]. By lengthy calculations, they arrived at a complicated nonlinearity which, by covariance and gauge invariance arguments, is tran­scribed into an effective Lagrangian (see Eq. (45a), p. 176) [56].

Perhaps more interesting than these calculations is the recitation of difficul­ties with which the paper concludes [57]. The authors are the first, I believe, to note (on p. 178) the existence of an electromagnetic contribution to the electron vacuum self-energy (nowadays called closed loop diagrams). They further remark that the fourth-order matrix element for Compton scattering diverges, as does the sixth-order contribution to light-by-light scattering, and conclude that: "The theory of the positron and present quantum electrodynamics are un­doubtedly to be considered provisional." (No.7, p. 180: "Die Theorie des Posi­trons und die bisherige Quantenelektrodynamik sind zweifel/os a/s vorliiufig anzusehen. ")

So it was in the 1930s, everywhere. The tools were available to push the theory much further, but experimental incentive was lacking, and there were so many other interesting things to investigate in theoretical physics: nuclear problems, P-radioactivity, cosmic rays, mesons, etc.

One final remark about those days. Already at that time one finds brief com­ments which prefigure what was to develop in the late 1940s. Furry and Oppen­heimer remarked in 1934: "Because it is in practice impossible not to have pairs present, we may redefine all dielectric constants, as is customarily done, by taking that of the vacuum to be unity". ([30], p. 261, footnote 12) In referring to this procedure, Serber introduced the expression "to renormalize" the charge in 1936 ([45], p. 546); Weisskopf noted in 1936 that "a constant polarizability would be in no way ascertainable" ([56], p. 6: "eine konstante Po/arisierbarkeit in keiner Weisejeststellbar ware"); and Kramers insisted that in the theory to be formulated, "the quantity which is introduced as 'particle mass' is from the out­set the experimental mass" ([58], p. 108). There may be other similar observa­tions.

Changing Scientific Outlook

Heisenberg's work on the positron theory strikes me as marking a period of transition in his scientific outlook. There is the man who with great persistence

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goes after the consequences of the theory. There is also the man who believes, or at least suspects, that this work is but a prelude to a new, quite possibly revolu­tionary chapter in physics. It can hardly be a chance coincidence that he starts groping for "a theory of the future" during and shortly after his work on positron theory. As early as 1934, he expressed the belief that "the road from the hole theory to [the determination of] e2/hc is not too far" [59]. In 1938, he proposed that there should be one more universal constant, a length, in a paper which, incidentally, is dedicated to Max Planck! (See paper No. 6 of Group 8.) His S-matrix theory of the early 1940s was designed to extract such observables as will survive in a "future theory". (See Nos. 2 and 5 of Group 10.) All these visionary themes largely originated, I believe, from his reflections on the com­plexities and apparent paradoxes of positron theory. Even the roots for this final efforts at a unified theory can be traced back to 1936, when, in a letter to Pauli, he referred to a nonlinear wave equation of the form

and commented (just as he would do in the 1950s): "The problem of elementary particles is a mathematical one, to wit, simply the question how one can con­struct a nonlinear, relativistically invariant and quantized wave equation without any constants of nature whatever." ([60]: "Das Problem der Elementarteilchen ist ein mathematisches, ntimlich einjach die Frage, wie man eine nichtlineare, relativistisch invariante und quantisierte Wellengleichung ohne irgendwelche Naturkonstanten konstruieren kann. ")

I can think of no better way of concluding this essay on Heisenberg's encoun­ters with the positron theory in the 1930s than by quoting from a letter which he wrote shortly before Pauli's death: "Our world is, we might say, the simplest of all possible worlds! But all that is pie in the sky, and before that a lot of mathematics needs to be done." ([61]: "Unsere Welt ist sozusagen die einfachste aller moglichen Welten! Aber all das ist Zukunjtsmusik und vorher mu.P noch vie/ Mathematik getrieben werden. ")

I am grateful to Res Jost, Robert Serber, and Victor Weisskopf for discus­sions and to the Pauli Estate for making available to me hitherto unpublished parts of the Heisenberg-Pauli correspondence [62].

References

See W. Heisenberg: Der Teil und das Ganze (Piper Verlag, Munich 1969}, especially p. 131; in English: Physics and Beyond (A.J. Pomerans, trans!., Harper and Row, New York 1971}, p. 92.

2 See the Annotation to the papers of Group 3 in Gesammelte Werke/Collected Works A I, pp. 329-343.

3 W. Heisenberg: Ober den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172-198 (1927}; reprinted as paper No. 3.7 in Gesammelte Werke/ Collected Works A I, pp. 478- 504

4 N. Bohr: Nature 121, 580 (1928} 5 P.A.M. Dirac: Proc. R. Soc. London Al17, 610 (1928} 6 P.A.M. Dirac: Proc. R. Soc. London Al18, 351 (1928} 7 Ref. (5], p. 618

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8 P.A.M. Dirac: Phys. Z. 29, 561, 712 (1928) 9 Wolfgang Pauli: Wissenschaftlicher Briefwechsei/Scientific Correspondence, Volume I:

1919-1929 (Springer-Verlag, New York, Heidelberg, Berlin 1979) 10 See the Annotation to the papers of Group 4 in Gesammelte Werke/Collected Works A I, pp.

507-515. 11 See Ref. [8], p. 562, footnote 2. 12 W. Heisenberg, letter toW. Pauli, July 31, 1928; published in Scientific Correspondence I, Ref.

[9], p. 467 13 A date refers to the time of receipt by a journal of a paper in question. 14 H. Weyl: Z. Phys. 56, 332 (1929) 15 P.A.M. Dirac: Proc. R. Soc. London A126, 360 (1929); Nature 126, 605 (1930) 16 H. Weyl: The Theory of Groups and Quantum Mechanics (H.P. Robertson, trans!., Methuen &

Co., London 1931) pp. 263-264 and Preface 17 R. Oppenheimer: Phys. Rev. 35, 562 (1930);

I. Tamm: Z. Phys. 62, 545 (1930) 18 P.A.M. Dirac: Proc. R. Soc. London A133, 60 (1931) 19 C.D. Anderson: Science 76, 238 (1932); Phys. Rev. 43, 491 (1933) 20 See the Annotations of Group 10 in this volume and of Groups 12 and 13 in Volume Alii. 21 R. Oppenheimer: Phys. Rev. 35, 461 (1930) 22 I. Waller: Z. Phys. 62, 673 (1930) 23 V. Weisskopf: Z. Phys. 89, 27 (1934);

V. Weisskopf (and W. Furry): Z. Phys. 90, 817 (1934) 24 H.B.G. Casimir: Z. Phys. 81, 496 (1933) 25 C.N. Yang, D. Feldman: Phys. Rev. 79, 972 (1950) 26 G. Kallen: Ark. Fys. 2, 187, 371 (1950) 27 W. Heisenberg, letter to Pauli, July 21, 1933; published in Wolfgang Pauli: Wissenschaftlicher

Briefwechsei/Scientific Correspondence, Volume II: 1930- 1939 (Springer-Verlag, Berlin, Heidelberg 1985) pp. 206, 207

28 W. Pauli, letter toW. Heisenberg, September 24, 1933; published in Scientific Correspondence II, Ref. [27]. p. 212

29 P.A.M. Dirac: in Structure et Proprietes des Noyaux Atomiques. Rapports et Discussions du Septieme Conseil de Physique (Gauthier-Villars, Paris 1934) p. 203

30 At about that time W.H. Furry and J.R. Oppenheimer independently made the same calculation: Phys. Rev. 45, 245 (1934); see especially their footnote 11. R. Peierls also did similar work: Proc. R. Soc. London A146, 420 (1934).

31 A numerical error in his coefficient of the second term has been corrected. 32 E.A. Uehling: Phys. Rev. 48, 55 (1935) 33 R. Serber: Phys. Rev. 48, 49 (1935) 34 P.A.M. Dirac: Proc. Cambridge Philos. Soc. 30, 150 (1934) 35 W. Pauli, letter toW. Heisenberg, February 6, 1934; published in Scientific Correspondence II,

Ref. [27], p. 277 36 W. Heisenberg, letter toW. Pauli, February 8, 1934; published in Scientific Correspondence II,

Ref. [27], p. 279 37 W. Heisenberg, letter toW. Pauli, April25, 1935; published in Scientific Correspondence II, Ref.

[27], p. 386 38 W. Pauli, letter toW. Heisenberg, June 14, 1934; published in Scientific Correspondence II, Ref.

[27], p. 327 39 W. Pauli, letter toW. Heisenberg, January 21, 1934; published in Scientific Correspondence II,

Ref. [27], p. 354 40 W. Pauli, V. Weisskopf: Helv. Phys. Acta 7, 709 (1934).

In those times, Pauli's attitude to the Dirac theory showed a marked ambivalence. In June 1934, he wrote about the theory of spinning electrons as "my old enemy" (meine alte Feindin). In 1935, he spoke of the Pauli-Weisskopftheory as the "anti-Dirac theory": in The Theory of the Positron and Related Topics. Report of a Seminar Conducted by W. Pauli, notes by B. Hoffmann (The Institute of Advanced Study, Princeton, NJ, 1935 -1936), unpublished.

41 For the case of no interaction, this formulation was also advocated by V. Fock: Dokl. Akad. Nauk 1, 267 (1933).

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42 W. Pauli, letter toW. Heisenberg, December 11, 1933; published in Scientific Correspondence II, Ref. [27], p. 238

43 The first self-energy calculation O(e2 ) with the help of Heisenberg's new method was in fact completed in March 1934, in Zurich [23], where Heisenberg's ideas had been known since the previous December.

44 The density matrix approach was put into modern language in a series of papers by J. G. Valatin: Proc. R. Soc. London A222, 93, 228 (1954); A225, 535 (1954); A226, 254 (1954). More recently, off-diagonal distances have made their appearance in operator product expansions, see e.g., K.G. Wilson, W. Zimmermann: Commun. Math. Phys. 24, 87 (1971).

45 R. Serber: Phys. Rev. 49, 545 (1936) 46 M. Born, W. Heisenberg, P. Jordan: Ober Quantenmechanik. II. Z. Phys. 35, 557-615 (1926);

reprinted as paper No. 3.4 in Gesammelte Werke/Col/ected Works A I, pp. 397-455 47 A. Pais: Subtle is the Lord: The Science and the Life of Albert Einstein, (Oxford University

Press, Oxford 1982) Chap. 21 48 Cf. J.J. Gonzales, H. Wergeland: K. Nor. Vidensk. Selsk. Skr. No.4 (1973) 49 Heisenberg discussed only the case b :.>him c. The case b <C hlmc was treated by R. Jost, J. Lut­

tinger: unpublished; cf. also E. Corinaldesi: Nuovo Cim. 8, 494 (1951); Suppl. Nuovo Cimento 10, 83 (1953); W. Pauli: Selected Topics in Field Quantization (MIT Press, Cambridge, MA 1973) pp. 41, 42.

50 W. Heisenberg, letter toW. Pauli, March 22, 1935; published in Scientific Correspondence II, Ref. [27], pp. 381 - 382

51 W. Heisenberg,letter toW. Pauli, April25, 1935; published in Scientific Correspondence II, Ref. [27] p. 386

52 0. Halpern: Phys. Rev. 44, 855 (1933) 53 H. Euler, B. Kockel: Die Naturwissenschaften 23, 246 (1935); cf. also N. Kemmer, V. Weisskopf:

Nature 137, 659 (1936) 54 Many more details are given in Euler's doctoral thesis: Ann. Phys. (5) 26, 398 (1936). For

). <C hlmc one has a -(e2!hc)4 ).2, a result first obtained by A. Achieser: Phys. Z. Sowjetunion 11, 263 (1937).

55 This paper also contains a few corrections to paper No. 4. 56 Their calculations were simplified shortly afterwards by V. Weisskopf: K. Dan. Vidensk. Selsk.

Mat.-Fys. Medd. 14, No.6 (1936). 57 Of less interest is the comparison they made between the pair-induced nonlinearities and those

contained in the still-born nonlinear electrodynamics of M. Born, L. Infeld: Proc. R. Soc. London A143, 410 (1933); M. Born, L. Infeld: Proc. R. Soc. London A144, 425 (1934); A147, 522 (1934); A 150, 141 (1935).

58 H.A. Kramers: Nuovo Cimento 15, 108 (1938) 59 W. Heisenberg, letter toW. Pauli, June 8, 1934; published in Scientific Correspondence II, Ref.

[27], p. 326. See also Ref. [51]. 60 W. Heisenberg, letter toW. Pauli, May 23, 1936; published in Scientific Correspondence II, Ref.

[27], p. 443 61 W. Heisenberg, letter toW. Pauli, December 14, 1957 62 Most of these letters have meanwhile been published in Wolfgang Pauli: Wissenschaftlicher Brief­

wechsei/Scientific Correspondence II. See Ref. [27].

105